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A digital image preprocessor for optical character recognition Lunscher, Wolfram H.H.J. 1983

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c  A DIGITAL IMAGE PREPROCESSOR FOR OPTICAL CHARACTER RECOGNITION by WOLFRAM H.H.J. LUNSCHER B . A . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1980 B.Sc,  The U n i v e r s i t y of Toronto, 1974  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE  STUDIES  (Department of E l e c t r i c a l E n g i n e e r i n g )  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1983 © Wolfram H.H.J. Lunscher, 1983  In  presenting  this  requirements Columbia, for  for  I agree  reference  copying  granted  by  thesis  written  of  the  representatives.  advanced  that  and  extensive  this  an  thesis  for  partial  degree at  the  Library  study.  I  this  head It  in  further  thesis of  my  for  the  gain  University  make  that  scholarly  that shall  be  or  of  Electrical  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 17,  1983  Engineering Columbia  British  available  his  may or  publication  allowed  without  permission.  Department  the  permission  by  Wolfram  October  of  purposes  or  copying not  of  i t freely  agree  department  i s understood  financial  shall  fulfilment  Lunscher  for be her of my  ABSTRACT  Optical character recognition requires from a camera d e v i c e f i r s t  that  data  be r e c o n d i t i o n e d i n t o a s u i t a b l e  T h i s t h e s i s presents a design study of a preprocessor accomplish  this  task  acquired  under  the  constraints  intended to  of  real-time,  autonomous o p e r a t i o n as p a r t of a reading machine f o r the The preprocessor c o n t a i n s reduce  the  influence  b i n a r i z a t i o n stage pixels  by  to  of  three  noise  and  identify  a single b i t ;  components:  the  enhance  a  blind.  filter  to  the t e x t seen; a  character  and  background  and a segmention system which  i n d i v i d u a l c h a r a c t e r s f o r d e l i v e r y to the  form.  recognizer  isolates  in  proper  causal order. Filtering  of  the  acquired  video i n f o r m a t i o n i s performed  with the L a p l a c i a n of a Gaussian, developed acter  by  D. Marr.  edges and from  models  promises  that  the  of  the print  design s t r a t e g y i s models.  This f i l t e r  of  detection  operator  i s shown to l o c a t e the  the human v i s u a l system, t h i s  preprocessor  resolution capability.  enhancement of  edge  2  char-  to c o n t r o l the amount of d e t a i l seen o p t i m a l l y .  Developed  thresholding  V g,  could  Binarization filter's  presented  of  a  i s a l s o reduced  output.  structures  attain  To  known  incorporating  filter  human to a  achieve  simple optimal  dimensions a two  text  filter  periodic  edge  Since the f i l t e r must be d i g i t i z e d , the design method i s  f u r t h e r extended  to  include  an  ii  analysis  of  the  effects  of  sampling  the  continuous  for a d i r e c t - f o r m f i n i t e  f i l t e r and  impulse response  To v a l i d a t e the c l a i m superior  to other  compared  to others  remarkably noise  that  this  implementation.  filter's  of V g  filtered  2  C.T.  published  and  The  from  the  t e x t images  are  incoming  more than two  the  image scan l i n e s  all  the  a scan, and  through  a  local  changes  to  data  and  test  o b j e c t i v e can architecture.  images, be achieved  a  immediately in  structure.  chainexternal  raster-acquired  the  All as  data. closed  they occur  image's E u l e r number  A  following  pointers  simulation  implementation of the segmentation system, o p e r a t i n g filtered  as  with  hybrid  internal  v e r i f y i n g g l o b a l c l o s u r e by  linked-list  a  need be s t o r e d .  w i t h i n the t e x t are detected  monitoring  during  with  accomplished  reduced  boundaries concurrent  by  results  binary-image d e s c r i p t i o n method of  of  borders  The  found s u p e r i o r , as w e l l  code-and-coordinate d e s c r i p t i o n  No  is  immune.  adapted  Zahn.  performance  t e s t images.  Segmentation of the b i n a r i z e d t e x t i s technique  its coefficients  edge d e t e c t i o n methods, a chapter i s devoted to  quantitative evaluation are  quantizing  i n d i c a t e d that the r e a l - t i m e  on  of  an  fifty  V g 2  performance  through a combined s e r i a l and  parallel  TABLE OF CONTENTS Page Abstract Table of Contents L i s t of Tables L i s t of F i g u r e s Acknowledgements  i i iv vi vii xi  I.  INTRODUCTION  II.  OPTIMAL EDGE DETECTOR DESIGN  10  2.1 2.2  Introduction The Case f o r O p t i m a l i t y  10 15  2.2.1 Marr O p t i m a l i t y 2.2.2 Dickey and Shanmugam O p t i m a l i t y 2.2.3 U n i f y i n g the F i l t e r s  15 18 21  Predicting F i l t e r  22  2.3  2.4 2.5 2.6 2.7 2.8 2.9 2.10 III.  1  Performance  2.3.1 S t a i r c a s e Edge Response 2.3.2 Square-Wave Edge Response  24 27  Performance i n A d d i t i v e Noise Multiband F i l t e r i n g Sampling the V g F i l t e r C o e f f i c i e n t Quantization F i l t e r Design Example Test-Image Examples Design Summary and C o n c l u s i o n s  30 43 46 51 67 70 83  2  OPTIMAL EDGE DETECTOR EVALUATION 3.1 3.2 3.3 3.4  3.5 3.6  88  Introduction Kitchen-Rosenfeld Evaluation E v a l u a t i o n Experiment Procedures Evaluation Results  88 95 102 106  3.4.1 Rings Image E v a l u a t i o n 3.4.2 V e r t i c a l Step E v a l u a t i o n 3.4.3 Pure Noise E v a l u a t i o n  106 116 123  Comparison to other Edge Operators Conclusions  126 128  iv  Page IV.  FAST BINARY-IMAGE SEGMENTATION  130  4.1 4.2  Introduction P r e l i m i n a r y Concepts  130 133  4.2.1 4.2.2 4.2.3 4.2.4  133 135 136 137  4.3 4.4 4.5 4.6  4.7 4.8 4.9 4.10 4.11 4.12 V.  Connectivity Borders and Edges Chain Codes Trace D i r e c t i o n  Review of Past Work Zahn's Binary-Image D e s c r i p t i o n Method Border Linkage and C l o s u r e Closure Detection  138 144 150 163  4.6.1 Region Counting Approach 4.6.2 The E u l e r Number Approach 4.6.3 E u l e r Number C l o s u r e D e t e c t i o n Procedure Summary  163 168  Border Point R e p r e s e n t a t i o n Object R e c o n s t r u c t i o n Touching C h a r a c t e r s Segmentation Summary Segmentation S i m u l a t i o n s Conclusions  178 182 190 195 200 231  DISCUSSION AND CONCLUSIONS  REFERENCES  175  233 238  v  LIST OF TABLES Table  Page  I.  Values of maximum 7  19  II.  r and j / 2 7 a g a i n s t unsigned word s i z e t  60  III.  Expected in-band r e j e c t i o n s a t fl f o r 6- and 8 - b i t q u a n t i z e d V g f i l t e r s  78  2  IV.  Segmentation s i m u l a t i o n r e s u l t s "checkerboard" worst-case  vi  including  207  LIST OF FIGURES Figure  Page  1.1  Preprocessor block  2.1  S t a i r c a s e edge model magnitude response  26  2.2  Square wave edge model magnitude response  29  2.3  C a l c u l a t i o n of the zero c r o s s i n g standard i n the presence of noise  2.4  structure  P r o p o r t i o n of a l i a s e d f i l t e r the normalized h a l f - s a m p l i n g  2  deviation  energy p l o t t e d frequency  33 against 49  2.5  V g normalized s p e c t r a l energy d e n s i t y  49  2.6  Additional bits, At, r e q u i r e d to produce a given in-band r e j e c t i o n change, A„, due to q u a n t i z a t i o n ....  57  2.7  Filter half-width, N, r e s u l t i n g sample spacing 8, f o r 4, 8, c o e f f i c i e n t word s i z e s  61  2.8  2.9  2  from normalized 12, and 16 b i t  Quantized in-band r e j e c t i o n r e s u l t i n g from unsigned word s i z e t f o r unquantized in-band r e j e c t i o n s of (a) 20, (b) 40, (c) 60, and (d) 80 db  62  Minimum unsigned word size required given the quantized in-band r e j e c t i o n f o r At of (a) 0, (b) 1, (c) 2, (d) 3, and (e) 4  64  2.10 V g(n,m) f i l t e r c o e f f i c i e n t s f o r sample 5=1.25 and 8 - b i t t o t a l word s i z e 2  Ideal V g f i l t e r e d 6.4, and (c) 1.6  r i n g s image with a  2.12 8 - b i t V g f i l t e r e d 6.4, and (c) 1.6  r i n g s image with a  2.13 6-bit V g f i l t e r e d 6.4, and (c) 1.6  r i n g s image with a  2.11  2  f  spacing  69  of (a) 16, (b) 73  2  f  of (a) 16, (b) 76  2  (  of (a) 16, (b) 77  2.14 Unbiased 8 - b i t V g f i l t e r e d r i n g s image (a) 16, (b) 6.4, and (c) 1.6 2  2.15 Unbiased 6-bit V g filtered (a) 16, (b) 6.4, and (c) 1.6 2  vi i  with  a,  of 81  r i n g s image with o of f  82  3.1  Edge p i x e l neighbourhood  3.2  Minimally  3.3  a, =16.0 V g f i l t e r e d r i n g s image: (a) edge magnitude histogram; e v a l u a t i o n measure a g a i n s t (b) t h r e s h o l d l e v e l ; (c) edge p i x e l f r a c t i o n  3.4  o =6.4 V g f i l t e r e d r i n g s image: (a) edge magnitude histogram; e v a l u a t i o n measure a g a i n s t (b) t h r e s h o l d l e v e l ; (c) edge p i x e l f r a c t i o n  3.5  a, =1.6 V g f i l t e r e d r i n g s image: (a) edge magnitude histogram; evaluation measure a g a i n s t (b) t h r e s h o l d l e v e l ; (c) edge p i x e l f r a c t i o n  3.6  a, =16 V g f i l t e r e d r i n g s image e v a l u a t i o n results: (a) SNR= 1, 2, 5, 10, 20, 50, and 100 (from bottom curve t o t o p ) , 6=0.8; (b) peak e v a l u a t i o n scores a g a i n s t SNR  3.7  a, =6.4 V g f i l t e r e d r i n g s image e v a l u a t i o n r e s u l t s : (a) SNR= 1, 2, 5, 10, 20, 50, and 100 (from bottom curve t o t o p ) , 6=0.8; (b) peak e v a l u a t i o n scores a g a i n s t SNR  3.8  a, =1.6 V g f i l t e r e d r i n g s image e v a l u a t i o n results: (a) SNR= 1, 2, 5, 10, 20, 50, and 100 (from bottom curve t o t o p ) , 8=0.8; (b) peak e v a l u a t i o n scores a g a i n s t SNR  3.9  a, =6.4 V g filtered vertical SNR = (a) 1, (b) 2, (c) 5, (d) 10, and (g) 100  curved  neighbourhood  2  2  f  2  2  2  2  2  step edge f o r (e) 20, ( f ) 50,  3.10 a =6.4 V g filtered v e r t i c a l step evaluation r e s u l t s : (a) SNR = 1, 2, 5, 10, 20, 50, and 100 (from bottom curve to t o p ) , 8=0.8; (b) peak e v a l u a t i o n scores a g a i n s t SNR 2  f  3.11 a =1.6 V g SNR = (a) 1, and (g) 100 2  f  filtered vertical step edge for (b) 2, (c) 5, (d) 10, (e) 20, ( f ) 50,  3.12 a, =1.6 V g filtered vertical step evaluation r e s u l t s : (a) SNR = 1, 2, 5, 10, 20, 50, and 100 (from bottom curve t o t o p ) , 6=0.8; (b) peak evaluation scores a g a i n s t SNR , 2  3.13 V g f i l t e r e d random noise (a) a, =6.4; (b) a =1.6 2  f  viii  evaluation  results for ,  Figure  Page  3.14 Superimposition of the V g maximum e v a l u a t i o n scores upon those of Kitchen and Rosenfeld : (a) r i n g s ; (b) v e r t i c a l step  127  4.1  Modular image r e p r e s e n t a t i o n  160  4.2  E u l e r number v a r i a t i o n image r e p r e s e n t a t i o n s  4.3  Touching c h a r a c t e r s e p a r a t i o n  194  4.4  Segmentation system s t a t e diagram  198  4.5  Sample t e s t  2  in  Is  topped  and  0  topped  170  images: (a) a, =0.8; (b) a, =1.6;  (c) a( =2.4; (d) a, =3.2; (e) af =4.0  203  4.6  "Checkerboard" worst-case image  205  4.7  List-1  209  4.8  a =0.8 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row o =1.6 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row  213  4.10 a, =2.4 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row  214  4.11 a =3.2 histogram: (a) number of p o i n t s traced row; (b) number of p o i n t s t r a n s m i t t e d per row  215  4.9  occupancy a g a i n s t row count  f  212  f  f  per  4.12 a, =4.0 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row  216  4.13 T o t a l combined histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row ....  217  4.14 a, =0.8 stack occupancy: (b) i n t e r n a l border stack;  (a) e x t e r n a l border stack; (c) t r a n s m i s s i o n stack ....  224  4.15 a =1.6 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack ....  225  4.16 o, =2.4 stack occupancy: (b) i n t e r n a l border stack;  (a) e x t e r n a l border stack; (c) t r a n s m i s s i o n stack ....  226  4.17 a =3.2 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border s t a c k ; ( c ) t r a n s m i s s i o n stack ....  227  f  (  ix  Figure 4.18  Page  a =4.0 stack occupancy: (b) i n t e r n a l border stack; t  (a) e x t e r n a l border stack; (c) t r a n s m i s s i o n stack ....  4.19 T o t a l combined stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack  x  228  229  ACKNOWLEDGEMENTS I  wish  offering valued  the  to  my  thanks  to Dr.  opportunity to explore this  M i c h a e l P.  project,  and  support  from  Beddoes f o r for  much  guidance.  The  author  assistantship Canada,  program, keyboard,  number  1982-83  Special her  acknowledges  1980-82 a w a r d e d  grant  fellowship  for  extend  thanks  by  67-3290.  the N a t i o n a l  Research  A University  of  a  research  C o u n c i l of  British  Columbia  i s gratefully appreciated.  go  unfaultering and  financial  out  emotional  w i t h o u t whose  preparation  t o my  of  lovely  fiancee  support  throughout  frequent assistance this  tedious.  xi  thesis  Cindy  would  and  K.Y.  this  skill  have  Chan degree  at  been  the truly  1  I.  This  thesis  presents  development of a d i g i t a l recognition.  INTRODUCTION  a  design  image preprocessor  number  preprocessor,  and  of  approaches  to  studied  No attempt  character  developed  E l e c t r i c a l Engineering p a r t of  by  processor design  (OCR).  machine-printed  i s kept  for  The are  preprocessor  the  blind.  pixels  characters  into  of  existing  the  as pre-  so t h a t i t may  systems.  d e v i c e i s to accept a d e s c r i p t i o n of a  d e v i c e as input and  c h a r a c t e r as input and then make a The purpose of an  video i n f o r m a t i o n  from  an  then transform i t i n t o a form the  OCR  Among the t a s k s performed by the  removal  problem  However,  as p o s s i b l e  i s to accept the raw  background  the  The dominant reason f o r  as to the i d e n t i t y of the c h a r a c t e r .  noise  configuration  a  Beddoes of the Department of  as g e n e r a l  or handwritten  d e v i c e can a c c e p t . are  of  at the U n i v e r s i t y of B r i t i s h Columbia  The purpose of an OCR  imaging  this  e n v i s i o n e d t o complement an  serve as an input to most other OCR  OCR  in  Dr. M.P.  a reading machine  decision  development  i s made to address  recognition  t h i s i s t h a t t h i s system was system  the  to f a c i l i t a t e the f i n a l assembly and adjustment of  the p r e p r o c e s s o r .  OCR  for o p t i c a l character  then endorse a p a r t i c u l a r c o n f i g u r a t i o n .  o p e r a t i o n a l parameters i n c o r p o r a t e d  optical  p r e l i m i n a r y to the  As a design study, the o b j e c t i v e of t h i s t h e s i s i s  to examine a  then  study  (filtering),  i d e n t i f i c a t i o n of c h a r a c t e r  (binarization), individual  p r e p r o c e s s i n g systems perform  preprocessor  units  and  "packaging"  (segmentation).  of  and the  Not  a l l OCR  a l l of these t a s k s , but the  system  2 proposed  in  this  thesis  does.  Figure  1.1  i l l u s t r a t e s these  o p e r a t i o n s i n the order they w i l l be performed.  filter  camera  tm  binar i zat  ion  the •="0" •="1"  t  segmentat  Figure  1.1  Since the f i r s t by  h O C R  ion  Preprocessor block OCR  structure  system was commercially marketed i n 1954  the I n t e l l i g e n t Machines Research C o r p o r a t i o n , the l i t e r a t u r e  has come to abound with m a t e r i a l r e l a t e d to surveys  of  OCR.  the e a r l y machines and the problems they encountered  were p u b l i s h e d by the B r i t i s h Computer S o c i e t y and  Harmon  [3].  More  recently  Ullmann  presented s e v e r a l surveys of the f i e l d of  the past decade.  Schurmann [ 7 ] . exists  among  These OCR  purely  choose optical  holography, may This  to  surveys  systems.  omit  systems,  [4],  [5],  spanning the  [ 6 ] , has  developments  illustrate  the  diversity  that  Not only do many systems choose to  one  from F i g u r e  1.1,  but  or more of these s t a g e s .  utilizing  the  principles  some  Indeed, of  laser  do away with the e n t i r e system a l t o g e t h e r .  diversity  d i v e r s i t y of OCR  [ 1 ] , Auerbach [ 2 ] ,  The c u r r e n t s t a t e of the a r t i s reviewed by  o r g a n i z e the p r e p r o c e s s o r d i f f e r e n t l y systems  Comprehensive  in  designs i s l a r g e l y a consequence of the  applications.  These  range  from  magnetic  ink  3 character  readers,  binarization document  and  employed to i d e n t i f y bank cheques, f o r which segmentation  preprocessing  r e c o g n i t i o n i s very d i f f i c u l t .  Most of these  material  rigidly fixed  for  trivial,  which  input  readers  is  to  be  f i x e d format. locations  presented  in  a  to  hand-printed-  and  particularly  systems r e q u i r e the  s u i t a b l e and  This includes characters  on  hand-printed  the document, or machine-printed  organized  i n t o w e l l d e f i n e d l i n e s without  non-text  material.  sometimes  the  at  characters  interspersion  However, modern r e s e a r c h i s making  of  progress  towards r e l a x i n g the r e s t r i c t i o n s on the input to permit v a r i a b l e text  formats,  [8]).  and  random placement of i l l u s t r a t i o n s  T h i s added s o p h i s t i c a t i o n r e q u i r e s the  components  to  the  the t e x t regions and The  system  component  of  a  by  addition  p r e p r o c e s s i n g system of F i g u r e 1.1 to d i r e c t t h e i r  developed reading  c e r t a i n c o n s t r a i n t s on accompanied  (Wong et a l .  here  targeted  for  motor  to  to locate  serve  the b l i n d .  i t s operation.  various  more  acquisition.  is  machine  of  Blindness  disabilities.  as  T h i s imposes is  frequently  T h i s demands the  system's o p e r a t i o n to r e q u i r e l i t t l e or no manual d e x t e r i t y . order  to  L a s t l y , s i n c e the complete system i s e n v i s i o n e d to autonomously  by  a  single  user,  it  D e v i a t i o n s i n p r i n t q u a l i t y , or minor operator generate  a  Of course, for  In  produce a n a t u r a l reading pace, r e a l - t i m e o p e r a t i o n i s  required. operated  a  situation  errors  robust.  must  not  r e q u i r i n g i n t e r v e n t i o n by a second p a r t y .  the p r e p r o c e s s i n g  fullfilling  must be  be  system alone cannot  be  responsible  a l l these c o n s t r a i n t s ; these are i s s u e s that must  be addressed  by the OCR  acceptable,  the  system  preprocessor  as must  a  whole. be  However,  compatible  with  to  be  these  4 constraints. The  image a c q u i s i t o n d e v i c e , or camera,  forms.  These  include  flying  scanners, and charge-coupled currently array.  sport  device  on  many  scanners, v i d i c o n s ,  laser  (CCD)  machine-directed  operation.  designed t o accomodate a and  suitable.  other  to  Therefore,  two-dimensional  handheld  the  or  simple  preprocessor  was  device.  r e q u i r i n g l e s s d e x t e r i t y and The primary  constraints  camera a r e p i x e l r e s o l u t i o n , and f i e l d of view. pixel  size  f o r most  l3 Mm, though Schurmann [7] c l a i m s n  book p r i n t .  The  noise  into  filtering  a  styles  i s necessary f o r  50Aim  include  expected. and b i n a r i z a t i o n stages a r e f r e q u e n t l y  i n a d i f f e r e n t order  combined  Ullmann  print  The f i e l d of view must, of course, at l e a s t  the l a r g e s t c h a r a c t e r  arranged  system  a c q u i s i t i o n systems a r e a l s o  i s scanned e l e c t r o n i c a l l y ,  [5] notes t h a t the o p t i m a l covers  The  linear-array-type acquisition  t r a i n i n g on the p a r t of the b l i n d user. the  arrays.  These permit the camera t o remain s t a t i o n a r y while the  document  on  take  i n use by Dr. Beddoes u t i l i z e s a 64 element l i n e a r CCD  T h i s d e v i c e i s r e a d i l y amenable  Vidicon  can  single  from  Figure  operation.  1.1,  and  Binarization  sometimes  reduces the  o r i g i n a l p i x e l data r e p r e s e n t a t i o n t o a s i n g l e b i t with c h a r a c t e r pixels  receiving  value  "1"  and background the value "0". The  reasons behind t h i s o p e r a t i o n a r e images  require  less  operation.  economic.  storage space, and allow both  and r e c o g n i t i o n to be performed Binarization  largely  always  Binarized segmentation  with simple boolean o p e r a t i o n s .  involves  some  form  of  The success of the p r e p r o c e s s o r c o r r e c t l y  thresholding identifying  5 the c h a r a c t e r p i x e l s threshold.  depends  on  the  proper  simplest  such  operation  i n t e n s i t y l e v e l and apply notes and  that  impossible  is  to  an i n t e r v e n i n g f i l t e r .  pick  a  fixed  i t throughout the image.  intensity  levels  vary  threshold  Ullmann  continually  making  floating  threshold  a c c o r d i n g to l o c a l g r e y - l e v e l measurements.  most s o p h i s t i c a t e d such methods was developed IBM  [4]  it  to s e l e c t a s a t i s f a c t o r y l e v e l f o r a l l circumstances.  An a l t e r n a t i v e approach i s to adopt a  the  this  t h i s i s not a very u s e f u l o p e r a t i o n s i n c e background  character  adjusted  of  In the m a j o r i t y of systems reviewed the t h r e s h o l d i s  a p p l i e d d i r e c t l y to the input data without The  setting  1975  influenced  optical  by  character  local  line  page reader. grey-level seen.  One of the  Bartz  [9] f o r  The t h r e s h o l d was not only  averaging,  but  Furthermore,  by  the  analysis  r e v e a l e d that t y p e w r i t t e n and machine p r i n t e d documents  differed  i n t h e i r noise content.  T h e r e f o r e , two d i f f e r e n t ,  user s e l e c t a b l e , t h r e s h o l d i n g  algorithms  however,  with  is  constraint. requires  inconsistent  our  considered  This,  Another, l e s s common, t h r e s h o l d s e l e c t i o n  procedure  of the document's g r e y - l e v e l histogram. of  modes  an  optimal  identified  partition  with  Such a method was o u t l i n e d by  successfully  applied.  operation  requires selection  pixels.  were  autonomous  analysis  distribution  a  also noise  substantially  widths  by  level  by  Tou  et a l .  to be i n s u f f i c i e n t l y Since a l l  of  character Otsu  separate  and  [10],  the  background and  applied  [ 1 1 ] . However, the method was not  s u i t a b l e here because the  accumulated by prescanning  to  This  grey-level  statistics  are  the e n t i r e document c a u s i n g the method  s e n s i t i v e to l o c a l c o n t r a s t v a r i a t i o n . the  above  binarization  techniques  operate  6  directly  on the input image without an i n t e r v e n i n g noise  a c e r t a i n amount of  extraneous  inappropriate l a b e l . in  the background,  rough  character  sufficiently system  detail  l i g h t h o l e s w i t h i n the c h a r a c t e r  an  pixels  regions,  or  Removal of t h i s n o i s e i s c o n s i d e r e d  important that almost  c l e a n i n g stage.  receives  T h i s may take the form of s t r a y dark  outlines.  employing  invariably  filter,  every  developer  of  an  OCR  t h i s form of b i n a r i z a t i o n has i n c l u d e d a noise A g e n e r a l study of t h i s procedure was  published  by R o s e n f e l d and Park [ 1 2 ] . In  view  binarization approach, justified. their  of  the  unsatisfactory  procedures,  founded  on  i t was  the  principles  Edge d e t e c t i o n procedures  primary  objective  variations  while  variations  of  i s the  maximizing  interest  preprocessing  a r e , of  of have  edges  d i s c u s s another  to  edge the  course,  Use of  direct  a  contour  scanner  between  a  that  intensity intensity  caused by the i n OCR  discontinuity  trace,  and  where  the  beam  (an  defocused  developed  at  and the BCS [ l ]  method  edge  has  advantage  The f l y i n g spot scanner  located character  research  d e t e c t i o n , was  detection  approximation to a L a p l a c i a n ) detector  most  alternative  The  those  edge  control  focused  of  of l o c a l  [13] sensed the l o c a l  f l y i n g spot  signal difference  an  rejection.  i s uncommon, but not new.  used by Greanias e t a l .  that  detection  noise  presence of p r i n t e d c h a r a c t e r s .  character  felt  performance  edges.  a c l a s s of edge  Recent filters  which  simulates  visual  system.  because  the range of d e t a i l r e s o l v e d and n o i s e r e j e c t i o n achieved  would  be  similar  the o p e r a t i o n of the lower l e v e l s of the human Such  to  a  filter  that  of  holds  a particular  a human reader.  attraction  T h e r e f o r e , any  7 document t h a t a human c o u l d read, a machine i n c o r p o r a t i n g such an edge  filter  detection observed  should  was  not  that t h i s  do t h i s .  be  able  to  developed new f i l t e r  Furthermore,  this  read  also.  f o r image b i n a r i z a t i o n ,  modification  2  introduces  these  address  the design of the f i l t e r  any given image p r o c e s s i n g t a s k .  filter the  resolution  filters  The r e s t of the t o render  I t w i l l be  and  chapter  i t suitable for  seen  that  depends on a t l e a s t one parameter.  time r e j e c t i n g unwanted n o i s e ,  i s given t o the r e s o l u t i o n response filter  under  variations  filter  is initially  point  spread  sampling  and  of  specified  optimal Since the  and noise s e n s i t i v i t y of the  this as  parameter. a  Digitization  coefficient  t h e r e f o r e a l s o examined a t  i n Chapter  3 2  A l s o , s i n c e the  continuous  will  takes the  quantization. length  sample spacing and c o e f f i c i e n t word Chapter  considerable attention  two-dimensional  f u n c t i o n , i t must be d i g i t i z e d before  i n t o the p r e p r o c e s s o r .  attempt  to  to  installation  form  These  determine  of  spatial  operations are the  necessary  size. s u b s t a n t i a t e the c l a i m s made  t h a t the performance of  t h a t of a l l other edge f i l t e r s  t h i s objective, a purely q u a n t i t a t i v e method  edge  must be capable of r e s o l v i n g a l l c h a r a c t e r d e t a i l while a t  same  exceed  class i s  and forms the s u b j e c t  optimal  d e f i n e s the nature of t h e i r o p t i m a l i t y .  filter  i n v o l v e s the  two c h a p t e r s .  Chapter  will  simply  T h i s new edge f i l t e r  c a l l e d the optimal edge d e t e c t i o n f i l t e r s , the next  i t was  c l a s s c o u l d r e a d i l y be m o d i f i e d t o  a p p l i c a t i o n of a f i x e d t h r e s h o l d .  of  Even though edge  an  optimal  filter  will  i n common use.  To achieve  edge  evaluation  detector  w i l l be a p p l i e d t o an optimal edge f i l t e r .  When compared  8 to p u b l i s h e d  evaluation  performance, clearly  hence  the  optimal  suitability  filter's  f o r the  b i n a r i z e d image serves as the input  preprocessor, i s  of  the p r e p r o c e s s o r .  t o the  the  image  recognition received  segmentation  Segmentation i s viewed as necessary  because i t separates the i n d i v i d u a l c h a r a c t e r s of  for analysis  from the remainder  and r e c o g n i t i o n .  In t h i s way, the  system can be reasonably c e r t a i n that the data i t has represents  a  single  character.  However,  researchers  agree on the need f o r segmentation.  and  [14], maintain that segmentation i s n e i t h e r  nor  Parks  d e s i r a b l e since  matching r e c o g n i t i o n without  prior  technique  is  consistently  Clayden,  to  While  i t is  segmentation.  largely  a  the  i t i s also  f o l l o w a given  l i n e of t e x t .  r e c o g n i t i o n d i d not seem to  character  results spacing  necessary  show  and  running  text  true  that  some  true  that  this  response to the need f o r the camera to They a l s o  c o r r e c t segmentation may not always be p o s s i b l e .  own  Clowes  Instead, they a p p l i e d a mask  directly  p o s i t i o n a l c o n t r o l i s necessary, necessity  not a l l  i t r e q u i r e s a c e r t a i n degree of p o s i t i o n  o r i e n t a t i o n c o n t r o l over the input.  their  superior  seen.  The stage  and  scores,  offer  a  useful  claim  that  However, d i r e c t  alternative  since  r e c o g n i t i o n accuracy to d e t e r i o r a t e as  decreases,  a  situation  in  which  most  segmentation schemes a l s o break down.  In an attempt to a v o i d the  d i f f i c u l t i e s posed by v a r i a b l e spacing  and  the  IBM 1275 reader [15] p l a c e s  of s h i f t  r e g i s t e r s which i n turn  correlation-type characters  recognizer.  t o every p o s s i b l e  touching  characters,  the b i n a r i z e d data i n t o a s e r i e s are  connected  to  a  template  The s h i f t r e g i s t e r s then move the vertical  and  horizontal  position  9  within  the  correlator  Ullmann  [ 5 ] , [6] notes two shortcomings with t h i s with  as  though  has  difficulty  certain  "rn"  which can be confused with "m".  mounted  on a r o t a t i n g drum. approach.  It  types of c l o s e c h a r a c t e r s such as A l s o , t h i s method  is  slow  of  seg-  and c o s t l y to implement. Chapter 4 mentation this  will  methods  system.  and  a  process.  constraint  will  component.  In and  have an  review  of  then d e s c r i b e the  Segmentation  time-consuming  requirements  present  can  For its  attempt  both  reason,  greatest  impact  estimate  processing  i s performed on the  segmentation  binarized  images  input.  serving  as  both  delay  simulation  number  method  this  to  expected  be  a  endorsed  for  complex  and  a the on  real-time this  the  hardware  times, a software system  with  as  a  complex problem,  t h i s area are presented.  fifty  The problem of s e p a r a t i n g  t o u c h i n g c h a r a c t e r s i s , however, addressed o n l y l i g h t l y . seen  system  This i s  but suggestions f o r f u r t h e r work i n  10  II.  2.1  Optimal  Edge D e t e c t o r  Introduction  Recently appeared. was  a  way.  developed  approach  Two  such  D.  Marr  can  and  that  The  E.  the  Hildreth  local  be by  a  optimality.  edge  is  later  given  be  by  shown  approximation  more amenable  The others  primary  published  the  regions  of  they  localize  Marr  Dickey  filter  filter;  i s that  the  they  filter  to  edge o p e r a t o r s ,  about  the  these  Dickey  of  an  and  edge  as  and  a  step  differing,  but  that  the  ideal  about  function.  the  It  will  essentially  an  form  operators from a  desired, necessary  however, of  a  also  makes  the  many  study.  form  properties  spatial  facilitate  by  i t s simpler  response a  to  edge  is  were d e v e l o p e d  the  of  an  a p p l i c a t i o n of  show  wave  first,  through  i t s response  spheroidal  d i f f e r e n c e between  constrained  observations  an  clearly  The  range  published  defining  t o a p p l i c a t i o n and  describing  Most  the  and  has  response  in a  shows,  its definition  intensities,  a prolate  assumptions  response.  adjacent  that  to  in  By  to optimally  [16],  response  first  for  filter  edges  operation  criteria  bandlimited  of  whose  Laplacian  i s more r i g o r o u s  uniform,  filter  through  [17],  individually  edge d e t e c t i o n  minimized  second,  between  image  of  have been p u b l i s h e d .  Shanmugam  discontinuity  problem  enhancement  filters  followed  detection.  the  the  optimally  filter,  to  i n v o l v e s an  optimize  argument,  variation  edge  approach  to  by  heuristic  Gaussian  new  This  derived  defined  it  Design  begin  edges  by  i n the  to  and  narrow  set  of  which  then  provide  this  making input  certain image  and  11 then  seek  to  isolate  background. assumption the  A  image.  simple  gradient  Roberts  [18].  properties  whose  gradient  to  the  square  of  reduces This  A the  regions  edge.  by  Hueckel of  applied. measure  the  average  of  i t  soon  and  will  [21],  was  by  orthogonal  from  produce  enhancement  areas  little  the  in  to  to  then  response  distance  square  size  and  according  [20]  avoids  characteristic  the of  smoothed  a Gaussian. of  be  the  is  when  image image.  from  the  over  these  to a  model  s e r i e s of  centered was  region,  templates image  on  developed  representing  a circular the  by  resolved.  formulate  of  a  For  input  output  methods  templates,  f u n c t i o n s over the  the  estimate  final  the  can  gradient  nine  reduce  control i s provided  that  step,  that  used  two,  Macleod  s o p h i s t i c a t e d of  w h e r e up  Hilbert  to  [19]  The  a maximum r e s p o n s e  most  basis  ancestry  dynamically  seen  image d e t a i l  A f t e r combining the  but  intensity  the  the  in adjacent  w i n d o w e d by  large  image d e t a i l  an  of  d i f f e r e n c i n g image a r e a s  smoothing  true  of  included to  intensities  determined  gradient  apparent  Thurston  difference.  detail,  [22]  be  them.  the  noise  became  resultant "ringing"  approach  Perhaps  of  well-known  between  absolute  edge as that  an  set  the  different  templates  pixel  noise  i t s  the  with  development  owes  must  from  start  i n the  d i m e n s i o n s were powers of  a  amount of  maxima  the  average  at  amount  ideal  to  operation  the  and  by  operators  largely  displaced exponentials  reduces exact  which  Rosenfeld  operators  the  edge  properties  s t i m u l a t e d the  gradient,  spatial  region  edge  idea  smoothing  response  superimposed both  pure  greatest  wideband  the  the  between  orientation  of  these  signified  This  edge d e t a i l .  regions, the  class  operator  of  difference  with  However, owing  of  sort  spurious  large  t h a t edges are  input  some  pixels  sample  into to  a are a the  12 ideal  edge, a d e c i s i o n  Like  t h e g r a d i e n t methods,  the  presence  choosing  the  template  also  gradient  and  fitting regions. have  when  edge  methods and  objects provide  connect  feature,  passband.  such  these  unwanted  primarily  of  of  both  i n the results; the regardless To  combat  of r e l a x a t i o n  edge  segments  image  of this  or  into  be shown,  encloses  i s that the spatial  e.g. Hale  fields  [23],  passbands  like  line  closed  does  not  regions i n  as the " r i n g i n g "  of  frequency  features a  operator  contrast,  threshold  fora  involving  well  frequency  to  averaging  noise  the  reveal  principal  i s not e n t i r e l y in  the  edges, appear.  be  o f some  - Thurston,  beyond  attention  level  particular  local  features not present of c e r t a i n  i s  One e x c e p t i o n i s  filter  the Rosenfeld  careful  response  Fourier transforms  those  extending  Also, additional  image,  In  u n d e s i r a b l e smoothing  some f o r m  As a consequence, h i g h  suppressed.  suffers in of  noise.  I t naturally  made.  The p r o b l e m  irregular  or  the  operators, particularly  frequency  resultant  edges.  i n edge d e t e c t o r d e v e l o p m e n t .  discontinuous  selection  also  i s seen  t h e d e t e c t o r i s t o a c t as a matched  over high  to  objection  considered  these  approach  i s  boundaries.  edge o r l i n e of  by  shortcoming.  Another seldom  edge  The o p t i m a l edge d e t e c t o r , i t w i l l  this  unbroken  an  the g r e a t e s t problem  disconnected  most a u t h o r s  technique  to prevent  However  template  generated  shortcoming,  size  remains.  edges a r e g e n e r a l l y whether  the template  of  of noise or non-ideal blurred  the  edge d e t a i l  as t o the presence  i s  applied  To  given  input remove  to  the  against  the  output.  the  optimal  i n the frequency  edge  domain.  detectors  The M a r r  were  designed  and H i l d r e t h  filter  13 evolved  from  a model of  mammal v i s i o n channels [24].  were o b s e r v e d  to f i l t e r After  response  of  human  showed  each  that  two  and  produced.  constitutes  the best  Hildreth  to the L a p l a c i a n  limiting  condition  of  again  developed  function  o p t i m i z e s the edge the  Another the  controlled of  the  The  filter  by  Marr  of  the  product  Shanmugam  This  optimal  filter.  which  are  filters  will  be  filter  adopting  an  asymptotic  filter.  This to  noise at  will  of o p t i m a l  of  case  of  approximation only form  one of  a  edge  the  of  the wave  strictly  is  Instead i t i s i n the  case  and  Dickey  form. the  the and  detection The  form  of  bandwidth  two the and  Dickey-Shanmugam  parameter, the  on  filters  the  with  the  their  manner,  bandwidth  common to  the  outset.  fixed one  t o be  form  many o t h e r e d g e  into  c h o i c e of  the  the  a mask o f  and  from  fixed.  by  from  Marr  spheroidal  the Gaussian  and  the  combined  leave  determine  in  the  i s not  what  filter,  in part  [26]  to  as  the  a  zero  analytical  prolate  results  filter,  as  through  Gaussians  of  by  the  as  show  Shanmugam  a  filter  differs  specified  Marr-Hildreth  deviation,  that  response  Hildreth  two  frequency  Poggio  lead  they  direct,  standard deviation  and  the  through  which  of  resolution  and  filter  and  feature  resolution-bandwidth  filters  edge  to control  their  Marr  of  frequency  approximated  consideration  fact  distinguishing  form  an  i n a more  serious  of  considerations  Dickey  response.  be  levels  spatial  that  edges  of a Gaussian  frequency  bandlimiting  detect  of  The  involved  showed  can  the d i f f e r e n c e  converge.  h a n d , was  that  [25]  Further  form  lowest  bandpass  distributions,  channel could  i t  The  at v a r i o u s degrees  Giese  Gaussian  crossings  but  include  these bandpass channels of  other  to  image d e t a i l  Wilson  difference  variances  vision.  filter.  a The  standard c h o i c e of  14 the  appropriate  standard  for  the f i l t e r ' s  accuracy  addressed  here.  Shanmugam  provide  Hildreth  edge  The D i c k e y  will  another  edge  resolution  input  image and a c c u r a c y  done  by  blurred  with a  noise  necessary the  will  remaining  signal  two d e s i g n  implementation: the  stages  sampling  i s  very  robust  will  of  selection  meet  specified  This  the  under  coarse  to  image a c q u i s i t i o n  be a  the  input  filter;  one with  image. of  of that  minimum The  digital  and q u a n t i z i n g  however,  operations  that  the  allowing a  and a q u a n t i z a t i o n word  systems.  of  deviation  the  the problems  these  set  resembles  as  models  of a d d i t i v e  standard  well  be s e e n ,  both  spacing  will  be  positional  t o be a c h i e v e d  as  will  p e r i o d i c edge  filter  address  filter  involve  Later,  in  It will  comparatively match most  two  i n part,  ratio  within  found.  of noise.  accuracy  other  t o the s t r u c t u r e of the  the continuous  sample  interval  to  The r e s u l t  presented,  resultant coefficients.  filter  to  twice the the  stages  two w i l l  maximum  The e x p e c t e d  resolved  on  c r o s s i n g i n the presence  the  noise  be  be  four  according  be c o n s i d e r e d .  to  and  not  deviation  t o r e s o l v e a n image w h i c h ,  tolerable  Marr  feature w i l l  response  d e v i a t i o n i s then  matter.  resolution  The f i r s t  zero  and  filters,  a  problem  Dickey  wavelength exceeds  distribution.  outlining  edge models.  standard  edge  design  nor  could  i n the presence  Gaussian  Gaussian  rules  of  standard  the  o f an i s o l a t e d  design  central  requirements  accuracy  this  s t a t e of a f f a i r s  filter  examining  in  a n d Shanmugam  be c o n s i d e r e d .  an a p p r o p r i a t e  Hildreth  edges  specification  i m p r o v e on t h i s  design of  parallel  i ti s unlikely  To  and  guidance  when t h e f i l t e r ' s  involved  which  Marr  much  that  spacing.  hand,  c o n s t i t u t e the p r i n c i p a l  Neither  claim  accurately  d e v i a t i o n and t h e r e s u l t a n t consequences  size  15 This for  design methodology  the selection  outset First  to  be  we w i l l  should provide a comprehensive  and implementation  optimal review  at  how  of a f i l t e r  designed  the task of resolving  t h e form  edge  of the optimal  guide at  the  features.  filter  can  be  found.  2.2  The Case  Marr  and H i l d r e t h ,  arguments an  used  by Marr  filter's Dickey  an edge  in  claim  have  rigorous  cost  turn  and frequency  Hildreth  changes  illumination  changes  and changes  intensity  range  then  now  the  proportion interval  each  the results  the  domains.  resolution  examine  combine  and  the  in such  of  committing  of  these  a  single  into  wavelike.  natural the  phenomena  visual  as shadows,  are  giving  rise  to  include  of  visible  One o b s e r v a t i o n i s  localized  these  a  t o draw  These  the orientation  Furthermore,  to  instead  world.  spatially  i.e., spatial  processes  themselves  of an edge p r e f e r r i n g  i n surface reflectance.  changes  of s c a l e s ,  avoided  model  to the variety  intensity  extended  and  a specified  will  of  Optimality  mathematical  surfaces,  We  function.  the localization  the spatial  within  found  filter.  and  attention  identify  energy  independent  of having  in a different  measures  both  feature.  in  Marr  Marr  wide  respective  i s rooted  and H i l d r e t h  filter's  near-optimal  that  a n d Shanmugam  a n d Shanmugam, on t h e o t h e r h a n d , a d o p t e d  approaches  2.2.1  their  Each  influence  the  about  and Dickey  to substantiate  optimal f i l t e r .  That  of  for Optimality  changes  frequencies.  rather occur In  than over  order  r e s p o n s i b l e f o r t h e image d e t a i l s  a to  within  16 this  range  the  corresponding  The  to  a  To  the  filter Ao.  influence  m u s t be  optimal  filter,  only  blurring coincide This in near (Marr  and  variation be  can  original.  that  minimize  an  The  to  edge  of  the  to  the  Hildreth's  of  condition  hold, one  of  of  the the  gradient  the side  image  zero of  the  filter's  of  the  this  standard Gaussian  function  i n the a  of  alone  detail  i s defined filtered zero  by to  image. crossing  if intensity variation  the  true  by  [27].  edge  crossings  of  minimal  a  is locally  linear variation).  position of  standard  two  range  by  of  with  these  form  is characterized  line  filtered  minimized  product  reducing  filtered  be  a Gaussian  position  steepest  an  this  image w i t h  operation  l i n e s of  to  function  i n the  a  the  positions  filter  the  displaced  cost  therefore  bandwidth  those  Only  smoothing  does not  about  in  image.  are  accurately  product  a  parallel  minimal  bandwidth  output  filter  scale  i s the  Laplacian  and  edge  of  a  i n the  edge p o s i t i o n s  The  filtering  with  restricted  therefore,  g(x,y),  indicates the  Ax.  AxAco.  However,  range  concentrated  deviation  the  possess  localize  standard  distribution,  on  the  a  resolution  placed  must  To  deviations,  must have  c e r t a i n edge  restrict  deviation  is  filter  constraints  conflicting. image  optimal  linear  If  linear  zero  crossing  edge  (i.e.  will  gradient  maximum).  There are edge  number of  detection.  therefore final  a  It  producing  image.  edge phenomena  The  is  only  advantages an  a  to  using  orientation  scalar  zero  crossings  totally  enclosed  value  for  produced by  the  the  Laplacian  independent each  image.  operator,  pixel  form c l o s e d  for  in  curves  This  is  the for an  17 important  consideration  segmentation combined  s c h e m e s be  with  process  into  the  one  V [g(x,y)  spread  Marr  filter  = -[1  2  spatial  described  to  with  - (x  the  condense  *  or  object  Laplacian  the  I(x,y)  Hildreth  can  image  be  filtering  i s the  the  .  (2.1)  argue,  Laplacian  resulting  the  of  zero  + y )/2o, ] exp[-(x  2  2  frequency  a  optimal  edge  Gaussian  crossings  point  defining  the  filter's  meets  the  claims  one  that  i t s zero  have  2  response  a  half  + v ) 2  power  octave one  2  2  this  octave  crossings.  image  described  2  (2.2)  2  filter  that  edges  signal  i s bandpass  found  2 f  ]  .  as  (2.3)  octaves.  This  almost  theorem  [28]  which  i s completely  and  this  the  2  Logan's  Poggio  bandwidth  little  in  + v )c  2  i s 1.2  Ullman  with  the  of  bandpass Marr,  detail by  exp[-27r (u  requirement  considerably  Therefore fully  of  bandwidth  found experimentally  relaxed  + j y )/2a, ] / ( ™,« ) .  2  by  2  by  and  apply  G"(u,v) = -4TT (U The  to  region  edges:  V g(x,y)  The  Gaussian  Finally,  2  function  image  applied.  I(x,y)] = V g(x,y)  Therefore, detection  subsequent  step:  *  2  should  [29],  however,  requirement  introduction  of  r e s o l u t i o n band of on  determined  application  can  be  error.  interest of  the  is V g 2  filter.  This weaken was  bandpass  i t s claim  found  that  to a  character  of  o p t i m a l i t y by lowpass,  the the  V g 2  filter  previous  Gaussian  filter  would  argument was  of  seem  since  the  to i t  correct  18 form. of  Furthermore,  the  edge  features  suitability  of  structures. heuristic  nature  filter  2  this  These  development. V g  s i n c e n o t h i n g has been s a i d sought,  filter  for  criticisms of  In  being  the  fact  c a n be s a i d  highlighting  arise  arguments  o p t i m a l up  largely used  the following  i s asymptotic  little  of the exact  a  edge  because  of  this  argument w i l l to  of the  specific  in  form  the  filter's  show t h a t t h e  predetermined  cutoff  frequency.  2.2.2  Dickey  By Dickey  defining  an edge a s a u n i t  a n d Shanmugam d e r i v e d a n e d g e  different defined the  a n d Shanmugam O p t i m a l i t y  measure  of  as maximizing  vicinity  resolution  of  local  the  edge  2  influence.  location  cost  space,  that optimized a The o p t i m a l  for  I n one d i m e n s i o n  of the following J  filter  the p r o p o r t i o n of output  requirements.  maximization  s t e p f e a t u r e i n image  image  given  this  very  filter  was  energy  in  bandwidth and  translates  into  function:  |g(x)| dx 2  -1/2  7  (2.4)  =  / where I  |g(x)| dx 2  g(x) i s the step spread  i s  filter  the  resolution  i s also  function  interval  of the optimal f i l t e r ,  c e n t e r e d on t h e s t e p e d g e .  c o n s t r a i n e d t o be b a n d l i m i t e d t o r a d i a n  and The  frequency  J2.  With that  this  information Dickey  the optimal  filter  transfer  a n d Shanmugam function  were a b l e  i s given  by  t o show  19 K 1^,(0,0)1/20),  |o)|<f2  =  H(CJ)  (2.5) 0  where  K,  prolate  is  a  elsewhere,  real  s p h e r o i d a l wave  bandwidth product, c  Note  =  product  function  defined  is similar  c.  designer  and  Table  Observe  i n concept  relates  to  of  Marr  Hildreth  because  interval  7 ,(c) m a  order  resolution  tables  [30].  either  the  is  -  and  by  7 attainable  g(x)  and  H(CJ)  a r e now  does  the as  Shanmugam's a p p r o a c h  integrations  bandwidth  H o w e v e r , H(co)  specified  maximum  -  given  than  of  to  a  restricted  frequency.  0.00858  0.06279  0.35564  0.91211  0.99988  the its  either Also two  of  filter  maximum  as  given  practical be  spatial  a  that  8  Values  in  provides  4  I.  not  filter  2  when  must  resolution  1  Representing  ^,(c,x)  the  zero  0.5  Table  problems  to the  c o n c e n t r a t i o n of  i n space and  c  c  the  that Dickey  measure  of  is  derivation.  Rather,  sharper  finite  c  degree  by  i n Marr's  to minimize  and  and  first  (2.6)  minimized  I.  \p, i s t h e  01/2  that c  seek  constant,  7  i n (2.5)  makes  implementation  n u m e r i c a l l y computed or c o n v o l u t i o n systems  dimensional  form  o f H(co)  be  is  for a  number  considered.  referenced  would  from  require that  transformed  to  the  20 spatial line  domain  spread  i/>,(c,x),  with  function  [31,  not  well  admit  more r e a l i s t i c  to a point  the spread  edge d e t e c t i o n  problem.  these  difficulties,  to replace The  ^,(c,x)  resultant  out  that  the applied  approximation  the  correct  (2.5).  Lunscher  form of the asymptotic R u  | C J | < Qc  Note  that  2  an  -  scaled  optimal  i s given  < S2  of a Gaussian  even  filter  and  extension  t h e form o f H(w)  distribution.  function  dimensional  that  by  2  '  7  )  and  filter  the step  response,  t h e edges  Furthermore, edge model  g(x),  i n an image  i s that  i s  i s an odd  also  filtered  o f ( 2 . 7 ) a r e i n d i c a t e d by z e r o  since  an  odd  by t h e two crossings i n  image. these  cutoff  identical.  pointed  2  i n the region  Therefore,  suitable  however,  / «  i  function.  From  was much more  elsewhere,  the resultant  output  asymptotic  e x p ( - c o j / 2 J i ) , |o)|<Q  2  function,  the  [33],  a  and Green  (  on  i s  form  do  given  '  the Laplacian  H(CJ)  which  was i m p r o p e r l y  0  of  the Abel  performance  a closed  the  rescaled  through  form of the f i l t e r  than  valid  that  Shanmugam, D i c k e y  with  to analysis  -  function  of the f i l t e r ' s  amenable  U  or  d e r i v a t i v e of a  t o an a n a l y s i s  approximation.  H( )  transform  p. 2 1 0 ] . T h e s e a r e c u m b e r s o m e p r o c e s s e s  Addressing attempted  H(w),  of  be c o n v e r t e d  transform  [32]  the discrete Fourier  results, frequency,  Therefore  i t can fl,  the  be  the Marr  V g 2  filter  concluded  that  a n d Shanmugam i s actually  up t o a  f i l t e r s are  suboptimal i n  21 the  sense  of m a x i m i z i n g  edges and 2.2.3  bandlimiting  Unifying  By  the  n o r G"(o>) cutoff in  result into  o f H(co)  agreeing  high  to  G"(w)  and of  H(CJ)  = c/S2  resulting  By  step-like  pp.16-20] n e i t h e r  because  the  high  order.  smoothed  filter  or  input  image.  then, i s to  force  their  can  frequency  therefore on  be  shown t h a t  be  G"(CJ)  standard convention the later  Particularly  be  filters  this  3 db  somewhat The form  related  by  which  will  half  power  occurs  at (2.8)  therefore  equivalent  i n form  when  = I/2G,  2  H(w)  frequency  would  spatial  filters,  the  .  t  are  [34,  sharp cutoff  The  It will  fi = 2 7 r 0 . 3 2 5 / a  about  passed.  than exponential  cutoff  fl.  i s chosen.  scales  realizable  the  agreement.  upon a  correspond point  , the  implementing  closer  of  Criterion  to zero faster  a c t of windowing of  range  image e n e r g y  Filters  are p h y s i c a l l y  falls  the  the  filtered  the  Paley-Wiener  the case  by  the  (2.9)  in  c  = n a, ,  ( 2 . 10a)  1/2  = Qa,  ( 2 . 10b)  z  Substituting c 1/2  2  .  (2.8)  into  (2.10):  =4.17, = 2.04a, .  Table filtered  2  I shows t h a t image  energy  this  v a l u e of c c o r r e s p o n d s t o 91%  being concentrated within  the  of  the  resolution  22 interval  I.  This  another  zero crossing  observation filtered [36].  is less  rigorous  2.3  These  Predicting  Filter  filtered controls  the  bandpass octave  been  image  description prediction than is  2CT,  evident.  next  of  that  about  image  It  argued  be this  real  variations  such  noise tend detail  respect . to  the  do  a variety  a  f  exhibit as  edge  i f more  structures passband  as  t o be range  Marr  not  narrower of  and  diffraction  in  of  they  smooth  consist  light,  detail.  Grimson,  of  Furthermore  abruptness  randomly by  detail  Hildreth of  of p e r i o d s . ideal  the  response,  consist  Rather,  examined  complete  frequency  resolved  one  However, beyond  the  such  i t s  being about  expected  the  and  through  image p r o v i d e a  to blur  was  maximizes  structures,  is unlikely  intensity.  Processes  s p a c i n g of  in a  filter  2  but,  images  in  V g  passed.  that  measure;  rarely  with  this  measure of  changes  images  these  spacing  intensity  the  the  detail  these  of  edge  p o o r l y seen,  are addressed  step-like  of  over  case  a  Performance  zero crossing  additive  be  of  section.  changes o c c u r r i n g  i n k and  would  then  2  Clark  i f the  wave,  V g  by  square  quantitative  function.  of  part  Furthermore,  out,  This  i f image edges a r e  questions  abrupt  of  corrected  the  point  feature.  [35], later  of  correctly  main edge  encountering  analysis  structure  detail  can  of  probability  zero crossings in this  provides  step  the  the  G"(u,v),  sinusoidal  , the  f  response  range  that no  a  established  response.  wide,  that  performance i n the  has  the  as  probability  of  f  imply  2o  manner  It  2o  by  such  than  low  of Grimson  also  structure  a l l .  within  noise  T h i s would  spacing  a very  i s supported  white  periodic  at  implies  of  a  diffusion  The  extreme  structured and  Clark.  23 However  n o t h i n g was  response  a g a i n s t edge  Since  real  structure, standard  what  these  invisible  predictions  The observe  filter image  two  .  equation  to predict  the  by  X-ray  edge  terms  performance  of  a  i f the  filter's  this  of  kind so  applying  blurring  the  degree  of  since  the  visual  V g 2  system,  performance.  sets  of  permits  was  to  periodic  a process  of  a,  for  resolving  were  an  ascending  model  non-ideal  suitable  an  [38]. ideal  to a Gaussian.  Gaussian, resolution  used  train.  To these  variable i s the most  diffusion  photographic on  at  performance  two  processes,  Gaussian  radiographs  approximation  Also,  filter  seen  the  human  result  the  response  results  models  wave p u l s e  of  measurable  regularity.  edges through  of  the  The  various  blur  of  of  relating of  the  c h o i c e of a  known  square  reported  influence  order  a  grains through  industrial  in  a  on  resultant  image.  identified.  modeled  the  form  of  these  be  [ 3 7 , pp.1 1 3 - 1 1 7 ] a n d  been  the  method  when p r e s e n t e d w i t h  with another  deterioration exposed  some  With can  periodic  Gaussian  also  of  blurred  convolved  exhibit  of  made c o n c e r n i n g human v i s u a l  through  and  original  structures.  structures  edges  b  c o u l d be  design  staircase,  a  f o r a,  originally  edge  i n the  is a  structures.  i t s response  The  do  magnitude  a, , t o t h e m a g n i t u d e  method chosen  non-ideal  the  spacing  i s needed  detail  was  of  images  deviation,  edges of  filter  said  standard  kernel  of  of  dye  model  Shanmugam  diffusion describes  through  Some  paper  success  to blur et  are  deviation  the  accurately  emulsion. this  structures  a l .  edge w i t h e s s e n t i a l l y  or has  found  in  examined a  first  Their conclusions, translated  show t h a t interval  the  filter  is  greater  retains than  optimal na /\/2 b  r  24 i.e.,  1/2  result  t o be  The only  = 2a, too  of  The  following analysis will  show  presented  the f i l t e r  2  will  variable.  V g , ( x ) = J° V g ,  be  This  line  considered will  spread  simplify  dependent  function:  (x,y)dy  2  2  2  2  2  (2.11a)  3  f  Fourier  G"(f)  transform  = -4TT f 2  Edge m a g n i t u d e s at  the zero  2.3.1  2  e x p ( - 2 7 r f a, ) 2  will  be  edge  staircase  e  .  2  defined  by  (2.11b)  the slope  of  the output  a  blurred  signal  Response  model  of ascending  (x) = g ( )  used  b  to  represent  infinite  magnitude i s ,  * Z u(x  x  S T  2  crossing.  S t a i r c a s e Edge  The  on  the a n a l y s i s  = -(1 - x /a, ) exp(-x /2a, )/(i/2*a ) with  this  optimistic.  spatial  use  .  b  edge models  one  through  1. 11a  >  - nT)  (2.12)  n;- oo  where  g (x) b  The  result  = Gaussian blur of  e o(x) S T  filtering  e  S T  = V [ g , (x) * e  function ( x ) with  2  = Vg, ( x ) * g  b  S T  = exp(-x /2a )//2;ra . 2  2  b  2  b  (2.13)  V g (x) i s , 2  f  (x) ]  ( x ) * I 5 ( x - n T ) ,  (2.14)  n i - oo  with  Fourier  Esio(f)  transform  =  j27rf Z n exp[-27r n f 2  n r - uo  2  2  2  (a  2  +  a, )] 6 ( f - n f , ) , 2  (2.15)  25 where  f  =  s  1/T.  Normalize with  respect  to  The  standard  the  blur  standard  deviation  deviation:  (2.16b)  (f)  into  (2.15):  = j27r/(/3 a, )I n e x p [ - 2 7 r n d + a ) / / 3 ] 6 ( f - n f ) . 2  2  2  2  2  (2.17)  2  s  magnitude  will  filter  and  = 0*i •  Substituting  S T O  the  T  ( 2 . 16a)  aa,  b  E  edge s p a c i n g  ,  o= T  the  define |Ve  S T O  of  the  the edge  ( x = 0) | =  slope  of  the  zero  crossing  at  the  origin  magnitude:  |j2irj  fE  S T 0  ( f )df I  — oo  =  8ir /(0 a, )2 2  3  n  3  e x p [ - 2 7 r n (1 + a ) / / 3 ] .  2  2  2  2  (2.18)  2  na 1  The  result  i s p l o t t e d in Figure  magnitude against Observe is  a and  that  for small  large  .  Outside  response decays  linearly  per  Therefore  0  >  5.5  decade  resolved  0.  in this a  b  < 0 .5 1 5.5  image  in  db  normalized  the  3 db  to  a the the  a  (a  < 0.2)  3 db  of  3 db  level  i n db  per  peak  region  bordering  decade  a,  and the  turn i s at the at  on a  >  a, ,  i t i s expected  that  i n the  negligible blur  T  > 2.7 5 a, .  case  of  =  a higher  edges are  for  and  spacing  plateau  a, ,  T  peak  /3.  At  for  0 = 2.75.  2.1  0.51 the rate fully  Figure  2. 1  Staircase  edge  model  magnitude  response  27  It  can  between  be  the  increasing confuse  steps  0  a  multiple  m o r e common  falling  blurred  e  S Q  (x)=  1/2  )T o f m a g n i t u d e of  edge d e t e c t i o n be  discussed  filters  this  zero  decreasing with  zero  crossing  for these later  of d i f f e r i n g  crossing  to  may  t y p e s of  image  resolve  this  bandwidths.  Response  image  intensity.  square  there i s another  existence  will  Square-Wave Edge  A and  The of  A method  using  ( n +  5.  process  structures.  2.3.2  shown t h a t  at  f o r /3>  the  problem  readily  structure  In  i s one  i t s simplest  of  form  periodically i t can  be  rising  modeled  by  wave,  g„(x)  * u(x)u(T-x) * Z  5(x  - 2nT)  .  (2.19)  n z — co  Here  T  r e p r e s e n t s the  rather  than  filtering  e  S Q 0  the  with  (x)  V g  t  6(x with  of  (x) the  result  (  (x) *  2  = Vg  the  period  2  = V [g  s p a c i n g of  f  e  S Q  square  of  the  wave  square  itself.  wave After  is  (x)}  (x) * g ( x )  { Z [ 6 ( x - 2nT) n s - oo  *  b  -  the  edges  (2n+1)T)]  }  -  (2.20)  Fourier transform, Esodtf) =  J27rf Z n[1  - exp(-jnff) ]  2  s  n z -oo  exp[-27r n f (a 2  2  2  2 b  +  of)]  5 ( f - nf ) s  (2.21)  28 where  f E  = (2T)" .  Substituting the normalizing  1  s  ( )  (2.16):  = j f / ( 2 0 a )L n [ l - e x p ( - j n r r ) ]  f  S Q O  factors,  2  2  f  n r - oo  exp[-7r n (1 2  The  + a )/20 ]  2  2  magnitude of the slope IVe  s o o  (0) | =  of the zero  | j27r/° f E  S Q 0  = ir /(0 a )Z 2  3  6 ( f - nf, ) .  2  at the origin i s  (f)df| n (1 -  3  cos(rnr))  2  f  crossing  (2.22)  n=.1  exp[-7r n (l 2  This  result Note  narrow  5.3  indicates  a  - large  db  that  the  higher blurred  response  above  The  i s of  model.  the  reaches  blur  falls  rate  off  per decade  linearly 0.  s q u a r e wave image  3 db  level  below  a  brink  the  a (a < 0 . 2 ) .  small  For large  a  f o r a=0.51 a n d 0 £ 1 . 3 6 .  i s identical  unchanging Again  The  f o r small  enhanced a t the  reached  f o r a<0.51 a n d 0S5.OO.  model.  s t a t e a t 0=1.81  f o r small  Finally,  staircase  strongly  i s first  3 d b maximum  (2.23)  more complex a n d s e n s i t i v e t o  steady  a t 0=1.15  ] .  2  2.2.  response  of response  staircase  response  i n Figure  t h e edges a r e most  this  achieved  2  i n the case  peak  0 plateau  level  Note that  the than  visibility.  this  is  that  spacing  strong  of  i s plotted  + a )/20  2  steady  i t  i s  to  that  state seen  of  response that  the  i n d b p e r d e c a d e a , a n d a t a much  Therefore i s fully  i t c a n be c o n c l u d e d  resolved f o r  a  < 0 . 5 ! a, ,  T  > 1.36 a, , a n d i n t h e c a s e o f n e g l i g i b l e b l u r  T  > 1 . 1 5fff•  that  a  Figure  2.2  S q u a r e wave edge  model  magnitude  response  30 2.4  Performance  The to  forgoing  when c e r t a i n  given  a,.  the  analysis  edge  the  remains  to  additive  has  since  filter be  one-dimensional  with  Noise  succeeded  structures  However,  deterministic, noise  in Additive  can  the  expected  signal  signal  address  models used  Gaussian  were  i n the presence  To  thus  noise  a guide  t o become  models used  performance  resolved.  white  be  in providing  this  of power  visible  perfectly  of  additive  problem,  f a r , and  as  retain  c o r r u p t them  spectral  density  170/2.  The zero  first  crossings  Consider zero a  an  with  + y (x)  will  0  zero crossing problem  and  the n o i s e at  y,(x)  2  Clearly the For  an  square  the  true  edge s i g n a l ,  filtered  Gaussian  noise,  nothing  can  said  cross  the  x-axis  i n the  about  be  form  assume a  the o r i g i n .  o  the z  The  0  positions.  c e n t e r e d on  where x=0.  the  Being exactly  Instead, a  then  be  sought.  f o r both  To  the  of make  signal  becomes  (2.24a) with  = n x 0  + n  y-intercept  n  0  and  slope n , 0  becomes (2.24b)  0  assumption  this  ideal  the  standard deviation  model  signal  of  n ( x ) , added. of  , must  linear  edge  y (x),  nearest  of  the o r i g i n ,  accuracy  = Ax  this  signal,  of  the  filtered  manageable,  the noise, y (x)  concerns  measure  description  the  and  a  process,  statistical the  as  isolated  crossing  random  n(x)  question to arise  range step  wave e d g e s w i t h  of  linearity  i s bounded edge, /3 >  by  the  has  a  signal  corresponding to 5,  this  limited  occurs at  peak at  ideal |x|  range. |x| =  staircase  = a, .  For A. and  Plausible  31 values  of  o  95.4%  of  This  in  ratio  i n both  filter  the  zero  turn  can  The solving  should  z  will  lead to a  filtered  expected  most the  c r o s s i n g s t o be  the  be  t h e r e f o r e be  bounded  contained  lower  and  bound  of  the  (A  + n )x  i n the on  linear  the  image  accurate  results.  to y i e l d  for at  f  unfiltered  s t r a i g h t f o r w a r d approach  variance  a /2  by  signal for  to determining  x - i n t e r c e p t of  the  sum  of  least  region. to  noise  which  a  the  involves  z  y,  and  y , 2  i.e., y,(x) yields  =  _  n /(A  a  z  2  and  forms  0  + n ) ]  variance  - E[n /(A  2  0  of  the  zero  and  crossing  2  =  0  the  0  independent  n )- ]  2  1  0  2  0  (2.26)  2  of  the n  0  [39,  filter  and  n  0  as  does not nl  pass dc.  and  Therefore,  h, 2  o o  2  0  first  are  0  n exp(-n /2n . /  processes  uncorrelated  + n )" ].  since  variance  are  - E[n ] E[(A +  2  0  E{n }  (2.25)  2  0  i t s derivative  = E [ n ] E [ ( A + n )' } E[(A  + n )] .  0  o o  The  the  uncorrelated Gaussian  2  writing  x  random v a r i a b l e  = E[n ] where  0  2  2  245],  0,  a:  0  a  =  0  h ).  of  = E[n /(A  Since p.  +  0  estimate,  2  + n  0  x - i n t e r c e p t of  variance  a  =  2  an  *o The  + y (x)  2  2  )dn  0  /2Tn7 V2imJ  integral  rexp(-n /2n ) d n / 0  0  (A ++ hn ) J^ (A 0  simply  2  yields  2  2  /27rrVJ  n ,. , 2  0  (2.27) .  However,  the  second  32 integral at  n  = -A.  0  which  cannot  In  intercept  integrating  n  |x |  effectively is  not  n  = -A  ignored  are the  A  the  a, ,  with  near  model  only  noise,  n(x):  the  for  simple  the  zero  the  singularity  model  with  of  n ,  assumption. x  that the  can  this  the be  solution  slopes  filter  an By  within  0  region  However,  since  applied  produces  0  this  reason  infinity  about  obviously  slope.  approach  x - i n t e r c e p t of  signal  linear  linearity  outside  reasonable that  the  intercept at  together the  of  region which maintains  the  perfectly  signal  the  h i g h l y improbable.  for  more w o r k a b l e  the  the  slopes  as  of  violates  f  over  satisfactory  passed  from  0  <  0  which,  0  j x |= a  only  0  n ,  because  i m p l i e s an  limitations  any  outside  evaluated  singularity  the  fact  bounds of  directly  This  underscores  here.  0  be  i s made p o s s i b l e by  noise  standard  d e v i a t i o n bounds  This  assume  crossing.  signal  estimating  within  time, < A,  |x|  and  add  a  a  z  about linear  to  this  the  a  constant  y ( x ) = A x + n ( x ) . Since  the  variance  noise of  n  is and  2  assumed standard  stationary,  y(x)  has  deviation  tfo = / n f  (2.28)  centered  on  through  the  upper and  to  estimate  be  used  the  superimposed bound  signal.  on  the  The lower a  z  .  signal  x - i n t e r c e p t s of b o u n d s on Figure model.  y(x) 2.3  The  line  the as  lines  defined  shows defined  passing  by  a  these by  the  Q  will lines lower  is y, ( x )  = Ax  -  o  0  (2.29)  with x-intercept  x  0  -  o /A. 0  (2.30)  33  Figure  2.3  C a l c u l a t i o n of the i n the presence of  zero crossing noise  standard  deviation  34 The  x - i n t e r c e p t of  and  opposite  standard of  sign.  Before filtered  the  i s o b v i o u s l y of equal  intercept,  power  2  from equation  |G"(f)|  is  n  or zero  will  serve  magnitude  c r o s s i n g , of as  the  the  estimate  c r o s s i n g , (7 . 2  necessary  density  to evaluate  i s given  a.  The  Q  by  (f)  (2.31 )  (2.11b),  16w"f " e x p ( - 4 7 r f a  =  2  i t  spectral  |G"(f)| S  (f) =  0  d e v i a t i o n of the zero  proceeding,  noise  n o  where,  Being  line  d e v i a t i o n bounds of y ( x ) , x  the standard  S  the upper  2  2  2 f  ).  (2.32)  and S„(f)  =  TJO/2.  Since  n ( x ) i s an  given  by  ergodic  the output  process  noise  the  filtered  noise  variance  is  power,  n | = J Sno ( f ) d f =  \6ir r) ? 0  f"exp(-4ff f o ) . 2  n  o  5  Substituting crossing  a  z  f  n|  into  standard  =  1  —  /  2  f  = 3 T J / ( 16^o ) 0  2  .  (2.33)  (2.28) and  (2.30),  the estimate  f o r the  d e v i a t i o n becomes  1/ 2  3r?<  (2.34)  A  The  magnitude  determined (2.18) and  earlier  zero  of  the slope for  (2.23) y i e l d s  at  the  zero  crossing,  t h e p e r i o d i c edge m o d e l s . o  z  for  A,  was  Substituting  35 The s t a i r c a s e  / 3j? a }  /•V ff  model:  0  ZST  16TT ' 9  The s q u a r e  1 / 2  f  03  2  wave  I  n exp[-27r n 2  2  2  (1+a  )/B  2  ]  2  (2.35)  nsi  model:  -  Z  n (1-cosn7r)exp[-7r n (1+a )/2^ ] 2  2  2  2  2  #  (2.36) As  was o b s e r v e d  edge p e r f o r m a n c e assume the  these  for a «  Therefore  ...  } = 4.0421  f o r the case  z  the zero  available  noise  a  2  case  the terms i n  to  2  resolved  edges,  .  estimate  standard  predicts that  deviation.  be u s e d .  This  o , t h e more  spatially  wider  would  f  and would  the standard  of noise Therefore  measure of t h e edge p o s i t i o n  should  smaller  However  further analysis  10- .  c r o s s i n g i n the presence  more a c c u r a t e  step  (2.37)  this  the f i l t e r  the  In this  ideal  = 24.74  Clearly  a  X  simplify  on  1/2  „  9  of  To  (2.36) a r e a l l e q u a l  of f u l l y  16TT '  of  models converge  t o be t r u e .  of (2.35) and  /  a  these  1, 0 > 5.  conditions  parenthesis {  earlier,  filter  seems  be more  t h e r e f o r e be t o l e r a b l e  v a r i e s as the i n order  root  to  achieve  the narrowest  filter  intuitively  local  deviation  the  filter  effective of lower  correct  since  influence.  i n smoothing the signal  to  noise  36 ratios.  To the  test  signal  to  this  hypothesis  noise  r a t i o given  As  i l l u s t r a t e d e a r l i e r the  a  breaks  z  down  find  (2.37), A  must  largely  which,  to  for =  a  be  95.4%  =  rj  0  noise image  , a  the  V'  must  used  fully  the  lowest be  i n the  resolved  a  filter  bound  placed  on  zero  on o  derivation edges  d i s t r i b u t i o n of  within  37,0  limit  model  used  z  . of to  crossings  standard  deviation  implies  2  <  0.5  (2.38)  \ l6ir ' o l 9  2  f  ideal  square  as  1.15a to  waves can the  f  can  >  This  result  ratio  2448  i n the  f o r the  [33],  refined  ratio  within  filtering  with  a  z  from the the  resolved with  be  criteria  meaningful readily  t h e r e f o r e bounded  Of /no  be  above  guarantee  adjustments  are  Therefore  establish  the  24.74  restrictive such  f  For  probability,  °f  little  .  /  —  > A.  z  o  linear  contained  £  Since  t  o  and  9  be  2.66  2  used  filtered estimate  edge be  spacing  made e v e n  positions.  (2.38) as  as more  However,  needed.  a  and  f  by  [ 3 / ( 1 6TT ^ ) ] =  can  could  edge  made t o  an  to  to  be  asymptotic  (2.39)  establish  resolution  Shanmugam e t resolution  .  valid. a l .  minimum  interval, As  was  [32], the  i n t e r v a l of optimal  the  an  filter  is  to  and  initial  by  Lunscher  shown signal  ideal  signal  step  to  noise  edge  after  37  7^ SNR  / w exp(-ccj /Q )do> 2  =  0  2  2  °— t  C TJO  (2.40) w exp(-cw /fi )dw . a  2  2  o  Since  the  V g  filter  filter  used  to derive  2  to  infinity.  was  shown e a r l i e r  and  (2.8) and (2.10a)  that  =  2.8a  representing filtered  f  7=0.91.  Therefore,  f o r 0 and c, (2.40)  to  limits  Q,  as the  may  extend  c u t o f f frequency, i t substituting  for  7,  becomes  2  2  f c j " e x p ( - a c o )dcj O 2  /(37rrj0)  the signal  zero  the integration  Q t o t h e 3 db high  2  7j0  no  bandlimited  J a > e x p ( - a a j )da> -  =  0  (2.40),  By f i x i n g  1.4 SNR  i s not sharply  2  .  (2.41)  to noise  crossing.  ratio  within  2o  t  Substituting inequality  of  (2.39)  the into  V g 2  this  result,  SNR  > 2.8(2.66)/(3ir)  0  provides  the  crossing  fora  figure  by  minimum z  shows SNR  simply  bounds  Of is  that  wave  signal  to  image 0  processing  increasing  interest  than  of the u n f i l t e r e d models  yield  noise  with  about  the zero  i s a very  l o w SNR  Furthermore, since o , f  limit  (2.42)  filter.  the signal  identical  this  standards.  directly  image,  ratio  Clearly  the narrowest usable  greater  image  (2.42)  t o be m e a n i n g f u l .  most  (2.41)  = 0.79  SNRj . signal  to noise The i d e a l power o f  ratio step  of  (2.41)  and square  38 Sj  =  Since  lim[ (1/2T)/ u(t) dt]  the  power  noise  is  lowpass  done  on  clearly  =  where T  sampled with  with  .  assumed  (2.43)  to  be  However, image  white,  since  the  effects.  The  input  noise  image p r o c e s s i n g  initial  maximum b a n d w i d t h  the  of  image half  noise  the  power  bo  Substituting  be  sampling  passed  is  of  (2.44)  1  sampling  period.  Normalize  T  to  the  filter  with  .  f  is  must  0  deviation  =  a  0.5  a maximum v a l u e  2 ( T J / 2 ) (2T)"  i s the  standard  T  a  aliasing  finite  0  was  infinite.  filtered  to avoid  N  spectrum  ideally  generally  rate  =  2  (2.45)  (2.45)  into  (2.44)  produces  the  normalized  noise  power,  N  T? /(25<7,: )  =  0  Together  0  with  SNR j =  S  this  Si /N  Substituting  .  0  =  (2.46)  shows  5a, / r ?  (2.39)  0  the  input  signal  to  noise  ratio  to  .  be  (2.47)  i t i s found  that  the  range  for which o < z  a /2 f  i s  SNR j >  As  the  signal SNRi  filter to  note  sampled  2.665  image  (2.48)  width  noise that  .  the  increases,  ratio.  To  highest  would  be  a  5 decreases  establish  frequency square  the  edge  wave o r  lowering maximum feature  pulse  the lower  required bound  possible  train  of  in  period  on the 2T  39 with  effective  aliased, the  and  filter  a  1.25a  e s t a b l i s h e s the  signal within and  to noise the  seen  and,  by  by  the  act  would correspond  to  f  The .  initial  increasing  the  a  ,  z  of  sampling,  that  of  with  this  o  f  ,  a for the  (2.48):  (2.49)  figure  (2.41),  without  is highly  detection spacing  Therefore, into  feature  .  This  decreases  f  a  most p e s s i m i s t i c  image,  to decrease a  3.325  ratio.  filtered  increasing  such  Substituting  SNRi(6=1.25) >  This  Since  represented  wave.  i s about  i s 1.25.  T.  expected  b l u r r e d square  6  of  therefore poorly  feature  maximum  spacing  sensitivity  somewhat such  edge  limit level  lower  i s higher  but as of  bound  is a  than  s t i l l  the  that  detail  input  required  remarkably  i s increased.  f  decreases  on  t h a t can  low  Of  be  course  resolved,  the  accuracy  of  the  can  be  used  directly  (2.37)  gives  edge  position.  When SNR| estimate  o  .  z  satisfies  (2.48)  Substituting  (2.47)  36o* 2  4  '  7  4  (  ba  adopting (  =  1.  the  pixel  \  edge p o s i t i o n  signal  to  , / 2  (2.50)  2  spacing,  T,  as  the  unit  of  measure,  Therefore,  /  The  \  into  — 16ff ' SNR,/, 9  On  i t  to  noise  3  ^  V'  l67r  9 / 2  accuracy ratio,  as  2  SNRi/.  is therefore expected.  improved  by  increasing  40  The zero  foregoing  crossing accuracy  resolution quality be  the  large  quality  ratio  and  interval.  outside  quite  the  analysis  SNRQ  case be  .  of  large  addressed For  the  to  infinity  the  signal  interval  is  ( 1 -  ).  The  the  within  a  2a, .  I  =  (  1 - 7  problem large In  )  this  in  integration for  c  7  and  more  case  limits 0  to  more  4(1-7)o,  2  diameter found  as  by  filter,  to  noise  step  edge  model  the  resolution  ideal  this  can  zero.  edge  To  centrally  D much l a r g e r  than  replacing  with  I/(D-I).  before  7  Allowing  and  the  substituting  2  (2.52) 3D  0  from  the  approaches  7?  0  SNR . 0  zero  u n i t y because c  ratio  place  infinity  narrow  i s therefore  m u l t i p l y i n g by  to  i s l e s s than  removed  noise  power  region  signal  outside  the image  -  SNRQ  approaches  signal  i s simply  Q  the  energy  finite  and go  = 3(D-I)T?  Clearly  SNR  of  the  a  the  the  gives:  4(1-7)o> SNRQ  of  (2.40),  case  to  this  and  of  within  as  Since  through  questions  ratio  remains  features  realistic  image  the  noise  question  extending  render  to  resolution interval.  best  there,  addressed  signal  The  i n the  can  has  zero  .  Also,  as  c r o s s i n g by  increases. for  this  becomes  increasing I , 7 rapidly  Therefore  regions  region  the  f a r removed  signal  to  from  the  zero  generalized  to  the  crossing.  Intuitively previous details for  this  conclusion  e d g e m o d e l s when here,  i t was  s t a i r c a s e and  found  square  fully that  can  be  resolved. SNRi  wave e d g e s .  Without  decreases The  repeating  inversely  presence  of  blur  with  the /J  serves  41  to  decrease  expected a  large  either  SNRQ , and  then,  0 filter  i s an  with  than  1.15o  dimensions image  noise  detail,  enhancing  these  filter guide  only this  edges  Since  a  >  any  of  0.51.  The  p e r i o d i c image  free area  edge  these  those  for this  about  2o  with to  f  by w i d e  areas  structures closer  noise  the  served  t h e edge  edges w i l l  minimum  to on  have  resolvable  t h e e d g e s o f known  image  The  forcing  through  edge  positions with with  reject  the r e s o l u t i o n  analyzed  t h e edge  of will  filter  an  From  intervals  the  widest  serve  as the  detector  maximum a c c u r a c y .  the narrowest  the  a l l noise  objects.  selection  models  However,  out again  the performance of  should  requires  achieved  point  detector  selection.  t h e edge  i s best  has  to overlap  possible.  pinpoint  should As  also  was  possible,  seen, hence  conflict.  One a  noise  o f t h e demands p l a c e d  Ideally  foregoing,  about  the  analysis  nature  detector.  this  a noisy,  resolved, with  for  p o s i t i o n s surrounded  edges.  be  1/a  detail.  conflicting  the  filtering  edge  noise  cannot  f  by  essentially  commensurate  This  edge  on  of the true random  further  0  result,  side  filled  SNR ,  way  narrow  filter  suitable edges  to resolve  level.  generated  magnitude because  and  this  then The  threshold premise  by n o i s e  than  those  where  likewise  incidences  below.  Therefore,  i s to f i l t e r  the r e s u l t a n t  behind  this  generated nature  the noise where  edges  weak  though  or  there  by of  image the  will  a  exceed  a  edges  at  i s that  lower  objects.  noise  corrupted are  t h e image  procedure  would g e n e r a l l y have  of the s t a t i s t i c a l  incidences  conflict  with some the  gradient However,  there  must  be  the threshold,  and  object variety  edges of  fall  threshold  42 selection  techniques  can  only  be  edge  segments.  guaranteed  statistical This  to  objects.  edge  image.  there  Again  concept  segments to  each  edges of  on  high  low  would  store  increase  qualitative the  the  filter then  The  are  chosen  for  object  out  since  of  the and  to  while  small over  gross  detail  are  will  of  of  edges.  the  edge value  the  image,  while  those Though  image-size  t i m e may  m u s t be of  also  executed  at  number  of  by  the  a  influenced such as  object  the  magnitude  another  failing.  [42].  subject  image d e t a i l  purpose  closed  a confidence  Processing  also  as  techniques  structures.  course  the  filtered  retaining closed  c e r t a i n parameters increment  out  the  requiring another  the  is  to  eliminated  algorithm  a  premise  passes over  closed  measures. a  screen  magnitudes  iterative  noise  condition  insure  assign  is  contained  relaxation  adapted  into  techniques  the  same  broken  filter  fully  this  progressively  image  of  the  be  performance  above  ones.  expense of  s e l e c t i o n of  the  larger  basis  formed  level  the  examine  pixel  the  the  observation  fine  i s to  confidence  confidence  Both If  the  in  iterations.  of  at  for  to  on  image of 2  applied  variety  successive  considerably  pixel  can  filter  edges  performance  V g  relaxation techniques  they  are  the  be  repair  a  confidence  since  to  must  are  by  In  i t d o e s so to  each  pixel  confidence  relaxation  array  a  their  image  founded  here  segment. of  edges,  and  [40],  r e s u l t i n g i n an  loop  the  to apply  [41],  basic  closed  approach, is  applied  best,  algorithm  thresholding,  be  unfortunate  segments and  Another  The  is  restore  post-processing  available  can at  produce  To  smallest  that  to  is desired  and  p r e c i s e l y locating that  also  appear.  Furthermore,  a  narrow detail, as  seen  43 in  the  s q u a r e wave r e s p o n s e  substantially  larger  dimensions coincide is  that  during  these  The  idea  the  image  of  detail  process is  gross  thereby  can  detail  turn-on. be  obscuring  The  given  the  conflict,  produce  gross  when i t s  net  high  a  result  confidence  detail.  while  avoiding  the  filtering.  multiband the  filtering  widest and  filter  note  the  progressively  in  close  i s simple. appropriate  narrower  the  to  edge p o s i t i o n s .  proximity  when  Initially  to  filters the  desired  the  level  Then  filter  r e t a i n i n g only  initial  level  filter  of  edges.  edge  This  precision  attained.  method  stages  techniques involves  three  the  averaging then  refining  differing as  of  cones  processing  at  each  images  of  resolution  low  original  followed  the  [43].  by  the  hierarchically  resolution characterizes a  processing  stages  edge p o s i t i o n s  i s given  of  extraneous  of  example  applied  In  of  known  typical  where  the  the  certainly  i s multiband  i s terminated  through  in  of  with  This  A  detail  after  solving this  desired  edges  peak  fine  Filtering  with  image  those  to  the  this than  would  difficulties,  Multiband  the  with  edges  approach  previous 2.5  response  post-processing,  The  plot,  image.  by  a  Often level  pioneering  A  are  simple  relaxation  this  of  work  of  approach  resolution. of  produced edge  class  Kelly by  local  detector  procedure  to  [44]  was  remove  detail.  contrast,  essentially gross  one image  the  V g 2  filter  step.  Use  of  detail  smooths  performs a  wide  out  this  filter  fine  image  operation matched  to  features.  44 Furthermore,  since  i t contains  no this  response  above  Q,  i t performs  averaging  which  does  possess  detection  is  these is  c r o s s i n g s are  such  the  frequency  o p e r a t i o n b e t t e r than peaks.  to a search  guaranteed  in  The  for zero  t o be  act  local  of  edge  c r o s s i n g s , and  closed  no  since  relaxation  stage  necessary.  The the  remaining  o b j e c t i v e of If  of  limited  peaks  the  gross  Begin  and  the  the  possible, equation  The (  within  are  = o  0  size  blur  for the  o  f  signal  <T . 2  level  a  of  detail  search  f o r the  resolution  f2  is  4a  edges w i l l  crossing.  Therefore  the  this  extra  f 2  process  precision  be  applied.  object  noise  ratio  2.2  of  will  image.  which,  the  of  which  and  r e s o l v e the  is  precision  assumed  .  After is  is lost  new  edge about Since  Gaussian  If with  zero crossing  or  until  in noise.  the  the  until a  22  a  position the  filter  old  the  search edges  a  the > 0(  2  crossing  with Z1  95.4% of  interval this  desired in  edges.  f 1  zero  in  of  i s performed  l o c a t e d w i t h i n 2<T  of  locating  by  distributed,  be  width  repeated  attained  i s determined  wide.  f 2  subsequent  4a  2.1  filtering,  interval  confidence,  =  dimensions  image  figures  to  the  D e p e n d i n g on  just  After  were  2 1  on  noted.  interval  4a  the  present.  , to  of  method would  of  image b e s t ,  locations  be  p r o c e s s i n g cone depends  following  the of  provides  T h e  z  the  this  also estimate  one  This  minimum  next 2  the  system.  of  degree  suits  (2.51),  positions  then  estimate  edge m o d e l s  width a  imaging  of  o b j e c t i v e i s p r e c i s e measurement  w i t h an  supply  the  image d e t a i l  interest the  structure  which  the  should interval  level  of  case  the  45  On is a  the  to  examine  is  z  be  other  too  hand,  and  i f the  classify  difficult  to  objective  a l l l e v e l s of  estimate,  then  applied  which  together  span  frequencies.  Since  the  filter  bandwidth  1.2  of  bands  make  allowance  some  octave  spacing  are  is  preferred.  until  the  maximum p r e c i s i o n  this  was  in  f 1  ,  by  of  from  set  1.2  [35]  the  filters  is  with  A  are  a if  However  implementation  widths  to  a  one  choosing  a  simply  method  halved  similar  to c o n t r o l vergence  binocular  must  achieved  after  if  spatial  bandpass  octaves.  ease of  filter  of  or  practical  is  is attained.  Grimson  depth  a  Therefore,  successive  system  disparity  for  to the  information  stereograms.  One is  o  imaging  image d e t a i l ,  overlap  by and  the  a l l  minimum  for error  large  determination  2  separated  suitably  used  V g  octaves,  successive  of  the  a d d i t i o n a l , important elimination  images. edge  model  located the  You  of the  will i t  exactly  true-edge  pseudo-edge this V g 2  zero  pointed  out  way  filter.  below  To the  decreases  with  well  established  for  parallel  .  can  confuse  filter  edges  true  staircase  i t a  quantization  for  zero  a  type  staircase  crossing  magnitude  imaging  attains  filter  the  edge s p a c i n g .  i t ,  filtering  edge p o s i t i o n s .  the  an  Consequently one  another  however,  minimum f  the  increased  2  o  that  with  V g  in  when c o n s i d e r i n g  eliminate  the  size  between  crossing,  multiband  pseudo-edges  that  pseudo-edge  falling  the  recall  decreases  system  two  was half  practical  into  of  a p p l i c a t i o n of  is  this  presence  system noted  finite level,  in  size and  eventually  progressively  using  that  pseudo-edge  size will  Unlike of  The  was  any  before  that  this  that  is  bifurcate narrower  46 filters. diverge  Further from  behavior of  progression  the o r i g i n a l  exactly  will  cause  opposite  2.6  system  Sampling  To been  to that  of  true  under  i t  sampled data the act  quantization. sampling discrete  signals. be  would  allow  a  will  objective aliasing  convolution  the  will  be  In this and  in  become  of  requires section  the  to determine  effects  useful  involves  processing  considered,  has  exception  processing  and q u a n t i z e d .  be  filter  of i t s performance  with  image  image  has proved  derivation  However,  sampled  sampling  that  This  Such d i s c r e t e s i g n a l  rate before  Recall  analytic  form.  processing,  The  Inclusion  edges.  2  circumstances.  filter  of  false  i n i t s continuous  signal  divergence,  i n our d i s c u s s i o n the V g  permitted  various  optical  the  point  represented  these  edges.  a  Filter  2  this  because  that  to ignore  the V g  two edges t o  low r e s o l u t i o n pseudo-edge p o s i t i o n ,  p r o v i s i o n s f o r the d e t e c t i o n of t h i s  multiband  these  a  next, minimum  excessive  in  a  implementation.  the  V g 2  two d i m e n s i o n a l  point  spread  function  is:  V g(x,y) 2  with  frequency  = -[1-(x +y )/2a, ]exp[-(x +y )/2a, ]/(7ra, ) 2  2  2  2  2  2  2  2  2  2  2  2  2  f  ideally  sample  6 (x,y) T  (2.53)  response:  G"(u,v) = - 4 r r ( u + v ) e x p [ - 2 7 r ( u + v ) a ] To  a  .  V g m u l t i p l y ( 2 . 5 3 ) by 2  = I If 5 (x-mT, y-nT) ,  (2.54)  47 which,  on s u b s t i t u t i n g  T = 5o  ,  f  yields  V g(m,n) = -[1 - ( m + n ) 6 / 2 ] e x p [ - ( m + n ) 6 / 2 ] / ( w o * ) 2  with  2  frequency  G"(u,v)  2  2  2  2  response,  = G"(u,v) * [ 8 ( u ) 5 ( v ) ] / T  T  (2.55)  2  T  (2.56)  2  T  where  6  ^ (x) = Z J ( x - k / T ) , '  T  Because  G"(u,v)  causes  a  certain  G"(u,v) .  The  T  is  most  are  contribution  the  worst  case  a  along  frequency  v  previous  can  be  Label  to  zero.  the harmonics  where  <  the  adequately the  f  of 1  )  harmonics  and  that that  represented axis,  and s e t t h e o t h e r  The s i m p l i f i e d  (2T)"  l e t ' s assume  i s negligible,  this  sections,  convolution  ( | u | , |v|  axis  harmonics  response  with  component  between  F o r the time being  axis.  sampled  for  spatial response  becomes  -4rr f 2  G"(f)  To the  u,  off-axis  sampled  frequency  t h e baseband  the  spaced.  from  the  of a l i a s i n g  with  single  consistency  then  degree  closely  the  bandlimited,  aliasing  severe  most  along  i s not  ( m , f o r x=u k =[ n, f o r x=v.  =  T  e x p ( - 2 7 r f of) 2  investigate  spectral £'(f)  2  energy  the  degree  density  * L 6(f-n/T)  2  of  of t h e baseband  = - 1 6 7 r f " e x p ( - 4 7 r f o )/T« a  Most of t h e a l i a s i n g e f f e c t s  aliasing,  2  2  2  .  come a s a r e s u l t  .  l e t us  (2.57)  examine  harmonic: (2.58)  of overlap with  the  48 first  harmonic  degree  centered  of t h i s  will  be  infinity,  £  overlap,  found  by  f  (2T)"  1/T.  t h e amount  To  of  integrating  (f)df = /  estimate  aliased  £'(f)  the  filter  from  f =  energy  (2T)"  ( f ) d f - J £' ( f ) d f .  O  1  1  to  respect  t o t h e baseband <2Tf  i=  1 ~  af  (2.59)  O  A p e r h a p s more u s e f u l measure with  =  ie.,  = X t  a  at  i s the fraction  of  aliased  energy  energy,  oo  1  X £'(£)df/X  o  £'<f)df  o  <2T)-  1  J o =  f"exp(-4Tr f o 2  2  )df  2  1 -  (2.60) r f"exp(-47r f o o 2  Normalize  (2.60) w i t h ( 2 5  S  /  ) "  f =  peak,  2  Figure  points  u  2  2  „9,2j  f  :  2  1  1  6-'«exp(-47r /a )d52  2  remaining  Note  response:  Figure  s  1  S  was  af  peak  of  i n Figure at  .  1  was  plotted  with  evaluated Figure  _  and  1  f  2.5 t h e k e y  The  in  f = (6a )  f = /2/(27ra  a n d 0. 325/CT, .  (2.61)  integral  £' ( f ) a l s o n o r m a l i z e d 2.5.  choice  T = 5 o  <*s>"  a t f = 0.139/a,  previous  2  and the r e s u l t a n t  with  so that  5- exp(-47r /6 )d5-  1 - _ _  frequency  (5a, ) " '  a  The  together  .  1  256  numerically  )df  2  5- exp(-47r /6 )dS-  / o  =  2  f  ),  latter  points and  of  half  leads  2.4 unit the power  to  the  fl.  2.4, t h e n ,  illustrates  how  the  energy  of  figure  49  us Figure  2.4  Proportion of a l i a s e d f i l t e r energy p l o t t e d the normalized h a l f - s a m p l i n g frequency  us Figure  2.5  V g 2  normalized  spectral  energy  density  against  50 2.5  is distributed.  0.14%  of  fs/2  the  = 0.5/a  when  10%  occur  at  shown one  filter  .  f  of  energy  the  filter  the  wide  sample  to  therefore  i s 1.25.  even  f o r the  just  the  On  filter  energy  narrowest  filter  become  f /2.  This  s  of  only  f  above extreme  i s seen  1.46c .  would  be  so  If  1  feature,  above  At  2.4,  the  filter  T^1.25a . f  i t i s seen  f /2  =  s  d e s i r a b l e , the  that  0.4/ff, . degree  be  such  a  a l i a s e d as  then  to As  f  edges.  Figure  lies  to  (2T)" . a  examining  a  to is 6  S  only  Therefore, of  aliasing  slight.  What o f and  second  The  next  the or  previous  higher  greatest  beyond  first  off-axis  the  foregoing,  the  nearest  frequency  u n i t s away  2.4,  i s clear that  negligible,  harmonics  influence  from  i t  assumption  on-axis  w o u l d come  is  =  such  =  r e s o l u t i o n d e s i r a b l e would pulse  f  to T  frequencies  spacing  spectra  at  in  i s above  filter  resolve  1 leading  S  sample  edge  impulses  sampled  of  to a  maximum  8 =  considered  energy  these  only  2.7%  be  f o r s q u a r e wave o r  spacing, of  for  i s contained  can  8=2.92, l e a d i n g  pixel  is  that  Aliasing  earlier,  consist  Note  from  that  off-axis  have  negligible  the  these  is  the  axis,  ie.,  amount of  therefore  centered  confirming  at  8 =1.25.  filter  energy  the  effect?  first  harmonic.  of  the  on-axis  harmonics  harmonic  From least  the 0.8/o  f  From  Figure  above  8=1.25  previous  assump-  tions .  In for  filters  filter, is  conclusion with  8=1.25,  slight,  resolvable.  then,  6 <  1,  the  compared  to  filter  and  response can  even  degree  of  that  the  of  for  the  aliasing finest  be  considered  narrowest i n the  filter  edge models  ideal  desirable response considered  51 2.7  Coefficient  Sampling filter.  Quantization  is  The  second  coefficients.  The  processing  system  mainframes,  to  for  special  [45,  p.  the  16 o r  114].  analysis  be  the  the  error  within  The direct  filter  by  The  can  be  a  the  the  and  64  video  of  minimum  word  size  for  the  large 6  bits  convolution  the  thereby develop a  a  resolution  objective  influence  by  even  discrete  The  of  resulting  bits  m i c r o s , or  form  considered.  the  i s determined  typical  direct  examine  of  from  f o r m i n i s and  of  the q u a n t i z a t i o n impulse  of  this  quantization strategy  for  n e c e s s a r y t o keep  Chan and  Rabiner  errors  response [46].  in  (FIR)  Their  for two-dimensional f i l t e r s  the  one-dimensional filters  techniques in general,  was  first  have  been  and  the  V g 2  particular.  spatial  frequency response  (2N-1) X  (2N-1)  two-dimensional  function  h(m,n),  symmetrical about  H(e  range  i n t e n d e d t o match  finite  here in  quantization quantization  and  representation  a c c e p t a b l e bounds.  form  adapted  8 bits  spectrum  of  analysis  performed  of  used  to  filter  selection  is  Again will  on  step  t o be  implementation  error  step for d i g i t a l  limit  systems  will  first  i 0 l  ,e  i < j 2  )  = z'z' h(m,n)e " e i  n:(N-l)ms(N-l)  = h(0,0) N-1  N-1  N-1  + 2Z  n«1  1  f o r an  FIR  arbitrary,  filter  with  the o r i g i n ,  non-causal point  i s given  spread  by,  i o > 2  h(m,0)cosu,m  41 Z h ( m , n ) c o s w , m c o s u n 2  N-1  + 21  n»1  h(0,n)cosu n 2  .  + (2.62)  m =1 n =1  On  quantization  error  term  e(m,n):  the  coefficients  differ  from  the  ideal  by  an  52 h*(m,n) = h ( m n ) + e(m,n)  (2.63)  -Q/2  (2.64)  r  where  The  < e ( m , n ) < Q/2  error  i n the  quantization E (e  j t J  L  .  frequency  i s represented  ' ,e  j U 2  ) = H*(e \e i u  response  i < i J 2  ) -  N-1  msl  N-1  H(e  The of  upper  bound  the components  |E (e  J U  L  \e  i W  this  i n (2.65)  (Q/2  error  including  H i  +  2Z  2  60,  =  OJ  2  -  i s excessively of  the  by  |e(n,m)|  + 4Z  I  msl  n»1  =  ( n a x  +  Q/2.  + I C O S C J ^  coso) nl  ]  2  .  (2.66)  pessimistic,  r e f l e c t i n g only  distribution  at  the  the  frequencies  distributed  Gaussian. E  L  deviation.  condition  causing  E  L  Therefore  is  possible  To  find  prediction  independence  assumption, coupled  theorem  of  response  statistical  Lindeburg  limit  description  for error  uniformly  the  essentially  standard  This  are  satisfies central  error  model  assuming  coefficient. e(m,n)  2  n7r.  A more u s e f u l formed  e(0,n)cosoj n  f o u n d by m a x i m i z i n g a l l  |cosb>,m|  |coscj n| 2  limit  n«1  N-1 N-1  = Q(2N-1) /2  extreme  N-1  + 2Z  (2.65)  c a n be  rial  bound  from  .  2  2L  error  resulting  2  )  i U 2  e(m,0)cosa),m  N-1  This  1  nil  on  ») | <  ,e  i u ,  4L L c o s c j , m c o s o j n nm1  (w ,w )  as  = e ( 0 , 0 ) + 2L N-1  at  this  with  the  over  a  [ 4 7 , pp. , a  sum  of  can  each  fact finite  262-264]  complete  through  i t s  standard  mean  error  that  the  interval, of  o f many e ( m , n ) ,  a  be  the to  be  statistical (=0),  deviation,  and  first  53 evaluate  t h e ensemble  E*(e  , w  \e  i W 2  )  mean  square of E  :  L  = ( Q / 1 2 ) [ 1 + 4L cos u,m 2  +  2  N-1  N-1 N-1  41 c o s w n  + 16L L cos o),m  2  cos cj n]  2  2  ns1  m=1  2  2  .  (2.67)  nil  Defining: N-1  W (O>,,<J ) n  N-1  = (4N-3)" [ 1 + 41 c o s u , m 1  2  + 41 c o s c j n  2  +  2  2  mr1  n>1  N-1 N-1  161 L cos co,m  cos a)|n]  2  2  1 / 2  m=1 n»1  =  ( 4 N - 3 ) " [ 1 + 4Z c o s a ) , m ] 1  1  2  1  ,  2  [ 1 + 4Z  m=1  = W (w, ) W ( C J ) , M  m  1  1  2  M  expression a  E L  (2.68)  N  the square root  f o r the standard  (cj,,aj ) 2  = [E =  W  into  2  of half  deviation  (w, ,CJ ) ] ' 2  (4N-3)QW  (CJ,,CJ ) a t t a i n s  multiples  2  N  / 2  2  (2.68)  found  W (a>) = [ 1 / 2 + (2M-1 ) - ( s i n M w / s i n w - 1 / 2 ) ] '  Substituting  2  M = 2N - 1  2  where Chan and R a b i n e r have  cos u n]'  nrl  1  .  (2.69)  of (2.67) p r o v i d e s t h e  of E  L  :  2  («, , u ) / ( 2 | / 3 ) .  (2.70)  2  a maximum v a l u e  of unity  the sampling frequency.  a t o>, = co  2  = kir, i e . ,  I t can also  be  shown  that  lim  W (CJ,,CJ ) = n  2  0 < C J , , CJ  1/2,  2  <  ir  .  (2.71)  N -» °°  Therefore a but  a <  E L  E L  lim  o «  by  (4N-3)Q/(2/3),  f o r large  N-»  i s bounded  E L  (2.72)  N,  (u,,w ) = (4N-3)Q/(4/3) , 2  0 < u  1  (  a>  2  < TT .  (2.73)  54 Since  equation  frequency of  this  may  (2.72)  bands,  i twill  section.  be  represents  a d e f i n i t e bound  be u s e d  to estimate  However, e q u a t i o n  pessimistic  by  up  a  over a l l  E L  i n the remainder  E L  (2.73) shows  to a factor  on o  that  the results  o f t w o i n some b a n d s o f  interest.  a that  represents  E L  c a n be e x p e c t e d  spacing of  an a b s o l u t e  Q.  interest, filter,  f o r an a r b i t r a r y f i l t e r  To be u s e f u l ,  the error  expected  b .  Since  k  measure o f t h e  this  be s e e n  t h e c o e f f i c i e n t word  statistics  quantized  with  r e s u l t must be f o r m u l a t e d  fora specific filter  Q will  error  to  level  i n terms  i n a c e r t a i n band o f  specify  N  s i z e c a n be d e s i g n e d  for from  the  this  V g 2  error  bound.  Prior filter given  peak  scaled  will  response.  aliasing  l f  u ) 2  -7ra (6j f  represent  the f i r s t  error  max|(T (e we b  =  The e r r o r  represents  After  to  the  unity  baseband magnitude  response  of  ( i e . , V g(0,0) 2  the  V g 2  = -1) i s  by  G"(u This  to quantization,  i c J  a  2  +  the  CJ  2  ) e x p [ - (OJ  model  2  + oj )a /2]/T 2  for  the  r e s u l t i n g from a l i a s i n g deviation  from  | 6 J  >) - G"(a> w )| 1 f  2  = 6  ideal into  (2.74)  2  V g  frequency  the  baseband  2  the i d e a l response, CJ i s d e f i n e d  i n band, bk, a t frequency \e  2  k  G".  The  by  .  (2.75)  k  quantization,  increases:  the  error  i n the resultant  response,  G*",  55 |G*"(e ' , e i W  0  j W  - G"(w  l f  w )|  <  2  | G*~" ( e ' , e'^ ) - G" i w  1  |(p(ei se *) w  <  iw  max|E cueb  -  (e' t , w  L  (e < , e  GMCJ, j W l  e  ,uj  i W l  ) | +  ,u ) | 2  )|  +  6  .  k  k  (2.76) Since  E  i s Gaussian  L  IE ( e  i < J |  L  Given  ,e )  size,  | < 2a  iu}x  that  d i s t r i b u t e d , t o 95.4% p r o b a b i l i t y ,  V g(m,n) 2  excluding  < (4N-3)Q//3  E L  i s peak  scaled  .  (2.77)  to unity,  with  a t-bit  sign,  0=2"'.  (2.78)  Therefore,  on  total  including quantization  error  word  substituting  (2.77)  and (2.78)  i s bounded  into  (2.76), the  to high  probability  by  |G*"(e  j W ,  ,e  i a , i  )  -  G"(w,,u )|  ^  2  2" (4N-3)//3  +  l  6 ,  cjfb  k  k  .  (2.79) Define  t h e maximum  e the  in-band  = 2" (4N-3)//3  k  i d e a l sampled  error  + 5  filter,  after quantization  as  ,  h  G",  (2.80) m u s t now  remain w i t h i n  a  tolerance  of  |C"(e "\e '*) i  if  the 2  terms  i n band  to design  A  - G"(w  quantized  G"(a>,,cj ) which  j u  more of  l f  w )|  filter b . k  This  2  in-band  k  an e r r o r  essentially provides sampled  representation rejections  in  (2.81)  - 2"' ( 4 N - 3 ) / / 3  i s to maintain  the p r e c i s e l y  common  < e  a  bound new  of e 5  k  to  k  about  of the response error  isin  filter.  decibels,  defined  for  the  56 quantized  The  and unquantized  filters  DL*  = -201og [max IG* '(e  DL  = ^Olo^JmaxlGMe^'  7  k  quantized  original  in-band  in-band  substituting DL*  rejection,  f o  minimum bound  DLk/2  2  (2.82b)  c a n now b e p r e d i c t e d  DL , and t h e  quantization  k  into  from t h e error  D L  .  %  m k / a 0  -  DL*m ,  (2.83)  for a  k  given  DL  k  i s  size  lO  D L k / 2  V(4N-3)]  (1 - lO" "°) ,  (2.84)  Ak/  2  by  (2.82a):  ° + 2~ ( 4 N - 3 ) / / 3 ]  v  = t«, - l o g  .  ju  by t h e minimum word  2  (2.82a)  2  on ( 2 . 8 3 ) ,  = -log [ /3(lO" "  m i n  ) - G" (OJ, , O J ) | ]  + 2"' ( 4 N - 3 ) A / 3 ]  k  fO  established  i a J i  as,  k  ,e ») - G " ( u , , u ) | ]  (2.75) and (2.79)  > -201og [6  The  ',e  rejection  = -201og [l0"  t  i a ,  ) c  inb  where t ^  = {DL*m  + 201og, [ ( 4 N - 3 ) / / 3 ] } / (201og, 2)  k  o  (2.85)  o  and A too  = DLk - D L * m  k  represents  attained filter  when i s  a. f  the 5  k  t  m i n  A , K  in  estimate  For  bound  (ie.,  DL  the frequency  N  and  linearly  i ti salso  possible  of the necessary  fixed  representing  Likewise,  lowest  approaches zero  (2.85) t o i n c r e a s e 6 db,  .  unbounded  more r e a l i s t i c small  k  A , t „ k  w i t h DL*m  m i  K  on  the  —> oo ) .  k  domain, word  i s seen  at a rate  Since  tmin  size,  word  the V g 2  serves  as a  at least  for  from o f one  (2.84) and b i t per  a halving  of the quantization-induced  seen  Figure  from  2.6  that  o f o n e b i t o v e r too p r o d u c e s a c h a n g e of 6 db, i e . ,the t o t a l  error  i s only  an  6 . K  error.  increase  i n in-band  double  size,  in  rejection,  57  0-\  1  Figure  1  4  0  2.6  1  1  1  8  A  k )  1  12  1  1 16  db  Additional bits, At, required to produce a given i n - b a n d r e j e c t i o n c h a n g e , A , due t o q u a n t i z a t i o n K  58 The  filter  the  selection  from  the f i r s t  in  in-band  of  a  harmonic  into  A  for  95.4%  system  available  simply  t o use equation  is  determines  (2.84)  to  K  path  i s  b  DL .  yields  the  word  of a l i a s i n g  determines  probability.  i s of a fixed  clear.  On  size  (2.83) t o determine  K  A .  Finally,  word  size  the other  t,  b  decrease  K  required  From  i n band  The a c c e p t a b l e  K  also  K  now  then  hand,  i t  to  i f the  i s  easier  i f the resultant  DL*m  K  acceptable.  The  design  parameter,  the  right-hand  side  complicates both  approach filter of  matters  i n turn,  just  However, filter  of since  This  i s because N i s dependent  q u a n t i z a t i o n p e r i o d 8a  also  determines  the  N  interested  V g(n) 2  Rounding  for  i n the extent  further  (2.86)  The  point  be d e t e r m i n e d  on  quantization  difficulty  spread  section  of equations  knowledge  low-pass  2  + n )8 /2]exp[-(m 2  2  symmetrical along  2  the axis,  will  2  be  2  used  2  about  = -(1 - n 5 / 2 ) e x p ( - n 5 / 2 )  of  .  f  this  of  the  and provided  optimal response  filters. of the V g 2  numerically.  form:  i s  2  in  The s o l u t i o n  K  recognized this  explicit  2  filter  DL .  requires  approximating  peak-normalized  the  The e x i s t e n c e o f N i n t h e  somewhat.  V g(m,n) = - [ l - (m Since  crucial  equations  i s known, N c a n r e a d i l y  In  one  the  Chan a n d R a b i n e r  method  N.  ignored  of  (2.83) and (2.84) suddenly filter.  outlined  halfwidth  many  t and t h e f i l t e r  period,  a  design  t h e maximum d e g r e e  f  rejection  substitution guarantee  implementation  + n )5 /2] 2  2  (0,0),  .  and  (2.86) we  are  s e t m = 0:  .  to  (2.87)  determine  the  filter  59 coefficients.  Therefore  where  first  is  (2.87)  most  N i s determined  drops below  2"'"'  from  as N =  the  2  1.  n ax + m  r e a d i l y f o u n d by s u b s t i t u t i n g r = n 6 / 2  value This  into  2  of  n  point  (2.87)  and  solving O-r)exp(-r) Given  N Table  value  of n  = TRUNC ( v / 2 r / 6 ) II lists  The  are  equation  (2.88)  equation  word  (2.89) p l o t t e d a g a i n s t  Figure  2.7. i n  the  DL*m  in  K  equation  f o r N.  Some r e p r e s e n t a t i v e Figure  This  2.8.  Substituting curves  The  f o r DL  inverse  process that only  involves tmin  i s  change  representation  of  represented  as  an  observations  i s that  K  including  of  I Iand  equations  a  t min  IS  integer. t  m i n  t oc The  (2.89),  produces  k  or  DL*m . K  = 2 0 , 4 0 , 6 0 , a n d 80 a r e shown o f s o l v i n g f o r tmin  steps.  descrete  in discrete  straightforward.  I I and equation  but s t i l l  two  i s  N, t a n d D L  process  a n d 5 i s more c o m p l e x  recognized therefore  solution  (2.83)  2.7  K  f o r the  (2.84).  figure  DL*m  8  The r e s u l t s o f T a b l e  t a n d 5, e i t h e r r e f e r t o T a b l e  K  15 f o u n d  method.  s i z e s o f 4, 8, 1 2 , a n d 16 b i t s ,  in  the  i s used t o give  of r and / 2 r f o rt = 2 through  Knowing  DL ,  integer,  (2.89)  ( 2 . 8 9 ) c a n now b e u s e d  Solving  in  an  (2.88) by t h e Newton-Raphson  shown  (2.83) and  N being  + 1 .  of  most p o p u l a r  sign,  numerically.  the values  curves  (2.88)  found through  m a x  solving equation  four  = 0 .  t_1  t , r c a n be s o l v e d  truncated  by  + 2"  steps.  given  straightforward.  First integer The  i t  must  be  v a r i a b l e and can extreme  which therefore consequence  - too = A t c a n r e a l i s t i c l y  minimum  should of  a l s o be  these  only  two  change i n  t 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15  Table  I I .  r 2.4532 3.8034 4.8010 5.7082 6.5693 7.4017 8.2144 9.0125 9.7993 10.577 11.348 12.112 12.871 13.626  r a n d j/2r a g a i n s t  •2r 2.2151 2.7580 3.0987 3.3788 3.6248 3.8475 4.0533 4.2456 4.4271 4.5994 4.7639 4.9217 5.0738 5.2204  unsigned  word  size  Figure  2.7  Filter half-width, N, sample spacing 5, for c o e f f i c i e n t word s i z e s  resulting from 4, 8, 12, and  normalized 16 bit  62  Figure  2.8  Quantized in-band r e j e c t i o n r e s u l t i n g from unsigned word s i z e t f o r u n q u a n t i z e d i n - b a n d r e j e c t i o n s of (a) 2 0 , ( b ) 4 0 , ( c ) 6 0 , a n d ( d ) 80 d b  63 integer  amounts.  Therefore,  D L * m , c a l c u l a t e AK t h e n  look  K  the  nearest  infinite  integer unless  i n which  Equipped appropriate Figure  2.9  equation  appropriate the  including  (2.84) of  curve  t round  (2.84).  a was  step  = 20f( t  DL*m  which  tmin check  effectively  These  plots  appear  the appropriate 4.  On  A  choice  t  can  then  integer.  be  into  K  the  select  read.  The t o t a l  in  of  to the abscissa, 5 will  K  t o an  For  word  a  size,  t + 1.  graphical because  approach  N  appears  to  solving  Instead  t  i s  m i n  on t h e l e f t - h a n d s i d e o f  N i sa function of t , direct  m i n  -At)log  l o  At and t m  solution  t h e approach chosen  2  - log  I 0  was  of  solution  [ ( 4N-3 ) / • ! ] }  = 2 , . . . , 15.  m  These  a great in  up t o  involves reference  (2.84).  up t o t h e next  results  2.9, i g n o r i n g DL*m  involved the  next  o n t h e same s i d e o f t h e e q u a t i o n  Since  be  and  K  problem,  straightforward. Figure  from  Since  the previous  appear  to  substituting  reference  largely  K  i s so l a r g e as  the  by  i snot possible.  DL*m for  plot,  the inverse  DL  2.6, r o u n d i n g  equation  indirect,  necessary equation  of  s i g n , i s then  This  K  i s , given  up A t on F i g u r e  f o r A t = 0, 1, 2, 3 a n d  appropriate  fractional  At,  calculated  (2.84)  A  step  case At^O.  with  plot  the first  this  found,  deal  K  too conservative  It  so s o l u t i o n  were  of rounding  and the other K  case  then  of  t  m i n  and N  DL*m  K  i s  p l o t t e d a s shown i n  < 0.  procedure,  on DL*m .  In this  (2.90)  intermediate  i t may p r o v e v a l u a b l e parameters  may e v e n  a n d may  of  be  into  be d i s c o v e r e d reduced  by  a  i s  to substitute  equation that  values  (2.83)  as  t h e tmin  found  b i t and  still  64  04 0  Figure  2.9  ,  ,  .  ,  . 20  ,  .  ,  .  .  ,  40  DL-m  .  ,  .  ,  ,  ,  „  60  j 80  k  Minimum unsigned word size required q u a n t i z e d i n - b a n d r e j e c t i o n f o r At of (a) ( c ) 2, ( d ) 3, a n d ( e ) 4  given 0, (b)  the 1,  Figure  2.9  (c o n t i n u e d )  Figure  2.9  (c o n t i n u e d )  67 maintain  2.8  an a c c e p t a b l e  Filter  We  will  previous size.  Design  now  consider  an example  to the design  choose  an  unquantized  in-band  rejection  implies  maximum  occur  to  pixel  i s 1,25a  Being  degree  energy  contribution  at  = 47r [a =  for  substituted  3  K  2 f  (1  as  word  that the  possible.  This  out e a r l i e r to If this  o  then,  Therefore  the  with pixel  2.7% o f t h e b a s e b a n d in-band  pass-band,  of concern,  energy i s rejection  b , cut o f f at K  i t  experiences  fl. the  Recall  of the a l i a s e d  corresponds  f  response  harmonic  at  along  fi,  consider  either  t o the baseband  axis.  response  ),  into  - 0 . 325/a  (2.74)  W,=CJ  and CJ = 0 2  )] exp[-2TT2 ( a 2  f  t  - 0 . 325)  gives  2  ]  0.208  a, = 0 . 8 .  i s  f  resolution  the resulting  of the f i r s t  u> = 2n( 1 - 0 . 3 2 5 / a  6  desired  was p o i n t e d  = 0.8.  f  filter  .  f  the magnitude  This  o  high  band  of a l i a s i n g .  the contribution  when  small  i s  f o r the system.  plot  narrow  frequency  only  which  found  To d e t e r m i n e  = 27r(0.325)/a  estimate  as  i t  of the  6=1.25.  an a r b i t r a r i l y  greatest  Q  giving  8.  the highest  be  chosen  i t was  the aliased  at this  examine  f  case,  a minimum o f one p i x e l  spacing,  From aliased  produce  i n the application  o f an a p p r o p r i a t e  of a l i a s i n g which f  spacing  To  extreme  f o r the smallest a  chosen unit  a  rejection.  Example  techniques To  in-band  Therefore,  the unquantized  in-band  rejectioni s :  at  68 DL The  ideal  = -201og, (0.208) = 13.62 d b .  K  o  baseband  |G"(0)|  response  = 47r o, ( 0 . 3 2 5 ) e x p ( - 2 7 r 0 . 3 2 5 ) , 3  = DL ,  therefore,  K  magnitude  increase  DL in  2  2  represents  a  response  quantization.  2  20%  increase  in  the  response  the ideal.  i salready quite  k  2  1.042.  a t fi o v e r  Since  a t ft i s  large,  magnitude,  permit  i e .  only  another  25% of the i d e a l ,  5%  after  Therefore  G*Mn) <. 1 . 3 0 3 , and  e  giving  DL*m  K  ^ 0.261, K  = - 2 0 1 o g , ( 0 . 261 ) = 1 1 . 6 7 d b . o  Therefore,  A  which  be r o u n d e d  must  K  referencing DL*m  = 1 3 . 6 2 - 1 1 . 6 7 = 1.95  Figure  11.67  a  total  db in-band From T a b l e  causing verify The  DL*m , K  result  7.5 w h i c h  word  size  K  curve  to  At=3.  5=1.25,  must be r o u n d e d  up  to  i s necessary  On  t for 8  bits.  for at least  a t 0 f o r a =0.8. f  t o occupy  i s DL*m  and  of 9 b i t s  I I and equation  substitute  correspond  f o r At=3  rejection  the f i l t e r  successive  (2.89),  (2N-1) X  N  i s  (2N-1) = 7 X 7  D L , t = 8 , a n d N=4 i n t o K  = 12.49  found  db.  4 To  (2.83).  through  error  be  pixels.  equation  Therefore,  rounding operations, the predicted  to  has  these improved  0.82 d b .  A to  2.9  = 11.67 i s a b o u t  K  Therefore  by  d o w n t o 1.16 t o  9-bit  be a d o p t e d  word  size  instead,  i s non-standard. N  11.48 d b ,  would  decrease  to  additional  penalty of the lost  still  f o r which  I f an 8 - b i t be  4  e =0.267. K  but  system DL*m  K  Therefore,  b i t i s o n l y 0.19 d b ,  were would the  producing  a  6S 25.6%  increase i n response  magnitude  over  G"(ft).  n3  1 0  To the  =  12  7  1  0  1  -13  15  7  0  0 -128-13  12  1  1.25  j A  a  passband, a =0.8  final  i 0  )| =  1  e  K  e  at  f o r the 8-bit  K  fi  f o r t=7.  V g(m,n) bits  values  spacing  equation  evaluate  quadrant  equation  plus sign  into  system,  One  from  2  of  the  (2.86)  i s shown  in  (2.62)  for  for  figure CJ,=J2  found  by  5=5.983 X  The  ,  those  is  will  purposes  1 2  be  f o r which can  be  observation that can  quantization  be  pessimistic  by  the  compare filter  ignored.  The  retained,  magnitude  here  14.42  6=0.625.  substituting 10"  0.190, or  let's  of  f  N=7.  .232.  exercise,  i e . , o =},6,  gives  earlier  of  is actually  to  filter  f  filter  these  error,  found  unity  ,e  prediction  just  of  t r u n c a t e d to seven  actual  As  and  the  response  |G*"(e  7-bit  3 m  gives  2  The  prediction  Substituting  CJ = 0  1 2  2  of  and  128  V g(n,m) f i l t e r c o e f f i c i e n t s f o r sample 6 = 1.25 a n d 8 - b i t t o t a l w o r d s i z e  this  spread  2.10. and  test  response  point 6  2.10  0  2  0 Figure  0  of  into  f  DL  but  K  The  effects  expected  therefore simplifies  to  octave  word  response  which  considered infinite. aliasing  rejections  size  lower of  6=1/1.6=0.625,  (2.74), t h i s  = 2 2 4 . 5 db  next  8-bit  now  for  the  db.  in-band  w i t h the  total  Therefore,  2.94  the  the a l i a s e d  a  db.  at  0  is  the  which again  time  resulting  for  a l l  This o  in-band  f  intents  reaffirms greater  rejection  in  the than  after  70 DL*m —  -201og [  K  = corresponding now This  K  4.169,  the  the  a  next  filter  (4N-3)/»/3  ]  Since  ideal  db  e =0.113.  represents  over of  18.96  to  is  2"'  ) O  e  represents  K  n e a r l y one  highest  octave  i s designed  t o meet  least  c o n s t r a i n t s f o r the  Test  To of  Image  examples of  were c h o s e n .  test  i m a g e s on  be  levels  contains  of a  replacing  each  of  interest,  word  design  principles  size  rejection  i t will  filters  Q.  accuracy  in-band  wider  115 of  meet  at  also.  4 X 4  grey  level  of  noise,  independent to produce  width  The  the  and  SNR  two  kinds  is  The  rings  more c o m p l e x  behavior.  image  and  The  permits  image  was only  image c o n s i s t i n g of  (dark)  and  140  32 is  64  and  alternating of  image  i s 128  block.  by  a  To Gaussian  fifty.  Rosenfeld:  i t s center  that  block  mean  at  (light).  SNR  a  single  include noise  The  light  is defined  by  dark  of  dark  produced  pixel  was  and  eye"  128,  the  image  surrounded  "bull's X  of  series  512  final  of  edge.  one  Rosenfeld  by  pixel  zero a  512  used  and  edge d e t e c t o r s : a  step i t  discussed,  Kitchen  had  various  radius  impression  average  Rosenfeld  filters'  as.a  center.  images of  single  the  r i n g s of  Kitchen  test  since  circle  The and  by  a  brightness:  background  variance  and  of  dark  intensity.  tightest  in  i f the  other,  to evaluate  originally  concentric  chosen  the  and  here  study  constructed  six  which  explored  systematic  two  Kitchen  concentric rings;  will  the  at  change at  improvement Therefore,  filter  tightly-controlled  [48]  of  response  response  Examples  provide  the  narrowest  magnitude  2.7%  case.  f o r the  2.9  a  magnitude  constraints those  the  by  having  the  effects  of  added w i t h i n the  a  manner  71 SNR  = h /a 2  where h  2 n  i s t h e edge c o n t r a s t , g i v e n  The  ring  exactly  s t r u c t u r e of t h i s  p e r i o d i c along  bounded  in  This  permits  the  previous  this image  should  model  o f edge  filter  be  able  to  =  6.96.  since  the f i n a l  local  averaging,  Gaussian, Clearly  this this  seriously  b l u r has a  recognition  of  i m a g e was is a  structure structure  of  noise,  on  blur,  the steady  of  o =1.6 f  for image  filtering  with  0=0.5.  this  than  is  expected.  o = 16. r  one  Though  of  However,  I t i s also  only  resolve  f  that  the  gross  and  little  f o r minimal  i s e s t a b l i s h e d a t 0=5. to fully  o =6.i.  to a periodic  revealed, seen  in  of t h e image, a  response  t o be  0.29.  f o r which  f  Therefore,  by not  about  0.51a =3.55  nature  no  the  rejection  i s chosen, producing  that  edge  steps,  present.  less  i s expected  expected  image,  d e v i a t i o n of  deviation  1.25  shows  state response  i s therefore  ideal  from a l a r g e r  blur  non-ideal  0 of  t o f i g u r e 2.2  the  some  deal  overall  i s expected  reduced  standard  somewhat more c o n s e r v a t i v e Reference  i n a standard  performance  the  128  test  filtering  2  I f t h e edges were  great  degraded  of the  p r e d i c t e d by t h e s q u a r e wave  resulting  there  is  V g  of  realistic  r e s o l v e them w i t h maximum n o i s e  = T/0  However,  applicability  I n t h e 128 X of  being  in structure.  periodicity  result  being  time  m o d e l s t o more  primary  The  T = 8.  1.15  t  two-dimensional  of the r i n g s .  most c l o s e l y  have a 0 of o  The  pixels.  spacing  just  would  spacing  i s eight  same  t o be d r a w n c o n c e r n i n g  structures.  i s t h e edge period  at the  unbounded o n e - d i m e n s i o n a l  two-dimensional image  while  and p r i m a r i l y  conclusions  25.  image h a s t h e a d v a n t a g e o f  the radius  extent,  here as  the r i n g  A  filter edges.  72 Furthermore, the  i t i s expected  poorest The  filter are  noise  test  with  shown  rejection  i n Figure using  coefficients  -point  precision. numbers  gradient. the  vertically  of  detail  the rings  borders giving  pixels  Since  absence  of any e x t r a  intervals  The  the  i ti s  with  claim.  structure.  o  f  Noise,  image  lies  results fast The  floating  by t h e Freeman  direction  of  maximum  to coincide  between  with  horizontally  corresponds  but the  ring  or  t o the borders  structure  image  lies  With  within  resulting  o f 6.4 p r e d i c t s  i s  however, does appear the outer  11 p i x e l s  from  w a s 32 p i x e l s  wide,  40  pixels  Note  i n t e r v a l of  Figure noise  the outer  The ring  removal.  of  the  2.11b  within  i n the form  edge.  resolution  noise  resolution  wide,  the complete  a resolution  i n perfect  no  structure outer  the overlapping  full  the  and  was  rejection.  there  only  inner  i s expected.  detail.  that  circle  that  The f i n e  extreme  dark  behavior  seen  apparent  i s resolved.  noise  beyond  2  2-D F F T .  double  a r e chosen  readily  maximum n o i s e  Note  structures  structures  this  entire  filter  the  The  V g  through  a r e marked  also  the central  of t h e edges  structure  This  to  scaled  regions.  a 0 o f 2, a n d t h e  f  in  edge p o s i t i o n s  i s not detected,  a 0 o f 2.5,  edge  edge  2.11a,  giving  this  The  pointing  done  floating-point  represented  of the test  are.  2cr =32,  precision  was  but  sizes.  o f 1 6 , 6.4, a n d 1.6.  The f i l t e r i n g  pixels.  precision  by t h e peak  also  image d a r k  Figure  filter  filtered  of the zero crossing  adjacent  From  three  were  [49]  side  the test  gross  2.11.  The a c t u a l  positive  of  deviations  a double  filter  direction  of these  image was t h e r e f o r e  standard  convolution  t o do so t o t h e h i g h e s t  ring  validates the  ring  of extraneous  closest which  of  these  i s close  to  73 (a)  (b)  "l.  I.  3  (c)  -..!---•- s:"-  =  !  ».  «".."'  i,,  t':  .-•,•-!..••  : •"•••V*;--I  !..«"•!...•-  J  ....  "v,  !": I •  -—..;—•«.  .--  t'°fi....8-".V if:vJ  *l" -:• .»" ii;  i.  •.  I.  J '  Figure  2.11  Ideal V g f i l t e r e d rings ( b ) 6.4, a n d ( c ) 1.6 2  image w i t h  a, o f  (a)  16,  74 the  2o  noise  f  The  free  a =l.6  filtered  f  have poor  resolution  noise  detail  r e s o l v e d as  the  noise  resides  inner  rings.  structure edges the  to  in  way on  the  systems.  three by  expected  be  third  the  decreasing  expected  a  for  fully  resolved,  the  bright  while  the  (  of  within  the  and  within  to  be  of  one  used  t h a t most n o i s e the  c l o s e approaches  increase  the  ring  the  noise  third  of  interval  but  in  multiband  structures  incidence  remain  of  in multiband  errors  systems  the  are  best  located to  while  pixels.  Perhaps  gradual  curvature  dark  light two  the  this of  the  image.  the  region ring  inner  has has  is  outer  a =6.4  ring  step.  is  dark  attributable  Also  thereby note  that  For pixels  three  pixels.  differ pixels  by  up to more  can  expand  four  w i t h i n 2.5  the  though  to  2.11a  wide  that  ones.  by  by  i n Figure  which  f  tending  the  i s too  a =1.6,  image,  f  widened  border  for  precision  precision  contracted  remains  behavior  one-dimensional  The  contracting  border  fact,  inaccurate,  rings resolved  outer but  i n edge p o s i t i o n In  quantized  central  The  a  Most  resolution  interval  i s however q u i t e  position  resembles  noise.  lies  i s a l s o observed.  regions  Interestingly,  correct  a  outer  accuracy.  pixel  r i n g s so  expected  image b o u n d a r i e s  the  the  low.  general,  instance,  the  This  but  shows  outer  appears  image  clearly  of  the  test  c l o s e s t approach  3.2  search  from  deal  pixels.  i t i s seen  pixels  be  The  the  indeed  beyond  feature  expected  pixel  great  space  i s two  the  resolve  2.11c  a  ring.  edges  two  as  noise  unacceptably  to  In  dark  one  However,  caused  with  ring  well  predicted.  should  Figure  i n the  the  within  exactly  beyond  Only  the  image  rejection.  ring  interval  of  in the  to  six  the  more  closely  a l l edges  form  75 closed the  curves.  noise  indicates  The  regions the  Since  boundaries,  the  infinite results  bits  result  the  0  the  to  the in  at  is  filter  response  eight bits  Figures  position  ideal  2.12a  up  noise  w i t h i n the  present  results border, the  Figure  there  image  run  an  sides are  presence presence  of of  i s the a  s t r o n g dc this  where W (CJ,,CJ ) n  midband  2  offset  has  amplitude.  a  at  six  for  seen  correct  Figure  for  bits. in  edge  precision  there  is  i s increased shows  the  become w e l l - e s t a b l i s h e d i n  the  the  dark  edges  corners to the  2.12b  beyond  image  response.  extent  i n the  of  frequency  this  dc  of  offset  The  the  the,outer  sides.  circularly-connected,  of  in that  f r o m 0.26%  six  structure.  these  expressed  precision  predicted earlier  result  t o e i g h t and  i t is  but  peak magnitude a t The  of  When  of  the  finite  offset was  to  f o r ^=1.6  negative  combined  set  same  rejection  f o r a =1.6,  considered  2.13  the  range  set of  region  the  expected  extra  of  show  Q,  In  this  by  III.  loss  essentially  and  magnitude at  2.13.  form a c l o s e d r e g i o n  with  quantized  Table  t h a t has  edges  edges.  2.12  (  ring  a phenomenon  of  region  in-band  However,  this  a s s o c i a t e d w i t h dark  Figures  9.1%  2.12c  dark  broken  2.12b, a  evident. in Figure  of  to  and  re-established  beginning  is  among  after  filtered  in  occurs  in actuality,  generated  expected  listed  this  regions  coefficients  The  are  were  i m a g e was  wide  are  i s apparently  test  with  change  In  edge p i x e l s  of  However,  pixel  coefficients.  at  f  one  precision  but  a =l6.0  of  2.11c.  results  proportion the  the  respectively.  images  Figure  previous  after  filters,  of  seeming v i o l a t i o n  formation  filtering.  The  only  then  If these  reason  filter  for  and  the  response.  in equation twice i s an  the  The (2.70) mean  expansion  of  76  Figure  2.12  8-bit V g filtered ( b ) 6.4, and (c) 1.6 2  rings  image w i t h  a,  of  (a)  16,  77  Figure  2.13  6-bit V g f i l t e r e d rings (b) 6 . 4 , and (c) 1 6 2  image w i t h  o  f  of  (a)  16  78  Of  16 6.4 1 .6  DL*m  |G"(u)|db  -52.4 -36.5 -12.4  Table  I I I .  ( DL*m  K  K  -  G"(w)|db  8-bit  6-bit  8-bit  6-bit  -0.868 -7.18 -18.96  -11.85 -3.71 -8.43  51.5 43.7 31.4  40.6 32.8 20.8  Expected in-band r e j e c t i o n s a t Q f o r 6- a n d 8 - b i t q u a n t i z e d V g f i l t e r s 2  )  79 the  regions  islands  with  of  the  this  same s i g n a s  sign  the  in large  offset,  regions  and  the  normally  creation  of  the  of  opposite  sign.  In  each of  offset  and  boundary  of  longer  the  2.13a no  i s so  especially  edges over  of  noise  tests  measured  by  then  This  i t  was  felt  would  have  the  least  filter's one  unit  to  point  be  said  offset remove  quantized of  the  spread  effect  on  location  the  in that  was the  filter  by  begun  filter  the  coefficient  of  of  a  The  first offset  unit  per  filter  radius  rounding  process  This  radius  cross-section  greatest  the  removed  i t was  most one  response.  would  quantized  effects  offset,  simple  of  magnitude  therefore  at  Figure  number  the  the  at  the  in  are  offset.  coefficients.  where  f u n c t i o n has  that  i n the  dominated  that d e v i a t i o n from a  the  change  out  they  the  can  To  Figure  that  have  The  of  mean  dc  of  narrow  E v i d e n t l y the  of  many  edge p o s i t i o n s  to note  reduction  2.13b  into  2.11c.  s u b t r a c t i o n was  where  corresponds  than  little  summing a l l t h e  coefficient.  is a  effects  repeated.  subtracted  interesting  in Figure  rejection.  i n the  outer  Figure  structure the  the  circular  Instead,  p r e c i s i o n of  there  i s less  examples,  and  the  those  offset  in-band  is also  its  center.  ring  somewhat, t h o u g h  2.13c  regions  decreased  was  It  the  of The  r i n g s have broken the  However, the  placed.  noise  dark  Interestingly,  evidently diminished  coefficient  with  structure.  intact.  dc  merged  ring  is  Since  i t  r e s o l v e s the with  result  i s more p r o n o u n c e d .  about  2.13c  the  the  ring  crescents.  and  size  2.13,  forms a  island  2.12c  Figure  distorted  associated  correctly  of  filter  longer  fully  edges  has  examples  smaller  images and no  the  slope.  of Here  generally s t i l l  the a  agree  80 with  the  the  two-dimensional  0.79a  .  f  The  center until  filter  of the  the  the  V g  four  filter axis  If  the  The  so  axis  radius  the  at  the  radius  that  the  i s odd,  that  a pixel  performed  22.5° t o  offset  c e n t r a l peak  origin.  are  along  at  half  filter,  2  subtractions the  reached. on  magnitude w i t h i n  one  decreased  eight  are  exhausted.  of  offset  does not  i s much s m a l l e r greatly distort  The  results after  Figures  2.14  little  difference  noise  and  present  indicated  to  Interestingly of  the  a =1.6 f  precision almost  of  that  two  the  sets  same.  the  of  of  area  are  i s done about  i s performed  so  as  the  this  is the  amount process  in the  the  in  have  edge p o s i t i o n s and  noise  edge  a l l  decreased  a  is  the  positions  sets  of  right  cases.  down  significant  are  those  2.11. It  in-band  of  images. of  Figure  in  very  degree  same a c c u r a c y of  shown  there  Even  both  coefficients  the  o f f s e t are  images.  Also,  about  not  removed.  vertical  Fortunately  filter  from c o e f f i c i e n t q u a n t i z a t i o n does  perpendicular  pixel,  qualitatively  accuracy  identical that  filter  one  the  filter.  Note  edges are  therefore  resolution  resulting is  the  the  removal  comparable  images are  resulting  the  i n the  i s about  infinite  concluded  video  2.15.  than  by  is  about  symmetrical  subtraction  For  slope  subtraction  remain  and  axis  mutually  further  the  point.  greatest  h o r i z o n t a l and  then a l t e r n a t e l y increased perpendicular  of  that  alternately  are  others  which  of  The  can  be  rejections to  six  e f f e c t on  r e j e c t i o n a f t e r the  dc  bit the  offset  82 (a)  Figure  2.15  Unbiased 6-bit V g filtered rings ( a ) 16, ( b ) 6.4, and (c) 1.6 2  image  with  of  83 2.10  Design  The of  the  foregoing  D.  design and  Summary a n d  Marr  V g  a n a l y s i s has edge  2  Conclusions  filter,  s t r a t e g y f o r the  coefficient  and  selection  word  size  contemplated.  The  performance  with  t h a t minimum  the  result  were  established.  many  levels  of  precisely,  a  individual found  that  deviation, apply  only  bands  To  the  these  review,  distinct  signal  t o be  periodicity  the  i n these  First  i t  This  necessary  i f  3  db  and  resembles  o .  filter  was  bounds  to  resolve  in  employed one the  and  quantized  the  It  was  filter  passband.  noise  where  octave.  on  task  examined  ratio  i n order  the The  of  standard  word  The  important  to  determine  size,  remaining  coinciding  with  the  increases  directly  with  a  f o l l o w s from  the  the  ideal  through  require t o be of  some  of  the  resolution .  of  the  the  interval,  the  edge  of  any  the widest  3 db  standard but  is  point  of  edges  at  maximum which  model  follows.  guesswork,  r e g i o n of  Therefore,  selection  i t s  of  the  edge  question  resolved at resolving  number  scale  present  result  f  the  S e l e c t i o n of  a  identify  blur  importance a  to  in  edges are  is  follows  feature  s t e p may  point  process  the  rejection  rejection  deviation  embedded  be  is  features.  deviation,  chosen.  that  frequency  design  degree  Of  noise  to noise  least  ratio,  comprehensive  edge d e t e c t i o n  edges  contraints  filter  the  the  i s at  highest  Estimate  b  seen  must  to noise  resolved  w h i c h most c l o s e l y  was  a  optimality  standard  additive  signal  dual  constraints automatically.  stages.  features  filter  f o r the  locate  spacing  provided  a  in  system  tightest  to  satisfy  or  multiband  input  of  input i t  detail,  the  has  appropriate  Also,  passband  e s t a b l i s h e d the  in  greatest allowable  the noise turn noise a . f  84 The above 2.1  s e l e c t i o n of information  and  3 db  2.2  for  filter  Square The  lower  the  limits  shows t h a t  On if  a  of  is  f  5 >  1.25  is  corresponds aliased For  5 <  of  the  that  1.0  of  noise  liberal bound  t o edge  (  only  T/1.36  <  o  f  <  T/1.15;  i f  is  a  a  Reference assert  falls  f  a  be  Figures T,  presence  the  of  filter  to  the  response  Figures  significant  within  may  number o f  o /0 . 51 <o  <a  /0 .2  before  the  b  necessary  2.1  influence  on  o  and  constrains  two  f  b  are  drops not  to  be  since  8  .  f  the  below  to  >  image.  2.666.  decreases  narrowest  that  point  then  and  the  filter  with  excessive.  i t is unlikely  undersampled  i n the  the  Undersampling  since  an  role  rapidly  0.14%.  permissible  SNRj  3 db  to  found  undersampled;  a problem  a  I t was  becoming  original the  = 8a  determine  i t s relation  grounds:  matched  f  i n the  the  T  2.7%,  6 does p l a y  found  l a r g e l y on  that  exceeds  therefore a  further constraints  normalized,  wave e d g e s ,  was  to  the  beyond which  f  state.  a l i a s e d energy  permissible  range,  -  structures  However,  ratio  T/2.75;  0.5la  the  range: <  These depend  energy  square  i n the  to  spacing  f  iteration  a  would choose  ratio  edge  a  practical  filter  one  basis  not  the  with  <  ceases  , where,  filter  structure. noise  T  falls  reference  of  found.  i t is practical. spacing  After  basis  T/5.5  steady  blur  s e l e c t i o n of  sample  to  of  amount  appropriate  the  -  of  Therefore,  f  certain  blur  3 db  0.2a .  on  a l l these models presume  of  and  possible.  deviation  of  variance  s u i t a b l e model  Edges  below  a  now  most  Wave  remains  below  filter  Edges  maximum d e g r e e  2.2  is  standard  Staircase  the  minimum  signal  Derived range This  of  image  on  the is  increasing  in a multiband  to the  signal a  very  a  (  this  system.  85 O n c e Of the  zero  has  of  the  This  accuracy  that  the  standard  i t was  such  rare  or  one  octave  down  to  can  a l s o be  an  signal  such  highest used  Multiband which  occurs  images  a  the  between  implementation filter  influence  this  was  was  response  shown  t o be  standard  peaked  in amplitude  frequency.  considered  implementation.  Twice  the  into  for The  not  standard  Two  known  in  the  i t is  system,  system are  is  spaced  of  interest  S u c h an  approach  for  be  present each  zero  hardware  must  of  advance,  a given  staircase  a  knowledge Since  pseudo-edge  in  band.  crossing  edge  model.  system  is  quantized.  The  direct  form  finite  i n f l u e n c e of q u a n t i z a t i o n frequency  uniform  integer multiples  used.  detail  point  i n the  e r r o r i n the  d e v i a t i o n was at  3 db  be  details  range of  coefficients  a Gaussian  error  full  wider  bands  desired.  a  a much  latter  largest  a l s o remove the  the  impulse  the  t r u e edges  the  with  not.  by  the  of  explicit  frequency  edge model  the  associated  processed  precision the  show  i s always  level  of  (2.51) which  other,  image, the  made.  system  the  ratio  can  an  be  rejection  requiring  from  considered, of  noise  and  an  ratio  i f  c r o s s i n g i n c r e a s e s as  accuracy  t o be  system,  can  final  zero  d e v i a t i o n of  determined  to noise  ranging  to  systems  When  to noise  to c l a s s i f y  image a c c o r d i n g  one  ratio,  across  apart  the  the  be  that a multiband  to noise  In  the  standard  (2.37) and  the  were p r o p o s e d ,  uniform  recommended.  signal  equations  retain  f o r a l l the  even  or  attain  suggested  the  uniform  To  but  signal  that  power by  expected  t r u e edge can  d e v i a t i o n of .  f  systems image  and  o  filter,  filter  the  noise  the  i s given  root of  narrow  s e l e c t e d the  c r o s s i n g about  estimate  square  been  across of  d e v i a t i o n of  reponse. the  spectrum  half  the  the  peak  The but  sampling error,  in  86 db,  was c o n s i d e r e d  t h e minimum e x p e c t e d  quantization,  DL*m .  in  by f i x i n g  large  The q u a n t i z a t i o n  K  part,  this  quantity  advance.  Furthermore,  the  r e s p o n s e w h e r e DL*m  filter  required. since  In  the  the f i l t e r  frequency solely  behavior  to aliasing  with  less  0.625, D L  With  this  simple minimum of  necessity,  this  quantized  equation  series of test presented  predicted  difference  by  the  r e j e c t i o n due  example  DL , K  that  essentially  outlined  f o r6  infinite.  sample p e r i o d ,  5, a  f o r determining  c o e f f i c i e n t word rounding  size, takes  tmin  .  the Since,  place  as a check  fi  high  coefficients,  normalized  (2.83) can serve  image  i n this  3 db o  during  on t h e r e s u l t -  radii  of a f i n i t e  s u c h an image even  square  wave  radius.  lies  I t was a l s o  increases  with  quantization  effect  was  W (CJ ,6J ). 1  in  2  accuracy,  seen  that a  of the  filter  f  The o f f s e t  served  a n d , i n some c a s e s ,  the noise  a strong  to  the  will to  the  crossing decreases.  most  t h e edges  edge into  a  along the  notable  dc o f f s e t p r e d i c t e d  decrease  t o break  to the  resemblance  rejection  the that  rings  of the zero  coefficients, of  matched  of the cross-section  the accuracy  of  f o rexample,  edges  i t s only  while  appearance  many  series of concentric  portion  decreasing  the  wave  though  a  substantiated  I t was shown,  f o r square  f  in  results  chapter.  resolve  n  was  t o be n e a r  K  ideas  On  a s t o be  i s  DL*m .  A  the  precision  a c e r t a i n amount o f  process,  ant  i d e a l in-band  the  procedure  unsigned  the  i s so l a r g e and  constrained  determined  K  in  K  was c h o s e n  I t was s h o w n t h r o u g h  information  two-step  b  value  band, b , w i t h i n  tightly  this  after  i s determined,  largely  infinite  be c a l c u l a t e d . K  size  a t an a c c e p t a b l e  i s most  R  i s  Within  must a l s o than  word  rejection  s e l e c t i o n of a frequency  example given,  cutoff.  in-band  by  position fictitious  87 closed  regions.  filters six  After subtracting out the offset,  performed  bits.  Only  almost  the  as w e l l as the p r e c i s e  noise  rejection  observation  the  edges  formed c l o s e d  regions  without  was  particularly  evident  when  implies that  there  image d e t a i l . further  This  relaxation post-processing  When D. M a r r detector, should  f  major  This  providing  a  bits,  fast no  filter  convolution object.  prove  i ti s felt  suitable  filter  real-time video providing  using  V g  such  the V g 2  f o r a wide v a r i e t y  apply  a  segments.  the  V g  edge  that  this  2  i tc l e a r  r e s o l v i n g edge  that  filter  2  filter  this  this  in a  i s i t s size.  by  means  detection  pixels. performing size  i s  i s required,  for  filters  to  At eight  49 X 49  rate processing  edge  general  objection  FFT, t h e f i l t e r  large  was a  situation  i n r e a l - t i m e and  the  a  f o r the selection of  requires  a two-dimensional  that  published  One r e m a i n i n g  not operating  i s rapidly  convolution  Therefore,  2  to  t o r e p a i r edge  methodology  of the V g f  using  Where  new t e c h n o l o g y direct  design  a = 6.4  for facilities  need  was t o remedy  implementation.  previous  was m a t c h e d t o  I t was f e l t  the  object  implementation  the  However,  chapter's  breaks.  was o u t l i n e d f o r s e l e c t i n g  a t hand.  applying  systematic  Of , a n d f i n a l final  to  An  that a l l  spurious  f o r unambiguously  procedure  f o r the task  disincentive  manner.  stage  no  a n d e x a m p l e s made  of choice  H o w e v e r , no c l e a r  appropriate  the  arguments  i s the fact  any  i s  even a t  slightly.  the f i l t e r  a n d E. H i l d r e t h f i r s t  be t h e d e t e c t o r  detail. a  their  images  quantized  filters,  decreased  important  This  of a l l the test  the  performing  [50],  [51],  filter  will  o f image a n a l y s i s t a s k s .  88  III.  3.1  Introduction  In  the previous  Hildreth  V g filter  that  than  most  edge magnitude  isolating these  detectors  this  in  a  operators  proved  a  Before  choice:  o r whether  successful. Besides  choose  literature  The  evaluation  presenting  on t h e s e  t o design  by  has  edge d e t e c t o r  the  base  this  other  perform  be t h e f o c u s Rosenfeld  choice,  evaluation  i s  already  that  was c h o s e n  of  procedures that  the  o f most o f repeated.  this  section  [48].  l e t us f i r s t  schemes  one  has  scores  of  test  evaluation  work n e e d n o t be  and  of  of  t o a l l edge  advantage  the evaluation  Kitchen  behind  latter  an e s t a b l i s h e d  will  set  a new e d g e  course  that  If  experiments  the  so t h i s  noise.  series  t o be a p p l i e d  I t  procedure  the reasons  a common  one t h a t  detectors,  of  to  a  a published  contain  method  i t s advantage  i s  comparison  measure(s)  approach  claimed  outperform  but lose  incorporating  was  edge  claims.  detector  existing  developed  unreliable  chapter  these  embarking  detect  I t was a l s o  t o adopt  this  edge  number o f o t h e r  of this  whether  would already  popular  that  there  from,  the  is  evaluation  tested.  with  procedure,  be d e s i g n e d  to  should  comparision,  edge  the Marr and  the background  2  and s u b s t a n t i a t e  fixed  an  the V g f i l t e r  The o b j e c t  facilitate  images and  faced  then  was  edges from  controlled  e x p e r i m e n t s must  w a s made t h a t  edge d e t e c t o r s .  object  thresholding.  To  the claim  thresholding  are true  comparison  here.  other  preferred  claims  after  chapter  was more r i g o r o u s l y d e s i g n e d  2  features  to  O P T I M A L EDGE D E T E C T O R E V A L U A T I O N  Before  review  published.  a  89 The  earliest  performance image  and  was t h a t  consisted  regions.  of  differing  two r e g i o n s ,  intensities  of  test  were  36 X 36  contained  wide  output  was t h e n  evaluated  something pixels  generated  weighted  center scores  ratio  due t o t h e p r e s e n c e  measure  panel range  over  output  fall  row.  Later,  Macleod, operators  the  output  [53], the  three  of  center  between t h e Gaussian  Ninety-seven levels  p a n e l s ) , and  After  of the  filtering,  P, a n d P .  image P i was  2  found  edge  o f rows  pixels. edge  image t o u n i t y with  feature  i n t h e image.  orientation bias  to  P  2  a  was a  within  These  the  evaluation  features  uniformly  i f a l l the pixels'  at least  one  pixel  o f edge d e t e c t o r s  by r o t a t i n g t h e c e n t r a l edge p a n e l  respect  A  t o produce a binary  random  panel  right  i n v o l v i n g the proportion of  signal  for totally  the  The  and l e f t  level.  the fraction  contained  within the center  investigated  Only  which from zero  distributed  60° w i t h  representing  feature  ten different  two p a r a m e t e r s ,  to noise  test  statistics  feature.  by a d o p t i n g  sum o f a l l t h e e d g e p i x e l s  continuity  panels  intensity  i n the right  using  The  containing  t o s i x b i tp r e c i s i o n .  means  detector  [53].  by i n t e r p o l a t i n g  was t h r e s h o l d e d  of a signal  edge  d e v i a t i o n o f 24.  t e n images p e r c o n t r a s t  resultant  which  vertical  c o n s t i t u t e d t h e edge  (differing  approximately  array  Gaussian  and found  truncated  of  [52],  pixel  as three  images were g e n e r a t e d  contrast  the  a  b u t a common s t a n d a r d  s i x columns  other  such  of  which  mean  evaluation  o f Fram and D e u t s c h  These were a r r a n g e d  left  ramp,  quantitative  per was  1 5 ° , 3 0 ° , 45° a n d  to the v e r t i c a l .  edge  operators  and Rosenfeld-Thurston. was c h o s e n h e u r i s t i c l y  were  examined:  The t h r e s h o l d  the  Hueckel,  f o r each of  by i n s p e c t i n g a s m a l l  these  sample of  90  outputs a  from  level  "well the  the test  at which  found  This  i s perhaps  Fram and D e u t s c h a p p r o a c h . this  judgement  by  thresholding the  user  does not f a c i l i t a t e comparison  and  design  component  detector  of  when a p p l i e d locations  Though  later  and  evaluation  of t h e i r  correct  and  with  an  ideal  the  edge to  with  performed  i n the absence of n o i s e .  noise.  minimum e r r o r maximize whether  The is  the  edge  The  The  decision  the a given  evaluation figure  rule  f o r the of  pixel  procedure  of merit  qualitative  makes  comprehensive The  of three  filter  an  of  and a  analysis  edge  parts:  comparison  the figure  involved  output  a  amplitude  orientation  and  I d e a l l y an  edge  declining  The a b o v e a n a l y s i s  was  conditional probabilities  additive used  a  zero  by  of  correct  assuming  mean  to formulate  selection  making  which  and  were e v a l u a t e d  r e s u l t s were a l s o  probability output  with  method i s  and a r a p i d l y  from c e n t e r .  corrupted  a more  center.  show no o r i e n t a t i o n b i a s  with  edge q u a l i t y  differing  and f a l s e edge d e t e c t i o n  ideal step  Gaussian  edge  at  response  correct  a  the f i l t e r  displacement  a  a  difficult.  consisted  sensitivity  of  respect  their  f a l s e edge d e t e c t i o n ,  The  produce  methodology.  analysis,  measurement to  developed  paper  should  an  claim  of the procedure  [54]  operator  for  they  acceptable  automation  computation.  deterministic  selecting  difficulty  procedure does r e q u i r e  sensitivity  probabilities merit  the greatest  concerning  total  Pratt  edge d e t e c t o r  edge  and then  of t h e r e s u l t s between r e s e a r c h e r s  Abdou  evaluation  operator  t h e number o f p i x e l s a b o v e t h r e s h o l d  edge".  unbiased,  of  images f o r each  white a  Bayes  threshold decision  to  as t o  c o n s t i t u t e s an edge o r n o t .  of p r i n c i p l e i n t e r e s t here  comparison.  To p e r f o r m  this  however  evaluation  91  the  edge o p e r a t o r  vertical of  step  a single  the  of f i x e d  height  column of h e i g h t  addition  deviation,  a.  e d g e map.  The  F  i s a p p l i e d t o an N X N t e s t  h s m o o t h e d t o a ramp by  h/2  at the step,  of  white  Gaussian  The  output  i s thresholded  figure  = max{  of m e r i t  I,, I  noise  and  of  i s defined  1  a  inclusion  corrupted  varying  to  by  standard  produce  a  binary  as  [ 1 + a d (i) ]"  }" Z  A  image c o n t a i n i n g  2  1  i= 1  where  Ij a n d  distance  Clearly  and  i t h edge a  i s  edges.  F=1. The  and  This  intent  present  which  near  class  thresholding  of  type  [18],  template  operators  [57]  and  3-  5-level  were p l o t t e d  was  as  SNR These  exact  Rosenfeld  =  ( h / a )  extended  of merit  2  the  to  width  P,  Sobel  masks.  the signal  of  P,.  operators and  gradient), The  signal  explicit  [56, p.271],  (compass  of  the enhancement/  differential  template  to diagonal  d ( i ) a d d s a more panel  1/9. edge  the p r o p o r t i o n of  e x a m i n e d was  [55]  edge  ideal  i s similar  the fixed  and  against  figure  to noise  of the  Kirsch  of  ratio  .  same o p e r a t o r s who,  also  included the  evaluations defined  was  than  ideal  emperically set to on  edge o p e r a t o r s  of P r e w i t t  the  only  However,  [55]  to  fall  i s to estimate  t o F,  Prewitt  and  constant  figure  t h e edge.  which  Roberts  edges  detected  technique  goodness of f i t aspect  The  pixel  of t h i s  Deutsch  and a c t u a l edge p o i n t s , d ( i ) i s t h e  a scaling  i f a l l t h e image  position  pixels  are the ideal  A  of the  position,  Fram  I  merit which  (3.1) were a l s o  however, adopt a very  evaluated different  by  Kitchen  figure  and  of m e r i t .  92 The  principal  is  that  edge  analogy  to  produce  a  but  drawback  the  P .  Peli  higher  score  for broken,  They c l a i m  and  Indeed even  were  presented.  Two  b o t h of 0.  ideal  replaced added  at  a  with  10% noise  the  a  detected  step  qualitative observations broken,  was  and  "salt  were  discontinuity measures, as  the  perfect  type but  of  to  account.  edge  detector  results and  background  of  smoothing other  edge  contour  to  was zero  12  was  0 and  15.  The  two  broad  Though  seven  of  only  merit;  edge.  two  and  The  the ideal  coinciding with  hand,  or  noise  was  and  into  ideal  image:  texture;  results  the  given.  test  9,  of  of  square,  pepper"  figure  single  not  background  c l i p p e d at  the  broken),  perfect  into  the  6,  no can  F does  a  simulate  single pixel  before  on  and  the  from  for  their  qualitative.  Pratt's  to a  a  and  grouped  and  F  images were not  d e v i a t i o n 3,  edge p o i n t  defined  on  object  probability  devised  of  to corrupt  measures were c o n s i d e r e d  position  ideal  applied  standard  Abdou  subset  the  resultant intensities  presented: of  of  of  ramp; b i n a r y 20%  of  is  most c o m p r e h e n s i v e  superimposed  between  quantitative,  quantitative  variance  and  measures  categories:  edge  point  edge  study  i s the small  than  i s that  the  a  that  images were used:  dimensions  step  report  problem along  a  test  separately  five  5%,  performance  were  basic  The  mean G a u s s i a n added  though only  intensity  by  the  approach  g r e y - l e v e l 15  T h r e e methods were the  that  [58]  measures  Thus t h e r e  t h i c k edges  edge p o i n t s  their  published,  grey-level  Malah  Malah have a l s o undertaken  those  circle,  of  and  Pratt evaluation  considered.  fact,  evaluation.  a  Abdou and  In  2  distribution  Peli  the  c o n t i n u i t y i s never  t h i c k edges.  take  of  a  the  ramp.  The  involved  such  produced double  (perfect, edge,  and  93 distortion succeed the  through  in  of  filling  subjective  operators  of t h e edge.  of  structure.  On  for thresholding  Generally, extends  that  circular  object  all  qualitative poor  this  basis,  the  t h e r e s u l t s was  Peli  and P r a t t .  to their  Regrettably  with  work.  Malah approach  they  However  i s also  amenable  a prior  Malah  For instance,  evaluations,  of edges a t  as the  results  operator  in  could common  of the P e l i  A fully  similar  a  by i n c l u d i n g a  and P r a t t  one  and  by c o n s i d e r i n g  Also  drawback.  with  t h o u g h no  parallels  measure  the quantitative aspect  t o comparison  [ 1 8 ] , Hale  such observations  o f t h e Abdou  i t sgreatest  knowledge  given.  and  chose only  and  detection  were c h o s e n ,  o r i e n t a t i o n s c a n be o b s e r v e d .  aspect  measures scores  Edge  Roberts  t h e e f f e c t s on t h e e v a l u a t i o n  be n o t e d .  approach,  evaluation  quality.  [19] operators  continuity sensitivity  that  rigid  image  the approach of  o f Abdou  possible  qualitative  selected which d i d not require  and Rosenfeld-Thurston  system  The  t h e gap between  impression  were  t h e image  [23]  shift  work,  and  automatic  i s certainly  preferable.  Shaw  [59]  evaluation. ratio  This  and change  image  presented  when  included  i n edge  compared  examined, t h i s  little  information  edges  simple  t h e change of output  orientation  produced  image.  detector  signal to by  a  Compared  a p p r o a c h was n o t v e r y  on  edge  noise  distorted  t o the other  useful.  I t provided  t h e goodness of f i t or the c o n t i n u i t y of  found.  Three schemes  fairly  to the ideal  methods  the  a  basic  were  failings  noted  i n a l l of  by K i t c h e n  the  above  and Rosenfeld.  edge With  the  evaluation exception  94 of  Shaw, t h e y  position.  a l l required  While  this  statements concerning not  general but  lack  the  scores. change  Finally,  direction  of  a l s o noted  t o make  Edges t h a t  the true  Ideally  perpendicular adjacent  edge  edge,  a r e fragmented  receive but  schemes used  edges  i s the  similar similarly  consistency  i n their  evaluation  t o compare t h e  o f edge segments between n o i s y  and n o i s e - f r e e  the  edge  and  direction in  a  should  manner  be  everywhere  consistent  with  pixels.  these  undesirability  idea  are  but only  t h e edge g r a d i e n t  to  To a d d r e s s  developed  detected  Shaw n o t e d e d g e d i r e c t i o n s ,  in direction  edge  definite  techniques  by Mero a n d V a s s y ,  none o f t h e s e  the  true  images where t h e edge p o s i t i o n s a r e  of a c o n t i n u i t y measure.  edges.  Only  images.  the  failing,  the opportunity  the  from Abdou and P r a t t a s p e r f e c t l y c o n t i n u o u s  displaced  the  world  of  precision, similar  c o n s i s t e n t l y d i s p l a c e d from  scores  in  Another  knowledge  provides  spatial  applicable to real  unknown.  prior  a fully of l o c a l  criticisms o f human  automatic edge  and the e a r l i e r  ones  intervention, Kitchen  edge e v a l u a t i o n  coherence.  and  technique  concerning Rosenfeld based  on  95  3.2  Kitchen-Rosenfeld  The premise Edge  concept that  a  line,  chose  of  local  i d e a l l y edge  coherence  be a d j a c e n t  Evaluation  of  the  scheme.  wide.  these The  local  edge  i s ever  holds  adjusting  The No  edge  images  about  Threshold  selection  evaluation  technique,  and K i t c h e n  threshold  the  previous and  chapter,  Also,  because  edge,  there  that  edge  component of  simply  "true"  that  2  edges.  This  comparison the V g 2  edge  of  i s a major  that  filter  the  theme  interest  i n the  cleans  this  that  the  t h e edge evaluation  f r e q u e n t l y made i n t h e  returns only  filter,  strengths  noise  key  threshold  propose  maximizes  2  The  underlying  which  V g  similar  which  optimize  i s known.  and R o s e n f e l d  i s some c o n c e r n t h a t  of Abdou and P r a t t ,  terms, i t  to  perfectly  edge e v a l u a t i o n a p p r o a c h  with  filter  little  i s the claim,  the V g  In general  t h e p r o p e r edge d e t e c t o r  i s  detector  tuned t o the strengths  facilitates  therefore  determination  parameters  Of more s p e c i f i c  edge  2  continuous  well  setting  measure.  V g  like  one  in their  knowledge  which  setting.  of  into  thinness  operator  i s often  evaluation  should  be t h i n  and R o s e n f e l d  two a t t r a c t i o n s .  of i n t e r e s t  optimal  line-like.  required.  approach  parameter  agree  direction.  position  in  should  components  pixels  t h e l o c a l edge d e n s i t y .  performance  be  on t h e  component, c o n t i n u a t i o n , measures  measures  permits  Kitchen  two  first  to which adjacent  This  founded  locally  and t h e r e f o r e c o n n e c t e d ; and they  evaluation degree  should  is  has t h e r e f o r e two c o m p o n e n t s : edge p i x e l s  incorporate  the  coherence  features  i d e a l l y one p i x e l  to  edge  i s therefore  and  therefore  i n other  t o 2o  f  from  thin  the  filters. object  an e v a l u a t i o n measure, such  scores  any  noise  clutter  as  found  96 near  the  merits.  true The  The  Kitchen-Rosenfeld  Kitchen-Rosenfeld  neighbourhood in  this  image must  i s present  This  direction  central  Pixels  in  direction shows  a n d what  in  the  numbers  edge  filtered  neighbourhood  to  an edge n e i g h b o u r h o o d  the  3 X 3  the  Each or  of that  are  not  and t h e p i x e l  an  edge h a s . I f the the  neighbourhood.  identified  central  pixel  i s an edge p i x e l edge  i t s  pixel  i n 7r/4 i n c r e m e n t s .  i t are called  relative  for  bias.  image.  the gradient  neighbourhood  such  as t o whether  by r a d i a n s  3 X 3  surrounding the  edge  direction  filter  2  examines every  information  is signified  pixel  pixels  evaluation  contain  the V g  s c h e m e c a n show n o  i n the thresholded  edge  eight  edge h i g h l y w o u l d p e n a l i z e  pixel. labeling  by  their  . Figure  3.1  system.  3 2 1 4 X 0 5 6 7 Figure Scores  for  separately single edge  pixel  neighbourhood and  thinness  neighbourhood measure  for a perfect  and  whose v a l u e edge.  The  are  then  first  computed  combined  ranges  from  into  0 for a  continuation  a  poor  score  is  by  ( L(k) x + R ( k ) ma  L(k)  a n d R(k)  the  edge  are  this  evaluation  C =  These  Edge  continuation  over  to unity  given  3.1  m e a s u r e how  to the l e f t  functions  given  )/2  well  or right  are zero  explicitly  ma<  by:  ,  k  = 0,1,...,7  the p i x e l  .  at location  (3.2) k  continues  o f t h e edge a t t h e c e n t r a l  i f no edge  i s located  at position  pixel. k,  and  97  L(k)  ( {  =  a(d,d  )a(7rk/4 ,d+7r/2),  K  0,  K  (3.3a)  )a(7rk/4,d-7r/2) , i f k i s an edge  0,  •  pixel  otherwise;  a(d,d  R(k)  i f k i s an edge  pixel  otherwise;  (3.3b)  where  a(a,0) and  d ,  d  0  1  - |a - 0| ) / T T ,  = U  d  f  7  = edge  gradient  direction  at neighbour  d  = edge  gradient  direction  of the  a(a,0) measures t h e agreement between t h e angles of  t h e minimum a r c s e p a r a t i n g differ  it  when  they  pixel  k has an edge d i r e c t i o n  then  a(d,d ) K  by  will  the premise  requires  weighting  higher  than  the  a (7rk/4 ,d±7r/2) and  d±7r/2  the right  evaluation  Only  thinness  pixels  a perfectly center  L(k)  a n d R ( k ) max  The the  term,  of  L(k)  7rk/4  score,  from  edge, t h i s  i s simply  edge has o n l y fraction  i f neighbour edge  locally  score.  line-like  provided  edge  by  the  to neighbour  perpendicular  t o the immediate  which  pixel  to the central  i s  are  a=1;  k to  left  and  in  the  involved t h e maximum  values,  chosen.  i n the neighbourhood thin  be  direction  direction  R(k)  T,  equal,  a perfect  i s the direction  pixels  gradient  , are  are  to  weighting  three  and  R(k)  should  or right-ward  the  a and 0 i n terms  Therefore,  perpendicular  This  where  a=0.  and  an edge  rest.  When t h e y  center.  i d e n t i c a l to the central  neighbours  of the c e n t r a l  L ( k ) max  that  i s the l e f t -  center.  radians,  drive  However,  them.  k;  that  defined  a r e non-edge  two o r l e s s  i s given  by  as the f r a c t i o n  pixels  of  pixels.  Since  adjacent  to the  98 T  = max{  6,  no. of non-edge  C and T a r e then the  central  is  of  zero  ( 1 -  Finally,  One last  E i s averaged  images  the "bull's  by Abdou review,  115  between 32 to  pixels 128 X  pixel  of  proposed an  ring  by  on an o v e r a l l  128 b y  have been  for  explored image  was  a  r e p l a c i n g each  g r e y - l e v e l equal f o r study  of 7  vertical  was  an  series  of  and  pixel  dark  rings  line-like  with  similar  in  ramp  The  the image  noise.  remain  grey  alternating  i m a g e was  block  in a  of  115) o f  with  radius  width  reduced  a  This  means o f s u b j e c t i n g t h e edge o p e r a t o r s which  whole.  circle  because i t s r i n g s of d i f f e r i n g  but  i n the  constructed  (grey-level  background.  and i s  image o f p u r e  five  7  Rosenfeld.  step  t o that of the block.  orientation  of  found  introduced  originally  4 X 4  ranging  compiled.  by K i t c h e n  already  140) a n d d a r k dark  suitable  f o r t h e image as a  the single  image  values  a l l t h e edge p i x e l s  and t h e l a s t  (grey-level  conceivable  locally  was  E a  evaluation process,  I t c o n s i s t e d of a c e n t r a l  surrounded  excellent  every  eye" ring  and P r a t t , the  light  results  over  were  another  512 X 512 a r r a y . level  measure  Fortunately selection  to this  an e v a l u a t i o n s c o r e  test  chapter,  used  aspect  to give  Various  be e x a m i n e d .  a l l the other  to provide  was  evaluation  adjusted  t h i n n e s s and coherence.  after  Three  (3.4)  (3.5)  coefficient  the only q u a l i t a t i v e  image  the  .  ) T,  7  to unity will  performed  To  +  7C  7 i s a weighting  balance from  combined i n t o  } / 6  pixel:  E = where  pixels  single  image  was  provided  t o edges i n approximately  c o n t r a s t t o t h e Abdou  and  Pratt  99 image  (due t o t h e b l o c k  The where  vertical  the  left  grey-level Pratt  step  final  contained standard  sense  an  study  Gaussian vertical was  noise step  to  and  (3.1),  was a d o p t e d  derived  shown  Pratt  permitted  Basically, and  exact  provide  this  direct  the  to  signal  1, 2, 5,  noise  was  largely  128 a n d  edges,  in a  zero  mean  Kitchen  reduction) the  to  noise,  noise  and a,  ratios  10, 2 0 , 50, a n d 100. definition,  step  height,  ratio  equation  h,  i s  i s exactly  where t h e s i g n a l  25.  double  power  was  image.  detection and  schemes used by Abdou and Rosenfeld  of the r e s u l t s ,  the  but  detector  ratio  to noise  step  evaluation  f i n d i n g s were  P r a t t i n showing  o f mean  (after  deviation of  chapter  edge  of  pixels  independent  rings  seven  signal  comparison  merits their  noise,  values:  fora single  same  64 X 64  i n s p e c t i o n o f edge  where t h e i n t e n s i t y  i n v e s t i g a t e d by  relative  to  i n the previous 2  were  of  signal  that  t o be h / 2  The  the  Pratt's  note  and  images.  The s t a n d a r d  spaced with  Abdou  However,  world  added  images.  logarithmically  of Abdou  c o n t a i n i n g no w e l l - f o r m e d  influence  was  also  random n o i s e  allowed  image  the  adjusted  that  This  s i m u l a t i n g some r e a l  To  image was  independent Gaussian  in  half  g r e y - l e v e l o f 128.  d e v i a t i o n 16.  performance  array  g r e y - l e v e l 115 a n d t h e r i g h t  t h e ramp d i s c o n t i n u i t y  random n o i s e  only  pixel  column a t the j u n c t i o n of the d i s c o n t i n u i t y  an i n t e r m e d i a t e  The  was g i v e n  To s i m u l a t e  the single  given  i m a g e c o n s i s t e d o f a 64 X 64  half  140.  averaging).  This  and c o n c l u s i o n s  schemes  i n accordance with  t h e same m e r i t  also.  to  be  about drawn.  t h o s e o f Abdou  ordering  of the  edge  100 detectors  tested.  However,  convergence at high Kitchen-Rosenfeld  merit.  low  scores  seems t o i n d i c a t e attribute  and  that  SNR,  tends  local  monitoring,  whereas F  fraction  present.  The  edge  estimated.  total  area  ideal  drawback  has  an  the  premise  inherent  However, high  that  this  scores  illustrate, 3.2  128  below.  of  premise  sides  consider  acts  the  of  the  edges a r e , an  overall  degree  r i n g s image  2.11c,  the  t o remove  curved  against  test  epf  the minimally  how  noise to  perfectly epf can  be  With  a  i s f o u n d t o be epf  i s  simply  centered  on  the  i t s influence).  approach  This  should  be  i s a  locally  the rings test  edges  of  i s difficult  ideal  ideal  edges.  features  pixels  i s 1218 p i x e l s .  the  pixel  of  the nearly  the ideal  image  image  indication  the  by c o u n t i n g  pixels  step  edge  figure  structural  the  the Kitchen-Rosenfeld  ideal  individual  i s t h e edge  as an  and  circumference  bias against  when  the  Figure  2  This  i s a  indicating  ( e x c l u d i n g the s i x columns  edge a t t h e image  One  for  ring  For the v e r t i c a l  6 4 / ( 6 4 X 5 8 ) = 0.017  can serve  However,  of  since E  the  constant.  and P r a t t  throughout  resolved,  of  of  i s a more  the proportion of  epf  pixels  The  image  0.074.  ideal  fairly  measure  a l l show a  spread  well-formed  used  This  are  analytically.  resolved  total  edges  the  t h e Abdou  how  scores  edge.  quantity  that are edges.  the test  evaluate  than  locally,  (epf) representing  examined well  t o remain  i s more o f a s t a t i s t i c a l  important  whereas  i s not s u p r i s i n g  goodness of f i t t o the i d e a l  An  and P r a t t  edge c o h e r e n c e  o f an edge d e t e c t o r  In a sense t h i s  measure  t h e Abdou  are perfectly  curved  image  i s that i t result  line-like. reducing  resolved.  neighbourhood  of  of  To  Figure  101 TT/4  V//A  Figure  The  arrows  pixel  3.2  Minimally  indicate  i s a t k=3 w i t h  L ( k = 3)  curved  neighbourhood  t h e edge g r a d i e n t direction  directions.  7r/4 g i v e s  leftward  The  leftward  measure  a(k7r/4,d+rr/2),  = a(d,d ) 3  = a(0,*r/4) a ( 3 7 r / 4 , 7 r / 2 ) , = 0.75  The  rightward  the  center  pixel  pixel  X 0.75  i s a t k=6 w i t h  giving  rightward  6  = a (0,0)  1.0  The  continuation  C =  a  measure  a(37r/2,-7r/2) ,  1.0  X  measure  7  of  =  1.0 .  of this  sufficiently  0.8  neighbourhood  (the preferred  i sperfectly  neighbourhood gently  neighbourhoods, neighbourhoods analyzing  with  i s therefore:  0.78125 .  neighbourhood this  0 i n agreement  ( L ( k ) + R ( k ) )/2 = ( 0 . 5 6 2 5 + 1.0 ) / 2 ,  =  Given  direction  a(k7r/4,d-7r/2),  R(k=6) = a ( d , d )  =  = 0.5625 .  would  they  present. image  the final  be E=0.825.  so as not t o  but  the rings  thin),  be  do form  This fact scores.  value)  and  T=1  evaluation  The r i n g s  test  dominated  by  a substantial s h o u l d be k e p t  (this  score  for  image  curves  such  curved  fraction in  mind  of the when  1 02 3.3  Evaluation Experiment  Since  Kitchen  evaluation  scores  reproducing results. at  and  a l l of  one  Rosenfeld  these  i t was  of  results  the given  that  the technique  coded.  Only  the three-level  for  this  experiments Rosenfeld process for  were  published  these  to the  V g 2  re-evaluate and  properly  and  matching  complete for this  experiments  seen i n  t o a c t as a reference understood operator  set  of  in  part  reproduction  three-level  to  could  be  evaluation  operator.  and t o determine  of  produced  published.  I t was  Kitchen explore  the ideal  and the  choice  The  Marr  could  and  Hildreth  resolution  through  chosen  V g 2  the  V g 2  fast  represented  across  deviations  The  were  then  the  to double  entire  zero  was  crossings  i t  correct.  published.  fast  precision The  test  convolution The  filter  floating-point filter  standard  chapter.  detection  generates.  of  confidence  transform.  the  those  a p p l i e d to the  of the l a s t  for  with  t o those  of  image a r r a y .  designed  essentially  was  use  Fourier  were e x a c t l y t h o s e  filter  evaluation  the implementation  proceed  filter  s e c t i o n through  two-dimensional  coefficients  could  be c o m p a r e d d i r e c t l y  images of the p r e v i o u s the  that  e v a l u a t i o n m e t h o d was filter  2  template  i n e x c e l l e n t agreement w i t h  therefore concluded  of the V g  the r e s u l t s  the  results  Kitchen-Rosenfeld  Evaluation  via  the  was  to fully  of t h r e s h o l d s e l e c t i o n  experiments  that  since  only  designed  template  value  complete  7.  The  the  task  been  the  comparison  prudent  operators had  little  for  considered  indicate  chosen  published  o f t e n edge o p e r a t o r s ,  However,  least  Procedures  of  However,  edges this  103 evaluation  scheme a l s o  assigned  to  conventions and  each of  m a g n i t u d e and this  is  dimensional  of  must  be  made.  array  N  m a g n i t u d e s and  positive  to  of  i n the  Since  a  a  will  border  adjacent  set  pixel  may  taken.  formed,  the  was  pixel  was  intensities chosen response  an  h a v e up  a t t r i b u t e s was gradient of  because  of  reasonably  such,  of  special  i t is  X  In 2N  the  assign  with  the  four  junctions  on  is  borders  edge This  pixel  this  to  output  array.  the  a  not  array.  to  output  is  between  s i z e 2N  r e s u l t of  a l l the  the its  to  unconventional Once  zero  zero  a  approach  such  calculated  calculated using across  The  two-  the that  of  dark  with  the  image.  gradient  slope  As  adopted  the  the  Though  negative  between p i x e l s i n was  as  crossing  output  arrays  associated  background,  the  zero  the  defined  certain  to  the  continuous  edge a t t r i b u t e s t o  negative  First  in  of  crossing. be  a  be  magnitude  segment.  pixels.  p i x e l s of  the  a  positive  pixel  the in  The  was  array  convention  zero  input  concept  junctions  assign  the  at  direction  requirement,  retained.  First,  any  d i r e c t i o n s to  information  objects  the  X N),  was side  with  another  represent  (size  edge  vertically  creating  convention  gradient  from  this  segment  discrete  intensity  associated  explicitly  the  a  and  were  zero-crossing  in  of  inherently  a  fulfill  chapter  plane,  horizontally  lieu  of  a m a g n i t u d e and  To  straight-forward  accomodations transition  previous  d i r e c t i o n of  a  that  edge p i x e l .  the  direction  requires  i n the  crossings simple  approximates  that  two  and of  border  solving i t s point  following  adjacent  crossing.  simplicity,  a  to  to  was  manner.  the  border  point  d i f f e r e n c e of  the  This  calculation  was  because  i t s frequency  a d e r i v a t i v e across  the  104 V g  passband.  2  a  I f two j u n c t i o n s a r e found  clockwise or counter-clockwise  diagonal  gradient  theorem.  Finally  pixel  will  greatest  be  their  The  that  which  them  sense,  i s estimated  of  the gradient  This  provides  are identified  Freeman d i r e c t i o n  above approach proves  image.  In t h i s  or  l e g i t i m a t e edges of which  consequence,  occur  f o r very  expected  one p i x e l  seen  The  low standard  choice  i n the subsequent  are produced  pixels only  could  deviation  such  thin  filters  and  chapter.  The r e a s o n s  and  too small.  Therefore,  the evaluation t r i a l s  to  the  and  1.6.  and  the  should  image  are  r i n g s images o f F i g u r e  i s  test  image,  a n d 1.6.  ( i t s zero only  narrow  i n the  output  by  to  two As a  suffer.  features so  only  are  not  similar;  only  two  a, =16  was  64 X 6 4 ) .  observe  properly  matched,  will  be  applied  a, = 1 6 ,  standard  6.4,  deviatons  rejected since  Actually,  width.  to  2.11, where  crossing occurs  be r e s o l v a b l e by a n y f i l t e r  parallel  (  which a r e too l a r g e ,  too large  very  deviations, a , will  of f i l t e r s  hopelessly  images  problem.  standard  a, = 6 . 4 ,  gradient  output  expected  influence  applied,  border  returning the  be bounded  the  were  the  one c a n be r e c o r d e d .  chapter,  of f i l t e r  the v e r t i c a l  the  when  of the l a s t  For  by  calculation  that  filtered  in  the Pythagorean  adopted  m e a s u r e c a n be  i n the last  t o be a g e n e r a l  with  of  up t o e i g h t p o s s i b l e  i n width,  the border  the continuity  H o w e v e r , a s was  magnitude  ambiguous o n l y  features,  case,  adjacent  numbers.  object  more  the  the gradient attributes  magnitude.  directions by  between  immediately  i t i s  at a radius of a single  However, t h e  step  32 edge  expected  105 decrease  in  noise  reflected  i n the  connection  of  the  filter  vertical  evaluation the  stripes matched  which  is  even  o =6.4  filter  five can  which  I,  was  of  6.4  optimum n o i s e  To  The by This  evaluation trials and  Rosenfeld  facilitates  changing  of  a  the  signal  infinite and  spacing o =16.  for the  T=32 h a s  ratio.  a  for this  20  the  and  a  image the  with with  of  o  =25.6  f  0  kind  value  and  (  as  of  the  filter  of  image. interval,  from  above experiments,  of  the  the  edges.  o =6.4,  well  spacing  is isolated  three-level  as  of  resolution  test  the  used,  series  be  smallest 0 f o r which  be  circular  interestingly,  f  analysis  edges produced  edge  would  performed p a r a l l e l e d  detailed  to noise  than  i n f l u e n c e of  also filtered  the  an  intensity  0 of  t h a t most of  of  as  edge response to a  should  f  c o n v o l u t i o n method  s q u a r e wave of the  o  the  for this  compatibility  i m a g e was  Kitchen  thinness  step  rejection  maintain  pure noise  size  t o be  correspond  indicating  image  impractical  shown  ideal  would  f  more  Because  by  alternating  filter  decreasing  scores.  this  applied to a  p r o d u c e an  a =1.6  sees of  The  with  image b o r d e r s  actually  T=32.  t  rejection  the  1.6.  those template  published operator.  the  continuity  the  sensitivity  and to  a  106 3.4.  Evaluation  3.4.1  Rings  Image  Figures filtering test of  Results Evaluation  3.3 t h r o u g h  the rings  image h a s z e r o fifty.  3.5 d i s p l a y  the  image a t t h e t h r e e mean G a u s s i a n  Analysis  of  results  standard  noise  added  derived  deviations.  t o produce  edge  images of F i g u r e  The  an  the results follows; reference  made t o t h e u n t h r e s h o l d e d  from  SNR  will  be  2.11, chapter  2.  o =16: f  The the  edges produced,  extreme  detailed is low,  inner  ring  seen  and outer  expected  that  and that  thinness  scores  histogram  t h e edge p i x e l  uniform  distribution  level.  The a b s e n c e o f l o w i n t e n s i t y  Figure  3.3b  induced  shows  of  pixels  that  the  7=0,  with  the observed  remains constant edge t h i n n e s s .  the scarcity  curves on  drop  breakage  continuity-only  curve,  indicates  a  grey  consistent  with  measure below 54%  i n c l u d i n g , 0.9 f o r a l l i s consistent  T h e maximum e p f v a l u e  o f edge d e t a i l .  otherwise  remain  5 4 % maximum  a t u n i t y , which  3.3c, o f 0.027, a b o u t  i n a complex manner of  i s  evaluation  For  reflects  I t  edges.  7.  Figure  always  3.3a,  above  pixels  remains above, and approximately  region,  The  smoothed away.  fraction  Figure  threshold  E  to  perfectly.  o f edge m a g n i t u d e s ,  absence of noise  only  of t h t e s t pattern.  are totally  nearly  the  2.11a, c o r r e s p o n d  boundaries  s t r u c t u r e and noise  therefore  The  i n Figure  one t h i r d Above  reflecting  perfect  7 = 1 ,peaking  of  of that  54%,  this  expected,  the  remaining  the loss of c o n t i n u i t y  edges.  Note  a t 0.9, r e f l e c t s  that the  the  inherent  107  Figure  3.3  a, =16.0 V g filtered histogram; evaluation l e v e l ; ( c ) edge p i x e l 2  r i n g s image: (a) edge m a g n i t u d e measure against (b) t h r e s h o l d fraction  108  Figure  3.4  o =6.4 V g filtered histogram; evaluation l e v e l ; ( c ) edge p i x e l 2  f  r i n g s image: (a) edge m a g n i t u d e measure against (b) t h r e s h o l d fraction  109  f r a c t i o n of  Figure  3.5  max.  mag.  a = 1 .6 V g filtered histogram; evaluation l e v e l ; ( c ) edge p i x e l 2  4  r i n g s image: (a) edge m a g n i t u d e measure against (b) t h r e s h o l d fraction  110 bias  of t h i s  method  against  curved  edges.  a =6.4: (  The  edge  structure  image,  Figure  2.11b,  of the rings pattern  indicates  was d e t e c t e d .  no n o i s e  i s seen.  A small  patches  i s  i n t h e image c o r n e r s .  seen  amount o f n o i s e  that  Within  the  full  the pattern,  i n t h e form of  closed  A l l edges a r e p e r f e c t l y  thin.  The  histogram,  bimodal  distribution  maximum  level.  indication  The but  thinness  on  Of  remain of  epf value  exactly  that  empty  the  the  band between  small  mode  of Figure  i s again  56%  expected  a  9% a n d 5 0 % o f  below  3.4b h a v e  9%  i s  an  seemed  slightly  The  perfect  i n the constant  t o have  thresholding a t 0.0741  spread  threshold.  reflected  The n o i s e  after  of  noise.  regime of t h e e v a l u a t i o n just  beginnings  little  measures. out  despite  unity  the  effect Finally,  noise  the error  was  i n many  t h e edge p o s i t i o n s . = 1 .6: The  resolved  edge  image o f F i g u r e  t o high  Reflecting  accuracy  noise  mode  the pervasive  a t 9%.  a l l t h e edge  2.11c shows  b u t embedded  h a s become s t r o n g l y b i m o d a l  of  reveals  0.88 u p t o  edges  found  of  scores  above  the  an  of  of the presence  o f t h e 7=0 c u r v e .  almost of  The o n s e t  t h e low t h r e s h o l d  the  3.4a,  with  evaluation  still  value  Figure  noise,  with  in a field  ring of  the histogram,  t h e peak p i x e l  T h e many z e r o s  magnitudes having  the f u l l  count  i n the histogram  integer  values  pattern  noise.  Figure now  in  are a  between  3.5a, the  result  0 a n d 62  111 w h i c h do  not  The 38%  to  44%  18%  the  be  this  with  same a s  3.5b  for  the  the  found  time  occuring  of  Kitchen  the  and  optimal  of  score  of 1.0,  Interestingly,  the  histogram  mode  a  Kitchen-Rosenfeld's 3.5c  epf=0.075  Rosenfeld:  dramatically  just  C l e a r l y the  level  range  drop below  94%.  Figure  at  the  peak  to  in  evaluations.  over  lowest  v a l l e y between  that  100%.  peak  first  rings themselves.  that  thresholded,  Figure  0 and  returning  the  with  peak  from the  7=1  seen  template  illustrates  of  between  a t t a i n i n g a minimum of  coincide  three-level  expected  with  is  consistent  level  scores  threshold,  peaks  result  every  threshold  Thinness  below  is  to  evaluation  0.865.  the  map  above  that  conclusion  i f the  coicides  here  image  must  the  peak  with  evaluations.  In  general,  threshold seen  remain  i n the  concluded quality  i t i s observed high  that  the  next  V g 2  to  setting for  2,  20  10,  setting. Kitchen a  The  The and  factor  reason  50,  was  thinness  information  the  and  in contrast  This  compared  required  the  at  local  low  divergence  therefore  the  be  continuity  was  7=0.8,  but  not  the  the  tend  to  noise  one. be  therefore  ratios  (1,  fixed  7  used  results The  the  by was  principal  perfectly  provided  bias  against  that  published  to  upwardly  a  same a s  a dominant  score  scores  choosing  Compatibility with  of  scores  can  edges e x h i b i t a  edges were o b s e r v e d  would  It  to  seven d i f f e r e n c t s i g n a l to  choice,  component  evaluation  results.  tests  s e t t i n g chosen  in this  the  thinness.  of  100).  Rosenfeld.  that  filter  their  series  threshold 5,  a l l 7  three-level template  comparable  The  for  that  little  thin. new  continuity  1 12 component.  Continuity,  particularly  for  continuous score  applications  for  region  to  compensate  general  edge q u a l i t y ,  of  continuity.  edge  and provide  concern  which  require  envisioned  The p u r e  edge b i a s .  for  serious  the  good  A  7  bias,  continuity  of  0.8  was  acknowledge the  indication  of  the  degree  variation: These  dropped  images  below  thoroughly  were  only  0.02, a t w h i c h p o i n t  3.6a  shows  the  scores  a r e seen  shape  and small v e r t i c a l  t o be l a r g e l y  same f o r a l l t h e c u r v e s , edge  detail  was  expected  because  resolved  rings  The  noise  largely  a, =6.4 SNR  SNR  at  the resolution covers  to the level  test  for  Since  Figure  where epf are  quite  a t 0.027  each  about  exhibiting  the  SNR.  interval,  The e v a l u a t i o n similar  t h e e p f r a n g e was t h e  This  same noise  level  image.  The maximum  of  imunity i s  I , o f 64 a b o u t  the  two  evaluation  a t 0.925 a n d was a t t a i n e d a t maximum  SNR c u r v e  of Figure  3.6b r e i t e r a t e s t h e  found.  results  occurs  of Figure i n both  3.7a  are  a t low e p f .  epf=0.74 produced  3.7b, d i s p l a y s t h i s  The i n c r e a s e  by t h e r i n g s .  SNR  s t i l l  shape and v e r t i c a l  r e s p o n s i b l e f o r i n c r e a s i n g the epf range.  occur  images  =16.  independent  the entire  constant  independent  some d i v e r g e n c e  results  beginning  max. E v e r s u s  immunity  The  the  variation.  resolved  m e a s u r e was r o u g h l y epf.  thresholded  broken.  Figure  is  of  segmentation.  does, however, have a curved felt  was  the  edges  therefore  SNR  however,  independence  seen spread,  in  edge  The peak The  to  be  though detail  scores a l l  max. E  by r e m a i n i n g  curve, flat  at  11 3  50  Figure  3.6  100  a =16 V g f i l t e r e d r i n g s image e v a l u a t i o n r e s u l t s : ( a ) SNR= 1, 2, 5, 1 0 , 2 0 , 5 0 , a n d 100 (from bottom curve to top), 6=0.8; (b) peak e v a l u a t i o n s c o r e s a g a i n s t SNR 2  r  1 14  1.0  06-I  as  10  20  so  100  SNR  Figure  3.7  a, =6.4 V g f i l t e r e d r i n g s image e v a l u a t i o n results: ( a ) SNR= 1, 2, 5, 1 0 , 2 0 , 5 0 , a n d 100 ( f r o m b o t t o m curve to t o p ) , 5=0.8; (b) peak evaluation scores a g a i n s t SNR 2  11 5  1.0  OA  i  0.2]  0-I  -r  1.0  .  1  ,  0.16  0.40  1  epf  1  ,  a06  -r-  .  0.03  1  0.01  IJO-  (b)  09-  08 E 07-  ae  as  1 1  -I  1  — I  2  i  S  ,1  10  ,1  20  ,1  50  1 100  SNR  Figure  3.8  a =1.6 V g f i l t e r e d r i n g s image e v a l u a t i o n r e s u l t s : ( a ) SNR= 1, 2, 5, 1 0 , 2 0 , 5 0 , a n d 100 (from bottom curve to top), 6=0.8; (b) peak e v a l u a t i o n s c o r e s a g a i n s t SNR 2  (  1 16 a b o u t 0.9.  When a  was  f  independence horizontal definite  decreased  vanished.  However,  spread.  and  peak  the observation  that  indicates  that  the  noise  magnitudes as t o render without  also  SNR c u r v e SNR=1 also  to  Figure  the The  3.8a,  curves  high  about  the  a  SNR  curves  critical  t h e new SNR  structure.  dependence  As w i t h  t o resemble  those  of  show  wide  a  range of  removing  T h e max. E ranging  3.5 t h e s e  the  broad  e p f a t 0.074.  in  by  Figure  SNR  peak a t h i g h e p f  sufficiently  the rings'  the  now s h o w a  t h e l o w SNR c u r v e s has  0.90 a t S N R = 1 0 0 .  beginning  in  thresholding ineffective  damaging  reflects  1.6  Instead,  and v e r t i c a l increase  to  i t  versus  0.77  at  results are  three-level  template  operator.  It  i s  structure high.  o f t h e image, I f ,  thresholding as  g e n e r a l l y seen  witnessed  However,  then  however,  can only by  that the  o  i s  f  safely  the  too  narrow,  general  threshold that  f  noise  rejection  peaks  i fthe noise  i t i s perhaps  will  than  remove t h e n o i s e  that  then  i s matched  smaller  evaluation  i t i s a l s o seen  filter  i fthe a  can  be  very  required,  then  i f t h e SNR i s  near  high  the c r i t i c a l epf.  level  is.high  impossible  remove t h e n o i s e  t o t h e edge  and  and  the  to select  also  a  guarantee  edge c o n t i n u i t y .  3.4.2  Vertical  The that  for  principal  i tshould  rings  Step  tests.  Evaluation  interest  be d e v o i d This  p e r f e c t edges.  of the curved  permits Since  i n processing  the vertical  edge b i a s p r e s e n t  i n the  p e r f e c t c o n t i n u i t y and thinness decreasing  a  f  increases  the  edge i s  scores  influence  1 17 of  noise  edges,  on  the  filter  i t i s expected  over  the  Tests  analogous  constant  small  that  response  the curved  vertical t o those  performed.  is  adopted here also permitting proceed  i n turn  edge  f o r small 3.4  concluded  that  produces  bias  will  a , high  t o 3.6 w h e r e level  the optional  SNR  images. i s  held  changed were  7 i s  the signal t o noise  curved  dominate  noise  f  7 and t h e t h r e s h o l d  not  to  Having  edge  of Figures  a t f i f t y and both  which  0.8,  ratio  i t  trials  immediately.  o =6.4: (  The  printout  threshold), devoid  of  perfectly  middle  Figure  The  and  feature  of  the  plot  left  is  curve  presence of  features Only  panel.  E  of the noise  i s  out  there in  an the  images,  only the  influence of the noise  appears  of t h e edge.  against  epf, Figure  the evaluation  SNR=1  almost  stand  "island"  For a l l the other  due t o t h e s l i g h t  of  SNR=1  edge  (before  an image  always  for  deviates island  3.10a,  plots are nearly  a p e a k o f 0.994 a t t h e maximum  not perfect  The  edge  The p r i n c i p a l  of  SNR=5 a n d g r e a t e r , reaching  two  directions  a l l SNRs r e v e a l s  connected.  be on t h e s t r a i g h t n e s s The  for  and  i n the form of a small  edges a r e present. to  3.9,  noise. thin  auxiliary  o f t h e edge p o s i t i o n s  winding markedly  but s t i l l  shows  that f o r  identical  each  e p f o f 0.018.  The  of  observed.  the  from  edge  score  t h e r e s t due t o t h e  a t t a i n s an e x c e l l e n t  score  0.958 a t e p f = 0 . 0 ! 8 .  The ideal curved  max. E  evaluation edges  versus  SNR  f o r a l l SNR. bias  i s  plot  of Figure  Clearly, the  responsible  for  3.10b shows a  elimination this  much  of  near the  improved  118 (a)  <d>  <b)  (c)  (e)  (f)  (g)  Figure  3.9  a = 6.4  g filtered step edge vert ical f o r SNR = ( a ) 1 , ( b ) 2, ( d ) 10, ( e ) 2 0 , ( c ) 5, !  (  (f)  50,  and  (g)  100  1 19  0.2-1  1.0  0.40  0.16  epf  aoe  0.01  ao3  0u6H  as  10  20  50  100  SNR  Fiqure  3.10  a =6.4 V g filtered vertical step evaluation results: ( a ) SNR = 1, 2, 5, 10, 20, 50, and 100 (from bottom curve to top), 8=0.8; (b) peak evaluation scores against SNR 2  120 performance.  o = 1.6: f  The  results  a, =6.4. of  a  The  edge  is  By  equation  1.6625,  or  edge becomes  edge,  The shows a seen  at  from  SNR=100,  approaches  diversity  with  of  to  the  poor  SNR  quite  high,  f o r an  ideal  3.12b  also  approach  score  of  a  0.77  to The  that at  max. of  E  SNR=10 a n d  for  o =1.6 f  background  noise,  and  background pixels  noise  of  the  pixels.  seen the  epf,  of  Figure  of  0.77.  at  SNR  is  3.11.  The  thresholding  is  The  peak  epf=0.0l8,  about  curve  of  Figure  t h r e e - l e v e l template steadily  are  This  operation.  occur  3.12a,  scores  in Figure act  versus  the  previous  the  this  cleaning  0.978, and  eye.  SNR^5,  maximum  show t h a t  noise  up  edge.  a  image q u a l i t y  curves  the  B e l o w SNR=10, t h e  from  absence  For  w i t h i n two  three  scores.  uniformly  in performing  are  beyond  used here. the  of  untrained  t o l e r a b l e SNR  e v a l u a t i o n measure a g a i n s t  of  that  a  even  the  lowest  definition  those  shows t h e  a n a l y s i s of  plot  highest  scores  the  from  noise  stays  decline  effective  the  and  3.11,  generally  consistent three  the  (2.48),  However,  great  to  by  3.325 by  significant  but  different  e d g e b e l o w SNR=5 t o  increasingly isolated  straighter. remains  markedly  image p r i n t o u t , F i g u r e  substantiated  chapter.  with  are  discernable vertical  This  is  here  climbing  to  operator 0.978  at  SNR=100.  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Figure 1  3.11  0..4:171. .0 4. ...1..3.4.1. 0«44.47. .Mlll. . 4 1 1...7 .4...74 ..111.,3 0 ...4.0..40....4O O41 0434..0O M ..3%440 4 1B 1..M 1 41 .14O 0M ..O .O. . *41 4M 4.40 . 4.191 • . . 0 4 0 . 4 . 1 1 M 4 . . . 0 0 . . 4 . 7 . 4 4 0 . 1 . . 1 . . 1 4 . 0 0 0 *4-.0. .0441.1.*.4 ..31.M l 4 . 0 4 0 4 0 . 1 . 1 1 . 1 1 1 1. .1 4. .0 4.0.30. .33. .3 7 . .0. . -4T. , 8 0 . . . M M 4.  4BM7. . . 4 .  4.  3 . . .444.11.4 ..M.l 4.  o =1 . 6 7 g f i l t e r e d vertical step edge f o r SNR = ( a ) 1 ,(b) 2, ( c ) 5, ( d ) 1 0 , ( e ) 2 0 , ( f ) 5 0 , a n d ( g ) 100 2  r  1  122  100  Figure  3.12  o = 1 .6 V g filtered vertical step evaluation r e s u l t s : ( a ) SNR = 1, 2, 5, 10, 20, 50, and 100 (from bottom curve to top), 6=0.8; (b) peak evaluation scores against SNR 2  f  123 SNR.  The  contort was  principal  i t from a pure v e r t i c a l  the test  Even was  edge o b s e r v e d  though  particularly  matched  then  readily  isolated  isolated  3.4.3  Pure Noise  was n o t e d image,  accidental  The  highly the be  this  edge  unlike  the side  a pure  step,  the f i l t e r .  I t i s seen  the  intermediate  test SNR  edge i t  I t i snot  i sparticularly  the background.  at  SNR=1  strong  t o become  edge,  becomes  values  f o r both  by K i t c h e n will  and Rosenfeld be a c e r t a i n  due t o s t a t i s t i c a l  that  even  occurence  fluctuations  in  of  a  pure  well-formed  i n image d e n s i t y o r  alignments.  smoothing  observed. random  Being  t o design  and,  at  f  Evaluation:  there  edges a r i s i n g  f o r a =1.6  was t o  broken.  t o observe.  that  even  Only  on t h e edge  of the experiment,  used  from  SNR  straight  variances.  noise  model  t o observe  filter  It  t o be  interesting  a t a lower  perfectly  line.  i t was o u t s i d e  the original  surprising and  influence of the noise  property  A tendency regions  curved.  filter's clearly  on t h e o r d e r  therefore present  performance  defined,  2  filter  has a l s o been noted  of diameter They  of the V g  to  has already  clump  of o . (  noise  been into  These edges a r e to  measure  i n n a t u r a l images where borders  may n o t  e.g. i n remote  an o p p o r t u n i t y  sensing.  a =6.4: (  As to  f o rthe rings  overlap  maximum e p f .  with  image,  changing  The p e r f e c t  the curves, o . f  score  They,  a t 7=0  Figure  3.13a,  however,  tend  not  peak  early at  indicates perfect  thinness  1 24  0.2  i.<H  epf  Figure  3.13  V g (a) 2  filtered  a =6.4; f  random noise ( b ) a, =1.6  evaluation  results  for  125 throughout. indicate that  The  decline  with  t h e breakage o f edge  there  declining  i s  no  scores  epf  of  features  optimal  with  threshold  indicate that  the  remaining  curves  thresholding.  point  a l l t h e edge  Note  indicated here, the  features  are  equally  well-formed.  o  f  =1.6: Again,  the  curves,  Figure  changing  7 , b u t t h e y do c o n v e r g e  due  the  to  loss of thinness,  above epf=0.01. maximum  epf  threshold  At high  level.  previously  epf,  general,  also,  then,  i t i s apparent  regions  in  increasing the  pure  o  widest  continuity  as  is  that not  as  an  of any peaks This  apply  that  Also, an  level  optimal  peak  at  the  i s  equally  no  f  optimal  well-formed,  fora l l 7 at  produce  scores  thinness  level  a r e produced exhibit and  of  by  similar  decline  in  However, t h e r e i s because  measure a f t e r  the claim  well-formed  i n maximum e p f w i t h  regions  threshold  large  the  of  the  thresholding last  chapter  an a r b i t r a r y image whose p r o p e r t i e s a r e  or are nearly  any t h r e s h o l d  again  i s raised.  i n the evaluation  with  o ,  does  a l l the in  i s  this  the decline  increase  This  with  scores  the widest  substantiates  when c o n f r o n t e d well-known,  filter  with  thresholding.  initial  From  the threshold  i n d i c a t i o n of  applied.  this  noise.  filter.  such  absence  that  i ti s clear  f  tendencies  no  the high  overlap  epf.  there  A l l of t h e edges a r e  from  not  observed  once  though perhaps a b i t c l u t t e r e d before In  do  somewhat a t h i g h  7, the curves  Here  indicated.  3.13b,  random,  at a l l after  then  i t  filtering  with  i s  best  the chosen  not o . f  to  126 3.5  Comparison  Figure on  to other  3.14  a l l of  above  the others.  standard  the  produced  The n e a r - i d e a l most c l o s e l y i s  f  However,  shows t h a t  not the  fact  t h e good  are  a t t r i b u t a b l e to the noise the  image e d g e s . curves  o f an i d e a l  general  the  shape and amplitude characteristic reasonable further  that  where e x t r e m e l y match  with  the V g 2  fine  ((7,^1.6)  i s  outstanding.  by  desired,  any  the  of  t o image d e t a i l ,  line-like  of  closely  results  filter  offers  of  the  little  o f edge s p a c i n g  greater  performance  of  wide  of the test a =1.6 f  Evidently, the published  of a  not  but rather  the  operators.  further reduction  other  those  character  response curves  t o be  are  f  the  filter  scores the  o f t h e o =6.4 c u r v e  influence  well  the  f  shape o f t h e o t h e r s .  the  riding  t h e a =16 curve  edge  plots  t h e image d e t a i l s  I t will  t  other or  r e s o l u t i o n s are concerned.  t o image d e t a i l  The  s e e n when  t o n o t e how  a l l narrow  to predict that  matches  cleaning  of the noise  of  agreement  indicating  general  superimposed  filter  2  match  I t i s interesting  match  by t h e V g  that  local  chapter  Rosenfeld.  response  results  much t h e r e s u l t  and  and  approached  so  filters  of t h i s  of Kitchen  curves  o =6.4,  operators. highest  results  deviation  resolved,  Operators  shows t h e r e s u l t s  the evaluation  show  Edge  no  i s  a  appears produce  operators, advantage  However, where a than  the  two  V g 2  pixels  filter  i s  1 27  100  1.o> = 16.0 2. Of = 6.4 3. Of = 1.6 4. t h r e e - l e v e l Figure  3.14  5. 6. 7. 8.  five-level Kirsch Sobel sqrt Prewitt sqrt  9. 10. 11. 12.  S o b e l sumabs P r e w i t t sumabs compass g r a d i e n t Roberts sqrt  13. R o b e r t s sumabs  S u p e r i m p o s i t i o n o f t h e V g maximum e v a l u a t i o n scores upon t h o s e o f Kitchen and Rosenfeld : (a) r i n g s ; (b) v e r t i c a l step. 2  128 3.6  Conclusions  This filter  chapter  provides  edge o p e r a t o r s that  operator.  The  may  in be  superior  may  lost  threshold  point  However,  c a n be  t h e a, =16  evaluations  without  guarantee  that  disrupt  closed  edge  as  noise  a  observed  Kitchen should  and step  an o p t i m a l V g  method.  the  wide  4a,  edge  considered  There  with  an edge o p e r a t o r  position.  In  would  to confirm  serve  crossing, and  Pratt,  image  to  a  z  .  the  However,  at  This i s  2  clearly  superiority  sought. an  The  optimal  scores  peak.  and both  noise  image  maximum  edge  pixel  therfore, there  i s no  strongest  asset,  the formation  i s therefore  Multiband clean would  their  the  last  word  value  an  the V g 2  the predicted  that  not  which  predicted  be m o r e  effective.  in  edge  judging  distribution  method  an  i t s  such a of  the approach  weigh every  and  edge  the accuracy  about  operator  of  takes  strip  edge e v a l u a t i o n  of adopting  will  recommended  filtering  i n understanding  localizes  o r P e l i and Malah,  which  judge  g o o d n e s s o f f i t , a more  the o v e r a l l  V g  found  that  instead  other  unnecessary  was  evaluation  note  of  2  c a n be  features  context  of  V g  point  noise  i s a certain  an  i m a g e s show t h a t  occur  threshold  cleaning  operator. which  edge  Thresholding  and R o s e n f e l d  n o t be  pixels  In general,  filter's  2  strong  two  to  i s  sections.  rings evaluation  peak  the  the majority  edge o p e r a t o r s  of the l a s t  threshold.  segments.  about  other  below  that  the performance of the  found where t h e  this  the  advantage of  over  filtered  showed  fraction  degrade  resolution  of the ring  over  thresholding  simply  superiority  where  two a s s e r t i o n s :  performance that  the comparisons  evaluations  not  t o prove  used; and  operation  seen  sought  known  measure  the  zero  of  Abdou  edge p i x e l  i n the  restrictive  1 29 test  i s recommended  usually  fully  noise. output about  object the  to  edge near  image,  of  the  future serve  performance.  to  recognizes and  isolated  sophisticated  keeping  deviation  indication  should  resolved  T h i s more  standard  left  which  of  the  work,  the the  edge of  the  (  data  the  picture  edge  search and  then  for  edge.  will  then  serve  of  the  Such a  2  the  The as  an  method,  presented V g  of  follow i t  true  results  is  sea  the  edge o p e r a t o r .  combined w i t h the  of  test  background  would  t r u e edge p o s i t i o n  the  complete  from  method  t o w i t h i n 2a  precision  that  here filter  1 30  IV.  4.1  range  purpose  of  scale  features  isolate  of  threshold  filtered  printed  characters  unity,  background, background  from  to  the  the edge and  the  a l s o be  is  process, as  represented the  In  objects  image,  the by  complete  image, as  the  will  i n the  o b j e c t must by  at  to  the  found.  the  A  of  form  identifying  image.  To  present  least  one  be  row  level  "black" Since  a  light  m a g n i t u d e on  a  i s to present  a  characters  seen  of  those  positional  internal  opposed  segmentation individual  considered image,  and  borders  distinguish  or  to  their recognition. of  those  i n the  result  zero. on  b i n a r i z e d image, as  of  the  pixels.  description  facilitate  task  the  zero  to  unit  individual  the  in  positive  chapter  a  Rather  done  objects  e x t e r n a l and  The  be  minimum.  the  the  those  "white"  of  to unambiguously  of  enhance  regions  this  process,  take  of  operation  image b o u n d a r i e s  of  reduce  i s t o use  uses  regions  to  was  and  dark  as  produced  borders.  "1"s  to  segment,  essential  these  design  "white"  purpose  or  information  character  defined  appear  The  description  relational  the  was  and  for translation  negative  will  zeros.  will  image  "black"  correspond  background.  This  image  separate,  characters  stage  image e s s e n t i a l l y  and  they of  output  into  output  to  input exceeds  the  image  regions  filtering  a more u s e f u l o p e r a t i o n  the  the  the  edges of  the  2  spacing  chapters,  Binarizing  V g  in  edge  the  binarize  system  the  found  whose  previous to  SEGMENTATION  Introduction  The  than  FAST BINARY-IMAGE  to  of an  between, can  be  regions  segmentable, separated  column  of  from  zeros.  131  Segmentable Segmentable their  objects  outer  A this  borders  number of system.  real-time.  the be  image  For  The  of  The  foremost  relayed  to  raster-scan.  The  raster-scan  vertical  be  of  the  the  referred  be  t o as  is  considered  closed  to  m u s t be  since  in force  found  be  detected,  r e c o g n i t i o n system,  be  be  that a l l processing  Those b o r d e r s  will  Segmentable  loops.  i s , closed borders  ignored.  not  considered  and  with  segmentable  from  left  must  along to  in  their  concurrent  to proceed  p r i n t e d page, advancing  done  in  the  right.  example:  direction  advancing was  form c l o s e d  totally  local  also  constraints will  That  description  will  Not  along  chosen the  of  the  scan  proceeds  r.  This  scanning  for consistency with  acquisition  methods  along  method a CCD  seriously  and  c,  with  coordinate  image s c a n n i n g considered.  new  rows system  device,  The  one  vertical  1 32 extent  of  spanning again also  c  currently  about  to  two c h a r a c t e r  be  consistent  will  line.  However,  be c o n s i d e r e d  with  implicit  in  the  finite,  however  define  a moving  t h e window.  different would  may  note  and bound  F o r example, bound  periods  of the scan,  Obviously flexible best  outlined  0  line i s  scan, The  than  c  and  scan bound  i s clearly  solution  is  to  dimensions  N.  i t s row  one  per no  r to the fixed  t h e window  N/2  0  Then  at  coordinates  N  design equal  decision.  the extent  to  operate  the extent  loosely defined.  to  terminology,  this a we  segmentation  However,  The  r extent  i t will  of r chosen  be s h o w n  of c f o r the segmentation  correctly.  Generally,  that  method  i t will  is  a  i t must to  be  be  assumed  used  rather  of r and c a r e e q u a l .  proceeding  With  ON/2  r  r i s r e c y c l e d m o d u l o N+1.  Before  the  more  r between 0 and  r  that  It is  have t h e appearance o f :  0  at  considered.  there  indefinitely.  image window  used,  of a s i n g l e  the  columns, be  include  of  64  frequently  that  extent  extend  However,  device  the case  Also  row-wise  r may  t h e CCD  of c  only  here.  undefined.  heights, will  p o s s i b l e that the extent  print  of  stands  further,  point, solid turn methods  some o f  such  as  endorsed  terms  "border",  understanding to a review  the  of  of past  for this  must  be  rigorously  this  image  work  and then  system.  analysis address  1 33  4.2  Preliminary  4.2.1  Connectivity  Define points It  a  binary  (i,j).  [40],  properties  of  now  used  and  in  of  body  Let >  1  A  in  this  {  studied,  paper  His  field most will  a  binary to  at  to  (i ,, j i ) , . . . , ( i , j ) t  }  of  0 or  1s  background. length,  the  describing  the  points  analysis.  the  1.  picture,  background  picture  outlined  array  value  the  great  and  relevant  t  the  definitions  of  be  integer  belong  among o b j e c t  the  properties  such  Os  images.  an  assume o n l y  in  and  as  ideas  are  Those  presented  here.  be  a  set  of  (i,j)s  where  .  A  can  4-path  -  ^  1  (  ir , j  be  defined  i f for  .  In r  ),  Such p o i n t s as  =  that  has  properties  widely  definitions the  [60],  picture  w h i c h may  points  such  interconnection  of  understood  object  Rosenfeld  digital  Each  is generally  represent  t  Concepts  each other  or are  illustrated  i s one called  as  one  of  r,  1 <  r <  words, of  the t,  that  f o l l o w i n g two | i ( i  -  r  r  r+  are  ] j  said  i ,j /////  w  of ( i , j )  to  r  —  either  vertical  m  4-neighbours  +  r  below.  d  ,|  +i , j *i)  i t s h o r i z o n t a l or  4 - n e i g h b o u r s and  i  paths: j  r+  i |  equals  neighbours. be  4-adjacent  134 8-path  - i f for  I Jr  Jr*i|  (  _  i ,  j , ) or  r  diagonal said  )  each ^  1 .  i s  one  neighbours.  r , 1 < r < t , we That of  i s , ( i  Such  points  The to  - i f  there  exists  - i fa similar  dark  object  r  horizontal,  of  r  +  , | ,  equals  vertical,  8-neighbours  or  and a r e  ( i , j )  elements.  a 4-path  8-path  exists  Define  with  (h,k) and  (h,k) as the  between  The  (h,k) and  background  as 4-connected.  A l l points  are  not connected t o the border of the p i c t u r e  in  S.  S i s c o n n e c t e d a n d h a s no h o l e s ,  first  i n S;  o f an image a r e c o n s i d e r e d  8-connected.  t o S and i s defined  If  either  i )  +  h a v i n g a l l i t s members  points  S and a r e considered  belongs  j  i  being:  t e r m a n d (m,n) a s t h e l a s t , 8-connected  i ,  ( | i ,-  below.  S be a s u b s e t o f p i c t u r e  4-connected  +  max  are called  8-neighbours  (m,n) o f S a s  r  i t s eight  t o be 8 - a d j a c e n t a s shown  Let  have  are  (m,n).  to belong therefore of S  that  called  holes  i ti s called  simply  connected.  The  principal  connectivity  in  S  advantage and  connected  across a thin  following  image  array:  S  i s  diagonal  in  using  that o f S.  this  different  types  p r e v e n t s S from  For example,  of being  consider the  1 35  S  If  considered points. the  were d e f i n e d as connected Therefore  similarly  4-connected, A.  This  as 8-connected, by v i r t u e  B could  8-connected B  would  i s i n keeping  regions  o f t h e two  8-adjacent  n o t be c o n s i d e r e d S.  region  become a h o l e  with  our  i f S  intuitive  inside  i s defined  i n s i d e S,  disjoint  notion  be  diagonal  t o be a h o l e  However,  then  A and B would  of  as from  holes  in  through  two  be d e f i n e d  as  images.  4.2.2  Borders  The  outline  features: a  everywhere  of  4-adjacent  object  t o S.  defined  elements,  regardless  one  by  it  edge lies  those that the  regions.  within S This  a l l edge p o i n t s case  with  border  in  will  the  thin  An  objects,  Rosenfeld  [60] i n S.  closed  i s distinct  nor S but rather results  are uniquely points.  which i s  one  pixel  be one a n d t h e same.  a simple  edge  S,  object  o f t h e c o n n e c t i v i t y o f S.  o f an o b j e c t . neither  described  i n S and the other  edge e l e m e n t s a r e l i n k e d i n t o the  8-path  may  be  A border  For very  and o b j e c t  was  can  or i t sedge. closed  4-adjacent  edge  applies  an  bounded,  the border  An  Edges  i t sborder,  simple,  across,  and  a  pair  The  of  defintion  When a l l a d j a c e n t curve,  they  from a border  form  i n that  a t the j u n c t i o n between  i n the rather defined.  as  desirable  This  is  not  property always  1 36 4.2.3  Chain  Codes  Once the  border  question  remains as  storage  in  chain  code  curves.  for  the  how  3X3  of  the  best  of  to a  8-connected  the  point  found,  the  points  use  of  arbitrary  rectangular  curve  are  these  proposed  coding  applied  rectangular  represent  [49]  efficient  Each  picture objects  to  Freeman  method, as  simple.  following  to  memory.  The  fairly  points  a  for simple  shapes,  or  image a r r a y ,  is overlaid  is  with  the  points  in  array:  3 2 1 4 X 0 5 6 7 Chain The the  numbers  8-neighbourhood  direction  numbers  between next can first can  be  point  points  then  of  is only  Further  reduction  known  be  chain  to  code per  curve  direction  since  into  specified  representation storage  X,  in  the  be  of  slowly of,  point.  squares,  are  called the  45° this  their  bits  are  in  storage In  than  this  ±45° n e e d  be  the The  the  numbers. the  line  be  remaining  numbers.  represent  realized  case,  only  recorded,  segments  coordinates  point of  the  points  The  list  code.  straightforward to  These  each curve  forms a chain  needed can  of  Since  direction  derived  three  say,  directions  recorded.  compact  varying.  direction  manner, o n l y  n e e d be by  representing  increments.  numbers so more  Array  the  quantize  quantized  connected  bits  inside  Code  of This  coordinate each  i f the changes  requiring  point.  curve  is  in  the  only  two  137 4.2.4  Trace D i r e c t i o n  A concept idea a  of  point  that  an  edgein  P on  the  will  be  employed  and  edgeout  border  (or  as  in  finding  defined  edge)  of  by  an  a  g l o b a l , s e q u e n t i a l m e t h o d was  proceed  from  right-handed, Now  the  will  be  which  we  defined  f r o m P.  must d e f i n e  clockwise  edge or  the  border  as  trace  the  point  edgein  proceeds  You  of  the  may  here.  This  Zahn.  In  lie  the  edgein,  notice  i s to maintain  fact,  these  object's  P.  find  P,  and  Consider  later  for this  shown a b o v e , w i l l this  trace  Likewise,  the  defined  numbers w i l l  be  used  the  [70],  the  trace  be  trace. be  A  used.  a r r i v e s at  next as  to  to  point  the  P to  edgeout  describe  the  points.  term  "edge"  consistency  edgein(out) border.  as  is  S:  from P w i l l  edgeout  that  to  from which to  of  a direction  direction,  Freeman d i r e c t i o n  locations  on  P,  used  Zahn  object  direction  If  borders  i s used  with  points  will  the  rather  loosely  definitions  later  be  defined  of to  138 4.3  of  of  In  simplest  the  Past  Work  character segmentation  b o r d e r , edge, and  such a  Review  as  that  pattern  for such  of Mason and  similar  blank  scan  rows a r e  the  Clemens  to that  rows.  to the  process.  McCullough  Clemens-type  fonts. sever  The  expected. of  partitioning  information completes  the  These first  lies.  object portions  of  the  window.  If  a l l  was  the  net  i n the  columns  rate  image,  approach two  between  completing method  in  f o r 12 p i t c h techniques  i s the  between  which  a  to  may  Mason  optimally found  same: t o p r o d u c e  transmission  of  sherif-type  i s not  the  two  characters  t o 35%  column  in  searching  circuitry  o f up  result  not  satisfactory  address  the  image window.  c h a r a c t e r s from For  bounded  systems  a  when pair  desired character  of  this  information  process.  are not  the  sophisticated  i f a blank  Column-wise  t h e y do  found  concepts  Some  introduction,  recognition  to develop  columns  approaches  points  i n the  failure  segmentation  i s that  simply scan  the  role.  [62], i t i s recognized that  assemblies  However,  [61]  In t h e more  segmentation  solution  character  little  Object points  touch or o v e r l a p c a u s i n g a and  play  proposed  transmitted  segmentation  Hoffman and  connectivity  systems,  f o r two  problem These can  adjacent print  of  reasons.  unsegmentable  readily  lines  The  occur  when  protrude  into  transmitted,  the  example:  containing  the  "a" are  t o be  1 39 protruding  ascender  This  p r o b l e m must  or  narrowing  by  and be  the  descender  dealt  vertical  guiding  the  to  a l l responsibility  place  stage,  scanner  and  to  requirements, second deal  The  neither  objection  and  separate  along  the  with  of  print  A  i s to  line.  losing  the  is  i t is desirable into  one  satisfactory. i s that  characters.  For  from at  shortcomings  A  i t cannot  example:  H o w e v e r , no  image window  stages,  tracking proficiency  segmentation  the  well.  precisely  segmentation  information  s o l u t i o n to  and  Since  solutions  segmentable.  as  processing  scan  operator  overlapping  search  transmitted  the  for character  these  clearly  be  subsequent  extent  column-wise  without  those characters. segmentation  of  to  " t " are  them  by  r e l a x hardware or  satisfactorily  "a"  with  must  column  can  l e a s t one  of  of  column-wise  for closed  boundaries  instead.  The two  approaches  broad  categories:  techniques,  and  techniques. randomly then of  execution  border,  followed  former  addressable of  by  linkage  The  edge, of  border,  to  for  fall  edge,  edge, of  processing  the  boundary  method  found  image pixels,  those points  involves  during  into closed  the  into  following  systematically follow  latter  point  or  or  storage  search  algorithm  found.  or  a  information  border,  involves  array,  an  border  sequential  non-sequential  The  each boundary  each  for acquiring  the  a  and curve  recording  raster  borders  in  scan,  during,  140 or  after,  address  the raster  the issue  Hoffman will  and  of  two  sequential  categories,  Once  the  adjacent  show  and  first  to  so  on  circular  transition.  [13].  seen  feature  applied  is  of the  to character  their  acquiring  simplicity.  the  the  border  of  arcs  Later,  adjacent  spot  scanner  centered  Clemens  border  recursive  arithmetic  explored  this  were  a consequence,  to f i r s t  and then on  through  pixels  next to  was  point  produce  border  was  a  inconsistent  not a l l t h i n ,  i t so u t l i n e  with  boundary  following.  search  of  or 8-connected  one p i x e l  potential  method  by  The  set  of  simple  Lunscher  found  [64]  i t s principal  border  connectivity.  depending  under  wide,  a and  and  was  a  a purely digital  Beddoes  view  diagonal directions  locate  by  method f o r  light-dark  guided  technique  explored  analog  follow  the  expressions.  c o n s i d e r e d 4-  two p o s s i b l e  idea  [63] proposed  segmentation  t o be a n  this  They d e v i s e d a l a r g e l y  character segmentation  Borders  often  so i t  chapter.  methods a r e the o l d e r  for  c h a r a c t e r boundary,  shortcoming  in this  around  application  of a f l y i n g  handwritten tight  characters.  trace.  et a l .  the c o n t r o l  touching  however,  p o i n t has been a c q u i r e d , a m u l t i t u d e of  be c o n c e i v e d  and  A pioneering Greanias  most  of  approaches,  t o be a c o m p l e x i s s u e ,  following  notable  border  i t  this  much l a t e r  those  T h e i r most  right-handed  be  separation  border  methods can r e a d i l y  the  the  McCullough  segmentation.  for  N e i t h e r of these  n o t be d i s c u s s e d u n t i l  The  for  scan.  on w h i c h  consideration.  diagonal lines  of As  could  followed.  Both 8-connected  border  and  borders  edge  following  were p r e s e n t e d  methods  by R o s e n f e l d  for [60],  4The  and edge  141 follower edge  examined a  points  larger  3X3  current eight the  along  pixel  border  pixels next  method  t o be  the  technique image  to  replaced  central  labeling  of  the  borders  of  an  of  new  produce of for  this  sequential  of of  the the  pixels  pixel  novel,  of  modified  this  found.  This  ignored  during  this  memory.  fast  border  operator  over  following the  binary in turn  the  input  image.  This  of  object  pixels  the  read  and  method  the a  as  c o u l d p o s s i b l y be  any  modified  to  technique  The  the  complete,  borders  output.  code  adjacent is  t a b l e lookup  along  code d e s c r i p t i o n  neighbourhood  image  a  labeling  of  of  outer for  neighbourhood code which  recoding  the  search  eight-bit  a  number  techniques. of  pixels. system,  window  to  [40]  i m a g e t o be  the  Foremost  input  This  of  the  i s used  present, two-pass  t o make  drawbacks  i s the  requirement  image and  obviously  especially  evident  provide  i n c r e a s e s the  when a  Sobel-type  H o w e v e r , memory management a l s o b e c o m e s moving  Kak  border  3X3  w h i c h was  a  i t  and  nature  suitable  applications.  are  complete copy  of  an  After  technique  There  a  involved applying a  a chain  pixel  t r a c e r e q u i r e s the  and  per  successive  follower required  Unfortunately,  bits  developed  guide  real-time  borders.  e x t r a two  neighbourhood codes are to q u i c k l y  center  right-handed  arrangement  pixel.  border  Rosenfeld  each p i x e l the  the  locate  pixel.  produce  described  The  to  i n a c l o c k w i s e manner  [65] which  area  examined  r e q u i r e s an  Sobel  The  known  in search  process  pixel  border.  point.  i n c l u d e the  permitted rescan  a  2X2  neighbourhood,  border  to  simple  of  a  dynamic  image  more  common t o a l l to  store  random a c c e s s memory system  to a l l  requirements is  complex  i s considered.  a  employed. when  the  Clearly  some  142 form  of  the  window  top  two-dimensional  and  which  in turn  bottom of  methods  cited  never a  circular  the  i n the  sequential  these  techniques  recognition follower  a l .  rate  in  unspecified  systems can  be  adapted  to  They a d d r e s s  problems  simply  image a t only  a  one  r u n s of  line.  graph  i s mapped  completely  The points [68].  In To  image d a t a ,  using  proposed a procedure  rows,  search  into  current  a  list  data  of  input  of  second  therefore,  these  operate  the  end  management of  the  input  [67]  stores  coordinates  entered  an  very  image  into a  scan,  of line  the  LAG  boundary  points  which  structure to  connect  border  image.  was  the  also  explored  input  i m a g e be  r e q u i r i n g storage  segments.  constraint.  border  2 lines  are  of  string  s p e c i a l data  that  a  whether  per  Pavlidis  line  completion  for 8-connected  straight line  special  the  to  by  sequential  methods  more t h a n  the  operator, to  no  encode  found non-sequentially He  finding  method  On  references  a b o v e m e n t i o n e d memory  into a connected  of  Evidently,  the  (LAG).  as  speed  a  was  needs.  the  i n the  The  guided  character  uses  fact,  describes  idea  type.  storing  object-points  adjacency data  time.  100  current  system.  frequently  real-time  that  the  memory  i s some c o n c e r n  a  store  memory m a n a g e m e n t  frequent  pixels,  report  border  differently.  the  the  our  non-sequential  by  there  system  an  The  so  of  to  of  images  [66]  of  communication  static  meet  a  employed  following  algorithm,  H o w e v e r , D'Amato e t  be  border  for border  could  must  the  Because  search  small  to  on  consideration.  involved  requires  list  operated  list  object  These  structure.  line  of  at  by  Chakravarty  scanned least  p i x e l s that  can  segments are  Concurrent  with  the  by  a  3X3  two  image  be  linked  stored scan,  in  a  these  143 lists  are linked  together  through  description  of  a l l borders  chain  No  effort,  code.  borders  since  segments.  the  segment p i x e l s The and  Mar low  as  i t i s raster  the  system.  been  [69].  problem  but t h i s  t h e system  the  Therefore,  scan,  even  image a r e s t o r e d , t h e i n i t i a l subsequently  lost.  All  the  of  above  systems,  i n operation.  image  i s  and  must  description  image  be p r o d u c e d  Modifications drawbacks; that  C.T.  their  lines  seeks  to  Zahn  of Batchelor image  a r e i g n o r e d by process  purpose  in addition During  a moving  the techniques  object  hardware i n  that  by  had  the  detects a l l borders  of t h i s  to the  the  During This  not  points'  i s  shortcomings  first  pass,  the  pass, the  i s unacceptable  with may  system  t h e second  window.  cited  t h e answer  description  line  ambiguity  i s generated.  b u t i t was f e l t  inherently  generating  thin  "on t h e f l y " c o n c u r r e n t  to  define the  t h e c h a i n code of t h e border  processed.  incorporating  line  t o the input  memory a d v a n t a g e  a r e two pass scanned  closed  though o n l y two rows o f t h e i n p u t  cited,  dynamic  applied  time  of a  unique  i n the approach  r e p r e s e n t an  During  as  a  them.  i n t h e 3X3 w i n d o w w i t h s p e c i a l lines  achieve  lines  would m u l t i p l y  i salso  form  i n t h e form  to  i s s t o r e d i n a RAM m e m o r y a d d r e s s e d  coordinates.  image b o r d e r  made  thin  i s found  i s because  and t h i n  found  follower  scanned,  resolved.  points  stores  A 3X3 window  This  found  real-time,  i s  present  a s i t t r a c e d i t s way a r o u n d  opposite  borders  and l i n e s  however,  method  A true border  the use of p o i n t e r s t o  in a  Border  descriptions  the  raster-scan.  remove  this  l a y elsewhere, and p r e s e n t s  and other  in a  system  no b a r r i e r s t o  on t h e f l y .  [70] devised a binary-image  description  procedure  144 founded The  on t h e d e t e c t i o n a n d  result  was  a  two-dimensional provided from  complete,  patterns  t o s h o w how  the  linkage chain  seen.  the object  description.  Though  f o r real-time border  that  d e s c r i p t i o n method  meet t h i s will  objective.  be  4.4  images  and  p r e l i m i n a r y method  was  even  no  could  suggestion  processing  incorporated  be  reconstructed  as t o the method's  was  made,  enough  i t was  flexibility  sections,  i t s application  developed  patterns  a  which  a right-hand  this  information  description  of  totally  provided  Zahn's  to  felt to  method  closed-object  trace. can the  immediately  within  storage  o f no more t h a n  already  defined,  edge p o i n t  current  raster image  on a 2 X 3 w i n d o w :  This  pair  consistent  a  of any  description line  window.  are located  complete  internal,  scan  image l i n e s  the edgein-edgeout  on  i t s background  provide  e x t e r n a l and  edge p o i n t s  based  such  arithmetic relationships  to  two  is  and  direction  a  describing  t o segmenting  technique  t h e image.  after  explicit  To d e t e c t  the  both  object  elements.  This  processed  borders,  contained  into  adapted  Through simple be  formally  between an o b j e c t  complete  As  of  them an e d g e i n - e d g e o u t  with  object  way.  t h e edge p o i n t s  assigning  method i s readily  i n a non-sequential  identifying  each  any  explored.  Zahn  binary-image  be  of  Zahn's B i n a r y - I m a g e D e s c r i p t i o n Method  C.T.  and  A  present.  description  In the subsequent  presented  segmentation  code  patterns  suitability this  of a l l edge-points  admits  can the  Furthermore,  i s required.  between p i c t u r e  directions,  Zahn  centers  145  D1  D2  <*>  D4  The  edge-point  By can  be  were  readily  found  by  found Zahn  of  It edges is  t h e window,  edgein  AD1 A D 2 A D 4 A D 6 A D 5 A D 3 A D 5 A D 6 A D 3 A D 5 A D 6 A D 5 A D 2 A D 5 A D 1 A D 2  s h o u l d be  1 3 4 5 7 0  noted  along directions  overlain  near  A  2-6.  similar  ( C i + 0 . 5 , r j ). defines  For t h i s  neighbourhood  edgeout  this To  These  particular  remedy  D6  7 0  window c a n n o t  detect  an  window  additional 90°:  but r o t a t e d  r  i +  r  (  i  D8  edge-point  i s  w i n d o w , a new  the edgein-edgeout  this,  above  D3 nn  1 3 4 5  AD1 A D 4 A D 5 A D 2 A D 4 A D 5 A D 4 A D 5 _ A D 2 A D 3 A D 6 A D 6 A D 2 A D 3  to that  D7  the  functions.  D2 D1 D 2 D 5 D 5  that  D5  time  directions  boolean  D 3 A D 5 A D 2  D2  This  the edgein-edgeout  be:  neighbourhood D5 D 2 D 2 D 2 D1 D 4  i+  t h e c o o r d i n a t e s ( c , , r, + 0 . 5 ) .  using elementary  to  r i  D6  D5  at A occupies  examination  D3  centered table  directions:  of  on  B at  boolean  location functions  146 neighbourhood  edgein  D6AD7AD5 D6AD8AD5AD7 D6A D8A D5 A D7 D3 A D5 A D6 D2AD5AD3AD6 D5AD3AD6AD2 Since  the  neighbourhood  1 2 3 5 6 7  two  edgeout  D 6 A D 3 A D2AD5 D3AD6AD2AD5 D6AD2AD5 D5A D6A D8 A D7 D5AD7A~D6A D8 D8 A D5 A D6  windows  1  2 3 5 6 7  together cover a l l of the p o s s i b l e  edge d i r e c t i o n s , the two windows can be combined  into  a  single  window of the shape: D1  D2  D4  D5 (l  D3  P) D6  D7 The  positioning  of  the B p o i n t between D5 and D6 w i l l  seen to be necessary  edge-points  p r i n c i p a l advantage of Zahn's approach l i e s  i n the f a c t  of  linkage  scan.  that edge-points processing  t o achieve dynamic  l a t e r be  of  during a raster The  D8  are  uniquely  detected  for every edge-point.  defined.  This  simplifies  the  edges and i m p l i e s equal p r o c e s s i n g  time  The p r i n c i p a l disadvantage  here  i s that an  e x t r a b i t i s r e q u i r e d to s t o r e the one h a l f c o o r d i n a t e d i f f e r e n c e marking the edge p o s i t i o n . of  curvature  significant that  90°  points  Since Zahn advocates  (edgein  ?  edgeout),  i n l i g h t of the storage saved. corners  and v e r t i c a l o b j e c t  resulting sides  only the storage  this  not be  Another o b j e c t i o n i s  from the i n t e r s e c t i o n of h o r i z o n t a l  are  not  detected  c o r n e r s a r e represented by two 4 5 ° c o r n e r s . smoothed  directly.  These  T h e r e f o r e , these and  sharper c o r n e r s are  essentially  contour  r e p r e s e n t a t i v e of the border  not e n t i r e l y  may  into  a  rounded-edge  shape.  This i s  147 illustrated  As  well  i n the following  as the rounding  any  type  cause  than  border  technique what  detection  objection  i s  not  i s  seen  that  corners  o f more e d g e - p o i n t s  largely  generate  wanted.  A  modification  directions  an  border  simple  generated  i s  to  of  (black  dots)  one.  This  aesthetic  p o i n t s which,  modification,  detection  acceptable-.  The  associate  the  not with the between-pixel  the nearest object border  multiple  i t i s also  at heart, i s however,  can  this.  The  with  the  does  accomplish  effect,  points.  Trie l a s t  really  example  of  point.  certain  border  This  border  edge-points,  initially points,  points are s t i l l  edgein-edgeout  causes but  but some  that's  o n l y d e t e c t e d two a t a  time.  As  Here,  an example, c o n s i d e r t h e f o l l o w i n g  edgein  associated following  = 4  with  and the  edgeout pixel  neighbourhood,  at  = 5 D2.  and  neighbourhood:  these On  the  values other  will  be  hand, the  148  associates  the  modification that  the  would  directions  be  the  That  original  some  i n the  become  pixel  border  example.  B.  associated  c o o r d i n a t e word  redundant  following  with  done w i t h e d g e - p o i n t  edge d i r e c t i o n s  allowing  seen  edge  D5.  A  The  net  with  similar result  border  l e n g t h s t o be  points  maintained.  point detection occurs  Consider  the  is  image  can  be  segment:  1  t '///  §  1 i1  The  t  edgeout  c h a i n code generated  for this  edge  curve  would  read: 2 2 2 1 7 7 7 5 5 6 , with  pixels  following  1  and  3  scheme w o u l d  being  recorded  r e c o r d each  pixel  three once  times. and  A  border  generate  the  code: 2 Brief  examination  edgeout net up  n e e d be  result pointing  look  2 2 7 5 6  of  stored at  these  at  shows  each  pixel.  first  proceeding,  defined borders  at  the  i t would  be  above windows would  expedient be  edgein  and  the  last  that  the  i s that a pixel  ends  A l s o , i t i s seen  v i a i t s edge d i r e c t i o n s  Before how  that only the  multiply  itself  .  and  to  implemented  linkages.  take  a  brief  i n hardware.  1 49 Assume,  first  delivered  in serial  the  image  video The  shift  provide The  proposed  would access  basic  of  wide  at  would  The  other  would  bits;  would  regs.  for i l l u s t r a t i o n ,  that  64  three  then  being  3  tall.  consist  be  to a l l three  is  least  would  of  bits be  of  three  these  two  types  of  long;  two  of  bits  long  and  would  be  shift  regs.  D1  D2  D3  Z2  D4  D5  63  • • •  3  2  1  1  D6 i  64  1  S1  D7  D8  64  63  • •  3  2  1  S2  All rate.  of  The  the  Binary  three-bit  neighbourhood  of  would access shift  shift  points  e i g h t of  registers  configuration left  maintained.  The  extensive  detects  the  delay  results  to  the  in a  right.  use  border  of  points.  hold  bits  in this  the  video  3X3  the  the  In  data  no point  order  The  by  sweeping true  video b i t  Zahn-based The two  two rows.  across  image  linkage  i n which  the  c u r r e n t window,  window.  window  border  clocked at  examined.  Clearly,  following  are  registers t o be  •  Video  Hardware  registers  shift  used.  be:  64-bit  Non-sequential  make  and  image d a t a  Z1  Z3  from  binary  which  used.  shift  the  implementation  implementation  3-bit  that  A l s o assume,  pixels  One  be  all,  fashion.  i s 64  registers.  these  of  this  frame  the  or  method 64-bit This image  buffer is  procedure scanning  will window  1 50 4.5  Border  The  Linkage  Zahn of  However,  i t was  local  closure needed.  operator  s h o w n by  simple  The  the  border  In  this  a  points  every  Papert  Some f u r t h e r ,  just  pixel  [71]  and  to determine  global  up  in  the  with  the  direction  lines.  is  represents  the  character's  the  available  of  of  directed  to  the  was in  directions.  for using  this  closure.  system  used,  image rows w h i c h  local  nor  i s provided  coordinate  new  the  local  collection  processing  presented  i n the  direction  character  be  image.  connectedness  c o n n e c t e d n e s s and  that  i n the  edgein-edgeout  will  both  to note  the  outlined  that a  global information for this  method  c  and  method  neither determine  point coordinates  i s important  coincides  applied to  Minsky  curves.  section,  detection  could  necessary  information  r  local  point  operations  of  It  Closure  border  consists  of  a  and  horizontal right  along  a  also  of  the  row  and  vertical.  Linkage:  Using of  connected  linked  by  border  using  the  r , c,  points,  edgeout =  edgein  (2)  r  for constant  a  j  +  1  >  This ie.,  is new  larger the  For  r,  a  b  c, of  and  the  values.  o r d e r i n g of  successively  a  edgeout  i  +  nature  b  b  ^ Cj  1  of  This  by  be  pairs  properly  Zahn:  points.  the  always detected  linked  detected  c  values,  ( r , c ) , can  p r o p e r t i e s observed  points are  coordinate  proper  a  between a l l connected  statement  border  and  (r , c ) ,  following  (1)  edgein,  property  for  constant  scanning  r.  process,  at p r o g r e s s s i v e l y will  be  central  to  points.  border  points,  ( r , , c,)  and  151 (r , 2  c  2  ) ,  with  edgeout =  edgein  a  along  b  the  following  di rect ions:  (3)  0-4: r , = r Cj  such  and there  2  that  such p o i n t s (4)  c,<Cj<c . along  2  the  2-6:  c,=c  2  and  with  ri  such  automatically  1  with  c a n be no  other  row  with  occurences of linked.  point  r,<r,<r  ( r , , C| )  and c,<Cj<c .  2  2  o f t h e same d i f f e r e n c e s a r e  that  c a n be no o t h e r r,<r,<r .  such  points  point  on t h i s  Therefore,  2  in  the  column  consecutive  same  column  are  linked.  + 2  c  2  and there  c a n be no  t h e same sum s u c h t h a t  consecutive  consecutive  row a r e a u t o m a t i c a l l y  occurences  there  of  3-7: r + c , = r  in this  linked.  occurences  (6)  point  d i f f e r e n c e such that  consecutive  automatically  (5)  t h e same and there  2  same  Therefore  Therefore,  2  1-5: r , - c , = r - c with  c a n be no o t h e r  other  r,<ri<r , 2  occurences  of the  (1) simply  states  same  point  c,<Cj<c . are  (  ,c i )  Therefore,  2  sum  (r  automatically  linked.  Condition points (6)  t o the next  provide  points  succeed  The curve  the  border  point  means  for  one a n o t h e r  process  of l i n k i n g  i s accomplished  Conditions tables  which  explicit  use  the fact  i n succession. detecting  along  t h e same  a l l the border  through a series of  the pairing  of c o n d i t i o n  of  each  border  Conditions  when a g i v e n  (3) t o (6) a r e implemented  perform  that  linked  pair  point (3) t o border  border. points  into  a  closed  lists.  through a s e t of points.  ( 2 ) i s made t h r o u g h o u t ,  eight  Note i e . , that  that new  152 points  arrive  Direction  c  2  r  r  2  = 1  • •  i s  =  only  edgeout 1st  one  edgeout 2nd  entry  r ,  C,  c  2  sorted  by  in this  direction.  entry  difference  entry  -m  the  2  virtue  of  c  Therefore,  each.  -m  -min  1 / c,  0  f  2  , c  *  Cl  2  points  m  r  automatically  o f t h e same d i f f e r e n c e .  o r f r o m new  rows  in  i s provided these  (i.e.,  tables  data  much  modulo determined of  Also,  hash-coding,  c  i  C  access  l a r g e r than  i.e.,  entries.  be s u b j e c t  using  such to  r  •  m  r  2  on  these  the  entries  t h e 0-4  they  2  i  C i  m  *  m  2  , c,  must  direction.  I t will  be  possible  be One  shown  arithmetc  i s comparatively  the difference result  c  consecutive  Therefore,  i s  -m  2 f 1  directions point  t o modular  a table  entry  -m  2  not yet a r r i v e d ) ,  by t h e image w i d t h .  images,  r  sorted  Since  2nd  2  , c  f o r each p o s s i b l e d i f f e r e n c e .  d i f f e r e n c e s must  largest  ,  1  r,  edgeout  —m  m  max  / c,  are  —m  • •  • m  = 5 =  entry  t\  »  *  m  1st  • •  •  r  edgein  r-c  c,  Cl  C2  r  • • *  table  constant  = 4 =  entry  fir  *  n>  manage.  1st  c,  contain  -m  accumulated  the  r  entry  occurences  that  values  1-5:  These  slot  edgein  automatically  while  -m  the  are  t a b l e s need  2nd  coordinate  entry  r 1,  edgein  the  1st  points  Direction  to  edgeout  2  increasing these  = 0 =  entry  r , These  ascending  0-4:  edgein 2nd  with  even  with for  easy t o through  as the address t o  153 Direction  2-6:  edgein 2nd  =  2. =  entry  r 2 2 i r  c  c  edgeout  edgein  1st  entry  coord.  1st  r  r°  0  r  2  r °  0  _ m  r  2 f C  2  These  r  points  occurences  of  the  time,  maximum number o f  Direction  = ;3 =  entry  -m  -  r 2  c  edgeout 1st  m  entry  c  2  -min  t a b l e s , these  too  corresponds  case  is  ordered  rr°i  to contain  construction  of  r-coordinate  the  according  to  their  these  image d a t a .  as  implicit  tables.  However,  1-5  to  The  the  /  2nd  entry  r 2 t c2  c,  • •  'i  r° C|  r  2 f C  1  i  rr 2 '/ cr 2 m  C !  coordinate sorted.  the  modular  assumptions most  same  r  represents  r  must  an  of  only  m  here  sum. The  would sums.  underlying  the  i s that  values  as  be  recycled  the the  u n r e s t r i c t e d number  therefore  the  Again,  table  range of  fundatmental  range  2  • •  direction,  automatically  number of  is limited  c-coordinate. rows of  a  r  to  many s l o t s  entry  •  similar  as  edgeout  •  max  are  7  *  m  entries  =  •  0  r? '  There are  new  1  r  •  r?  , c2  are  have  C2  i  consecutive  entries  , 1st  *  consecutive also  2  r  on  edgein  *  This entries  c7  1-5  number o f  r+c  • •  r 2  the  sum  r 7  , C2 i  2  ,  2  values.  •  r  2 r C  r  sorted  Like  the  0  entry  3-7:  edgein 2nd  r7  automatically  This  2nd  •  max  same c v a l u e s .  h a v e many e n t r i e s . to  1 f c,  are  the  c  edgeout  •  •  m  6 =  entry  0  • r  =  in  of a  154 modulus  at least  equal  t o t h e maximum c v a l u e .  that  a l l of the previous  1-5  and  3-7,  assumption first. the  still  i s that  This  and  i s  and  of  A  representation full  of  information direction store  sorting,  and  other  list-1  - This  for  later  for  detection  in  this  is  list-2  - This  list-1  since  intimately  first  are  maintained:  list  would  to  i s simply  list  Also,  processor  can  with  there  into  This  after  in sequential  be c o n s i d e r e d number  of  since  t h e 2-6  one l i s t - 1  need  entry.  to  automatic  entry.  chain  tables,  data store  codes a r e  detection.  Data  as i t a r r i v e s .  a horizontal  of e n t r i e s ,  the  and edgeout  order  be  some  i s the primary  closure  In a  for lists  the linkage  and/or  to the  may  i s no  the second  points  proper  Finally,  pointers  B  points  deceptive.  r , c, edgein,  the border  stored  7.  structures provide  of points  Also,  guarantee  instance,  to store  contain  i t h a s t h e same  associated  data  down i n  i n the slots  instead. for  entered  b e t w e e n D5 a n d D6  3 and  stored  and processing.  until  d o e s A.  i s somewhat  redundant;  to the next  list  be s t o r e d  no n e e d  important  l i e farther  For instance,  a l l these  along  l i e between A  positioned  tables  the entry  Another  than  diagonals  directions  with  data  r.  i n increasing c values,  acquisition  transmitted  ±45°  was  along  i s really  lists  points  shown  particularly  B values  the border B  n e x t may  since  there  older  character.  shown  Concurrent two  into  be  a t B i n t h e window a r e  the information  i s stored  r,  test  the given  different  be d i s c u s s e d  a modular  because  edges along  implementation, a very  for  necessary  linkage  relations,  detected  points.  ensure c o r r e c t  to  points  s o m u s t be l i n k e d  linkage  of  hold  image and t h e r e f o r e  coordinates  linkage  It will  extension  each of which  List-2  is a list  of i s of  155 pointers,  the pointers  entries linked  being  representing  successive  i n a right-handed  through closure  these  addresses  border  pointers,  to  border  trace  this  other  points  list-1-list-2  as they  i n t h e image.  entire  approach  would  By  be  tracing  achieves  global  detection.  The  physical  structure  of these  Address  lists  i s therefore:  list-1  ri  c  ,  e,  1 ,  list-2  ,  ei , c / ei , 3  eo  —> e n t r y  a  e  —>  entry  b  eo  —> e n t r y  c  e  —>  x  0  * • •  C  where in  signifies  resolved).  a  This  ei  r  "—>" r e f e r s t o " p o i n t e r  list-2  been  N  ,  to".  null  0  It will  pointer  i s why t h e l i s t  entry  be assumed  (i.e.,  that  the linkage  addresses  are  a  0  has not  started  at  1.  The In  fact,  linkage  entries only  generating  in  i s  of  lists  image  list-1 points  list-1  and  really  i s  be  a  chain  needed  b e made t o a d d r e s s  the  end  of these  lists  to  this  be  stored  store  arrive.  Also,  redundant.  For  code,  only  (usually edgeout).  complexity  will  by t h e l i n k a g e  need  used  as they  i s indeterminate,  the  address  may  and processing  edgeout these  since  on b o r d e r  information  a r e generated  the l i s t - 1 - l i s t - 2  tables  information  of l i s t - 2  the some  one  question  general the  instance,  in  of  edgein,  The n e c e s s a r y  contained.  empirically  i s encountered,  i n the  of  a n d d e p e n d s on t h e s i z e  of the objects  tables.  An  of  size the  attempt  later.  the address counter  When will  156 simply this  cycle way,  back  t o 1 causing  objects  that  o l d e n t r i e s t o be o v e r w r i t t e n .  could  n o t be s e g m e n t e d  would  In  simply  be  forgotten.  The  complete  processing  (1)  Before  acquiring  and  list-2  pointers  preventing (2)  addresses  When  edgein  a  point  found  address linkage  (4)  deserve  the border  i s flagged,  into the  points  point  i s the f i r s t  of  slot  the l i s t - 2  into  linkage  place  list-2  as  table  entries  a general  reset  borders.  into the appropriate  t h e two l i s t s  If  under  ends a t a n u l l the trace  and transmission  of  entry,  the  address  the  of the  edgeout  its list-1  i t i s t h e second  of t h e edgeout  and  tables.  the list-1  slot  When  list-1  point.  address can entry,  the  point  i s found  i n the  of  (a process  defined  later),  trace  the d i r e c t i o n of the l i s t - 2 pointer,  the current  ends a t i t s s t a r t i n g of i t s points  points  border  point,  may now t a k e  concerning  this  pointers. i s not  the border i s place.  processing  sequence  special attention:  Step  following closure  serves  of closed  table.  A number  and  This  i n the linkage  the trace  closed  clear a l l linkage  table.  closed.  (a)  enter  0.  On a n t i c i p a t i o n o f c l o s u r e  through If  to  linkage  When t h e e d g e i n be  image d a t a ,  detection  On d e t e c t i o n ,  list-1 (3)  false  s e q u e n c e c a n now b e o u t l i n e d :  4 i s a very pointers  detection  the  memory  much s t r e a m l i n e d  in  memory,  this  border  approach  a t a speed p r o p o r t i o n a l reference  rate.  t r a c i n g method.  This  achieves  t o the border i s  also  By  global perimeter  the  only  157 s u b s t a n t i a l l y s e q u e n t i a l procedure i n the e n t i r e processor may  be  the most c r i t i c a l  i n determining  (b)  To ensure that u n l i n k a b l e o b j e c t s  the  "sides"  or  original  image  invoked  extremities. these  (i.e.,  "bottom" of the  s p e c i a l conventions must be the  i t s overall  The  scanning  extremities  are  image)  which  touch  are not  seen,  window  circuitry  instances  when  convention  i s that when the window i s p o s i t i o n e d with  on the extreme r i g h t or l e f t column first  row  (at s t a r t up),  image to b l a c k . the  prevent and  and  information  wasting l i s t of  A  the A p o i n t below  the  set those window p i x e l s overhanging  w i l l provide  the  the d e t e c t i o n c i r c u i t r y  f o r edge d i r e c t i o n d e c i s i o n s .  memory,  the  when  B point.  Therefore,  transmission,  flag  preferred  on  The  the  net  right  result  if  list-2  entries  with  However, to  image boundary  column,  disable  i s that those border  p o i n t s which touch the s i d e s w i l l always p o i n t to, or from, entries.  at  must  image, or  the extreme r i g h t column from l o g g i n g the  detection  is  T h i s w i l l cause u n l i n k a b l e o b j e c t s t o merge with  image borders,  consistent  reached.  of the  so  speed.  those  when the Zahn  and  null  are always c l e a r e d  u n l i n k a b l e borders can never be d e t e c t e d  i n step  on 4.  E.g. ,  untraceable  In the  long  could be a c c i d e n t a l l y l i n k e d i f l i s t - 2 e n t r i e s are not f i r s t cleared  run, the u n l i n k a b l e borders simply  get o v e r w r i t t e n  as  1 58 the  finite  (c)  On  are  seen  that  list  a  similar  Anticipating of  (d)  the  Due  there  note,  is a  however, a l l l i n k a b l e , when c o m p l e t e . no  when  border  is likely  a  next  list-2.  I f an  along  i t cannot  the p o i n t e r s  current way  of  at  If scan  the  entered  an  To  counter infinite  On  10,000  new  closed  list-2.  will  be  the  an  this,  was  pixels  be  loop while list-2  entries  and  during  border  of  to  infinite can  on  an  extent  the  lists,  more  a closed  a counter  the  such  point  form an  of  recycling  may  recent  border loop  be  on  the  with  no  maintained  each  trace  length,  the  stopped  without  closure  trace  with  the  border  a  has  being  t o have o c c u r r e d once  width  in  step.  list  observed in  infinite  slots,  though  incremented  can  memory  in  list-1-list-2  entries.  of  to another  closure,  circuit  Transmission  tracing  through  as  list-1  coordinates  list-2  and  list-2  outlined direction  entries  points  for reconstruction  recognition.  and  in  fact  even  entries  and/or  transmitted,  the  occur  before  level  and  this  128  verification  transmitted  can  encounter  l o o p and  lines  of  entering  value exceed  6400  of  will  trace,  In p r a c t i c e ,  length  t o be  borders  from  entries  list-1-list-2  closed  prevent  flagged. scan  be  recent  trace  with the  Should  follows  object i s u n c l o s a b l e or  these the  of  with  i t s deepoest  stopping.  parallel  the  This  cleared  transmission.  entries.  of  remote p o s s i b i l i t y  that  null  or c l o s e d  section.  recycling  are  r  This  leave  through  pointers  recycled.  borders  to the  tracing  are  immediately  closed  topic  slots  is  a  the  simple matter  in step numbers  are cleared  of  4,  except are  must  also  immediately  be  object of  again  that  the  read on  and  being  159 read.  It  simple  comparison  any  i s here  border  the  last It  the  that  circuit  or  will  now  pairing  assured.  of  will be  be  Lines  calculations  implementation,  To  maintain  symmetry  question the be  now  above  two  linked  by  show t h i s , of  2)  and  The line  falling  figure  will  The  back  arithmetic.  subsequently  For plane  the  where  be  the  onto  For  correspondence of  by  be  used  camera  example,  overflow  this  rows  in  off  in  presently region  r+c=K, t h e r e e r r o r s would  case. 64  modulo  and  pixels.  points  along  r=-32,  s t i l l  i s yes.  behavior  of  is  region  by  virtue  in region  5 share  6.  To  power  the  A l l points outside  i t  r The  (preferably a  bold  In  64.  answer  the  the  lengths, the  border  The  If  trivial  the be  is  together  these  would  r=+31  illustrate  region  a l l but  sums r + c .  word  modulo M  within  i f  directions  i s not  linked  scanner.  points  written into  equation  to  squared  with points  r - c , and  expressions?  points  characters  on  arbitrary  region  seen  map  portions  same s i m p l e  4.1.  physically region  but  are,  linked  forced to c y c l e at  lines,  an  3-7  are  coordinate  topologically  use  and  c dimension  can  will  1-5  but  arises:  Figure  r+c=K.  the  t h e r e f o r e be  the  we  in  the  A  a r i t h m e t i c i s employed,  infinite,  problem,  removed.  transmission.  the  were  tentative  I f they  directions  available  be  a p p l i e d to determine  i n the  difference,  w o u l d p o s e no  will  be  i f modular  these  our  dimension  p o i n t s can  repeated.  points along  along  lengths  are  that  common c o o r d i n a t e  register  can  ignored  shown  border  border  test  point coordinates  repeated  through  redundant  of  the  this  modular  a  one-to-one  Therefore,  unscanned  residing  in  region  5  will  regions  in  the  numbers  are  6.  will  be  occur  certain because  the  160  Figure  4.1  Modular  image  representation  161 too  large  ( l a b e l l e d Y///X  1  in  the  f igure).  These  regions  are  defined as: In  region  1:  r  In  + c > M/2  region  + c <  Careful generated a  i n region  those  region  segment the  the  mapping;  generated  line  B i n region  which  line  limit B.  l i e on t h e l i n e r+c=K  Also  exactly  represent  modulus  onto  i n region  i t might  new  line  parallel,  of  m o d u l u s M,  line  line  A  is  line  the  even  i n region  4  i n region 2 are  segment  mapping. of l i n e  non-overflowing lines  will  Furthermore,  into  region  6  A  line satisfy  points in outside  and  again  ignoring overflow,  points  1 will  s t i l l  satisfy  the  5.  that  unlinked, This  i f a line  i t may will  always  detected.  be  B,  already  i n a d v e r t a n t l y be  linked  happen under  lines  i n both  completed, This  segment,  i n fact  Since  - 1 , and separated  B will  first  map  circumstances.  slope  region  by o n e - t o - o n e  4, a n d y e t b o t h  will  s e g m e n t , A.  tolerable  onto  o f m o d u l o M.  be s u p p o s e d  to  in  t h e sums  the c o n t i n u a t i o n of the l i n e s  r+c=K  and i s c u r r e n t l y  special,  3, a g a i n  Therefore,  exists a  generated  p o i n t s on t h e o v e r f l o w i n g  of  directly  equation  i n region  r+c=K by v i r t u e  reveals that  a n d sums g e n e r a t e d  map  5  regions  1 are e x a c t l y those  will  positive  which  of these  1  relation  region  The r e s u l t a n t sum h e r e w o u l d be r e p r e s e n t e d a s a p o s i t i v e number.  examination  Therefore, in  T h e r e s u l t a n t sum h e r e w o u l d be r e p r e s e n t e d a s a n e g a t i v e number.  -M/2.  one-to-one  exactly  1.  2:  r  with  -  or  i s because  A  and  dimensions linked, line  B  only are  by t h e before  B leaves the  162 visible also  region  can  a t r=X  earlier  than  m e n t i o n e d c o n s t r a i n t s on t h e  lines.  The  start  only  there  are  image  (using  bottom, the  no  i f i t i s not f i r s t  exception  and the bottom  i s not a serious  clearly one  unlinkable,  large  entries  are  modular, time  a large  like  In  summary  argument point  as  unlinkable a n d be  then,  be  ignored  case  also  line  at  since  i s  edge  points  at  the  B a t r=X a n d  r=X.  However,  both polygons are  will  (This  will  where  link  them  guaranteed  i f list-2  are  entered.)  i s finite  become  into  and  overwritten  also in  forgotten.  modular of border in  the  i s t h e same a s a b o v e  linkage,  A  A  the above  o r i e n t a t i o n ) a n d one  o f t h e two l i s t s  object  of  line  one a t t h e t o p o f t h e  the algorithm  new  and  special  present:  because,  object.  and p r o c e s s i n g  can  i s the  line  situation  the length  any other  labelling flags  since  along  at worst,  cleared  because  one h a s an e d g e o u t a l o n g  unlinkable  Furthermore,  objects  the c u r r e n t l y depicted  t o p one h a s an e d g e i n  this  r=X  t o t h e above  two u n l i n k a b l e  connected,  and y e t c o r r e c t  arithmetic points.  can  be u s e d  Overflow  i n the  errors  or  c a l c u l a t i o n o f r+c and r - c ( t h e  by s y m m e t r y ) results are  f o r determining assured.  border  163 4.6  Closure  Detection  Actual outlined  detection  in step  3  time-consuming, to  border  such  some  techniques  "region  object scan  counting"  regions  is  implement. detect topology  the  within  a  Euler  closure a  list  less  by  i s done.  Both  of  first,  colors  longer  a l l  seen  The but  i t  local  is  trace  The  or  approach,  row.  step  discussion  reliable,  detecting  trace  this  section.  i s no  trace  number  given  The  this  When a c e r t a i n l a b e l and  this  physically labels  s o p h i s t i c a t e d and  Called  probable  of  list-2  supplement  is likely.  topic  technique,  seen.  less  the  the  since  method must  detection  forms  requires  However,  reliable  i t i s assumed c l o s e d  approach  closure  previously.  anticipate i f closure  two a  of  on  second  easier  strives  changes  a  in  techniques  to to  image  operate  non-sequent i a l l y . 4.6.1  Region  This  technique  effectively regions  as  in  able  to  given  i t s name  counting It  was  connectiveness ability  closure,  to  and  the  because number of  adapted by  count  i t could  regions  i t is in this  connected  from  Rosenfeld  be  a  and  used object  method  for  Pfaltz  requires  connection  that  that  [72]. i t is i t  is  image  map  here.  The (no  very  detect  applied  was  image.  region  its  Approach  a means of  the  determining However  Counting  larger  detection)  technique than of  cross  reference  map  (16  in  the  requires two  shift  a certain table this  maintenance  as  of  r e g i s t e r s needed  f i x e d depth,  with  case)  the  say  four  many e n t r i e s a s  is also  needed and  a  small  for  non-sequential  bits  per  pixel.  l e v e l s i n the a pair  of  A  image  flags  per  164 level  entry.  1.  On  2.  Image  The  method  operates  startup a counter points enter  before 2X3  b u t now  i n the f o l l o w i n g  i s initialized  t h e i m a g e map  t h e image  manner:  t o some n u m b e r ,  in shift  i s a l s o viewed  say i=0.  register  through  form  as  the f o l l o w i n g  window: r L  <— 3.  last  A  B  C  i s considered  point, I f L=0  (i)  and A*0,  If  shift  c  o f new  the current  points.  point.  B  a new  object  Now  examine  the  and  register  C=0,  increment  a t A.  reference  Also  cross  and  region-active  flags.  (ii)  I f B o r G*Q,  give  and  set  the  look  t a b l e and  has been  i and  detected.  store  i t i n the  up t h e v a l u e  set a pair  A t h e number  region-seen  reference  I f L*0,  point  C:  the  cross  image  L:  Examine B and  (b)  <—  i n d i c a t e s the source  Point  (a)  A  bit  of  found  for that  of  i  in  region-seen  in  B  number  or  C  i n the  table.  set A to the value  of L and a l s o examine B and  C: (i)  I f B a n d C=0,  (ii)  If  reference set  or  C*0*L,  on. look  up  t a b l e and s e t t h e i r  the C or D c r o s s  table both  B  carry  and A's numbers  flag  as belonging  i n the  region-seen  reference  alongside  B or C  flags  C o r D.  t o t h e same  flag.  alongside This region.  A  cross Also i n the  identifies  165 4.  On  the  completion  i s systematically  examined  seen  during  When a s e t r e g i o n - s e e n  the scan.  trace i s conducted  their to  region-seen  prevent  the  seen but  i,  The example.  the l i s t - 2  completion,  a certain  since this  operation of t h i s Consider  to set  When  complete, examined t o  regions  i s  the  were  case,  i s made t o f i n d  the  (in this  not  cross  then  technique  the closed  reference  table  case  the region  counter,  16),however,  i s perhaps  best  seen  (1s) f i r s t  four possible,  Cross time,  another  region  enters Cross Ref.  Region  Image  skip  w i t h an  and c r o s s  regions:  | f  a  label.  a n image- w i t h a t m o s t  One r o w o f i m a g e d a t e  At a later  i s found,  t o be c l o s e d .  1 2 3 4  2.  were  be e x e r c i s e d  i s again  On i n c r e m e n t i n g  i s the null  flag  must  trace.  If this entries  modulo  regions  by a t r u e a c t i v e - r e g i o n f l a g  clear  t h e image s c a n .  referenceable,  ^  flag.  reference  references  Caution  flags  as indicated  t h a t w e r e now f o u n d  observe  i=0  true also.  region-seen  region-seen  On  Continue  i t s cross  loops during t h i s  of  scan,  through  entries  t o l o c a t e which  i fany of t h e p r e v i o u s l y a c t i v e  false  border.  flags  set  in this  trace  through  infinite  entire  determine  1.  of a row, t h e c r o s s  table  a  5.  of t h e scan  X 0 0 0  0 X 0 0  0 0 X 0  t h e image: Seen A c t i v e  0 0 0 X  Reference  1 0 0 0  1 0 0 0 Table  row a p p e a r s  i n t h e image:  166 Cross Ref.  Region  0 0 0 0 X 0 0 0 0 X 0 0 0 0 X  1 2 3 4 Image 3.  With  with  region  X  Cross  the a r r i v a l  of  new  4.  Cross Ref.  a  region 1 with 2 setting  o b j e c t now  closed,  X 1 0 0  Cross  of  2 and  The  1 X 0 0  breaks  away  from  Image a w a y , no  regions are  flags  this  the p r o c e s s o r w i l l  by  by  list-2  will  The flags and  a  to remain  cross  then  main  the  be  attraction  arrival  of  principal  complexity.  To  be  0 0 0 X  of  to  1 X 0 0  0 0 X 0  cross  referencing  image b o r d e r  Seen  forming  seen A  on  Active  0 1 0 1 0 0 0 0  0 0 0 X  Reference  reveal  find  Table  this  method  ambiguity  of  with real  scan,  subsequent that  two  seen.  the c l o s e d  object.  disadvantage workable  1 1 0 0 Table  top  Cross Ref.  reset.  this  a closed i s no  1 1 0 0  by  r e f e r e n c e , were not  ordered  thoroughly, there  The  merge  flag.  the  X 1 0 0  Cross  "region-seen"  linked  to  object: 1 2 3 4  broken  0 0 X 0  i s recorded  i t s "region-seen"  simply connected  table  1 i s seen  Seen A c t i v e  Reference  Region  Having  Table  2:  Image  region  0 0 0 0  image d a t a , r e g i o n  1 2 3 4  merger  1 1 1 1  Reference  Region  The  Seen Active  is  causing' the  examination  active A  this  regions,  trace  through  border.  that  i t  immediately  When i m p l e m e n t e d about  of  properly  closure.  technique  c h a r a c t e r images,  is  i t s  many  more  167 features  will  On  the  surface,  of  shift  also  have  there  counter.  find  which  operation,  Tracing  active  made t o p r e v e n t There immediately  infinite  are  also  obvious.  counter.  background  reference  were  time-consuming. loops  during  number  table  This,  i n  t o become q u i t e  detection  of c l o s u r e  has  been  scanned,  there  find  the connected  may  contain.  This  keeping  a third,  already  examined.  somewhat lacks  outer  circumvent  less reliable,  in reliability  a  table  sequential  t o f i x a modulus  complex, and  turn,  for i ,  d i s t o r t e d object merged  would  the  with  regions  force  and the borders  to  many  scanning record  can  the cross  until  a full  implementation  in  simplicity.  list-2  of any  a l l of  those  m e t h o d was d e v e l o p e d .  i tgains  be  which are not  i s n o o p t i m u m way t o s e a r c h  list  to  P r o v i s i o n must a l s o  i s not confirmed  border  these  t a b l e and  long.  necessitates  binary  i s  i s  a trace.  t h e number o f c o u n t e d  large.  There  reference  seen  here.  a number  reference  disadvantages  It i sdifficult  Since  To  of  shown  numbers.  cross not  been  to utilize  the cross  the  In a s u f f i c i e n t l y  noise,  quite  through  a  have  of the region  regions  and therefore  than  i s a need  t o store and maintain  region  become  we s e e t h a t  r e g i s t e r s f o r storage  t h e need  region  t o be i m p l e m e n t e d  row to  holes  i t  list-2  and  list-2  entries  problems  another,  However,  what  i t  168 4.6.2  The E u l e r  An and A  the  number  i s a topological  i s one o f t h e few s u c h p r o p e r t i e s very  found the  Euler  Number A p p r o a c h  thorough  i n Gray  discussion  [73]. Basically,  d i f f e r e n c e between number o f h o l e s , E  This  number  countable  image  n  0  however,  countable.  number e v a l u a t i o n c a n be number,  E,  represents  i n a n i m a g e , B, a n d  also  be  found  using  readily  elements. i e . , = n  - n, +  0  = t h e number o f  n  2  vertices,  = t h e number o f o b j e c t line  segments, regions  contained  by t h e  segments.  vertices meet.  i s locally  images  = B - L .  n  segments  of binary  L. i e . ,  = t h e number o f l i n e  The  Euler  an E u l e r  n, 2  that  t h e number o f o b j e c t s  can,  E where:  of  property  The  are  defined  following  as places  two  concept:  examples  where  two o r more  should  clarify  line the  y.  - vertex - 1 ine - region  Since expected.  there  is  only  one  object,  without  a hole,  E=1  as  169 Eg.  2:  n  Note  that  The found  line  Euler for a  0  =  23,  E  =  23  values  locally of  B  or  have appeared  AE  detection,  i s considered  Os  below  are  the  top  which  of the  illustrated  Setting done  as  to  influencing  the  to  droplets the  two  objects Euler  be  used  to  i n the  two  the  in  later  approaches  regions  break  i s considered from  Euler  detect the  the Euler  locally. a  new  image.  1s  framed  be  from a l l  s i g n a l that  be  appears  left  of  I n m e t h o d A,  image  object  change  image c a s e .  that  in Figure  the  prevent  dynamic  cannot  can  readily determined  may  this  then  that  property,  be  and  contributions  However, the  image can  property  i s known  hole  the  topped,  It  turn,  for  appears  summing the  in  developed  object  countable  countable  L.  12  0  can,  implement  image  =  =  2  criss-cross holes.  i m a g e by  i n a dynamic  or  n  image segments.  any  information  To  12  is a  complete  object  +  locally  like  closed  35  number  number,  This  35,  not  exclusive  number, AE,  =  s e g m e n t s do  mutually  individual  -  n,  free.  1s  been  topped,  from which In method  to c o n s i s t  "nowhere".  have  of  These  a  the B,  row  0 of  methods  4.2.  and  right  which  number.  sides  touch This  of  the  Figure  4.2  image  sides  i s because  to  such objects  1 is from are  170  Method A  Method B  1 topped  / /  ////// / /  /  / /  0  E=0  E=0  E=0  E=1  /  /  yXA  /  HE=-1 /  /  topped  /  / /  /  /  \  / E=0  E=0  E=1  y y  ; y  Figure  4.2  E=1  /  /  number variation Euler image r e p r e s e n t a t i o n s  /  /  XX  in  s  s y y y y  E=0  CJ  E=1  E=0  L0  E=1  1s  topped  and 0 topped  171 considered various  unlinkable.  steps  behave  the  detects  same.  the  not d e t e c t A,  detects  o f a new  merger  o f an  i t detects  In  The  0  method  o f a new into  the  body  the  first into  interior  the Euler  shows t h a t  topped  i t s separation  separation  as  the example  appearance  however,  the  in  E x a m i n a t i o n of  the  B,  feature, frame.  hole  image  with  two  The  at  instance, or hole,  Is  hole,  It  the outside  does  method and  also  in a  not only  but  topped  o f a new  frame.  the  m e t h o d s do  for body  appearance  the  number  the  detects  similar  way  separation.  summary, t h e E u l e r  number c h a n g e s  as:  AE 1 topped new o b j e c t a p p e a r s new h o l e a p p e a r s hole broken hole established object separation 2 o b j e c t s merge In each  comparison  other.  the  other  is  unique.  reduce  during  the  net  The detecting B  the case  one  a  local  a l l but  row  that  at a  time.  AE  observation  Of  these is  methods  can  same m a n n e r  two  the  is  entering be  that  established  changes examined:  complement  i n the of  course,  method  AE  we  the  A  which  have  image when  as  pairs  pairs,  confused  or p o s s i b l y separated,  in detecting  neighbourhood  only  of m u l t i p l e o b j e c t s  separated,  detect  two  i t i s these non-unique  the method.  is  overall  is useful  To  then  of  the  number c h a n g e s  Unfortunately,  result  evaluated  Euler  +1 0 0 -1 0 -1  that  e v e n t , and  reliability  considered The  an  topped  0 -1 +1 0 +1 0  i t i s seen  Neither's  0  not  field. i t  is  i s useful in  object,  and  method  holes.  i n the Euler  number, a  2X2  square  172 A C  c  b B  In each method, the e v a l u a t i o n of AE pixel  A,  found  and  the  total  by summing a l l the  considering  i f a vertex  is  done  with  largely  in  to  AE f o r a row, or p o r t i o n of a row, i s local  values.  AE  is  calculated  by  i s present a t v, or v , whether a l i n e i s 2  present at a, b, or c, and whether A i s 0 or 1. differ  respect  how they c o n s i d e r a and v,.  c o n s i d e r s there to be a l i n e and v e r t e x at a  Methods A and  B  Method A always  and  v,.  If  A=0,  method B never c o n s i d e r s there to be a l i n e at a and c o n s i d e r s v, to be a vertex only i f B=1.  O c c a s i o n a l l y , a l i n e or vertex  be c a n c e l l e d i n order to avoid being counted Since  there  are  sixteen  twice.  p o s s i b l e boolean  expressions f o r  t h i s window f o r each method, the d e t a i l e d a n a l y s i s presented.  The f i n a l r e s u l t ,  must  will  not  be  however i s :  Method A ( 1s topped ) AE, = +1 f o r AE, = -1 f o r  Method B ( 0 topped ) 77Z  AE  0  = +1 f o r  AE  0  = -1 f o r  The beauty of the method r e s t s with the f a c t above c o n d i t i o n s can be determined with simple AE accumulated using a simple  up-down  counter.  that a l l of the  boolean  t e s t s , and  173 Since  many o b j e c t s , or segments of o b j e c t s , can be detected  on a s i n g l e row scan, i t i s best  to stop and t e s t the accumulated  AE at c e r t a i n s t r a t e g i c l o c a t i o n s . For corner  the  1s topped case, t h i s p l a c e was chosen as the r i g h t  of an o b j e c t ' s  boundary.  T h i s would be s i g n a l l e d by  /////  T h i s c o n f i g u r a t i o n must separated  object.  be  present  on  the  right  side  of a  I f at t h i s p o i n t , the accumulated AE,=+1, then  such a s e p a r a t i o n  may be p r o b a b l e .  Likewise f o r the 0 topped case, A E  0  would be t e s t e d  i f the  f o l l o w i n g c o n f i g u r a t i o n appears:  ( X i n d i c a t e s don't c a r e ) . upper r i g h t corner t h i s point Since and  T h i s c o n f i g u r a t i o n must appear at the  of a c l o s e d h o l e .  An accumulated A E  i n d i c a t e s that a c l o s e d hole  0  i s probable-.  the two AE counters a r e bounded t o the range of -1, 0,  +1, p r o v i s i o n must a l s o be made to ensure that they  incremented  or  happen with the holes.  of -1 to  decremented presence  of  outside  of t h i s i n t e r v a l .  multiple  unseparated  are not T h i s can  objects  For example,  1s topped:  A AE, of -1 i s accumulated a f t e r t h i s unseparated has been scanned;  hole  or  174  0 topped: fy  A A E of +1 i s accumulated a f t e r has been scanned. 0  In  each  correctly  the  above  examples,  the  overlooked.  true  objectives  That i s , accumulate  encountered  during  a  of the  scan  accomodate the presence of m u l t i p l e made  to  reset  boolean t e s t  will  this AE  method be  object  i s working accumulated.  approach  individually  are  being  for  each  l i n e to d e t e c t c l o s u r e . objects  provision  the E u l e r number c o u n t e r s whenever an  o b j e c t or hole has been encountered.  Method A  the  i n that the c o r r e c t AE per scan  However,  object  of  t h i s unseparated  This w i l l  To  must  be  incomplete  r e q u i r e one  more  f o r each method.  (1s t o p p e d ) :  Reset AE, f o r  This  indicates  that an unseparable o b j e c t has been encountered,  and that an unseparable hole may Method B  be encountered-.  (0 topped):  Reset A E  0  for  T h i s i n d i c a t e s that an unseparable o b j e c t has been encountered.  Before summarizing it  would  be  expedient  the complete to show how  with the border d e t e c t i o n scheme.  c l o s u r e d e t e c t i o n procedure, t h i s method would i n t e g r a t e  The c u r r e n t  2X2  o v e r l a p the border d e t e c t o r ' s window as f o l l o w s :  window  would  175 D1  D2  D4  D5 (1j) D6  D3  D7 The  current  D8  window i s o u t l i n e d i n b o l d .  The p o i n t  to be noted i s  that each of the two above stop and t e s t c o n d i t i o n s  occurs  there  I f e i t h e r AE  i s a p i x e l 4-adjacent to the previous  counter i n d i c a t e s the presence of a  A edge.  segmented  object  or  when  hole,  then i t i s a necessary c o n d i t i o n that t h i s A edge a l s o belongs to where edgein i edgeout.  a corner  border r e p r e s e n t a t i o n s in the  linkage  available condition at  the  in  previous  i t s list-1  register.  Therefore,  A  initiate point  1.  a l l of the  i twill  be  Since  recorded  address  can  be  held  when  closure  test  a  beginning  this fast trace i s a  i t may be expedient t o keep a short  table  of  s t a r t p o i n t s which a r e t o be t r a c e d at the end of the row  simple  method in.list-2  4.6.3  in  the t r a c e through l i s t - 2  entry.  when more time i s a v a i l a b l e .  point  later,  and  i s flagged,  s e q u e n t i a l process, list-2  t o be d i s c u s s e d  tables a  Consequently,  of  E i t h e r way, we have  indicating  Detection  On beginning the scan of  a  a  p o s s i b l e c l o s u r e , and f i n d i n g the  where the v e r i f i c a t i o n  E u l e r Number Closure  established  new  t r a c e i s to be begun. Procedure Summary: image  row,  clear  two  AE  counters ( A E = 0 topped, AE, = 1 topped). 0  2.  On  scanning  with border p o i n t  the row, t e s t the 2X2 E u l e r window simultaneous detection:  (a)  On D 5 A D 1 A D 2 A D 4  (b)  On  D 4 A D 2 A D 1  t r u e , increment AE, .  decrement AE,.  176  3.  (c)  On D2AD~1AD4AD5  increment A E .  (d)  On D 1 A D 5 A D 4 decrement A E .  (e)  On D1AD2 r e s e t AE,.  (f)  On D 1 A D 2 A D 5  0  0  reset A E . 0  the l i s t - 1 - l i s t - 2  Record  address of the p r e v i o u s  point  just  d e t e c t e d a t A i n a short stack when: (a)  D 4 A T T 1 A D 2 A D 5  (b)  D 1 A D 5 A D 4 AND A E  then  reset  AND  AE, 0  =  +1,  = -1,  the AE r e s p o n s i b l e f o r one of these e x p r e s s i o n s  r e t u r n i n g TRUE. 4.  At the end of a row pop the l i s t  and,  one  at  addresses  a time, t r a c e through l i s t - 2 .  from  the  stack  Transmit the border  p o i n t s when c l o s u r e i s d e t e c t e d . It method  should now be c l e a r that the p r i n c i p a l advantage is  i t s simplicity.  this  The a d d i t i o n a l hardware r e q u i r e d by  t h i s method i s two c o u n t e r s of only two b i t s each, of  of  maybe ten e n t r i e s , and some a d d i t i o n a l l o g i c .  a short stack The exact same  window r e g i s t e r s used by the border d e t e c t o r s are used here. One the  disadvantage of t h i s method i s that i t cannot  closure  of the borders i t has d e t e c t e d .  the l o c a l nature of the method.  I t only has  guarantee  T h i s i s because of a  memory  f o r the  topology of the row of p o i n t s i t has seen up to s t e p 3 above. can say nothing about yet  to a r r i v e .  the body of the image, or  the  row  A simple example would be the f o l l o w i n g :  It  points  177  The  scan  here  proceeds from l e f t  to r i g h t .  The 1s topped  d e t e c t o r would f l a g an outer border a t p i x e l X and the detector  would  f l a g an inner border at p i x e l 0.  obvious that n e i t h e r border i s c l o s e d , would  fail.  It i s d i f f i c u l t  would occur per row.  and  so  However, i t i s  a  list-2  list-2  images.  trace  can  be  done  There i s a p o s s i b i l i t y that t h i s detect  scan.  very  and  quickly,  but  the delay  also  to  important  check  whether  data being  scheme  transmitting  are c l e a r  the  border  transmission,  before  starting  I f they a r e c l e a r , the t r a n s m i s s i o n a l g o r i t h m can  entries  stack e n t r y .  F a i l u r e to  check  may lead to an i n f i n i t e loop, or erroneous  transmitted.  In c o n c l u s i o n , the E u l e r provides  on  they  proceed to the next closed-border cleared  great.  detection  to not only c l e a r the l i s t - 2 e n t r i e s d u r i n g  transmission.  for  closure  i n any  the same c l o s e d border more than once i n a given row  I t i s therefore  points  false  However, c o n s i d e r i n g  r e s u l t i n g from f a l s e alarms i s not expected to be very  will  trace  An attempt w i l l be made to count the  that many o b j e c t s are not that complex i n s t r u c t u r e a  topped  t o say how o f t e n such f a l s e alarms  alarms produced by a s e r i e s of t e s t  case,  0  a  cheap  border i s c l o s e d .  number  closure  detection  method  and easy method t o a n t i c i p a t e whether a given  178 4.7  Border Point To preserve  avoided  Representation generality,  the q u e s t i o n  represented information  of  this  how  discussion  the  border  has d e l i b e r a t e l y points'  i n storage and d u r i n g t r a n s m i s s i o n . associated  with  border  points  are  Thus has  f a r , the  been somewhat  redundant, c o n s i s t i n g of both c o o r d i n a t e s and d i r e c t i o n In  the  interests  of  minimizing  bandwidth, a more compact border chosen.  memory point  to be  numbers.  s i z e and data-channel  representation  must  be  B a s i c a l l y , three methods a r e being proposed as f e a s i b l e  candidates: coordinate  storage, c h a i n code storage, and a  hybrid  of the two. Representing  the  border p o i n t s by t h e i r c o o r d i n a t e s  image window i s probably the ( r , c) c o o r d i n a t e s  the c l o s e d s t r i n g of  Here  these  i n a subsequent storage b u f f e r .  drawback of t h i s method i s the l a r g e For  approach.  only  of each p o i n t need be s t o r e d i n l i s t - 1 , and  once c l o s u r e i s d e t e c t e d , sequentially f i l e d  the most obvious  i n the  volume  of  points  is  The p r i n c i p a l data  involved.  a 64X64 b i t window, each r - c p a i r would r e q u i r e a twelve b i t  word s i z e .  Besides  being  long, t h i s s i z e i s intermediate  between  the 8 and 16 b i t word s i z e s i n common use today. The  idea of c h a i n code r e p r e s e n t a t i o n was introduced  fundamental concepts. r-c  coordinates  pixel  in  the  T h i s number by d e f i n i t i o n chain.  necessary  arbitrary location.  since  points  At most only the c o o r d i n a t e s  f i r s t p o i n t i n the c h a i n need be s t o r e d . really  i n s t e a d of s t o r i n g the  of each p o i n t , j u s t s t o r e the edgeout d i r e c t i o n  number a t each p o i n t . next  The idea i s simple,  i n the  the  Therefore,  However,  readout of l i s t - 1  this  to the of the i s not  can s t a r t at any  the s t a r t p o i n t , f o r i n s t a n c e , i n  1 79 border  r e c o n s t r u c t i o n , can be p l a c e d at some normalized  In e i t h e r case, the advantage of that  chain  code  position.  representation  each p o i n t i s simply c h a r a c t e r i z e d by a t h r e e - b i t  number. curved  I t can even be argued t h a t i f the only one  or two  d i r e c t i o n number. now  very  The  direction  is  gradually  b i t s i s needed to represent the change i n net r e s u l t  compact.  border  is  At  most,  i s that border p o i n t storage i s only  one  q u a r t e r the memory of  c o o r d i n a t e storage i n a 64X64 image i s needed. A  further  transformation  advantage  of  operations  this  approach  the  45°  simply  involves  increment.  However, one  increments  other  border. these  are  simply  a  matter  adding  of  by  increments  1 to each c h a i n code e n t r y per  should be cautioned  storage, on the  transformations 2X2  R o t a t i o n of the border  that  rotations  than 90° r e s u l t s i n s p a t i a l d i s t o r t i o n s of  Coordinate  involving  certain  c h a i n ' s anchor (the f i r s t p o i n t ' s c o o r d i n a t e s ) about  by the a p p r o p r i a t e amount. of  that  on the border are g r e a t l y s i m p l i f i e d .  T r a n s l a t i o n s i n the X or Y d i r e c t i o n s moving  is  be  tensor  done  other  to  operations  hand,  every  would  by the  require  p o i n t with r o t a t i o n s  (though  without  attendant  distortions). A  combination  of  hybrid representation. first  formulated  completely he  by  two  previous  techniques  T h i s method of border p o i n t Zahn.  He  observed  that  forms the  storage  was  borders  are  d e f i n e d by those p o i n t s where edgein * edgeout.  called  "curvature p o i n t s " .  which have edgein between  the  and  T h i s i s because a l l those p o i n t s  = edgeout l i e along a s t r a i g h t v e c t o r d i r e c t e d  curvature  coordinates  These  the  points. edgeout  Therefore, of  all  the  storage  curvature  points  of  the would  180 completely  characterize  b i t s per p o i n t , but  a  border.  T h i s would r e q u i r e about  i f the border forms a very  as many c h a r a c t e r s do,  1.5  bits  per  This  border p o i n t .  would  also  transformations  be  a  applied  h y b r i d of the two  to  yield  would  be  this  representation  preceeding.  p a r t i c u l a r l y complex  applied since  to  the  very  lost. would  T r a n s l a t i o n s would be  performed only on the c o o r d i n a t e component of the rotations  a  However, i f the border i s  n o i s y or c i r c u l a r , t h i s advantage would be The  polygon,  then c u r v a t u r e p o i n t s would comprise about  one-tenth of the p o i n t s i n a border. compact  angular  both.  data,  whereas  Rotations  coordinate  would  be  can  be  components  r o t a t e d by a v a r i e t y of degrees but the d i r e c t i o n numbers by increments of 45°. borders  This  would  probably  h y b r i d r e p r e s e n t a t i o n has one  the most a t t r a c t i v e of the t h r e e .  very  irregular  be  performed  at  the  lowest  the edge d i r e c t i o n s found  edgein  edgeout,  =  no  further  edge-point does not get logged great  d e a l of p r o c e s s i n g  strong a s s e t t h a t makes i t  The  monitoring  A  yield  only  f o r non-90° r o t a t i o n s .  The  can  15  edgein  level in  decision  by d e d i c a t e d the  processing  in list-1  * edgeout  Zahn is  hardware  window.  done  and  and the l i n k a g e  time and memory space can  If that  tables.  thereby  be  saved. Hybrid topological information  and  chain  information  coding  can  about  also the  i n the form of change i n  provide border  curvature  elementary  stored. point  This  direction  number with border l e n g t h c o u l d f a c i l i t a t e c h a r a c t e r r e c o g n i t i o n . By p r o v i d i n g i n f o r m a t i o n on c u r v a t u r e , different  the  degree  of  p o s i t i o n s on the border can be c a l c u l a t e d and  bend  at  compared  181 to  tabulated  values  from  i n f o r m a t i o n would be s i z e and Structural border i s a l s o points  in  characters.  the  hybrid  such as the area c o n t a i n e d from  coordinates  representation.  s i t u a t e d at the v e r t i c e s of a polygon. linked  sequence,  the  such  o r i e n t a t i o n independent.  information obtainable  Normalized,  contained  of  These When  the  within a curvature  coordinates  read  out  in  are the  area A can be c a l c u l a t e d with,  [66]: A = (1/2)2 (Ac, 8r,  - Ar; 8c, )  where:  If  area  evaluate  N  =  the number of c u r v a t u r e p o i n t s ,  AZj  =  Z j  - z  5z  =  Zj  - z,  i  information the above  0  , . is  vital,  d e d i c a t e d hardware can  readily  expression.  In c o n c l u s i o n , i t appears that the h y b r i d approach to border point  representation  utilization  and  of memory, and  storage  represents  the most e f f i c i e n t  the most u s e f u l scheme  o p t i c a l character recognition.  to  supplement  182 4.8  Object  Reconstruction  After  the  detection  of an e x t e r i o r border confirms  c l o s e d o b j e c t has been found, the subsequent preprocessor buffer.  goal  of  i s f r e q u e n t l y to r e c o n s t r u c t that o b j e c t  T h i s r e c o n s t r u c t i o n process  u t i l i z e s the  that a  a  binary  i n an image  border  points  as a guide t o r e s t o r i n g the dark o b j e c t i n t e r i o r p o i n t s as 1s. v a r i e t y of depending and  procedures  can  to a l a r g e extent  be  used  implement  restoration  on the border i n f o r m a t i o n  the means used to represent  presented  to  this  information.  r e f e r e n c e t o any i n f o r m a t i o n  source  other  than  available, The  u t i l i z e s both i n t e r n a l and e x t e r n a l curves  A  method  and makes no  those  curves.  T o p o l o g i c a l p r o p e r t i e s of simply  connected o b j e c t s are e x p l i c i t l y  employed  three  and  any  of  the  previous  border  data  r e p r e s e n t a t i o n s can be used as i n p u t . The  procedure  intended  to supplement  border t r a c i n g methods, such as the non-sequential  schemes, which  provide  presented  here  is  complete e x t e r n a l and i n t e r n a l border i n f o r m a t i o n .  showed that such i n f o r m a t i o n completely to  be  object  r e c o n s t r u c t e d and so no f u r t h e r r e f e r e n c e t o the o r i g i n a l  image i s necessary. section  Three means were presented  f o r representing  presented all  c h a r a c t e r i z e s the  Zahn  the  border p o i n t s .  in  the  previous  The method to be  here was adopted from Zahn's proof and i s a p p l i c a b l e t o  three,  but  the  c h a i n code r e p r e s e n t a t i o n w i l l be used f o r  i l l u s t r a t ion. The  foundation  corrollary  of  the  for Jordan  this  reconstruction  other  point  is  Curve Theorem [74, pp. 13-16].  s t a t e s that i f P, i s a p o i n t enclosed some  method  i n the plane,  by a curve  then P  2  r  i s outside  and  a  This P  2  is  T i f a line  183 drawn from P, Likewise,  if  number of To  to P  P 2  intersects  2  is inside  r  an  odd  T, such a l i n e  number  of  intersects  times.  r an even  times. see  how  reconstruction,  this the  theorem  offers  corollary  is  a  means  rephrased  for  in the  object following  form: If P odd  i s e x t e r n a l to  2  T and  P,  i s i n t e r n a l to  r,  then  number of the l i n e segments of the l i n e , 1, between P, and  extend from number  r to P .  If P,  2  of  l i n e segments of 1 can  number of which i n t e r s e c t the s t a r t i n g p o i n t on A  i s a l s o e x t e r n a l to  simple  r an even number of  example w i l l  i l l u s t r a t e how  reconstruction.  connected  object  with  p o i n t s P,,  P,  far  displaced  intersects  P  2  r , then an even r to P ,  an  2  times,  odd  including  r.  a p p l i e d to o b j e c t  2  be drawn from  an  from  Consider  external  both e x t e r n a l to r.  A  this  contour  c o r o l l a r y can a  r.  closed,  line  f an even number of  segment  simply  Also consider  r with P, adjacent  to  be  r and  drawn from P,  to  two P  2  P  2  times;: Pi  la  "P This l i n e  i s subdivided  internal  to  o u t l i n e of cross  1  r and  1  2  i n t o two e x t e r n a l to  r in a clockwise at l e a s t once.  2  segments r.  fashion.  Now  1,,  1 ; 2  cause P,  1,  totally  to t r a c e  the  During t h i s t r a c e , P, must  At t h i s p o i n t , the l i n e segment from  P,  184  to  P  i s identically  2  adjacent  to  but  1 . 2  not  If t h i s t r a c e i s  occupying  'its initial  f a m i l y of a l l l i n e segments from P , to P  2  trace w i l l  2  If can  i n c l u d e 1 , e x a c t l y once and 1  an  complements  operator  a l l those  to  the  P i  -  P  When such an operator  trace,  that,  we  find  complemented  once  complemented  twice  the  when  yielding and  1 s and  so  2  exterior  T as 0 .  to  to  which  r  been  have  However, these  been  l i n e s are  generated  during  therefore,  r to 1 and leave a l l  those  r have t h e r e f o r e  does  not  i n t e r n a l to some other  T . 2  2  previous  anchor a t P . 2  if P  2  the  r  2  interior  i n s i d e the l a r g e r o b j e c t  T h i s then w i l l be the mechanism f o r r e c o n s t r u c t i n g  o b j e c t s from border The  curve  c o n t a i n P , then a s i m i l a r t r a c e about  t o 0, r e v e a l i n g i t t o be a hole  outline,  information. example f i x e d the l i n e from P , to a s t a t i o n a r y  In f a c t , the theorem and i t s r e s u l t s apply i s permitted  to move f r e e l y along,  edge of the image i n such a way that the l i n e s p a r a l l e l and v e r t i c a l downward. in  have  points  w i l l a l s o complement the p o i n t s i n s i d e i t r e s e t i n g  well  above  reconstructed. Note a l s o that i f r i s a curve  of  which  the  points  The o b j e c t p o i n t s i n s i d e  the  changes 0 to 1  The net r e s u l t produced by the o p e r a t o r , a l l points i n t e r i o r  2  segment  accompanies  1  remain 0 .  during  l a b e l l e d 0 , we  (i.e.  1,  P, i s  twice.  line  2  complete,  i s to convert  T  exactly  examples of an i n f i n i t e f a m i l y of l i n e s  trace.  been  generated  p o i n t s on the l i n e  and v i c e v e r s a ) .  when  p o s i t i o n , then the  now a l l p o i n t s i n t h i s plane a r e i n i t i a l l y  assign  simply  stopped  j u s t as  say, the bottom  from  P,  are a l l  In t h i s way, a l l p o i n t s below  the r e c o n s t r u c t i o n window are complemented.  This  permits  r a  185 feasible outline  proposal  for  i s written  object  reconstruction.  As  the  object  i n t o the r e c o n s t r u c t i o n b u f f e r , f o l l o w i n g  right-handed  trace  p o i n t s below  each  convention, new  point  complement until  all  border  of  the  the  buffer  reconstruction  is  complete. Before adapting  proceeding,  the theorem ignored  resolved. and all  however,  In  problems  in  in the example, which must f i r s t  be  the example,  r was  P, part of the background. points  on  the l i n e ,  c l o s e examination, we between  r and  P  2  now  is  that  In  two  considered  The  p a r t of the  convention was  1 , excluding  result  points  on 1  adjacent  2  the  event  that P,  to  r are  r by one  If  P,  was  would be that: only p o r t i o n s of  simply  problem  considered  become p i x e l s , The  other  was an  parallel  complemented an The  in  the problem cannot be problem was  to  1.  that P,  the  was  result object  when  was  points  ignored. never considered T forms a  i m p l i e s that p o i n t s on  is  top  behavior.  However,  i n f i n i t e number of times as P,  r e s u l t of t h i s operation  r , the  example because P,  That i s , a path where This  with a l l  t r a n s f e r r e d to the  to remove t h i s  infinitesimal point.  a path p a r a l l e l to L. line  ignored  net  these  F would be transformed to  There i s no c o n s i s t e n t way  is  transformed to  i s complemented along  allowed to r i d e on  f.  On  will  The  p o i n t over  portions  This  that  that when P,  number of times ( i e . , set to 1).  i s simply  points.  adopted  2  p i n t s of 1 , then the t h i c k e n i n g of  object  P, were complemented.  f i n d the c u r i o u s  object points, e s s e n t i a l l y thickening regions.  are  i n t e r s e c t i n g 1 , i t encounters p o i n t s that  be complemented an odd result  there  indeterminate.  moves  to  follow  straight  1 should along  be T.  186 Both  problems  can  be solved by  guide the r e c o n s t r u c t i o n .  The  imposing a set of r u l e s to  nature of these r u l e s  is  v i s u a l i z e d when using chain code border r e p r e s e n t a t i o n . the f o l l o w i n g example d i r e c t i o n number  displayed  alongside  our  readily Consider  coordinate  and  conventions:  c  r  To  reproduce the  downwards, apply  image segment on the r i g h t by  complementing  the f o l l o w i n g r u l e s to the c u r r e n t border p o i n t : For e x t e r n a l  borders:  complement the c u r r e n t p o i n t  i f edgeout = 1 to  4,  do not complement the c u r r e n t p o i n t  i f edgeout = 5 to  0.  do not complement the c u r r e n t p o i n t  i f edgeout = 1 to  4,  complement the c u r r e n t p o i n t  i f edgeout = 5 to  0.  For  The  policy  borders s t i l l The moving  internal  is  reversed  along  a  the  i n t e r n a l borders because the  T thinning  straight  problem of moving the t r a c e T h i s i s best  for  belong to the o b j e c t and  above cures P,  borders:  line  along  i l l u s t r a t e d with the  a  not  the  hole.  problem.  The  manifests  i t s e l f here as a  vertical  problem  border  f o l l o w i n g example:  of  segment.  187  r  Those p i x e l s marked i n bold correspond t o the p l a c e s where 1 i s tangent are  or  to  will  boundary. T,  T at a l i n e . be  These a r e p l a c e s that e i t h e r a l r e a d y  complemented  by  a  t r a c e f o l l o w i n g the upper  T can be f u r t h e r s u b d i v i d e d i n t o two  representing  convexities,  and  T  2  problem  areas:  corresponding  to  concavit ies. To e l i m i n a t e  the  problem  in  the  vertical  lines,  don't  complement when: edgein = 0 AND edgeout = 0 edgein = 4 AND edgeout = 4 The  T  2  concavities  pose a s p e c i a l problem  a l r e a d y complemented from above. lines  will  be  dealt  with  The i n t e r i o r s  s i n c e they a r e  of  the  vertical  by the p r e v i o u s method, however, to  prevent complementing below the corner p o i n t s i n  the c o n c a v i t y ,  don't complement when: edgein = 5 ,  Internal  6 AND edgeout = 4  edgein = 4  AND edgeout = 2, 3  edgein = 1 , 2  AND edgeout = 0  edgein = 0  AND edgeout = 6, 7  borders  are processed using the exact same r u l e s .  Since a l l of these r u l e s a r e simple boolean e x p r e s s i o n s , they can  188 be  readily  implemented i n hardwired l o g i c .  The only a d d i t i o n a l  memory burden i s the storage of the o l d edgeout edgein  considered, traces  reconstruction  in  the problem of  where  arises.  For  an  a  finite, to  physical  start  the  outer border i n h y b r i d  t h i s i s not a s e r i o u s problem.  are  be  used  as  buffer  is  above.  When  to  to  reconstruction representation,  The b u f f e r need only be  designed  accomodate the l a r g e s t expected c h a r a c t e r , and the c h a r a c t e r s reconstructed  pieces  of  information  p o s i t i o n of detector  the  and  transmission values  upper-left-justified  trace  the  are  in  required.  start  left-most  point  that  These  buffer.  are  supplied  by  the the  Two column  closure  column p o s i t i o n encountered d u r i n g  of the c u r v a t u r e  points.  The  difference  in  these  w i l l g i v e the column p o s i t i o n i n the upper row from which  r e c o n s t r u c t i o n may s t a r t . Since two  i n t e r n a l borders  complications  are  arise.  detected  The f i r s t  before  outer  borders,  i s where to p o s i t i o n them.  T h i s i s best s o l v e d by s t o r i n g i n t e r n a l border i n f o r m a t i o n an  external  border  reconstructed position  The  e x t e r n a l border can then be  followed by the i n t e r n a l border(s)  determined  cooridinates.  arrives.  by  the  until  difference  which s t a r t at a  in i n i t i a l  trace point  The second problem a r i s e s when the i n t e r n a l border  i s i n s i d e an o b j e c t which cannot be segmented. c o r r e c t outer border w i l l never a r r i v e .  In t h i s case, the  To a v o i d  this  internal  border being added i n c o r r e c t l y t o a subsequent o b j e c t , the values of i t s extreme upper, lower, l e f t and r i g h t c o o r d i n a t e s compared the  outer  to  those of the outer  border's  extreme  border.  points,  must  I f they- a l l f a l l then  be  within  reconstruction  can  189 proceed  normally.  discarded. fail.  Otherwise,  must  be  S i t u a t i o n s can be e n v i s i o n e d where t h i s technique  may  However,  expected  to  be  sophisticated How  in  character  too  rare  and  the  and  images,  to  reconstructed  match  justify  on  image  border  these the  situations  cost  of  is  handled  now  the image data and then f i n d  more  is  up  to  On the other hand, i t may  decide  more i s needed, eg., the body below the dot on an  some  transformation  on  the  image  such  a  i t satisfactory  leave the image t o have other p a r t s added l a t e r ,  order  a  are  For i n s t a n c e , a r e c o g n i z e r may perform  so c l e a r i t when f i n i s h e d .  something  internal  method.  subsequent p r o c e s s o r s . template  this  or  "i",  it  may  as r o t a t i o n or  scaling. Since d e d i c a t e d hardware can be devoted the  o p e r a t i o n promises to be f a s t .  the b u f f e r must be r e f e r e n c e d are  points  inside  the  to  reconstruction,  However, many more p o i n t s i n  to perform complementing than  object.  Therefore,  i t may be e s s e n t i a l  that the time i n t e r v a l between s u c c e s s i v e c l o s e d o b j e c t s be much  larger  than  o p t i c a l character  the  memory r e f e r e n c e p e r i o d .  not  procedure  will  would  been presented  Fortunately in  implementation of the r e c o n s t r u c -  not be presented.  i n v e s t i g a t e d through s i m u l a t i o n , i t  discussion  very  r e c o g n i t i o n , t h i s seems to be the case.  F u r t h e r d e t a i l s concerning tion  there  be  already.  Since such a system was was  felt  that  too s p e c u l a t i v e , adding l i t t l e  such  a  to what has  190 4.9  Touching  Characters  A serious investigation  of  the  segmentation  of  touching  c h a r a c t e r s d i d not form a part of t h i s t h e s i s r e s e a r c h .  However,  s i n c e Hoffman and McCullough [62] reported up to 40% i n c i d e n c e of touching was f e l t than  characters  experiments i n v o l v i n g 12-pitch type, i t  that the matter does deserve  attempt  separate  in  an  exhaustive  some  discussion.  Rather  examination of p o s s i b l e methods t o  touching c h a r a c t e r s , only one p o s s i b l e s o l u t i o n w i l l  presented  instead.  T h i s w i l l demonstrate that a s o l u t i o n t o the  problem c o u l d be i n c o r p o r a t e d i n t o the preprocessor  design.  A h i e r a r c h i c a l approach i s proposed f o r the segmentation the  text  address  in the  a  line.  The  segmentation  segmentation  of  concentration  on c h a r a c t e r s  However,  the  two notable elemental page  that  Secondly,  first  of  characters.  benefits.  l e v e l of t h i s h i e r a r c h y  words,  the  second  has  dominated  the  the  from the chapter.  of words as a u n i t produces at l e a s t  First,  words,  not  characters,  semantic components of language. reading  this  of will  level,  T h i s i s a major departure that  segmentation  a  be  machine  segmentation  must of  measurements whereby t o group and parse  the  I t i s the words on a  articulate words  are  accurately.  provides  a  the c h a r a c t e r s  set found  of at  the high r e s o l u t i o n l e v e l . Compared  to c h a r a c t e r s , words are easy to segment.  Even i n  the handwritten  case, words are set apart by a c l e a r space.  The  size  space  one  of  this  character-space. foregoing  typewritten  Therefore,  segmentation  unit provided  for  scheme  there  is  text no  is  at  question  least of  the  being able to segment words as a  that a l l i n t e r n a l d e t a i l s could be f o r c e d to  blend  191 together. spacing  From  qualitative  between  components  experience,  characters  within  within  a character  to  i f the standard  the word  separated  and  limbwidth  disconnected  of  a  i s chosen  2  are b l u r r e d  this  spacing,  together;  by more than two limbwidths,  but  are of a  character.  d e v i a t i o n of the V g f i l t e r  r e s o l v e a minimum of, say, twice  within  word,  (e.g., the dot on " i " ) ,  d i s t a n c e comparable to the average Therefore,  a  i t i s noted that the  a l l details  words as a u n i t ,  would be r e s o l v e d .  When p r o c e s s i n g t h i s b l u r r e d image, the segmentation may ignore  i n t e r n a l borders,  r e t a i n i n g only the e x t e r n a l  When a c l o s e d e x t e r n a l border i s d e t e c t e d , is  resolution  system  c o n s t i t u t e a word.  since  the  borders.  the r e c o g n i t i o n system  s i g n a l l e d t o i n d i c a t e that a l l of the c h a r a c t e r s  high  system  last  closed  found by the border  now  The primary u s e f u l n e s s of t h i s i n f o r m a t i o n i s  that the c h a r a c t e r s found can now be grouped with confidence word  units.  Such  a  system  could,  between a s o l i t a r y " I " r e p r e s e n t i n g pronoun,  and  an  the  separation  of  accumulating  beyond  segmentation  of  operating resolving  for instance, d i s t i n g u i s h  the  first  " I " o c c u r r i n g i n the f i r s t  such as " I n t e r n a t i o n a l " .  person  singular  p o s i t i o n of a word  Also, p o s i t i o n a l errors a r i s i n g  touching word  characters  will  boundaries.  words  must  be  in parallel  with  a  into  done  be prevented  This by  similar  during  filtering  a  dedicated  system  from and  system  dedicated  to  characters.  As the c h a r a c t e r boundaries are found by the high r e s o l u t i o n system, they a r e passed on touching should  or  poorly  to  resolved  the  recognizer.  characters  occur,  i n d i c a t e a low r e c o g n i t i o n c o n f i d e n c e .  In  the  event  the r e c o g n i z e r  Those borders  are  192 then  stored  for  l a t e r processing.  When a word i s f l a g g e d ,  s p a t i a l o r d e r i n g of the c h a r a c t e r s found becomes  known.  within  immediately  In the case of p o o r l y r e s o l v e d c h a r a c t e r s ,  knowledge c o u l d be used d i r e c t l y or coupled statistics  it  with d i -  to improve r e c o g n i t i o n c o n f i d e n c e .  c o u l d a l s o be a p p l i e d  to  assist  in  the  the  or  this  trigram  S i m i l a r knowledge  parsing  of  touching  characters. This  p a r s i n g c o u l d proceed in the r e c u r s i v e manner o u t l i n e d  by Casey and Nagy [75]. reconstructed the  in  characters  recognizer  a a  turn.  column If  confidence window  then  the  characters  are  send  result  each scores  statistical  segment high  information,  segmentation i s accepted.  split  to  the  confidence, for  both  However, i f low  i s i n d i c a t e d , then the column i s moved to i n c r e a s e  on  one  character  r e c o g n i t i o n process parsed  and  the  i n c l u d i n g the p o s i t i o n a l and characters  the  b u f f e r , s u c c e s s i v e t r i a l s are made to  at  in  For the case where  and decrease i t on the other, and  i s repeated  too narrowly.  u n t i l one  In t h i s case,  of  the  characters  the best guess must be  however,  one  The  column  method would then continue at  a  time being  as before  of  with,  s h i f t e d to determine the  position.  T h i s segmentation approach i s s u b j e c t to the same  criticism  that was  d i r e c t e d at Hoffman and McCullough.  There i s no  way  segment  To  to  is  c h a r a c t e r width must be employed to s e l e c t the m u l t i p l e  p a r s i n g columns.  optimal  the  taken.  If more than two c h a r a c t e r s are connected, then some estimate expected  the  criticism, o p e r a t i n g on  the the  overlapping reconstruction border  characters. step  address  is  abandoned  representation  directly.  optimal this  in favor of The  outer  193 border  can  readily  provide  the extreme h o r i z o n t a l and  dimensions of the c h a r a c t e r c l u s t e r . variation  of  the Casey and  With  Nagy method can  c o o r d i n a t e at a s u i t a b l e d i v i s i o n p o i n t the  narrowest  outer  character  with  row  coordinate  Once found, a second search  be  information, invoked.  (usually  from the l e f t  boundary of the c l u s t e r  point  this  vertical  the  or r i g h t )  The  row  width  of  i s noted.  i s then scanned to f i n d the g r e a t e r than or equal  The  nearest  to t h i s  value.  i s made for a s i m i l a r p o i n t , but  which the t r a c e i s proceeding  a  i n the opposite d i r e c t i o n .  for  This i s  i n d i c a t e d by the edgein-edgeout p a i r being d i r e c t e d i n t o o p p o s i t e half-planes  from  those  c a l c u l a t e d between limbwidth  these Should  segmentation row be  these  then  segmentation.  of  the  first  point.  points  is  points  represent  they  be  too  If the  sufficiently  far  a  close  good  apart,  guess i s not near the touching  distance to  guess then  p o i n t and  a for  this so must  shifted. To complete the segmentation, the compound outer  split  into  together head  the two  and  borders borders  tail  smaller  are  borders.  close points just of  on opposite  confidence below:  two  two  new  s i d e s of  This  i s done by  splicing  at  representing  region.  is  "splicing"  found by p l a c i n g them  border p o i n t l i s t s the  border  the the  These  then t r a n s m i t t e d to the r e c o g n i z e r to be assigned  measure.  This  process  is  outlined  in  Figure  new a 4.3  1 94  Parse row F i g u r e 4.3 If  After  Touching c h a r a c t e r  the segmented r e s u l t  splice  separation  i s r e j e c t e d by the r e c o g n i z e r , then  the o p e r a t i o n must be performed again, but with the guess  moved.  For t h i s reason,  of the outer border data for  subsequent  recursions  are  prescreening  i s s p l i c e d and  trials.  separated will  event and  that  form  in  this  place  the  at  The  intact as  because  some  spliced. touch  An  at  internal  between the two  two  widely technique  border  will  touching a r e a s .  This  i n c l u d e d in a subsequent  This  many  i n the judgement of s u i t a b l e  splicing pairs.  segmentation.  simultaneously  not  approach  characters  i n t e r n a l border must t h e r e f o r e be attempt  that  left  d i s p l a c e d p l a c e s , the above  the gap  row  that only a copy  the o r i g i n a l  however,  p o i n t s to be  vertically  reject a l l t r i a l  probably  from  takes  c l o s e n e s s between the two In the  Note,  expected  already  i t i s important  initial  time,  repeat  s p l i c i n g at both p l a c e s  i s required.  above has  been merely a p r e l i m i n a r y glimpse  address the touching c h a r a c t e r problem. serve as a f r u i t f u l  starting  point  at  how  to  I t i s hoped that i t w i l l  for  further  research.  No  doubt a more r i g o r o u s i n v e s t i g a t i o n , i n c l u d i n g s i m u l a t i o n t r i a l s , w i l l produce refinements,  or r e p l a c e these  suggestions  entirely  195 with a s u p e r i o r method. 4.10  Segmentation Summary The  complete b i n a r y - o b j e c t  i n t h i s chapter (1)  Before  tables, (AE , shift  scanning  to zero.  registers  bottom and current (2)  now  be  the  summarized in the  input, i n i t i a l i z e  a l l of the l i s t - 2  AE,)  0  can  The  are  sides  e n t r i e s , and  two  initialized  with  object  processing Maintain  scan  the  the  image  with the row  image width. the  from  first  and  A l s o maintain  or  last  unity  Zahn-window  to frame the  Also,  image  initialize  the  Using  D6,  the  and  the  appropriate  shift  a set  right shifting  the  list-1 linkage  registers.  r e g i s t e r S1  of  and  and D4,  Begin  is f i l l e d .  at p i x e l s  flags  at the  to  equal  indicate  For the last  D2,  first  column,  d i s a b l e d e t e c t i o n of a B p o i n t .  Zahn border p o i n t d e t e c t i o n procedure,  edgeout value of a l l p o i n t s where edgein  i n t o l i s t - 1 , simultaneously  p o i n t s are processed Log  D8,  modified  the c o o r d i n a t e  edgeout  of  to  image column enters D2.  and  entry.  the three  count c y c l i n g at a modulo at l e a s t  black out D3,  (4)  counters  column c o o r d i n a t e s  (set to u n i t y ) D1  log  the E u l e r number  left  sets  column, black out  (3)  linkage  image data only a f t e r s h i f t  a count of the row  and D6  when  the  points.  of  steps:  l i s t - 1 - l i s t - 2 address to u n i t y .  Raster  to the  to  following all  image-data and  b i n a r i z e d p i x e l s i n t o the two  D5,  segmentation procedure advocated  c l e a r i n g the l i s t - 2  slot.  B  before A p o i n t s . address table  of where  the it  current is  point  designated  into  the  as a f i r s t  196 (5)  I f a l i n k a g e i s f l a g g e d as a r e s u l t  also  being  of  the  a second e n t r y , then i f the f i r s t  current  point  entry being l i n k e d  to i s an: (a)  edgeout - l o g the c u r r e n t p o i n t ' s l i s t - 1  l i s t - 2 s l o t p o i n t e d t o by the f i r s t (b)  edgein  entry  - l o g the  list-1  A up,  entry.  address  found  just  flagged.  Increment the l i s t - 1 - l i s t - 2 address when p r o c e s s i n g of or B p o i n t i s complete.  each  When the a v a i l a b l e addresses are used  c y c l e back to u n i t y .  (7)  Perform the c l o s u r e a n t i c i p a t i o n o p e r a t i o n s  and  D5 t o a l t e r the E u l e r number counters  closed-border (8)  i n the f i r s t  i n t o the l i s t - 2 s l o t of the c u r r e n t p o i n t .  C l e a r the l i n k a g e t a b l e s l o t (6)  address i n the  If  a  on D1,  D2,  D4,  and/or d e t e c t p o s s i b l e  points.  possible  closed-border  point  is  identified,  then  r e t r i e v e the l a s t A p o i n t found from i t s delay b u f f e r and push i t onto a t r a c e stack.  I t may be a p p r o p r i a t e  stacks f o r i n t e r n a l and e x t e r n a l (9)  When  processing  of  p l a c e the c u r r e n t A p o i n t ' s  to  maintain  separate  borders.  the c u r r e n t ( i f there  Zahn-window i s complete, i s one)  list-1  address  i n t o the s p e c i a l delay b u f f e r used i n ( 8 ) . ( 1 0 ) Continue  s h i f t i n g the p i x e l s and r e p e a t i n g steps  u n t i l the f l a g  i n d i c a t e s the l a s t column has been  (2) t o (9)  processed.  ( 1 1 ) At the end of the scan, pop the t r a c e stack e n t r i e s and use each to i n i t i a t e a t r a c e through l i s t - 2 . number of e n t r i e s t r a c e d . I f :  Maintain  a count of the  197 (a)  the t r a c e leads to a n u l l  c u r r e n t t r a c e stack (b)  the  trace  (c)  the  then  discard  the  initial  address,  the  entry.  leads  back  c u r r e n t border i s c l o s e d . transmission  pointer,  to  Push  the this  trace  entry  onto  a  stack.  counter  exceeds the maximum l i s t - 1 - l i s t - 2  then the t r a c e i s caught i n an  infinite  loop.  address,  Discard  the  current trace entry. (12)  Pop  again. to  the  the t r a n s m i s s i o n  T h i s time r e t r i e v e and  transmit  the  t r a c e through information  list-2  in l i s t - 1  r e c o g n i t i o n / r e c o n s t r u c t i o n c i r c u i t r y while a l s o c l e a r i n g  the l i s t - 2 p o i n t e r s . before  stack e n t r i e s and  the  If the l i s t - 2 p o i n t e r i s found to  transmission  i s complete, abort  be  null  the t r a n s m i s s i o n  and  be b u f f e r e d and  the  f l a g the e r r o r . (13) On  r e c e p t i o n , the border i n f o r m a t i o n can  o b j e c t r e c o n s t r u c t e d when i t s outer border a r r i v e s , or the border i n f o r m a t i o n can be processed (14) Continue scanning (a)  directly  for recognition.  the document u n t i l complete.  Terminate the s h i f t i n g of image data  remaining  unclosed  Then  to ensure that  any  o b j e c t s are not c l o s e d a c c i d e n t a l l y .  (b)  Return to step  (1) to perform a complete system r e s e t .  At  first  this  sight,  segmentation  procedure  e x c e s s i v e l y complex to a t t a i n  r a s t e r - r a t e performance.  the  by o b s e r v i n g  complexity  processing parallel  is  reduced  stages are mutually execution.  l a r g e l y of simple  exclusive  may  However,  t h a t a number of and  thereby  seem  these  admit  to  Furthermore, t h i s e n t i r e procedure c o n s i s t s  logic operations  and  data  transfers  which  are  198 ideally  s u i t e d to implementation on f a s t ,  Once  the  processing  system  has  begun  to  d e d i c a t e d hardware.  deliver  image  of the Zahn window can be represented  data, the  by the f o l l o w i n g  s t a t e diagram:  edge-point d e t e c t ion and  linkage  F i g u r e 4.4 The  Segmentation s t a t e diagram  edge-point d e t e c t i o n and l i n k a g e , and c l o s u r e d e t e c t i o n  sequences, a r e the most c l e a r l y sharing  of  parallel  memory or p r o c e s s i n g  operations.  resources  between them.  the edge-point d e t e c t i o n and l i n k a g e branch, logging  the  address  into  concurrently.  edge data the  in l i s t - 1 ,  linkage  What  the A p o i n t , proceed  operations  of  (4),  can  also  proceed  the dashed l i n e attempts to show i s that i f  (3) t o (6) must be performed twice processing  the  Within  ( 3 ) , and l o g g i n g of the l i s t - 1  tables,  both an A and a B edge i s present  When  There i s no  i n the  current  window,  steps  i n succession.  of the window i s complete, the b u f f e r i n g of  (9), and the s h i f t of image data,  ( 1 0 ) — ( 2 ) , can a l s o  concurrently.  When  a given  row scan i s complete, the system must  dedicate  199 itself  to  the  transmission,  verification (11)  and  steps must be executed  of  (-12).  boundary  closure  and  its  I t appears unavoidable  that  these  sequentially.  o v e r l a p these o p e r a t i o n s  with those  It  has  during  the  step  one  may  to  depends  run on  to the  l i s t - 1 - l i s t - 2 , and The  become  in step (12).  delay must be allowed (12)  between row  size  of  the  scans to permit  image  of border data  the  memory  image  array,  (11)  there  i s an  rows,  operation  If new  can  border  be b u f f e r e d u n t i l  timings of t h i s stage  it  is  stages,  segmentable proceed, data  is  needed.  stages  i n v o l v e s no  of the e a r l i e r  processing  of  constitutes  Since t h i s stage  less  delay  length  It e s s e n t i a l l y  are  and  employed.  before p r o c e s s i n g or r e c o n s t r u c t i o n of previous data  at  and  objects a more  transmitted i s complete, The  exact  depend on the implementation of the border  a n a l y s i s / r e c o n s t r u c t i o n system and the o b j e c t s to be  during  sufficient  steps  the  i n step (13)  resources  l e i s u r e l y pace than b e f o r e .  simply  or  i n t h i s image a n a l y s i s p i p e l i n e , with  s i n c e i t i s expected that  then i t can  However,  What c o n s t i t u t e s s u f f i c i e n t  (2) to (12) c o n s t i t u t i n g the f i r s t .  than  before  the speed of the technology  the second processor  present  references.  overwritten  t o t a l l y e x c l u s i v e of a l l the o t h e r s .  to  to  the Zahn window  I t t h e r e f o r e appears that  completion.  processing  reference  possible  s e r i o u s drawback that c l o s e d boundaries found  (11)  transmission  be  of p r o c e s s i n g  through i n t e r l e a v i n g of the l i s t - 1 - l i s t - 2 this  may  processed.  on the expected complexity  of  200 4.11  Segmentation Throughout  Simulations  this  chapter,  discussion  of  the segmentation  system's hardware requirements has been p u r p o s e l y vague. this  i s due  dependent. maximum  to  That  the  Partly  f a c t t h a t these i s s u e s a r e implementation  i s , they depend on the input  image  width,  scan r a t e , and the bandwidth of the technology employed.  However, i t i s a l s o due t o the f a c t that the complexity and of  the  size  the o b j e c t s scanned can have a major i n f l u e n c e on the system's  requirements. it  Because so many c o n t r i b u t i n g f a c t o r s a r e p r e s e n t ,  is difficult,  i f not i m p o s s i b l e to a n a l y t i c a l l y p r e d i c t what  those requirements a r e . gained  through  Some e m p i r i c a l  simulations.  w i l l be performed  by  i n s i g h t s , however, may be  In t h i s s e c t i o n , such s i m u l a t i o n s  implementing  the  segmentation  software, and then a p p l y i n g i t t o a s e t of b i n a r i z e d The  simulation  involved  implementing  the  system images.  segmentation  procedure, e x c l u d i n g r e c o n s t r u c t i o n , i n PASCAL t o accept b i n a r y images as input and produce c u r v a t u r e p o i n t s as output. to  a list  in  128X128  of a l l segmented-border  Since t h i s s i m u l a t i o n  was  designed  measure such q u a n t i t i e s as the necessary l i s t - 1 - l i s t - 2 memory  s l o t s , and stack depths r e q u i r e d by these images, these resources were  made  overflow. slots  were  exceptionally In the case made  of  large  to  ensure  list-1-list-2,  available.  that they would not ten  thousand  memory  The s t a c k s were each given t h i r t y  slots. Concurrent  with  the  segmentation-related  activity,  the  f o l l o w i n g s t a t i s t i c s were a l s o gathered: (a)  Total  number  of  A  points,  B  points,  and  simultaneous  201 d e t e c t i o n s of A and B p o i n t s . (b)  T o t a l number of a n t i c i p a t e d outer verified  number  of  outer and  p r o v i d e s the number of f a l s e  and  inner  borders,  inner borders which together  alarms.  (c)  T o t a l number of c l o s e d border p o i n t s t r a n s m i t t e d .  (d)  Image  statistics:  total  the  image  area;  total  edge  points  present. (e)  Record  of  the occupancy  (number of memory s l o t s used)  l i s t - 1 - l i s t - 2 at the end of each row (f)  Histogram  record  of  the  number  scan.  of  t e s t i n g f o r c l o s u r e at the end of each (g)  Histogram  record  of  the  number  t r a n s m i t t e d at the end of each (h)  Histogram the  of  points  traced  while  row. of  segmented  points  row.  record of the occupancy at the end of each row  three  stacks used: stack of a n t i c i p a t e d outer  of  borders;  stack of a n t i c i p a t e d inner borders; and a stack f o r v e r i f i e d c l o s e d borders to be t r a n s m i t t e d . The  choice  difficulty. set  of  of  test  Foremost, these objects  as  images  possible  of the segmentation  more,  images  s c a l e and point  input image, appropriate  suitable.  that the V g  which filter  to process  2  in  turn  standard character  However,  s i z e s , so choosing any  printed one  some  to  provide  method's  a  a  comprehensive  operation.  Further-  a l s o to be i n d i c a t i v e of the range of  image q u a l i t y expected  required  designed  were  presented  images were to present as g e n e r a l  understanding these  initially  of the V g 2  operator.  This  last  operator be a p p l i e d to b i n a r i z e left  open  the  deviation. images,  question  of  Since t h i s system  these  initially  the the was  seemed  c h a r a c t e r s come i n many shapes and  would b i a s the r e s u l t s , and choosing a  202 wide  s e l e c t i o n of f o n t s , i n c l u d i n g a l l c h a r a c t e r s and a range of  print  s i z e s , would generate a vast number of  than  would  seem  necessary  for  the  test  issues  images,  addressed  Besides, i t would be d e s i r a b l e i f the r e s u l t s c o u l d system  performance  scenes.  for  any  contain  a  of  Only  These  were  produced  deviation  16.  of  the  Referencing  the  l a s t chapter.  square  3.0,  4.0,  for  during  Five V g  edge  evaluation f  1.6,  model, spacing  and 5.0 p i x e l s .  r e a l - w o r l d images.  2.4,  3.2,  and  4.0.  filters  will  0=1.25)  of:  these (for  These edge spacings present  observed that these spacings are i n f a c t  the  values  d u r i n g segmentation distinct  character,  Since these images are d e r i v e d from random  n o i s e , i t was average  128  standard d e v i a t i o n s , o ,  2  wave  the  as wide a range of o b j e c t s i z e s as can be expected i n or  2  these  Gaussian n o i s e of mean  used  r e s o l v e edge d e t a i l with a minimum 2.0,  V g  by i n i t i a l l y g e n e r a t i n g ten  were a p p l i e d to b i n a r i z e the images: 0.8,  1.0,  particular  These n o i s e s t a t i s t i c s were chosen  simply to be compatible with those trials  any  such b i n a r i z e d noise images were generated  standard  images  requirements.  128X128 images c o n t a i n i n g independent and  test  pure n o i s e b i n a r i z e d by a range of  f i l t e r s appeared a b l e to meet these  simulations.  the  complex set of borders e n c l o s i n g o b j e c t s of a  images.  Fifty  reflect  the  v a r i e t y of s i z e s , but with no p a r t i c u l a r b i a s to class  here.  kind of image, i n c l u d i n g r e a l world  In the i n t e r e s t s of g e n e r a l i t y , then,  should  more  obtained. can  be  Therefore  classified  as  close  the r e s u l t s arising  to  produced  from  five  l e v e l s of input image c o m p l e x i t y .  Figure  4.5  reproduces  each of these f i v e t e s t  two  r e p r e s e n t a t i v e candidates from  image c l a s s e s .  The  severe complexity  of  203  F i g u r e 4.5  Sample t e s t images: (a) a, =0.8; (b) a =1.6; ( c ) a, =2.4; (d) a, =3.2; (e) a, =4.0 f  204 the  image  a -0.B f  i s immediately apparent.  exceeds that expected i n text real-world  images.  At  images,  the  other  contains far larger, less structured text  images,  real-world  but  images.  representative  of  r e a l - w o r l d images. much  it  higher  a =1.6 f  object  objects  image  is  complexity  grouped  the  these  five test  according  number  a =4.0  most image  (  expected  in  resolution  probably  the  most  expected i n t e x t  and  still  i t contains  characters. image c l a s s e s present 6400 l i n e s  to  class  with  r e p r e s e n t i n g the r e s u l t s seen i n 1280 of  than  of  c u r v a t u r e s s i m i l a r t o those which  of image data to the segmentation a l g o r i t h m . be  the  images are l i k e l y to produce,  o b j e c t limb widths and boundary  Together,  extreme,  that  Though the amount of d e t a i l present i s  than text  compose p r i n t e d  and exceeds  i s c o n s i s t e n t with c e r t a i n low  The the  T h i s complexity f a r  of  each  The s t a t i s t i c s  will  grouping t h e r e f o r e  image l i n e s .  The  results  p o i n t s t r a c e d and t r a n s m i t t e d , and the stack  occupancies at the end of each row w i l l be combined to p r o v i d e an i n d i c a t i o n of the system o p e r a t i n g requirements a c r o s s a l l l e v e l s of input image c o m p l e x i t y . Before examining let's  examine  situation.  the  the r e s u l t s of the s i m u l a t i o n system  requirements imposed  by a worst-case  Such a s i t u a t i o n would be produced by a  image at the p i x e l l e v e l as shown below f o r even  experiments,  N:  checkerboard  205  C=  1  2  3  4  F i g u r e 4.6  . . .  N-2 N-1  "Checkerboard" worst-case image  Even though t h i s image has no c l o s e d outer maximum d e n s i t y observations (1)  Every  of  holes,  and  inner  borders,  borders.  A  number  of  j u n c t i o n of two p i x e l s i s an edge p o i n t with edgein * except  both  Therefore  an A and a B p o i n t to be logged.  (2)  i t contains a  can be made immediately:  edgeout, so a l l Zahn window p o s i t i o n s ,  logged  N  at  C=N,  produce  2N-1 p o i n t s are  i n t o l i s t - 1 - l i s t - 2 at the end of each scan.  Every h o l e , except a t C=N, i s detected  border;  therefore  the  internal-border  as a  possible  trace-stack  inner  contains a  maximum of N/2 e n t r i e s at the end of a row scan. (3)  With four border p o i n t s per hole, an average of ( 2N - 2.5 )  p o i n t s would have to be t r a c e d and 2N-4 p o i n t s t r a n s m i t t e d a t the end  of each row. A maximally d e t a i l e d  borders  would  be  image  containing  the same number of c l o s e d  border p o i n t s as (2) and (3) above. i s r e a d i l y p o s s i b l e that a given  close  a  border  closed  l e s s dense than the above worst case,  densest rows w i l l c o n t a i n  it  only  that  contains  outer but the  borders  and  I t i s important to note that row scan i n another image may  more  points  to  be t r a c e d and  206 t r a n s m i t t e d than permit  indicated  comparison  of  in  this  (3).  Substituting  worst-case  N=128  f o r the r e s u l t s  will  actually  obtained. Table IV presents some of those  of  complexity most  the  worst-case  the  decrease  constant dark image a r e a . to  obtained  The  alongside  decreasing  i s r e f l e c t e d throughout  (  in  results  situation.  with i n c r e a s i n g a  notably  the  image  the  table,  of edge p o i n t s d e s p i t e a nearby  The worst-case  i s revealed  throughout  be an u n r e a l i s t i c model about which to design the system.  is also  seen  that  the  a, =0.8  image  class  is  sizably  It  more  demanding of the system than the other c l a s s e s . It  i s observed  with approximately  that  equal  i n g e n e r a l , the A and B p o i n t s occur  incidence at  from 0.58 p o i n t s per column occurence  worst-case  rate  also  declines  from  ranging  Simultaneous  24.8%  of the  These r e s u l t s stand i n c o n t r a s t to the  f i g u r e s where 2 ppc a r e produced with e s s e n t i a l l y 100%  simultaneous The  combined  (ppc) down to 0.08 ppc.  of A and B p o i n t s  p o i n t s seen down t o 3.2%.  a  occurence.  curvature point a r r i v a l  some merit to p i p e l i n i n g the process.  In  the  first  r a t e s suggest  edge-point  that there may be  detection  and  linkage  stage of such a p i p e l i n e , the c u r v a t u r e  p o i n t s would be d e t e c t e d and logged  in  stage,  the c u r v a t u r e p o i n t s would be  the  list-1  addresses  of  buffered for linkage processing. process  an  average  p i x e l s h i f t s ), after  of  about  list-1.  In  the  second  Provided that t h i s system c o u l d 0.5 ppc  (or one p o i n t every two  a l l p o i n t s seen should be p r o p e r l y l i n k e d  shortly  the end of the row scan, minimally d e l a y i n g the subsequent  closure v e r i f i c a t i o n stages.  Such an implementation  would permit  207  Worst -case area  a  f  =0.8  1 .6  2.4  3.2  4.0  8192  8186.6  8196.7  8191.1  8190.9  8153.2  3251 2  14645.7  7710.7  5229.0  3930.9  3183.1  edges/row  225  1 14.420  60.240  40.852  30.710  24.868  A pts  1 28  37.275  15.194  9.009  6.238  4.905  B pts  127  36.887  15.156  9.094  6.433  4.921  Total  255  74.162  30.350  18.103  12.671  9.827  A AND B  1 27  edges  9. 198  1 .759  0.648  0.291  0. 159  anticipated outer  0  2.438  1 . 1 44  0.588  0.347  0.242  ant i c i p a t e d inner  63.5  6.038  1 .369  0.672  0.377  0.255  actual outer  0  0.271  0. 186  0.096  0.044  0.043  actual inner  63  3.484  0.347  0. 105  0.046  0.030  false alarms  0.5  4.720  1 .980  1.059  0.634  0.423  closed pts transmitted  252  56.796  1 0.670  3.932  2.053  1 .963  3  17.366  19.680  14.171  10.618  unclosable pts  Table  IV.  Segmentation s i m u l a t i o n r e s u l t s i n c l u d i n g "checkerboard" worst-case  7.863  208 use  of  slower  those few The number  hardware  c l o s u r e d e t e c t i o n method show that the  simultaneously.  b i a s i n favor of  a, =0.8  Euler  image c l a s s  i n t e r n a l borders.  T h i s may  anomaly s i n c e the b i a s r a p i d l y d i s a p p e a r s  be  f o r the  classes. It  i s observed that there  decrease  in  (8.5)  f  decrease percent  i s an almost  the borders a n t i c i p a t e d per  (63.5) to a =0.8  44%  B points a r r i v e  r e s u l t s f o r the c l o s e d borders a n t i c i p a t e d by the  statistical  other  would be more s u i t e d to cope with  i n s t a n c e s where A and  e x h i b i t s a 2.5:1 a  and  a =4.0  by  (a, =0.8)  Of  the  15%  (a, =4.0)  worst-case s i t u a t i o n .  magnitude  from the  worst-case  in  of  magnitude  borders a n t i c i p a t e d , the  a c t u a l l y be c l o s e d and  to  row  of  by a f u r t h e r order  (0.50).  f  t h a t can  followed  order  transmitted  contrast  to  ranges 99%  for  I t would appear that the e f f i c i e n c y of  from the the  E u l e r number c l o s u r e a n t i c i p a t i o n method d e c l i n e s markedly with a decrease i n image complexity. could  However,  a l s o be a t t r i b u t e d to the  points  in  increasing  i n c r e a s e of u n c l o s a b l e  causes a decrease i n the number of c l o s e d  to  total  20.0%  forming  curvature  (a, =4.0)  that the method's e f f i c i e n c y  points  which lends is  closed  curvature  a .  This  those  decline  to  f  to  f  apparent  proportion  border p o i n t s (a =0.8)  this  not  borders  ranging  from  support to the  strongly  related  with  76.6% premise to  the  the  dual  image's c o m p l e x i t y .  The  largest  data  l i s t - 1 - l i s t - 2 array. curvature before window  points  recycling. may  structure  in  the  system  is  I t i s important that t h i s a r r a y  likely  to  Otherwise,  be  encountered i n an  segmentable  become o v e r - w r i t t e n  before  objects  being  l o g a l l the image window within  detected  as  that  closed.  209  10  ROW  F i g u r e 4.7  List-1  occupancy a g a i n s t  row count  210 A r e c o r d of the occupancy all  of  the  test  of l i s t - 1  at the end of  images i s shown i n F i g u r e 4.7.  important p o i n t s to note about t h i s p l o t . representing  each  The  of  individual  the l i s t - 1 - l i s t - 2  requirements of a  given  This  s c a l e s d i r e c t l y with the image a r e a .  implies  Therefore,  a  for  a  requirement 64X64  image  memory space  images.  The maximum v a l u e s a t t a i n e d i n F i g u r e 4.7 size  class.  that  size  only one q u a r t e r of the l i s t - 1 - l i s t - 2  the 128X128  indicator  resolution  image complexity c l a s s , the l i s t - 1 - l i s t - 2  needs  class.  shows an almost e x a c t l y l i n e a r dependence  on the row count f o r a g i v e n c l a s s .  of  points,  per  t h e r e f o r e , serves as a r e l i a b l e  Secondly, the occupancy  window  two  the r e s u l t s of a given image, s c a t t e r very narrowly  average occupancy,  given  for  There are  about the l i n e s r e p r e s e n t i n g the average occupancies The  row  requirement  for  that  image  class.  represent the It  does  list  not mean,  however, that o b j e c t s are present with a s i z e that extends a c r o s s the  entire  length  of the image.  o b j e c t s were p r e s e n t , these required  to  guarantee  extend from about o =4.0,  for  list  their  sizes  represent  segmentation.  9500 memory e n t r i e s at a =0.8 f  with  f  Rather, i t means that i f such  3900  entries  The to  the  requirements 1300  The requirements f o r the worst-case  that  of  of  the  number  c h o i c e of which l i s t cost. the  That  about a  largely  The by  i s , the c o s t of the memory hardware, as compared to  from  the  occasional  narrow  scatter  segmentable  f  that  is  unlikely  object.  It  is  of the r e s u l t s i n F i g u r e 4.7  t h e i r average v a l u e s that the o =0.8  maximum  equal  present at 32,512 e n t r i e s .  s i z e t o adopt must be motivated  c o s t of l o s i n g the  evident  edges  entries  f o r that expected from p r i n t e d  c h a r a c t e r s at a =1.6. f  minimum  requirements represent  to be exceeded.  Indeed, even the  21 1 worst-case  requirements  are  deceptive  since  the  image  is  dominated by s i n g l e p i x e l h o l e s , so that segmentation i s a c t u a l l y guaranteed a f t e r only 383 e n t r i e s . The to  e f f i c i e n c y of l o g g i n g only c u r v a t u r e p o i n t s  a l l edge  or  border  as  p o i n t s i s evident on comparison of the  maximum values of F i g u r e 4.7 to the t o t a l edges present images  as  given  in  Table  IV.  in g e n e r a l , the intermediate  these  storage  This indicates  that,  requirements of t h i s form of  border r e p r e s e n t a t i o n are about one h a l f those that  in  As a percentage, the curvature  p o i n t s range from 65% of the t o t a l t o 41%.  representation  opposed  logs every p o i n t .  of a more complete  Consequently, a l s o , h a l f  the delay while v e r i f y i n g c l o s u r e and t r a n s m i t t i n g borders  can be  expected. In row  order  scan,  to estimate  the delay  the number of p o i n t s  completion in histogram  i n c u r r e d at the end of each  traced  and  of each row were recorded.  transmitted  trace  and  a  basis  for a  range from 1122 + 581 = 1703 Interestingly,  the  not much d i f f e r e n t The  worst-case  Since the c l o s u r e  border t r a n s m i s s i o n o p e r a t i o n s occur i n  sequence, the steps taken i n these o p e r a t i o n s provide  the  The r e s u l t s are presented  form i n F i g u r e s 4.8 through 4.13.  verification  at  delay e s t i m a t e . for  a =0.8 f  The maximum sums seen  to 298 + 130=428  steps counted f o r a =1 .6,  from the otherwise  should be summed to  f  for  o =4.0 . f  1177 + 411 = 1588, are  more complex a =0.8 f  image.  values of 255+252=507 are m i s l e a d i n g l y small due  to the edge-points being produced by a  large  number  of  single  p i x e l holes r a t h e r than extended f e a t u r e s . Examination of the histograms r e v e a l s that the maximum t r a c e and  t r a n s m i s s i o n counts are rare events of a magnitude f a r above  212  F i g u r e 4.8  o =0.8 histogram: (a) number of p o i n t s traced row; (b) number of p o i n t s t r a n s m i t t e d per row f  per  213  % max.  F i g u r e 4.9  count  o =1.6 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row f  214 10*T  ,  % max.  F i g u r e 4.10  count  a, =2.4 histogram: (a) number of p o i n t s traced row; (b) number of p o i n t s t r a n s m i t t e d per row  per  215  io\  F i g u r e 4.11  .  a =3.2 histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row f  216  F i g u r e 4.12  a, =4.0 histogram: (a) number of p o i n t s traced row; (b) number of p o i n t s t r a n s m i t t e d per row  per  217  F i g u r e 4.13  Total combined histogram: (a) number of p o i n t s t r a c e d per row; (b) number of p o i n t s t r a n s m i t t e d per row  218 the  mean  values.  In  than 10% of maximum. histograms  of  f a c t , the mean v a l u e s a r e g e n e r a l l y  T h i s i s most s u c c i n c t l y i l l u s t r a t e d  Figure  less  i n the  4.13 which combine the r e s u l t s of a l l the  image c l a s s e s .  The maximum count of p o i n t s t r a c e d  was  be 1177 but the mean was 64.5, 5% of the maximum.  found  to  after  a  row  S i m i l a r l y , the maximum p o i n t s t r a n s m i t t e d was 581 but with a mean of  15.1,  only  2.6%  a l g o r i t h m presented transmitted designing  at  of  the  required  the  end  maximum. that  of  a  However, the p r o c e s s i n g  a l l points  given  row scan.  traced  suggest that t o approximately be capable  be  1177+581=1758  2000 p o i n t s .  of p r o c e s s i n g  These a  simulations  safety  memory  references  in  scans.  verification  This  execution  pipelined.  these  scans.  must,  of  course,  be  detected  observation per  raster  complete  tasks between  involve  tasks with the c u r v a t u r e - p o i n t  earlier  concurrent d e t e c t i o n and  architecture  i s further  that a t most 0.5 c u r v a t u r e shift  permits  parallel  of t r a c i n g and t r a n s m i s s i o n with d e t e c t i o n and l i n k a g e  since l i s t - 1 - l i s t - 2 principal  and border t r a n s m i s s i o n  As a r e s u l t , the system The  will  execution  approach  of  linkage task.  objection  memory r e f e r e n c e s can  be  interleaved.  entering  The  of t h i s approach was that the e n t r i e s being  t r a c e d or t r a n s m i t t e d c o u l d be o v e r w r i t t e n and l o s t points  time  a l t e r n a t i v e approach a l l o w s advantage t o be taken of the  closure  points  margin,  the  much lower mean values by abandoning the requirement t o the  to  The hardware employed must t h e r e f o r e  2000  between s i n g l e row r a s t e r  o r , with  and  This requires  the system s o l e l y about the maximum p o i n t s expected  be encountered at the end of a given row scan.  An  be  list-1-list-2.  due  to  However, the l i s t - 1 - l i s t - 2  requirements a r e now understood, and seen to be very  new size  predictable  219 for  a  given  image  complexity  class.  I n c r e a s i n g the l i s t  above what i s r e q u i r e d for t h i s c l a s s can prevent loss during p a r a l l e l For  unwanted  data  execution. the o =1.6  example,  size  f  image c l a s s was  seen to r e q u i r e at  least  3900 l i s t - 1 - l i s t - 2 memory p o s i t i o n s with about 30 p o s i t i o n s  being  f i l l e d per row  and  transmitted  85+11=96. only one scans, row  scan.  per  row  Therefore  Even though the maximum p o i n t s t r a c e d was  scan.  However,  12.4X30=372 r e f e r e n c e per be  the  raster  fully  entries,  rare  scans.  or  for  processing  processing  system  g r e a t e r assurance that a l l without  c o n f l i c t may  The  only  execution  be  10%.  to to  between  stacks a r r i v e no  for  the  relatively  serious  this  increased  scans,  there  the a c t u a l memory  the c o s t saving of such  associated operations  with slower hardware will  by  the s i n g l e memory  be u n r e a l i s t i c a l l y slow, and  side-effect transmission  r e f e r e n c e d by both the t r a c i n g and even  require  accomodate be  Since  between  subsequent  would  run  to  resulting  a and  completion  from  a =0.8 f  parallel  i s that the s t a c k s  p o i n t e r s to p o s s i b l e i n t e r n a l or e x t e r n a l borders  Since,  were  considerable.  serious  of t r a c i n g and  needs  Therefore,  as  case  Therefore,  about  be much l e s s .  parallel  maximum  size  r a s t e r s h i f t may  time  i n c r e a s e may  pose  means  i t would g e n e r a l l y complete i t s task d u r i n g the  e v e n t u a l i t y the l i s t - 1 - l i s t - 2  the  the  even i f the t r a c e - t r a n s m i t system averaged  memory reference per s h i f t , with no  1588/128=12.4  may  1177+411=1588,  need to be  closure-anticipation  image c l a s s , new  t r a c i n g system r e q u i r i n g access  The to  difficulty the  oldest  systems.  e n t r i e s for  i n f r e q u e n t l y memory r e f e r e n c e  difficulty.  storing  these  conflicts  a r i s e s from the points  on  the  220 stacks  while  at the same time the a n t i c i p a t i o n  the newest p o i n t s .  There are b a s i c a l l y  three  system i s adding  solutions  to  this  problem: (1)  Ignore  the  problem and  process  the stacks as u s u a l .  Since  on average, the borders must be t r a c e d f a s t e r than detected, stacks  w i l l always be emptied e v e n t u a l l y .  of new  p o i n t s w i l l add  which  in  turn  may  to the delay  However, the a d d i t i o n  i n reaching  require a sizable  the stack  increase  memory to prevent these o l d borders from being interim.  A l s o , the borders w i l l  (2)  lost  be t r a n s m i t t e d  scan.  a l l out of havoc  The  recycled  set at the begining  closure v e r i f i c a t i o n  when  of  last  list-1-list-2  to  needed. sets  trace  memory  transmit  In the p r e v i o u s  a, =1.6  (rounded up from 12.4).  processing  was  stacks.  The  memory  the  equals  except  new the  row stacks  stacks The  same  number of manner  row  with  the  as  the number of e x t r a  maximum stacks  example, t h a t number would be T h i s method has usual  13  the a t t r a c t i o n of manner  without  f o r the a d d i t i o n of a system to change the  only drawback of the system i s that a great deal reserved  are  determined: the number of  the a n t i c i p a t e d borders i n the  modification,  the  and  each  in the s e r i e s i s f i l l e d .  image rows scanned while p r o c e s s i n g points  the  anticipation  The  e x t r a stacks needed i s determined i n e x a c t l y the extra  order  with  system then processes  from the o l d e s t to the newest. the  the  system.  to a new  in the usual way  the  during  Employ m u l t i p l e s e t s of stacks with the c l o s u r e  system switching  bottom  in l i s t - 1 - l i s t - 2  in r e l a t i o n to t h e i r c a u s a l appearance c r e a t i n g border p o s t - p r o c e s s i n g  the  for  the  stacks are seldom used to f u l l  stacks capacity.  will  be wasted s i n c e  of the  221 (3)  Replace the stacks  closure  anticipation  top of each l i s t old  pointers  and  transmitted  control  with  a  system  s e t of  will  while the c l o s u r e  from the bottom. will  the l i s t s  circular  verification  correct  and  must be c y c l e d i n a f i n i t e block  mean  f o r each l i s t  occupancy  of  points  removes  of borders t r a c e d  the  the  of memory.  system  used  to  lists,  and  these  The amount of memory  would be determined by the product of the  i t s corresponding stack and, as before, the  number of image rows scanned while p r o c e s s i n g maximum  system  i s s i m i l a r t o a stack except that two p o i n t e r s  are maintained f o r the top and bottom of  required  The  now place new p o i n t e r s on the  The c a u s a l order  remain  lists.  t o t r a c e and t r a n s m i t ,  accomodate l a r g e d e v i a t i o n s  the  row  with  the  plus a suitable buffer to  from the mean.  For example,  i f the  mean occupancy was 2 and the row count, f o r o =1.6, was 12.4, the f  requirement would be about 25 e n t r i e s . gives  a  total  s i z e of 35 e n t r i e s .  I n c l u d i n g a b u f f e r of  10  C l e a r l y , t h i s i s not a very  demanding memory requirement. The causal  drawback of t h i s system l i e s relationship  between  I n t e r n a l borders g e n e r a l l y closed  outer  border.  tracing  and  lists,  pointers stored the  highest  The  circular  within  to c o r r e c t l y  transmitting  the  and  the  correct  external  the  borders.  temporally  next  The r e c o n s t r u c t i o n or a n a l y s i s system may  o r d e r i n g was guaranteed. circular  internal  belong  r e l y on t h i s c a u s a l o r d e r i n g When  i n maintaining  process  the  borders.  a l l p o i n t s between scans,  However, t o ensure t h i s o r d e r i n g  system  i n those l i s t s .  must  monitor  The most  the  recent  values  this using  of the  pointers  have  l i s t - 1 - l i s t - 2 addresses, c y c l i n g a t a f i x e d modulus. list  f o r i n t e r n a l borders  must  therefore  be  read  222 first  until  the  next  p o i n t e r found exceeds the p o i n t e r of the  next e x t e r n a l border p o i n t e r . be  processed  until  next i n t e r n a l  border,  detected  the  in  The e x t e r n a l border l i s t  and so on.  correct  after  The c l o s e d borders  order  the  read  This  w i l l then be  for transmission.  U n l i k e the  p o i n t e r s were stacked  closure  read, t h i s system can transmit verified.  then  the next p o i n t e r exceeds the p o i n t e r of the  p r e v i o u s cases where closed-border transmission,  will  verification  the borders  t o await  s t a c k s had been  as soon as c l o s u r e  is  i s because these c i r c u l a r l i s t s cannot be f u l l y  i n a guaranteed i n t e r v a l of added.  Also,  time  since  always  being  borders  p r o v i d e s e x t r a assurance t h a t the data  w i l l not become o v e r w r i t t e n .  immediate  pointers  transmission  This s i m p l i f i e d  may c a n c e l out the e x t r a complexity  new  in  are  of c l o s e d  list-1-list-2  transmission  r e q u i r e d by the  system  verification  system. Which  of  approaches (2) and (3) i s favored depends  on the r e l a t i v e c o s t of seems  the  hardware  employed.  largely  Approach  (3)  the most elegant, and the immediate t r a n s m i s s i o n of c l o s e d  borders  i s very a t t r a c t i v e .  simple  extension  necessary  a  of the e a r l i e r stack p r o c e s s i n g method, but the  waste of memory i t would e n t a i l  economically of these  However, system (2) i s b a s i c a l l y  unattractive.  may  cause  i t to  be  The amount of memory r e q u i r e d by each  systems can be estimated  by the stack occupancies  found  d u r i n g the s i m u l a t i o n s . The  simulation  outer borders closed  used a stack each f o r a n t i c i p a t e d inner and  and a s i n g l e  borders.  stack t o s t o r e  pointers  to  verified  Each stack was read and emptied a f t e r each row  scan and the number of e n t r i e s found recorded.  The r e s u l t was  a  223 record  of  the  generating The The  stack  occupancy  i n terms of the number of rows  a given, number of e n t r i e s w i t h i n  r e s u l t s a r e shown i n histogram most remarkable o b s e r v a t i o n  very  long.  13  4.18,  i s that none of these  and  63  any  f o r the t r a n m i s s i o n  stack.  f o r the i n t e r n a l - s t a c k  The o, =4.0 f i n d i n g s , F i g u r e  of  This  small memory requirement  situation twenty  f o r each  i s not worth o p t i m i z i n g .  entries  to  be  f o r unexpectedly For  complex input  a simple  To  Optimization not  required  could  serious. an  classes.  processing occupy  system with m u l t i p l e s t a c k s ,  would  require  considered, of  parallel  stacks would only  the  780  locations.  but t h i s memory s i z e i s  circular  for  the  internal-,  was found to be 1.8,  roughly  comparable  f  the  buffer  list  structures requirements.  the combined occupancy r e s u l t s f o r a l l image  than the a =0.8 r e s u l t s . illustration  safety  of the average stack memory  means  transmission-stacks are  be  Design  estimate  The  given  now  F i g u r e 4.19 presents  results  For a p a r a l l e l  example  a  images.  of v e r i f i c a t i o n and t r a n s m i s s i o n , these  ( 2 ) , the  declare  needed f o r each stack would  three stack system without  60 memory l o c a t i o n s .  of the  f o r the stacks i n  c e r t a i n l y prove adequate and p r o v i d e a comfortable  still  positions.  f  realistic  case  is  the worst case i n the s i m u l a t i o n s , a =0.8, F i g u r e 4.14,  stacks.  minimum  stacks  transmission-stack  show only three p o s i t i o n s t o be necessary  three  class.  form i n F i g u r e s 4.14 t o 4.19.  r e q u i r e s only a maximum of 14 p o s i t i o n s and  image  The worst-case example would have r e q u i r e d up to 64  internal-stack positions However,  each  a =0.8 f  e x t e r n a l - s t a c k mean equal  1.0,  external-, and  1.0.  and These  t o the 0^=1.6 r e s u l t s but l e s s  Rather than adopt the o v e r a l l mean, f o r figures  will  be  used,  but with the  t o the i n t e r n a l - s t a c k mean s i n c e  their  224  F i g u r e 4.14  a = 0.8 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack t  225  4  F i g u r e 4.15  8  12  occupancy  1*  20  a, =1.6 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack  226  F i g u r e 4.16  o =2.4 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border s t a c k ; (c) t r a n s m i s s i o n stack f  227  F i g u r e 4.17  a, =3.2 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack  228  Figure 4 . 1 8  a =4.0 stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack F  229  occupancy F i g u r e 4.19  T o t a l combined stack occupancy: (a) e x t e r n a l border stack; (b) i n t e r n a l border stack; (c) t r a n s m i s s i o n stack  230 difference margin,  may  the  20 = 95.64  be a s t a t i s t i c a l  previous or  about  circular l i s t s .  anomaly.  calculation  With a 20 entry  will  require  12.4 X 6.1  +  100 l o c a t i o n s f o r the i n t e r n a l and e x t e r n a l  Since no t r a n s m i s s i o n stack i s used,  memory requirement i s only 200 l o c a t i o n s . produces an almost 4:1 system.  safety  the  total  Therefore, t h i s  system  memory advantage over the  multiple  stack  However, r e g a r d l e s s of the implementation chosen, i t i s  c l e a r that the l i s t - 1 - l i s t - 2 data  structure  will  dominate  the  memory requirements of the system. The  simulations  t h e i r primary operation  goal  and  of of  the  segmentation system have achieved  providing  hardware  empirical  requirements  of  o b s e r v a t i o n that the segmentation system can p a r a l l e l i s m was  unexpected, but welcome.  which  at  the  same  the  into  the  system.  accomodate  The  further  The net r e s u l t may  a more r e l i a b l e system i n terms of reduced borders  insight  loss  of  segmentable  time i s l e s s demanding of  hardware  technology.  Since a l l the t e s t  it  guaranteed that the system w i l l operate i n a s i m i l a r  is  not  images were d e r i v e d  be  manner on a p p l i c a t i o n to other images. of  the  test  images  be.  the  noise,  complexity  w i t h i n each r e s o l u t i o n c l a s s suggests that  these images were more demanding of the applications will  However,  from  system  than  most  real  231 4.12  Conclusions The  objective  the d e t e c t i o n and image data  in  of  t h i s chapter  The  i n p a r a l l e l with the performance  dynamic nature of the points,  and  monitoring The  data  in  input r a s t e r scan i n order  detection  linkage of  of  dynamic  binary-image to meet  detected  possible  the the  border  c l o s e d borders by  the l o c a l change i n E u l e r number.  Zahn  recommended  curvature  point  for  of  a  border  is  implementation  d e t e c t o r coupled  s t r u c t u r e to l i n k the c u r v a t u r e  Closure  a  I t a l s o took advantage of  image in the  the  within  proposed system processed  objective.  o v e r a l l design  modified  to develop a method for  i s o l a t i o n of c l o s e d borders  r e a l time.  real-time  was  uses  a  to a m u l t i l i s t  p o i n t s i n t o a c l o s e d border.  detected  (but not guaranteed)  by  an  E u l e r number c l o s u r e d e t e c t i o n system which f i l e s the p o i n t e r s to the  expected  borders  e x t e r n a l borders. row  scan.  on  two  s t a c k s , one  C l o s u r e v e r i f i c a t i o n occurs  Verification  of  border  storage  of  closed  a  border,  position  subsequent  processor. is  reconstruction  not  the and  of  curvature  Though  the  concern  procedure  alone.  In  points,  was  to  nature the  gathered  of  to  c o u l d be r e c o n s t r u c t e d  addition  a  from  a  the  curvature  On  discovery  including  the  their  show  system, a how  from the  preliminary  to a  subsequent  segmentation  investigated  o u t l i n e d f o r the s e p a r a t i o n of touching Statistics  of  edgeout d i r e c t i o n , are t r a n s m i t t e d  segmented o b j e c t s d e t e c t e d information  list-1-list-2  i n search of c y c l i c a l l i n k e d l i s t s .  coordinate  processor  at the end  and  c l o s u r e employs the g l o b a l  method of f o l l o w i n g p o i n t e r s w i t h i n the point  each f o r i n t e r n a l  the border  procedure  was  characters.  software  simulation  of  the  232 complete system provided the necessary prediction  of memory requirements  system  will  system  the  fact  architecture.  implementation  facilitate  The  final  will  that It  the  c o n f i g u r a t i o n of  be governed l a r g e l y by the incoming  r a t e and the image width i n underscored  to  and estimate the time r e q u i r e d  to complete each p r o c e s s i n g stage. the  data  pixels.  However,  the  video data simulations  much f l e x i b i l i t y i s p o s s i b l e i n the is  envisioned  that  the  final  i n c l u d e a r i c h blend of p a r a l l e l and s e r i a l  processes a c t i n g i n c o n c e r t .  233  V.  This  thesis  filtering,  DISCUSSION AND  has  presented  binarization,  and  proposed d i g i t a l preprocessor After to  be  the  the design  with  filter.  the  This  d e t a i l s of the edge  segmentation  components  for o p t i c a l character  camera d e v i c e has a c q u i r e d  filtered  detection  CONCLUSIONS  the  two-dimensional edge f i l t e r  was  image data,  V g  two  aspect  of  character for  the  be  filter  boundaries.  also  of  the  removed  r e j e c t i o n within  edges. any  character  The  smoothing  roughness  from  the  Both of these f e a t u r e s e l i m i n a t e d the need  r e s o l u t i o n response of the V g 2  edge models.  predicting "s",  deviations  in  an a d d i t i o n a l noise c l e a n i n g stage a f t e r b i n a r i z a t i o n . The  two  standard  edge  shown to be optimal  T h i s r e s u l t e d in' a l a r g e degree of noise  filter  i t is  optimal  2  a  recognition.  the sense of maximizing output s i g n a l energy about the edges.  of  when  The  square wave  the edge spacings  filter  was  is  most  model  The  common  background.  confuse the The matched  a different  "m"  "un",  segmentable,  or can  i t does  color  (a  However,  of a psuedo-object a s s o c i a t e d with  If  for  t h i s psuedo-object  the may  recognizer.  necessary f i l t e r to  like  in p r i n t a d v e r t i z i n g ) are r e s o l v a b l e .  i t a l s o p r e d i c t s formation highlight  like  s t a i r c a s e model i s not as u s e f u l , but  p r e d i c t that words l o c a l l y h i g h l i g h t e d by practice  suitable  within characters  or h o r i z o n t a l l y between m u l t i p l e c h a r a c t e r s resolved.  determined f o r  width f o r optimal  noise  rejection,  the c h a r a c t e r d e t a i l , has a o  equal  to the observed  f  but  234 edge spacing  in  pixels  Dr. Beddoes' CCD to  be two  model  be  predicts  less  Allowing a  also  1.25a .  found  sign.  resultant  dc  be at  bias  be  least  by  low  i t was  incorporate  proceeds  a  is  at  Since least  or  u n t i l a l l c o e f f i c i e n t s sum The  point.  The  filter  per  important  word  that  any  implementation  be  a  problem. a  unit  Bias  from the of  0.8a  i s a s u b j e c t which merits can  be  made  at  this  direct-form  S p e c i a l purpose hardware i s c e r t a i n l y t h i s at video a c q u i s i t i o n  rates.  To  the  supply  a growing demand f o r t h i s type of o p e r a t i o n , s p e c i a l purpose convolution support  integrated  a V g 2  circuits  up to a 26X26 operator  h e l d i n a s i m i l a r CCD filter  interesting  with a  r  f  to zero.  most obvious implementation i s a simple  only means to achieve  i s two  coefficient  slope at a r a d i u s  However, some suggestions  discrete convolution.  is  f  an e i g h t b i t t o t a l word  to  implementation of t h i s f i l t e r  f u r t h e r study.  a  should  The  bits  the  subtracting  c o e f f i c i e n t s near the maximum f i l t e r  wave  the best  to  six  seen to be  unlikely  adding  of  For  square  the maximum spacing  insensitive  removed.  s i z e , underquantization removal  Therefore,  a, =1.6,  largely  However,  will  1.25.  0.  g e n e r a l l y observed  i s no danger of undersampling.  to  q u a n t i z a t i o n down to  envisioned  For  f  excluding  appropriate  f u r t h e r found that the sample spacing  p i x e l s ; c l e a r l y , there was  the  f o r moderate b l u r r , the  minimum 0 of  I t was  than  by  scanner, the edge spacing was  pixels.  2/1.25=1.6.  divided  CCD  have been manufactured able to  [76].  With the c u r r e n t image data  l i n e such a system c o u l d r e a d i l y accomodate as l a r g e as 3.2  alternative  approach  at e i g h t b i t r e s o l u t i o n . was  developed by Orbach  incorporating hybrid d i g i t a l - a n a l o g c i r c u i t r y .  The  video  An [51]  delay  235 line  was  implemented with d i g i t a l RAM  data t r a n s f e r r a t e than was the  convolution  p o s s i b l e with a CCD  i s performed  line.  i s then summed by an  determined  with  a  single  line  whose  i s that  sign  comparator f o r b i n a r i z a t i o n .  can  matter  the  system  image has been f i l t e r e d , b i n a r i z a t i o n  of l a b l i n g the  negative p i x e l s  positive pixels  where  The  i s a simple  (dark r e g i o n s )  ( l i g h t background) "0".  passed to the segmentation  The  "1",  result  and  i s then  stage.  segmentor's task was  to  separate  characters  from  the  background p r o v i d e d they do not touch the image window s i d e s , are surrounded detected  by "0"  through  background  boolean  pixels.  operations  Border  performed  convolved over the b i n a r y image at the video  were  be  but the s i g n b i t l i n e are subsequently d i s c a r d e d . Once  the  the  This contrasts  with the e i g h t l i n e d i g i t a l output of an e i g h t b i t  the  operational  The p a r t i c u l a r a t t r a c t i o n of t h i s system  output s i g n a l i s c a r r i e d on a  all  However,  with m u l t i p l y i n g d i g i t a l to analog  c o n v e r t e r s (MDACs) whose output amplifier.  memory to achieve a f a s t e r  real-time logged  performance into  Simultaneously,  an  linked Euler  requirement, lists  were  on a 3X3 window  rate.  To  achieve  the border p i x e l s  immediately  number based  pixels  and  on  found  detection.  technique recorded those  p i x e l s suspected of p o i n t i n g to c l o s e d b o r d e r s .  Verification  of  c l o s u r e i n v o l v e d f o l l o w i n g the p o i n t e r s l i n k i n g the s t o r e d border p i x e l s at the end of each row external  and  internal  On c l o s u r e c o n f i r m a t i o n , a l l  borders were subsequently t r a n s m i t t e d to  the r e c o n s t r u c t i o n / r e c o g n i t i o n The  scan.  circuitry.  s i m u l a t i o n t r i a l s performed  i n d i c a t e d t h a t , even f o r the  236 most  complex  images  that  could  reasonably  be  expected, the  processing  load i s l i g h t enough t o allow new-found p o i n t s  to  processed  in  border  transmission.  parallel  image  reconstructed  of  a  method  the  was  segmented  from the border data.  Since  the  border data  c o n t a i n s , i t c o u l d be analyzed reconstruction shape  step.  Some  from  chain  analysis  Pavlidis  and  outlined  whereby  the  could  be  characters  However, i t should be noted  that t h i s i s not the only avenue t o taken.  verification  in serious delays.  transmission,  binarized  closure  Any r e s u l t a n t memory r e f e r e n c e c o n f l i c t s were seen  as u n l i k e l y t o r e s u l t After  with  be  recognition  that  could  be  f u l l y d e s c r i b e s the c h a r a c t e r i t  d i r e c t l y without  the  s p e c i f i c algorithms coded  borders  have a l s o p u b l i s h e d g e n e r a l surveys  t o perform such  were  [77, pp. 185-215] and Freeman [78].  intervening  presented  Both these  detailing  the  by  authors  progress  in  f e a t u r e e x t r a c t i o n and shape r e c o g n i t i o n from border d e s c r i p t i o n s [79], [80]. border the  The major a t t r a c t i o n of t h i s approach  data,  pixel  processed  positions,  rotation-invarient efficient  rate  matching a g a i n s t  that  the  t o remove redundant p o i n t s and normalize could  prove  representation.  i s demonstrated  recognition  is  by  reported  by  a  fully  scale-  and  That i t can a l s o be f a s t and  the  100  character  D'Amato et a l .  per  second  [66] where shape  f e a t u r e s e x t r a c t e d from a c h a i n code  facilitates  recognition. The  design  of  the OCR p r e p r o c e s s i n g  complete; c o n s i d e r a b l e o p p o r t u n i t i e s The  need  was  a c c u r a t e l y parse  already  outlined  remain for a  touching c h a r a c t e r s .  system i s by no means for further system  work.  component  However, a t the other  to end  237 of the r e s o l u t i o n s c a l e , an a c c u r a t e ,  autonomous method i s needed  f o r the t r a c i n g of p r i n t l i n e s .  This  causal  characters  acquisition  of words and  will  guide  subsequently, the c o r r e c t a c q u i s i t i o n of  connected  text  A  non-text m a t e r i a l such as Both  these  methods  unit,  represent The  blurring  f  the  require f i l t e r i n g  so t h a t p r i n t component  signalled  tracking  to  input.  the  image at lower  l i n e s would be r e s o l v e d  words.  Illustrations  level.  generated  by  Such an expanded system would seem to favor  new  high  resolution  stages  would  be  the  acquisition  resolution  data  imaging d e v i c e s capable of a c q u i r i n g more than a s i n g l e t e x t per  scan.  All  as  c o u l d then  ignore  and  gramatically  a l s o prove necessary.  non-segmentable o b j e c t s at t h i s low  character  non-text  o  print  of s i g n a l i n g and/or i g n o r i n g  i l l u s t r a t i o n s may  would  r e s o l u t i o n s , choosing a  method  correct  from a c u r r e n t  l i n e , and,  lines.  the  of these extensions  line  however would s t i l l  include  the same system components o u t l i n e d i n t h i s r e p o r t : a V g  filter  2  to  highlight  object-border  the  d e s i r e d l e v e l of d e t a i l ; and a  segmentation method to a c q u i r e  information.  Since  the necessary  such  an  satisfactorily.  image  these components have, i n t h i s t h e s i s , been  shown capable of r e a l - t i m e , autonomous o p e r a t i o n , that  non-sequential  expanded  preprocessor  will  it  is  also  likely operate  238  REFERENCES [I]  The British Computer Society, "Character recognition," Gresham p r e s s , Unwin Brothers L t d . , London, 1967.  [2]  Auerbach Info. Inc., "Auerbach on optical character r e c o g n i t i o n , " Auerbach P u b l i s h e r s , P r i n c e t o n N.J., 1971.  [3]  L.D. Harmon, "Automatic r e c o g n i t i o n Proceedings of IEEE, V o l . 60, No. 10, 1972.  [4]  J.R. Ullmann, "Pattern recognition Russak and Co. Inc., New York, 1973.  [5]  J.R. Ullmann, " P i c t u r e a n a l y s i s i n c h a r a c t e r recognition," Digital Picture Analysis (ed. A. R o s e n f e l d ) , SpringerV e r l a g , pp. 295-343, 1976.  [6]  J.R. 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