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Displaying human visual field data as shaded surfaces Jankowski, Richard 1984

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DISPLAYING  HUMAN V I S U A L  FIELD  DATA  AS  SHADED S U R F A C E S  by  RICHARD B.Eng.,  A THESIS THE  JANKOWSKI  McGill  University,  SUBMITTED IN PARTIAL REQUIREMENTS MASTER  FOR  1979  FULFILMENT  THE DEGREE  OF A P P L I E D  OF  SCIENCE  in THE  FACULTY  Department  We  accept to  THE  OF GRADUATE  of E l e c t r i c a l  this  thesis  the required  UNIVERSITY  OF  April  ©  Richard  STUDIES  Engineering  as  conforming  standard  BRITISH  COLUMBIA  1984  Jankowski,  1984  OF  In  presenting  requirements  this for  Columbia,  I  available  for  permission  her  advanced  agree  for  purposeScjnay or  an  be  thesis  allowed  of  without  Department  of  2  April  degree  at  fulfilment  the  Library  shall  reference  and  study.  I  extensive granted  by  this my  thesis written  Electrical  1984  copying the It for  Head is  my  financial  permission.  Engineering  Columbia  thesis  gain  the British  i t  freely  agree for  Department  understood  of  make  further  of' t h i s of  of  University  the  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date:  partial  that  representatives.  publication  in  that  that  scholarly  or  by  copying  shall  not  his or be  i i  Abstract  Human perimetry  visual  field  data  are three-dimensional,  presentation  have  of  The approach  the data.  earlier surfaces.  A  to  by  system  may  inspection  source  should  upon  shade  time  be a t l e a s t  light  12 b i t s  It  is  source wide.  data  map  with shown  from,  field  allows  t h e movement to  frame  and  contour surface  for  technique  movement,  shaded  continuous-  viewed  that  these  as  The o r i g i n a l  operation  in  behaviour  upon  of the v i s u a l  A memory  graphical  the  synthesizes  direction.  zoom  real  field  threshold  at  improves  surface  detail.  the surface.  unrestricted  a  any view A  efforts  visual  that  of  by k i n e t i c  i n conveying  herein  displaying  any s p e c i f i e d  of s u r f a c e  about  simulate  taken  interpretation.  surface  effective  images  be s u p e r p o s e d  enhance  vary  from,  and e a r l y  i s developed  grey-level  illuminated data  not been  renditions  tone,  collected  closer  i s used of a  to  light  effectively  buffer  pixels  i i i  Table  of  Contents  Abstract L i s t of Figures Acknowledgement  i i v v i i  Chapter I INTRODUCTION  1  1.1  C o n v e y i n g S u r f a c e Shape 1.1.1 M o t i o n P a r a l l a x 1.1.2 S h a d e d I m a g e r y 1.1.3 C o n t o u r e d , S h a d e d S u r f a c e s 1.1.4 C o l o u r - C o d e d I m a g e r y 1.1.5 S t e r e o I m a g e r y 1.1.6 L i g h t S o u r c e M o t i o n 1.2 S y s t e m O v e r v i e w 1.3 T h e s i s O r g a n i z a t i o n  4 7 8 9 9 9 10 11 19  Chapter II SURFACE REPRESENTATION  20  2.1 D a t a C o l l e c t i o n 2.2 T h e R e c o n s t r u c t i o n P r o b l e m 2.3 S u r f a c e R e p r e s e n t a t i o n 2.3.1 Triangulation Criteria 2.3.2 S u r f a c e C o n s i s t e n c y 2.3.3 T r i a n g u l a t i o n Procedure 2.4 S u r f a c e I n t e r p o l a t i o n 2.5 Summary  20 25 27 31 33 36 37 39  Chapter I I I IMAGE S Y N T H E S I S  41  3.1 3.2 3.3 3.4 3.5  Projection Transformations Hidden S u r f a c e Removal Surface Shading Contour Superposition R e a l - T i m e Movement o f t h e L i g h t 3.5.1 Optimal Quantization 3.6 Summary  Chapter RESULTS  Source  41 44 48 53 54 58 62  IV 64  4.1 V i s u a l F i e l d S u r f a c e s 4.2 G r a d i e n t Q u a n t i z a t i o n Chapter V CONCLUSIONS  AND  Experiment  PROPOSALS  5.1 C o n c l u s i o n s 5.2 P r o p o s a l s f o r F u r t h e r  64 78  86  Work  86 89  iv ,  BIBLIOGRAPHY APPENDIX A APPENDIX  B  ..94 97 1 0 2  V  List  1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10  of  Figures  C o n t o u r map 3 G r e y - s c a l e map 3 A s s o c i a t i n g v i s u a l f i e l d data with retinal nerve f i b e r bundles 5 Orthographic l i n e drawing 6 World coordinate system 12 Viewing s p e c i f i c a t i o n 12 Shaded s u r f a c e viewed from 0=160°,0=50° and i l l u m i n a t e d from 0 = 1 2 0 ° , 0 = 7 0 ° 14 Shaded s u r f a c e viewed from 0=70°,0=50° and i l l u m i n a t e d from 0=80°,0=70° 14 I n v e r s i o n o f s u r f a c e i n F i g u r e 1.7 15 I n v e r s i o n o f s u r f a c e i n F i g u r e 1.8 15 O r i g i n a l c o n t o u r s s u p e r p o s e d on s u r f a c e i n F i g u r e 1.7 16 O r i g i n a l c o n t o u r s s u p e r p o s e d on i n v e r t e d s u r f a c e i n F i g u r e 1.9 16 Zoom w i n d o w s u p e r p o s e d o n o v e r h e a d v i e w ( 0 = 1 8 0 ° , 0 = 9 0 ° ) o f s u r f a c e i n F i g u r e 1.7 17 W i n d o w e d s u r f a c e v i e w e d f r o m 6*= 11 0 ° , 0 = 5 0 ° a n d i l l u m i n a t e d f r o m 0= 1 3 0 ° , 0 = 7 0 ° ."...18 I n v e r s i o n o f windowed s u r f a c e v i e w e d from 0=160°,0=50° and i l l u m i n a t e d from 0=120°,0=70° 18 Convergent t a r g e t scan 22 Divergent t a r g e t scan 22 Scanning across a v i s u a l defect 24 A s s o c i a t i n g s u r f a c e f e a t u r e s with t r i a n g l e edges 30 Surface inconsistency along contour 34 T r i a n g u l a t i o n consistent with surface contour 34 Surface inconsistency along ridgeline 35 Triangulation consistent with ridgeline 35 A t r i a n g u l a t i o n of v i s u a l f i e l d data 38 Surface projection 42 Scanning a surface p r o f i l e i n the viewing d i r e c t i o n ..45 Scanning a surface p r o f i l e along a g r i d l i n e ....45 Viewing geometry 50 Geometry f o r o r t h o g r a p h i c viewing 50 F r a m e b u f f e r w i t h a t t a c h e d memory map 56 R e f l e c t a n c e map f o r d i f f u s e surface i l l u m i n a t e d from 0=67°,0=53° 61 Visual field 1 65 Visual field 2 67 Visual field 3 69 Visual field 4 71 Visual field 5 73 Visual field 6 75 D i f f u s e surface represented with 8-bit gradients 79 D i f f u s e s u r f a c e r e p r e s e n t e d w i t h 1 0 - b i t g r a d i e n t s ....79 D i f f u s e surface represented with 12-bit gradients ....80 D i f f u s e surface represented with 64-bit gradients ....80  vi  4.11 4.12 4.13 4.14 4.15 A.1 A.2 A. 3 B. 1  Specular surface represented with 12-bit gradients Specular surface represented with 64-bit gradients Graph of h(p) l i n e a r l y i n t e r p o l a t e d from h i s t o g r a m data f o r p Smoothed p r o b a b i l i t y d e n s i t y f u n c t i o n h ( p ) Smoothed p r o b a b i l i t y d e n s i t y f u n c t i o n h ( q ) Node N i n t r o d u c e d i n t o an e x i s t i n g t e s s e l l a t i o n N o d e N i n c o r p o r a t e d i n t o a new t e s s e l l a t i o n How a t r i a n g l e e d g e d e f i n e d b y v e r t i c e s I a n d J i s represented i n the TIN data s t r u c t u r e Triangle transformations  ...81 ...81 82 84 85 99 99 101 103  vii  Acknowledgement  The  author  guidance, J.  Little  framework James  Gunther  from  charts.  of  thank for  Steve  and Luis  Jim Clark,  This granted  of  displayed Vision  using  Interdisciplinary  the data  Nick  developed. and  from  the f i l t e r s  The author  by  Images  Jaeger,  a  of  produced  surface  that would  and  Drance  concerning the  visual  used  i nthe  enabled also  the  like  Brian  to  Maranda  postgraduate  scholarship  Graduate  Computer  Program  SMI-51.  for  this  Council  thesis  were  of the Laboratory f o r Computational  The l a b o r a t o r y  by NSERC g r a n t  was  S c i e n c e s and E n g i n e e r i n g Research  facilities  Columbia.  working  feedback  p r o v i d e d a program  supported  i n t h e Department  British  part,  was  Canada.  provided a  and suggestions.  by t h e N a t u r a l  (NSERC)  James  Dr. Stephen  and v a l u a b l e  i n the text.  for h i s  and  program  system.  Heiss developed  Marti  support,  triangulation  digitized  P a u l Freedman,  criticisms work  data  Wismath  Detlef  subscripts  their  field  the  J . Woodham  Woodham  display  the  to  Robert  his  Robert  contributed  the visual  photography, use  for  the surface  programs  project.  field  t o thank  F. Schrack  which  interpolation  the  like  f o rh i s enthusiasm.  Little  provided  would  Science,  i s supported, in  Remote  University i n part,  Sensing  of  by t h e  and,  in  1  I.  This visual  thesis  field  perimetry human  at a  light  positions  are  recorded In  position  a t which  the subject  a  a  a point  position.  grid  first  threshold  random  The  grid  need  While  of  on  vision.  to different stimuli  not cover  light  source  field  appears  of  at that  i s slowly  the stimulus.  on, and then  the entire  the e n t i r e  t o cover  This  of the subject's  until  a  field  grid  visual  of  fixed  view.  The  or disappears  threshold  sources  are turned  pattern  be p o s i t i o n e d  either  of such  detects  The t a r g e t s  in a  a  concentrates  the i n t e n s i t y of a target  sensitivity  time  of  i s measured.  his field  target  subject's  of v i s u a l  perimetry,  regular  subject  within  Threshold  sensitivity  the subject  responds  single  the  the target  marks  static  point,  presented  i s moved a c r o s s  subject  stimulation  human  chart.  a t which  In  may  reference  of  perimetry.  the threshold  light  k i n e t i c perimetry,  intensity  the  to  presentation  by t h r e s h o l d  whereby  stimuli  on a  the graphical  obtained  eye  fixed  The  within  with  i s a technique  detecting  the  data  subject's  staring  the  deals  INTRODUCTION  before  intensity.  light  source  increased  until  intensity  marks  eye a t that  target  o f f , one has been  field;  a  at  a  covered.  smaller  grid  sector  of  interest  to  the  targets  do  not  but the  ophthalmologist. In  static  intensity  of each  form  a  the  of  perimetry,  i s variable.  matrix  i n t e n s i t y of a  the  of l i g h t target  The d a t a  intensities.  i s held  fixed  are  move  collected  In k i n e t i c as  the  in  the  perimetry,  target  scans  2  along  many  obtained  by  intensity of  trajectories scanning  contour,  different  matrix  or  with  or  map,  a  sampling  of  variables,  where  positional  coordinates.  Intensity kinetic  that  produces  (1981).  A  i s sparse.  Either  interpolated  target  intensities  where  Intensity  matrix.  scale  image,  map  and  are  from  dense  where  each  i t s value  and  Figure  1.2,  where  dark  areas  an  attempt  to  improve  Wirtschafter corresponding partition  the  et to  of  as  of  a  a  described segments  two  from  system  by  Hart  i s shown  set  of  a  medical  automated  small  set,  independent  by  displayed  be is  a  in  standard  physiological field  of  of  into  in  grey-  grey  level on  a  i s given  in  visual  loss.  rendition, field  data the  fixed  a  a  pixel  static the  the  generate  as  a  grey-scale  structures domain  to  a  into  regions  the  group  between  g r e y - s c a l e map  indicate  a l . (1982)  assigned  mapped  of  upon  points  may  then  example  at  perimeter  interpretation  visual  a  data  obtained  static  sample  or  the  An  a  matrix  screen  objectify  code  interpolated  This  chart.  line  targets  function  An  is  equi-  sizes.  obtained  to  to  two  a  data  an  thought  manually  with  i s used  of  computer.  drawn  colour  values  samples  denser  to  a  be  samples  piecewise-linear contours  1.1,  In  contour  of  Either  can  is  The  additional  map.  function  between be  contour  proportional  with  contour  intensity  field.  samples  Scanning  a u t o m a t i c a l l y by  Figure  sparse  target are  yields  threshold  may  visual  a  values  or  the  single-valued  perimeter  technician  one  isopter.  intensities  contour  across  by  and  data, regions  eye. regions  They that  Figure  1.2  Grey-scale  map  4  project  onto  within  a  region. given  example  i n Figure  of  an  shown  enables  line  methods  1.1  When onto to  a  the  image  to  of  f o r that  They  as the contour  "view"  of the  and Hartz  the  visual  render  i s  field  curves.  of a v i s u a l  employ surface  field An  field  map  surface  (1982)  the v i s u a l  profile  drawing  be  The d a t a  effectively  c a n be  thought  investigation, therefore, surface  as a  example  surface i s  conveyed  to  of as l y i n g focuses  on o r  upon  human the near  effective  shape.  i s presented  The r e s u l t i n g in  whereby  Shape  surface  a 2D m e d i u m . ambiguities  a  Hart  of surface  can  Surface  3D  data  presentation  investigates a d d i t i o n a l techniques  f o r rendering  Conveying  field  such  offer  this,  viewing  line  data  The  Field  s i n g l e measure  of v i s u a l  they  direction.  drawing  ophthalmologist. surface.  retina.  1.4.  thesis field  a  presentations  Recognizing  that  i n Figure  visual  the  to yield  form  i s that  orthographic  The  a  with  map  an a r b i t r a r y  perspective  in  1.3.  only.  technique  layers  of t h i s  disadvantage  above  from  fiber  are averaged  the grey-scale  from a  region An  A or  nerve  t h e image.  help  the  graphically, i t i s projected  loss  of information  Visual  viewer  cues  better  often  are introduced resolve  leads into  spatial  relationships. In It  the current  i s important  from  one  application, a  t o convey  "locality"  t h e shape to  single  surface  of that  another,  to  i s  presented.  surface  as  depict  the  i t varies spatial  Figure  1.3  Associating visual f i e l d data with r e t i n a l nerve f i b e r b u n d l e s [ f r o m W i r t s c h a f t e r e t a l . (1982)3  Figure  1.4  Orthographic  line  drawing  7  relationships complex  surface,  difficult familiar in  objects.  Visual contour  1.4.  point.  These  that  perspective  providing  time  various  convey  very  many  important  obscured  drawn,  appear  shape  review  as  of  a  of by  to a  profiles in  lines  vanishing of  another;  how  profile  localities.  additional of a  and  hidden  notion  than  within  image  of Hart  parallel  t o converge with  as the  illustrated  makes  or f a r t h e r  i n t o an  such  the removal  projection  surface  cues  by  the viewer  i s a brief  the viewer of  helps with  views  i s rotated  i s  parts  to  of  the  frames  appears by  of the surface  views  surface  between  individual  produced  resolve  different  slightly  and the surface  depth  not  i s closer  technique  animation,  screen  become  of  3D  techniques  surface.  Parallax  sequence  surface  a n d more  composed  In the approach  profiles  are  provide  visual  abstract,  subtle  i n presentations  map.  the viewer  locality  Motion  This  of  The  follows  therefore,  surface  devices  be more  scene  are provided  viewer  introduce  1.1.1  A  of  and d e n s i t y What  cues  from  a  lacking  the  away  surface  shape  in  F o r an  process.  are sorely  depth  to  "farther"  those  V i s u a l cues,  Segments  Figure  than  or the grey-scale  visual  "closer"  one  cues  localities.  r e l a t i o n s h i p s may  synthesis  map  lines.  neighbouring  these  to depict  t h e image  Hartz,  between  image  o f t h e same  views.  are displayed smoothly.  the differences t o move  where t h e In  real-  r a p i d l y on a The  in velocity with  by  surface.  i s generated  successive  to rotate  appear  ambiguity  respect  effect a t which t o each  8  other.  The  rotation,  "farther" a the  faster  between  the  that  of  surface  can  resolve  relative which  1.1.2  in  viewer  move  elements  behind  the  d i r e c t i o n of  real  scene  the  depends  light  geometry, element shade in  in  illumination.  orientation  surface  surface.  upon  grey-level shade The the  source of and  depends  thus  moving.  in a axis.  his  axis  Surface  of  elements  d i r e c t i o n opposite Therefore,  detecting  surface  the  the  to  viewer  differences  elements,  experience  in  differences with  rotating  Imagery  variations  element  from  from  world.  Continuous-tone, exhibit  be  is  r e l a t i o n s h i p s by  appreciate  the  to  the  and  can  Shaded  and  element  i t appears  spatial  speed  he  objects  axis  surface  the  shape,  amount  a  of  light  reflectance  distribution,  surface  provide  in accordance  element.  only  images  on  For  a  reflectance,  while  cue  for  producing  with  surfaces  surface  reflected  properties  of  by  viewpoint,  fixed  viewpoint,  the  shade Local  inferring a  of  that  shape a  the  the  its orientation.  valuable  depict  surface surface, and  realistic  the  lighting a  surface  variations local  and  in  variations  image  of  the  9  1.1.3 C o n t o u r e d ,  Forrest surface  that  (1979)  may  arbitrary an  be  contours  for  a  to  Colour-Coded  depict  image, and  could  convex  and  dependent  on  1.1.5  Stereo  Humans images  from  advantage supply  contours than  levels  when  the  of  the surface He  upon  the shaded  on  image  experience.  overlayed  just  suggests  which be  may  image  surface  surface  a i s  suggests  the  shaded itself.  and  local  surface  be d i f f i c u l t  provide  could  saturation  of  of  be  as subtle  with  such  grey-scale  colour. with  colour  as  concavities  in a  coded  each  colour-coded  property,  to detect  more p r o m i n e n t  elements  the magnitude  p i x e l s be  v a r i a t i o n s , such  rendered  the  that  o f some  surface  surface  colours,  interpret  shaded  Imagery  the magnitude  convexities,  a  image.  further  Small  even  the viewer's  l i e at precise  l e s s ambiguous  curvature. or  to  beyond  depicting  Forrest  that  difficult  i s more e f f e c t i v e  The  1.1.4  Surfaces  observes  and f a l l s  image  surface  Shaded  Concave different  could  be  made  stereo  p a i r of  curvature.  Imagery  possess which  depth  o f human  a viewer  two e y e s ,  generated  from  viewpoint  of the  information  binocular  with  and these  stereo,  two s y n t h e t i c  the viewpoint right.  provide is  a  extracted.  s p e c i a l hardware  images  of the l e f t  o f t h e same  To  take  i s used scene,  eye, the other  to one  from the  10  Foley  and  generating from CRT  two  stereo  may  Dam  (1982) d e s c r i b e s e v e r a l  images.  separate  screen  rapidly  van  CRT  Special  screens  alternately  enough  t o produce  configuration,  two  each  of the viewer's  eyes,  the  CRT  to  so t h a t  expose  colour  the correct  filters,  eyes,  produce  1.1.6  Light  Source  images  of  illumination  analysis,  this  been  images  moved  present  in  attractive movement  idea  a  developed  a  or a  single  stereo In the  pair  single-  one p l a c e d  before  synchronized the other  Polarized  with  i s opened  filters,  and b e f o r e  can  be  frames,  single,  i f t h e images  of the l i g h t  technique  the  or  viewer's  multiple  fixed  direction  but under  Although  developed  f o r image  a  where  i s effective  shaded  in only  image a  synthesis.  light  in resolving  image.  c a n be g e n e r a t e d  source.  photometric  from  exploited  scene,  called  i s determined  from  conditions.  o f t h e same  between  eye.  images  effect.  s u r f a c e viewed  different  Viewing  shutters,  at the screen  surface orientation  the  effect.  i s closed,  to that  eyes, in a  are electronically  image  channel  Motion  (1980)  whereby  image  flicker-free  one s h u t t e r  similar  each  solid-state  positioned  a  Woodham stereo  when  may  to the viewer's  display a  CRT  optics  techniques f o r  This  is  in real  source  has  ambiguities especially  time  with the  11  1.2  System  An touch  ideal i t s  different the  Overview  to  surface,  presentation  some o f A of  the  visual  shaded  Hartz.  by  a  >  show  Viewing  space  z  0,  viewer  are  Figure  1.5).  where  based  6 and  an by  can  clockwise  from  measured  above  elevation  between  surfaces  in  be  viewing  the  positive xy  the  0°  and  90°  thesis  are  i t  image.  from is  Tactile  "manipulated" viewing  realistic  and  by  using  s y n t h e s i z e shaded  images  field  It  drawings from  is  conveyed  of a  Hart  and  specified  source. by  from  that  g e n e r a t i n g images from,  anywhere  both  the  The  lighting  azimuthal and  the  viewed  from  an  and  1.6).  (see  is  i s measured angle  is  Although  any  most  0=50° a n d  the  azimuthal  direction  angle  specified,  half-  system  elevation  (see F i g u r e be  surface  by  of  different i n the  coordinate  is specified  The  axis,  data.  effectively  illuminated  <j>.  may  to  thesis  different  line  "world"  plane  be  view  be  illuminated  0. y  may  more  direction  6 and  the  this  is  performed  angle  2D  made  to  light  (x,y,z)  to  in this  a  visual  and  would  discussed.  i t i s assumed  i t s own  be  i s "manipulated" from,  object  and  under  can  with  distant  elevation  specified  shape  surfaces  i n an  A  object  developed  than  viewed  directions.  was  by  object  previously  surface  surface  the  images  surface  single,  given  the  an  i t ,  object  but  of  addressed  underlying  imagery  Images  direction  angle  system  that  of  cues  shape  problem 3D  The  surfaces  demonstrated  the  a  images  software  A  of  the  manipulate  The  conditions.  the  with  to  is impractical,  generating  lighting  examine  directions.  inspection by  way  of  the  illuminated  12  Figure  1.6  Viewing  specification  13  from  0=70°.  The a z i m u t h  360°.  A  Figure  1.7, w h e r e t h e v i e w  same  surface  i s specified  surface, The  which  but viewed  prominent  l i e hidden  features pits  depicting  more  become  the  inversion half-space  features  below  "peaks". in  feature z <  surface.  Figures  1.11  of is  and  system  surface Examples  i s  shown  in  1.8  shows t h e  surface  and and  appear  area.  may  be  t o be  "pits"  T o make  these  "inverted"  so  1.10  show  1.8,  respectively.  permits  viewing  that  inversions  from  of This  the  lower  images  i s aided  visual  of contour  by  field  superposing chart  superposition  the  onto  the  a r e found i n  1.12. provides  (see Figure up  also  of  Examples  i s superposed  blown may  1.9  1.7  of these  also  size  the surface then  Figure  surface  surface  of the o r i g i n a l  shaded  specified  the  effectively  1.1  0° a n d  0=70°. of t h i s  Figures  between  0.  contours  The  i n Figure 0=160°.  the v i s i b l e  Figures  Interpretation colour  i s from  from  visible,  surfaces  the data  by an a n g l e  to  be  zoom  function.  on a n o v e r h e a d  1.13).  full  inverted  these  a  The  screen and  operations  A  view  surface  within  are  from  shown  of  (0=180°,0=90°)  resolution.  viewed  window  t h e window  The windowed  any  direction.  in Figures.  1.14  and  1.15. Finally, simulate surface. single are  a memory map  real-time This  movement  i s done  i m a g e i n much  used  attached of a  i n photometric  light  i n an a t t e m p t  t h e same  way  stereo.  to a  frame  source  buffer about  to resolve  that  multiple,  a  i s used  to  stationary  ambiguities shaded  in a images  Figure  1 .8  Shaded surface viewed from i l l u m i n a t e d f r o m 0=80°,0=70°  0=70°,0=50°  and  Figure  Figure  1 .9  Inversion  of  surface  in Figure  1.7  1.10  Inversion  of  surface  in Figure  1.8  16  Figure  Figure  1.11  1.12  Original contours Figure 1.7  Original in Figure  contours 1.9  superposed  superposed  on  on  surface  inverted  in  surface  17  Figure  1.13  Zoom window (0=180°,0=90°)  superposed on overhead of surface i n Figure 1.7  view  Figure  1.14  Windowed surface viewed from i l l u m i n a t e d from 0=130°,0=70°  0=110°,0=50°  and  19  1.3 T h e s i s  The  Organization  problem  divided  into  two  reconstruction second  with  surface.  of  by  process:, surface the  of  3  current  surfaces were  moving  from four  experiment 5  hidden  involved  in  sparse  of  data  these  issues  representation  in  and  an  are also  in for  discussed, simulation  surface.  field  the  charts.  examined  results  and  used  methods  also  I t contains  from  images The  s i x times  of  charts over  the light  a  source  included.  the results  system  Two  are  stationary  patient  Some  removal,  i n the real-time  six visual  single  synthesis  algorithm  described. surface  image  surface  surface  pictorial.  years.  and f u r t h e r  a  the  hidden  involved  from a  is  about  discusses  the current  improvements  steps  shaded  the problems  reconstructed  Chapter evaluate  on a  4 i s mainly  of about  movement  the  The s i m p l e  source  obtained  period  surface  and the  parts.  the  Consideration  the  reconstructed  issues from  i s  with data,  of the  of these  the  transformations,  contours  Chapter  image  surface  a  implementation  with  a light  of  outlines  shading.  together  of  sparse  data  scheme.  viewing  superposing  some  field  dealing  from  t o each  perimetry.  selection  interpolation  shaded  consistent  kinetic  the  Chapter  of a  visual  first  surface  i s devoted  a  human  the  a complete  2 ' discusses  reconstructing  3D  parts:  A chapter  collected  viewing  the synthesis  Chapter  guides  of  design,  experiments.  of Chapter and  makes  4, a t t e m p t s proposals  to  for  20  II.  2.1  SURFACE  RECONSTRUCTION  Data C o l l e c t i o n  The  visual  obtained This  from  field  a  human  apparatus  mechanism the  for  inner  medical light  data subject  consists  projecting  surface  technician target,  of  a  a  bowl.  the  this  a  thesis  kinetic  of  light  mechanism  size  and  t r a j e c t o r y along  are  perimeter.  hemispherical  spot  The  in  with  large  circular  controls the  with  examined  of  the  who  and  dealt  bowl  and  anywhere  i s operated  intensity  which  on by  of  i t moves  a  a the  on  the  bowl. The  subject  end  of  the  the  bowl  bowl  opposite  would  be  under  test  responds  fixed  target  positions  of  bowl.  onto  the  of  on  visual  fixation.  visual  target At paper  rectangular  eccentricity  target  of  threshold  fixed  a  The not  With  for  point  on  chart  attached onto  of  in  graduations  the  eye  subject  the  the  outer  chart, to  The  data whose  the  surface  as  well  facilitate are  and  i n t e n s i t y i s used  as the  marked  origin  axes  size  side  the  marks  fixation  one  and  two  technician  the  On  the  or  from  contour.  outside  field  the  along  open  visual  to  system  sits  bowl,  order  locations.  the  bowl,  his  the  i s projected bowl,  at  covered  the  within  times  coordinate  the  eye  these  the  support  technician  one  appears  threshold  The  If  reference  surface  point  A  a  against  inside.  other.  target  a  of  a  i t .  against  degrees  subject.  the  The  inner  recording  of  the  upon  from  head  peers  each  whenever  his  and  facing  disappears  the  places  is  the  represent  point.  of  the  to  obtain  contour,  one the  21  target  is  invisible.  visible  to  Contours  are  obtained  across  the  subject's  trajectories which the  the  subject  target  manner  are  controls can  l i e  the  number  with  are  linear  connected, Some curves  In  the  the  An  the  illustration data,  is  additional  data,  such  where  visual  view.  change  in  the  the of  points  target  the  The  the  that  raw,  each  points  collected The scans  data  along at  visibility  contour.  the  i t is  target  of  in  this  technician such  that  he  to  obtain  a  points  implementation,  from  are  traced  subject's of  such  shown a  the  side  visual  digitized  contour  field  data  is  a  points set  of  segments.  contours  outside  other  of  current  straight-line  the  field  Data  of  on  moving  between  reconstructed  of  a  spacing  and  by  contour.  samples  and  resulting  vision,  detects  i n t e r p o l a t i o n , so  from  center.  a  subject,  interpolate  contour.  contours  on  considered  manually  smooth  first  the  by  field  of  peripheral  in  Figure  contour  moving  may  sensitivity  the  target  view  towards  scanning,  2.1.  In  represent i s reduced  along  and  the  a  of  the  absence  "plateau"  outside  the  the  of of  plateau  rim. Another  scanning  sensitivity  loss,  pattern  usually  target  is positioned  within  moved  out  the  with the  a d d i t i o n a l scans region's  Figure may  towards  2.2.  represent  i s used near  the  field  the  This  In  the  absence  periphery.  the  rim  of  a  visual  out of  region A  pattern  is  a d d i t i o n a l data, "pit".  regions  the  field.  of  contour  in different  scanning of  trace  center  suspected  r a d i a t i n g out  interior.  to  loss is  The and  obtained  directions illustrated such  of  a  from in  contour  22  V  \ N  Figure  2.1  Convergent  target  scan  \  Figure  2.2  Divergent  target  scan  23  Figure exhibits target at  2.3  a  central  follows  the  top.  He c o n t i n u e s into  the  subject  illustrates  The s u b j e c t  b l i n d area.  I t remains  time  the target i s lost  contour with  leaves  the  scans  i s  represented  function  of  higher  visual  of f i e l d  the  position.  a measure o f v i s u a l  Finally, should is two a  time  the error  delay  times from  to the  and a r r i v e s a t D,  at  f o r that fall  which  intensity,  on  becomes  a  single  evident  only  After  this  field.  that  A.  loss the  i s probed defect  i s  field  i s  a  with  a contour  the behaviour  continuous,  In the remainder  sensitivity,  of  this  single-valued  of the  as the f u n c t i o n  map, a n  text,  z = F(x,y),  the where  and x and y a r e p o s i t i o n  i n degrees.  subjective.  reaction  at point  invisible  of v i s u a l so  starting  entry  C to point  made a b o u t  be e x p r e s s e d  be c o n s i d e r e d .  highly  that the  before  region  the  region  visual  threshold  z  given  across  being  field  coordinates  will  becomes  contour  intensity,  visual is  this  just  that  detail.  i s implicitly  phenomenon:  B,  A, B, C, a n d D  the central  representing  point  field  Assume  the target  the b l i n d  from  Points  i n greater  assumption  over  spot.  point  then  visual  i n the figure,  detects  until  a r e performed  of  a  the v i s u a l periphery  shape  traced,  •targets  By  indicated  The t a r g e t  visible  to sight.  but  more  a blind  first  i t has passed  C.  after  or  defect  t o see the target  until  isopter,  across  the trajectory  point  and  a scan  arising Visual Two  from  the data  threshold  collection  examination  individuals are involved  determine t h e moment  the accuracy the subject  by  process perimetry  and,  hence,  of the data.  There i s  first  a change  detects  Figure  2.3  Scanning  across  a  visual  defect  25  in  target  visibility  technician. technician arrests is  There  i s  acknowledges  target  moving  t o t h e moment  away  initial  detection.  kinetic  perimetry  During  its  Therefore,  surface data,  problem  i s  contours. not  problem  line  endpoints. of  data  Actually,  A  then,  contours  surface  may  collected  of  using  error.  c a n be r e g a r d e d  case  surface  hull raw  et a l .  in may  by manual be f i t t e d  visual from  field  a  set of  contours  a typical slices  one  in  i s  being  medical  approach to  i s composed  turn,  of  are defined  be r e c o n s t r u c t e d  set  a by  from  set their  a  set  (x,y) are scattered  i n t h e xy p l a n e . of the  as a  (1977).  chart  data,  scattered  of  from  whose p r o j e c t i o n s  a domain  original  are just  he  instant  graph-theoretic  field  these,  (x,y,z),  data  planar  A  by F u c h s  of t h e convex  the  technician, produces  and  a  from  in a visual  throughout  interior  data  applications,  (1983)].  The s u r f a c e ,  irregularly the  practical  i s outlined  points  the  reconstruction  to reconstruct  segments  t h e moment t h e  systematic  In the  surface  et a l .  contour  3D  problem.  reconstruction  [Cook  Each of  in  object  imaging this  presenting  of  The need  uncommon  solid  of  one  the  t o t h e moment  at  field  small  to  intervals, the target  position  to a  from  this  Problem  reconstruction i t  time  visual  are subject  delay  response  these  correct  2.2 T h e R e c o n s t r u c t i o n  The  an a d d i t i o n a l the subject's  motion. from  he communicates  of  This  domain i s  (x,y)  points.  a s c o l l e c t e d by t h e m e d i c a l  points.  I t i s the technician  who  interpolation. to a  s e t of data  points  such  that  26  it  either passes  called  through  interpolation;  be made b e t w e e n  these  Interpolation confidence when  i s high  and a  faithful  groups:  reconstruct a l l  from  points  about  or  height points  when  are  A choice  must  accurate  and  through  the data  through  them  the data, the  or  original  are subject  would  not  to  be  a  behaviour. for  These local  performing  are c l a s s i f i e d methods.  in  the  fall  point  domain.  relatively within  a  either into  Global  z a t an a r b i t r a r y  a t P from  whose p r o j e c t i o n s  methods P = (x,y)  Local  nearby small  two  methods  data  points,  neighbourhood  P. Global  methods  surface.  global local  Local,  trend  method  a r e best subtle  t o which  features,  defects,  which  i s used  surface  points  indicate small  importance  the  contours  error  scheme are  is  surface.  c a n be m a s k e d contribute.  or subtle  visual  so that  with  will  by  a  Because  visual  field a  local  process. in  employed  This  smoothing of  t o an o p h t h a l m o l o g i s t ,  inherent  consistent  an o v e r a l l  features  i n the reconstruction  interpolation  reconstructed  to achieve  a l l the data  may  are of great  Despite  field  The former i s  to preserve  available  methods  the height  passes  reasons  of t h e i r  are  data  the data  the surface  or approximation.  given  reconstruct  when  passing  the surface  them.  approximation.  i s performed  methods  global  or near  schemes.  f o r other  surface  interpolation  the  that  representation  Many  from  two  Approximation  error,  the l a t t e r  i s performed  i t i s desirable  data.  the points  the  enable  field  data,  the o r i g i n a l contours  correct  of  an  visual the  superposition  27  of  the  original  reconstructed  2.3  Surface  P  (x,y)  small in  reconstruction the  in  the  domain  about  Two  over  data  irregular  irregular  network  disadvantages  the  rectangles,  or  interpolated do  grid  spacing  regions  Mark should  Hartz  the  in this  describe  fall  of  within  a  neighbourhoods  of  a  of  are  grid  congruent,  surface defined  heights  o r an  advantages  and  The t i l e s  heights data  at  In  forces  i s varying  a properly  phenomenon i s t h e human  this  phenomenon  be  grid  slowly.  that  triangles, points  which  order  data  at the  tiles  to  capture a  redundancy A  are  generally  representation,  i s not storage  that  may  points,  in this  which  are specified  polygonal  locations.  surface  case,  that  point  f o r surfaces  grid  There  i s performed  arbitrary  selection  regular  features  the  points  regular  at given  therefore,  reflect  (1982)  a  The  used,  (1978) a r g u e s  phenomenon,  of  grid  be  the  representation,  a  heights  surface  must  where  in  with  varying  the  of the surface.  from  an  representations  hexagons.  not coincide  quickly  of  representation.  points,  domain  data  patches.  either  at  The o r g a n i z a t i o n  upon  are:  heights  or grid  from  common  of  with  Surface  cover  image  interpolation  height  P.  depends  representation.  vertices,  synthesized  by l o c a l  surface  domain  neighbourhood  the  a  Representation  determining =  onto  surface.  Surface by  contours  regular  fine in grid  efficient.  designed  being visual with  data  structure  represented.  The  field.  and  terrain  Hart  attributes.  28  They  use d e s c r i p t o r s  "trough".  If  irregular  should  Despite  then be  accessing  a data  from  the regular  i s  can  structure  storage  grid  be  "plateau",  and  conceptualized  consistent  inefficiency  enjoys  and programming  compatible  matrix  field  "hillock",  with  as  such  a  employed.  compatibility  structure  as " c l i f f " ,  visual  suffering  inconsistency, machine  the  terrain,  phenomenon  such  elements  with from  popularity  modelling  because  convenience.  linear a  and  The  of  matrix  memory o r g a n i z a t i o n ,  program  i s  both  i t s  simple  and and  efficient. A be  patch  representation  developed  by p a r t i t i o n i n g t h e domain  in  a  number  can  be c o n s t r u c t e d  interior with  is  o f ways.  of  polygon A  a  polygon,  a n d common  mutually  disjoint,  disjoint  i s meant  overlap.  When  of  irregularly that  common  called that  a surface.  surface  The domain  triangulated  adaptive;  or such  of scattered  of i r r e g u l a r l y  each  region  partition  data  that  hand, c a n  data  shaped  point  the data  lies points  of a set of scattered  i s partitioned shaped  the interiors  adjacent  an e n t i r e ,  sometimes  that  on t h e o t h e r  points  polygons in  the  coincide  vertices.  simple  The  A network  such  triangulation.  along  of a surface,  into  points  a network o f  triangles.  By  mutually  o f a n y two t r i a n g l e s do n o t  triangles  i n t e r s e c t , they  do so  only  edge. irregular  [Peucker  network, et a l .  i s , i t c a n be a d a p t e d Many s m a l l  or  TIN,  (1978)],  to the  whereas  i t i s  i s  resolution-  varying  complexity  t r i a n g l e s a r e necessary  of fine d e t a i l ,  as  a few l a r g e  t o represent  a  triangles are  29  sufficient  to  cover  resolution  makes  than  dense,  the  consistent medical  with  the  data  points is  in  these  consistent grid.  content  can  additional  of  the  more  field  target  i s changing regions.  This  adaptability  storage  structure.  visual  data  scan  regions  i t is  collection.  density  of  of  efficient  Furthermore,  r a p i d l y , thus  In  structure  Furthermore, be  encoded  storage to  exhibits  sudden In  surface  be  can  illustration Figure  of  in  The  regions  generating  little  more  change,  the  this  of  adaptability,  a  its  modelling  TIN  surface  implementation.  suffers  one  serious  t r i a n g u l a t i o n s of  goal  i s to  consistent  the  high  structure edges  along  can  as  ridgelines  surface  to  surface  structure.  model  made  a  actual  data  the  without be  which  such  a  information  structure  can  is  than  "physical"  t r i a n g l e edges  current  set  lines  network surface  of  data  slope,  the by  features  Triangle  in  way,  irregular  the  breaklines,  modelled  an  surface into  changes  how  of  triangulated  or  of  the  For  an  breaklines,  2.4.  Because  data  the  overhead.  surface  streamlines.  of  representation  correspond  many  grid  increases  irregular  regular  see  representation  method  field  region.  sparse.  The more  the  visual  TIN  uniform  regular  technician  where  data  the  a  may  have  choose with  a  the  consistency  and  representation  is  Despite  these  disadvantage. a  many  given  It  point  different  surface  unique;  Therefore,  produces  data.  choice  is this  in  to  a  TIN  there  are  a  surface be  the  the  representations.  triangulation that How  used  strengths,  i s not  set.  resolution  made?  given The most  30  Figure  2.4  Associating surface features with t r i a n g l e [from H e i l and B r y c h (1978)]  edges  31  2.3.1  Trianqulation  A  first  subject exist  step  the  and,  towards  general,  triangulations.  global their  the  few  upon  a  two  an  elongation  triangle's  partitioned resulting The  perimeter such  triangles  is  application  partition  which  resulting  criteria  of  are  its  noted  by  area.  maximum  to  criteria different applied  attaining  examined  two  the  is  l o c a l l y and  in  by  Many  produce  quadrilateral,  given  the  some  here  which  as  to  can  be  ways. Barnhill  ratio The  P /A  (1977)  where  2  P  quadrilateral  elongation  measure  is is  of  the  minimized.  of  a  max-min  maximizes  the  angle  minimum  criterion  selects  i n t e r i o r angle  of  that  the  two  triangles.  Application passed  through  fourth  vertex  partitioned opposite  that  A  may  defined  criterion  and  constraint.  triangles  measure  triangulation  objective  convex  adjacent  unique  to  be  the  local  triangulation  employs  other  with  a  criteria  c r i t e r i o n may  A  into  process  different  domain  effect  One  a  A  optimum.  partitioned  producing  triangulation  in  throughout  Criteria  of three lies  with  i t .  of  the  circle,  case  the  choice  Yet  another  is  criterion  the  within  the  the  diagonal fourth  unless  set  circle, joining  vertex  three  a l l four is  then this  lies  The  of  diagonal  requires  quadrilateral  chosen.  which  of  circle of  the  If  diagonal  irrespective  a  vertices l i e on  a  vertices. the  vertex  to  the  the  If  used  to  circle,  be the  the  is one  circle,  partition is  circle  quadrilateral  outside  same  points  that  the  results construct in  which  regions.  Each  arbitrary.  c r i t e r i o n makes  use  of  proximity  32  quadrilateral interior  points  vertices. the  vertex  vertices  regions  regions  within  will  regions  meet  circle  circle  The  Thiessen  intersect  equivalent adjacent  shows  that  partitions  resulting  from  a  circle.  Such  regions points  network a  triangulation,  meeting  in  connected.  partition  formed  a  line  regions  then the  partition  the  the other  two  Thiessen  l i e o n t h e same  applied each  f o r those  criterion  are  local, points a  regions  their  arbitrary lying  on  Thiessen  defining  i s the dual  themselves.  a Thiessen  a  Delaunay, or  neighbouring have  of  criterion yields  i s  triangulation  i s called  the  to a l l pairs  o r more  a l l  segment  criterion,  region  of four  where  by t h e T h i e s s e n  diagram.  When  angle  triangulation  The Delaunay  of Thiessen  Voronoi  Thiessen  except  a  I f two o f  i s arbitrary.  t h e max-min  groups  other  a l l four  vertices  in a triangulation, i s unique  the  segment,  surrounding  their  whose  of  to  If  region  regions.  line  point.  the  any  connected  f o r the quadrilateral.  triangulation  Dirichlet,  that  to  a  regions  then  proximity  Thiessen in  selection  and  triangles  common  than  are  i n one  a t one p o i n t ,  criterion,  i t  intersect regions  (1977)  a unique  are called  and the p a r t i t i o n  Lawson  to  these  quadrilateral. vertices  within  l i e closer  These  Thiessen  lies  data  of the Such  a  t e s s e l l a t i o n or  33  2.3.2  Surface  The to  triangulation  construct  While  Consistency  a  this  triangulation  is  representation After  consistent initial  criteria  triangulation. forcing  A  triangle  first  not  is illustrated  will  segments.  line,  not  which  l i e along  triangulation Figure  2.6  lies a  shown.  where  an  of  The  triangle  visual  to  greater  2.5.  field  to  are  surface  data. performed, alter lies  when  with  figure, field  is  in  this  is the  contour,  represented  aligned  the  coincide  the  visual  representation edges  is  arising  the surface  a  consistency  In  original  designed  triangles.  locally  inconsistency Figure  contour  thin  guarantee  triangulation  along  far are  triangulation  towards  An in  not  applied  i n the  thus  long  the given  be  step  line  dotted  with  of  does  Delaunay  contour done  i t  may  edges  discussed  devoid  desirable,  an  additional  criteria  by  the  corrected with  in  contour  segments. Further triangle  consistency  edges  discussed.  Even  inconsistent in  Figure  ridgeline, that  triangulation are  aligned  guaranteed  the contour  is  dotted  line  these  where  data  by  forcing  streamlines,  criterion  in  triangulation along  corrected  the  and  representation  terracing  with  Despite  when  the  exhibits  achieved  ridgelines  The  but  be  with  surface  2.7.  can  may the  as  as  of  previously  satisfied,  an  result,  as i l l u s t r a t e d  figure  lies  shown  the  in Figure  is  alignment  describes  dotted 2.8  where  along a  a  surface  line.  The  triangle  edges  ridgeline. measures, are  few  surface and  sparse.  consistency Lawson  cannot  (1977)  gives  be a  Figure  2.6  Triangulation  consistent  with  surface  contour  Figure  Figure  2.7  2.8  Surface  inconsistency  Triangulation  along  consistent  with  ridgeline  ridgeline  36  good  example  created no  by  data.  must  of a  The  data  a  reconstruction in this  2.3.3  scan  in areas  of  the  in a  current data  published Sibson  interest.  The  visual  l i t t l e  field  to allow  visual  contain only  200  or  data  accurate  field  about  algorithm, developed program,  implementation.  account  collecting  of  be  charts  points.  Procedure  computer  structure  r e c o n s t r u c t i o n can  density sufficient  typically  triangulation  implemented  surface  containing regions  technician  target  thesis  in a  distribution  Triangulation  A  the  artifacts  medical  ensure  used  how  A  i t creates of  the  was  by  obtained,  description  is  Fowler  of  contained  a l g o r i t h m may  and  i s used  in  algorithm  and  and  the  in be  (1977)  Appendix  found  A.  A  i n Green  and  (1978).  A  feature exists  produced lists  by  of  the  whereby  program  vertices  triangulation.  If  triangulation  is  to the  can  the  be  be  initial  altered.  joined  vertices  modified  are  Input  as not  Delaunay is  triangle already  locally  to  triangulation specified edges  by  in a  new  connected, satisfy  the  the  new  constraints. Specification triangle  edges  ridgelines  and  contour stream ridge line  is  or  segments  to  straight-forward,  but  high  is less  curvature  High-curvature  stream)  i f they  contour  streamlines  exhibits point.  of  are  on  adjacent  trivial. is a  sufficiently  may  close.  of  aligned  with  determination A  point  candidate  candidates contours  be  the  which  a  ridge point  or  same  at  of  type ( i . e .  be  connected  to  An  example  of  form  a  high-  37  curvature  contour  in  2.8.  Figure In  the  contours visual  current  are  contour  by  work  enable  automatic  contour  points.  map,  but  is required  of  the  visual  2.4  Interpolation  a  a  function  within patch  the  network for  surface  acceptable is  continuous  smooth  be  but  across  continuity  of  of  of  breaklines  the data  be  chosen.  streamlines by  of  criterion  a  from  on  against  files  a are  keyboard. to  high-curvature  and  shown  A  patch  breakline  in Figure  at  1.1  is  arbitrary  to  of  that  a  a  visually  planar  patches  surface  that  patch  using  different  the  interpolant  of  points  try  yields  composed  d e r i v a t i v e s along  constructed,  advantage  that  ensure  an  been  opportunity  one  To  seams,  first-order  heights  surface  smooth.  has  Another  i s the  until  found.  not  shown  candidates  contour  t r i a n g u l a r patches  must  is  into  is  2.9.  functions  image  ridgeline  displayed  development  field  representation  interpolating  and  entered  the  a  r i d g e l i n e s and  interpolating surface  patches  form  high-curvature  satisfying  in Figure  Once  and for  illustrated  Surface  the  extraction  triangulation,  criteria,  to  automatically  inspection  Further  A  connected  implementation,  determined  field  extracted  points  is  preserves  boundaries  may  used. A  computer  triangles  was  reconstruction  program also in  the  for  performing  obtained, current  and  smooth is  i n t e r p o l a t i o n over  used  implementation.  To  for satisfy  surface the  C  1  Figure  2.9  A  triangulation  of  visual  field  data  39  smoothness  criterion,  triangular pieces  of  s u r f a c e element data  are  interpolant. surface  triangle  with  values  The  at  scan  i n more  of at  A  scan  patch  of  line  are  at  the  but  a  parameter  to  raster of  next.  scan  of  discarded until  a  the  a  plane  The  to  each  interpolant  set x  by  or  of y  is  is  for  the  where onto  employed  scan  Such  interpolating  for triangular  raster  sampled,  axis.  surface profiles  parameters the  partial  derivatives  i t s domain,  coherence  The  on  obtained  computation  (or  B.  either  the  this  values  the  the of  of  vertices.  Nielson  operates  is  scan-line  i n the  and  Nine  the  to  exits  xy  avoid  patches  any  the  from  triangular the  patch's  domain.  Summary  The data  parallel samples  the  y  The  (1980).  functional  known,  i n Appendix  projections  of  to  not  projection  2.5  detail  curves  the  and  neighbours.  nine-parameter,  coefficients  vertices,  are  synthesis process  are  type  x  the  least-squares fitting  points along  repetition one  the  surface  lines  to  vertices  i t s immediate  profile  plane.  the  by  image  grid  respect  a  Nielson  to evaluate  the  and  surface  required  by  heights) at  discussed  a  developed  d e r i v e d from  estimated  surface  employs  are  function  vertex  program  They  derivatives  are  the  problem  is  scattered  of  illustrating  considered  one  of  irregularly  over  a  in  the  data  collection  an  approximation,  behaviour  surface plane.  process,  scheme  the  is  an used  of  visual  reconstruction  Despite  the  error  interpolation, in  as  from  field data  inherent opposed  reconstruction.  to  This  40  ensures  that  technician surface. local,  the  are The  consistent  subtle  surface are  into a  To  a  ensure  possible  the  not  the  constrained. and  streamlines; field  of  patches  acceptable Smoothness  visual  with  the  these map.  The  image  of  i s ensured  derivatives  surface  element i s used  by  are  [Nielson each  that to  surface  These  any the  trends. shaped,  triangles  are  Delaunay t r i a n g u l a t i o n . is  as  consistent  triangulation with  such  over  as  so  later  the  be  boundaries. which  visual  then,  a  is  a  visually  synthesized.  of A  and  triangles.  that  continuity  line  ridgelines  from  as  process  contour  irregular  seams  may  (1980)], patch.  so  irregularly  the  the  medical  reconstructed  representation,  surface  patch  the  importance  extracted  defined at  of  aligned  preserving  across  over  data,  the  global  that  are  joined  the  by  plane.  surface  elements  smoothly  are  a  by  locally  breaklines  breaklines  surface are  edges  of  small,  called  field  surface  partial  criterion,  over  over  network  Triangle  contour  patchwork  in  unique  which  smoothed  traced  contours  surface.representation  with  segments  contours  features,  sub-domains  organized  The  with  i n t e r p o l a t i o n i s performed  triangular  is  field  i n t e r p o l a t i o n i s performed  ophthalmologist, The  visual  first-order  nine-parameter  satisfies  this  C  1  41  III.  A  d i s p l a y program  reconstructed appear  from  i s used  visual  i l l u m i n a t e d by  complex  light  synthesized  a  source  by  IMAGE  generate  field  shading  visible  surface  elements.  divided  into  three  parts:  may  light  be  those  display  of  surfaces  the  surfaces  source,  modelled.  elements  only  The  Although  distant  projecting surface and  images  data.  distributions  plane,  onto  more  Images  pixels  in  pixels corresponding process,  p r o j e c t i o n , hidden  then,  can  surface  are a to be  removal,  shading.  3.1  Projection  Transformations  An  image  of  surface  onto  a  handed into  (x,y,z) a  system, axis.  view  is  plane.  such  line  that  by  points a  are  projected Figure  the  coordinate onto 3.1).  a  v  '  v v  by  The  defined  surface,  z v  system,  ^  of  sight  lies to  geometric  half-space  z  common  is first  along the  and  >  right-  transformed  positive  viewer  Sproull  the  coordinate  the  implementation,  transformation, plane  in a  z  y  coordinate  transformations.  i n Newman  current  projecting  viewer-centered  transformed  the  view Two  »  of  outlined In  to  x  are  series  viewpoint.  After  the  synthesized  coordinate  ^  constrained  (see  surface  left-handed  transformations given  a  Cartesian  Surface  system  is  to  single,  screen  and  SYNTHESIS  These  (1979) the  for  a  viewpoint  0. the  perpendicular projections  surface to are  the the  points line  of  are sight  perspective  42  Figure  3.1  Surface  projection  43  and  orthographic  expensive,  projections.  the p e r s p e c t i v e  p r o d u c e s more r e a l i s t i c If,  Although  projection  images than  is  popular  however, viewed o b j e c t s are a l i e n  Forrest  u n l e s s we  (1979)  can  perspective w h e t h e r we  t o human e x p e r i e n c e , field  of  an  are viewing  in  perspective  of  some  projection  the  or  the  given  (U,V)  U = x  coordinates  and  y  V = y  where  t o which the s u r f a c e p o i n t  These  (U,V)  coordinates are  screen  coordinates  then  tell an this  projection  are the view  (x,y,z)  just  a  p r o j e c t i o n i s simply  s c a l e d and  (X,Y), which are  to  For  orthographic The  since  viewing  shape."  i s used i n the c u r r e n t implementation. by  use,  cannot  merely  distorted  then,  surface  o b j e c t o f known s h a p e , we  as w e l l as f o r s i m p l i c i t y ,  it  such  data,  " p e r s p e c t i v e i s of l i t t l e  r e l a t e a p e r s p e c t i v e image of image  orthogonal reason,  says,  because  the o r t h o g r a p h i c p r o j e c t i o n .  as the a b s t r a c t s u r f a c e s r e p r e s e n t i n g v i s u a l as  computationally  is  transformed.  t r a n s l a t e d to  indices  plane  into  yield  a  frame  same (X,Y)  screen  of  points  buffer. Many  s u r f a c e p o i n t s may  location.  The  p r o j e c t onto the  problem i s to determine  appears i n the p i x e l at t h a t l o c a t i o n surface  points  the c l a s s i c deemed  hidden  visible,  computed.  is  which  so t h a t none o f t h e  incorrectly displayed. surface problem.  the  shade  these  After a  hidden  T h i s , of c o u r s e , surface  of the c o r r e s p o n d i n g  point  p i x e l must  is is be  44  3.2  Hidden Foley  Surface  Removal  and  Dam  algorithms patches.  van  patches.  These  objects;  composed  algorithms  however,  single-valued available.  The  to  Such  an  through  column  point  projected  = (XJYT)  the  initial  in value  of  H(X)  viewer,  i t must be next This  H(X),  frame b u f f e r . H(X)  =  Y  2  .  is  surface  polygonal  point The The  P  2  view  screen  is  =  line  of  sight  this and  point 2  point  is  P  2  than  and  horizon point  =  receding plane  i s updated 3  This  which  is  figure,  the  to  This  the  point  component Y ,  f o r scan the  c l o s e s t to i n the  P'  ( X , Y  2  2  =  2  current  new  P'  =  3  the  then 2  in  )  horizon  to t h i s  p r o j e c t s to  X  frame  is  2  is  plane  (x ,y ,z )  the  is  plane.  i s mapped t o a p i x e l  P  are  (x^y^z,).  p r o j e c t s to point  i s higher  next p r o f i l e  level  a  the  i s mapped t o a p i x e l  is visible  current  In  vertical  point  a  current  curve,  viewer  visible  Y  to the  to the  Since  Since  o f as  horizon  3D  by  the  thought  3.2).  The  planar,  surfaces.  Figure  the  of  algorithms  in  (see  current  surface  be  algorithms  representable  used field  surface  to d i s p l a y general  in a p r o f i l e  closest profile  image p l a n e .  level  planar  v a r i a b l e s , simpler  image p l a n e . the  = Y ^ .  The  examined. the  hidden  instead  of p i x e l s i n the  along  the  (i.e.  buffer.  surface  perpendicular  i n t e r s e c t s the  P,'-  a  algorithm  plane  a  surface  is  intended  two  sampled a t d i s c r e t e p o i n t s closest  are  employs what c a n  A vertical  plane  curved,  for displaying visual  algorithm  horizon.  of  display  f u n c t i o n of  implementation  point  several  L a n e e t a l . (1980) g e n e r a l i z e p o l y g o n - b a s e d surfaces  scan  survey  f o r d i s p l a y i n g s u r f a c e s m o d e l l e d by  to handle  passed  (1982)  in  the  level ( X , Y  3  ) ,  45  Figure  3.3  Scanning  a  surface  profile  along  a  grid  line  46  but is  because invisible  until Y  Y  the  and  the  profile  each  the  The  is  a  vertical  the  horizon.  During  from more  than  this  gap,  between surface between  the  H(X.),  horizon  can  be  in a  column  again point  gaps  sweeps  of  the  above  the  horizon, level  horizon,  is  the  may  to  next  Figure  i t s successor  p i x e l s e x i s t s between surface P,  must and  (x,y,z), and  be  P .  P,'  whose  (x,y)  (x ,y ), 2  and  any  as  in  horizon  the  current  back  across  rises  above i t .  column.  3.2.  If,  the  shaded  in  This  be  in  can  scanning  projection P ' 2  P '.  In  2  search  coordinates  a  order  along  iterative  from  changes  p r o j e c t i o n P,',  that  point  to  a  and  P  current of  the  gap  is of  to  f i l l  profile  i s made  l i e on  project  P  are  front  re-sampled An  2  2  the  At  abrupt  2  are  there  For  P ,  passing  there  p i x e l s are  within  p o s i t i o n above  If  then  the  surface  occur  plane  plane.  from  image.  considering P,  pixel  (x,,y,)  The  current  thought  scanning,  points  point  the  the  containing  profile  points  3  with  i = 1,...,P.  surface  the  to  P  processed.  horizon  image,  the  by  point  point  vertical  screen  the  columns,  This  each  whenever  one  unassigned  below  = X^. , w h e r e  vector  elevation,  sample  the  changing  top  understood  X  a l l image  surface,  profile  of  planes  visible  falls  for  of  for  to  projection  been  falls  and  the  from  compared  image  horizontal line  process,  bottom  the  in  scan  is  has  projection  pixels  levels  the  onto  a  horizon,  continues  point  i s performed  column  in a  This  projection If  current  examined.  process  through  the  profile  H(X).  If point  than  ignored.  is painted  updated.  the  of  level  point  pixels  is  farthest  coordinate  horizon  i s lower  3  the  onto  a  for line  pixel  47  located  between  For  Y,  clarity,  surface  of  surface  A  more  The  twice:  and  intermediate  the  point  is  from  The  with  samples used  the  surface  determined.  is  be  grid at  any  plots  grid  constant  one  (see  a  line  Figure by  y. now  3.3).  grid  line  of  original this,  f i l l  for of  a  the  the  again  columns, grid  value  of  grid  an  samples at  calculation,  patch  be  during  existing z  of  must  gradient  shading  bilinear  model  image  four  surface the  in  from  process,  surface  surface  then  gaps  performed  TIN  the  and  grid  whose  this and  corners  neighbours. a  gradient  the  or  rectangular  the  to  The  equation  gradient  current  a  i n t e r p o l a t e d from  four  surrounding  i n the  from  in order  is required  Surface  on  interpolation,  point.  the  3D  arbitrary  through x  image  current  rectangular  i n an  scans  along  scanning  the  a  scanning  the  Because  must  that  obtained  operates  during  In  on  constant  as  (1973).  gap-filling  point  to  in  Wright  is divorced  once  of  generating  surface.  interpolated  coincide  by  of  scanned  column  for  direction.  process  lines  points  pixel  re-sampling In  the  described  operates  Instead  along  Furthermore,  samples.  closest  one  field  been  viewing  surface,  d i s p l a y process  display. surface  the  is described  visual  scanned  heights.  algorithm  representation the  the  has  display process  surface  than  similar  scanning  of  2  algorithm  heights  Projections span  the  across  Y .  along  surface  direction of  the  profiles  implementation, grid  and  visible  visible must  again  sample.  implementation  grid  i s :  be The  sample  must  estimated gradient  also from  be grid  estimator  48  where  p(x,y)  =  [F(x+1,y)  - F(x-1,y)] /  2A  q(x,y)  =  [F(x,y+1)  - F(x,y-1)]  /  2A  gradient  the  [p(x,y)  point  q(x,y)]  (x,y,F(x,y)),  Other  gradient  errors  in  effect  may  warrant The  of  such  further  a  contains  at  most  horizontal  of  resolution current  of  Shading  After  a  depends element. source,  on  This the  orientation  the  spacing.  effect  by  the  in  as  Horn  of  local  (1981).  resulting  screen  not  be  image  The  are  screen.  features  may  image  for  space,  horizon pixels  The  whose  in  one  sample It  s i z e s are  detectable  and  vector  surface  resolution.  function,, however,  element be  in  may  of a  sampling  i s deemed  visible,  over  the low-  available in  finer  shaded.  intensity  intensity type  of  by  allows  must  the  surface  the  smaller  surface.  surface  i t projects  on  resolution.  display  zoom  implementation,  Surface  the  operates  elements  surface  The  3.3  which  the  spacing  image.  the-  visible  sample  suggested  estimators  screen  i s bound  sample  of  grid  reduce  are  algorithm  many  fine  sub-domains  at  experimentation.  therefore,  the  the  which  gradient  as  that  is  F(x,y),  specific  line  argued  order  A  receding-horizon to  be  where  height  conforms  spacing,  the  estimators,  surface  The  and  is  the  of  of  light  depends light,  surface  The  on the  shade  reflected  the  the  pixel  of  the  from  position  of  p o s i t i o n of  the  element,  and  the  the the  onto pixel  surface light  viewer,  the  reflectance  49  properties physical and,  of  the  surface.  properties  in  dependent  many  cases,  only  upon the  z  or,  outward that  the  point  is  c  of  denoted  N = [-p to  -q  the  S = t~P  [-F the  by  a surface  -F  x  and  F^  to  and  -  light  viewed  from  V  For  (x,y)  is  normal to and  x  F  respect  to  q,  the  then  -q  components  of V  source, "above"  the  q  An  relations  than  therefore,  is  placing  light  the  exist  orthographic  much g r e a t e r  the  vector  the  surface be  the  that  (i.e.  ] and y surface  are  y  surface  F x  also  to  so  the  at  If  F^  normal  is  assigned  surface  the  an  partial  viewer  the  from  normal  and is  half-space  e l e v a t i o n <t>,  are:  P  Similar  function  its  x and y .  may  1]  by  [F  > 0 ) . F o r a v i e w p o i n t a l o n g an a z i m u t h 6 and an  the  the  material,  by a  gradient.  z components V = [ P  the  modelled  given  F  by  surface  on  geometry.  point  function F with  depends  is  1 ] , where  y  Positive  n  the  the  element  its  at  directions  5  viewing  by  of  simply  positive z axis)  p,  1],  illuminated z  of  the  be  gross  gradient  (towards  derivatives is  can  equivalently,  = F(x,y)  reflectance  and m i c r o s t r u c t u r e  The o r i e n t a t i o n vector  Surface  the  = -sin0  /  tan</>  =  /  tan^  for view  -cos0  p  and  is  obtained with  dimensions of  parallel source  S becomes  v  to at  V-R a great  parallel  to  q  the (see  .  surface.  from  distance  The v e c t o r  F i g u r e 3.4).  distance S-R.  a viewing  the  The v e c t o r s  Also,  V, by  surface,  V - R and S-R  Figure  Figure  3.5  3.4  Geometry  Viewing  geometry  for orthographic  viewing  51  can  be  replaced simply  element  "sees"  direction. element is  Only  The  respect  the is  light  in Figure  source  viewer.  The  in  t h e same  plane  g  i s needed The  vectors  simple where  bright of  the  The r e s u l t i n g  3.5, w h e r e V a n d  incident with  and emergent  the surface  S, V ,  and N  geometry,  are  or  an  a t any p o i n t  function  on a  normal,  surface  R(p,q) dependent  both  incident  ray  i s  any v i e w i n g  direction  angle  angles  to  angle  be  the  i , e, a n d g .  The  modelled  by  a  i , e, a n d g,  surface appears  and has a  e  by  at the point.  matte,  from  reflected  specified  1] i s t h e s u r f a c e n o r m a l or  ray  them.  on t h e a n g l e s  diffuse,  system.  do n o t g e n e r a l l y l i e  only  perfectly  defined  and so a phase  can  to  geometry  The emergent  rays  by t h e t h r e e  element  coordinate  and a  then,  i n t h e same  from  S  between  surface  simplified  of the (x,y,z)  reflectance  equally function  form:  R(p,q)  where c o s ( i )  and  varies  the s u r f a c e normal  viewing  from  N  Each  source  t o d e s c r i b e t h e s e p a r a t i o n between  [-p - q A  light  and the s u r f a c e normal.  gross  reflectance  the  i i s the angle  between  the  respectively.  vector  the o r i g i n  angle  the angle  or  the surface.  to  incident  viewer  the normal  across  illustrated  with  the  b y V a n d S,  where  =  I  cos(i)  (3.1)  =  I  i s  the  source  radiance  (Note  that  the  viewer  52  direction A the  does  surface  enter  that  off-specular  direction  M  Figure  3.5).  equal  to  are  not  exhibits  angle  of  a  The  angle  the  into  s  the  reflectance  highlights  between  the  specularly  can  be  viewer  reflected  i ' between incidence  M  calculation). modelled  direction V light  using and  ray  and  the  surface  normal  i , and  the  vectors  S,  angle  of  Phong  (1975) d e v e l o p e d  the  N,  (see N  is  and  M  coplanar. Bui  Tuong  function  where  n  to  is a  simulate  surface  R(p,q)  I  =  positive  cos(s)  =  cos(e)  =  [X  /1  +  P  2  1 cos(g)  where  shown  in  this  (1981)].  =  +  q  +  p  + q  2  and  p  i s used  similar i s given  I  [X  +  (1-X)cos"(s)]  cos(g)  + q +  q P , 2  + q —  /T+  2  I are  and  -  /1  2  p  thesis.  derived A  consistent,  p  function  empirically  R(p,q)  +  cos(i)  reflectance  p  = /l  and  +  not  function,  2  q p  2  defined to  q  +  +  q  2  as  for  generate  Although does  reflectance  0^X<1,  integer,  1  following  highlights:  cos(i)  2cos(i)cos(e)  the  much  adhere but  to one  Equation  most used,  of  3.1. the  this  +  is  physical  law  which  physically  (1-X)cos(i)cos (s/2)] n  images  function  is  by:  cos(i)  This  /  cos(g/2)  [Horn  53  Specular surface  reflectance  gradient  surface  regions  defects  on  interest  a  than  of  to  matte  high  visual the  is  more  reflectance,  curvature.  field  sensitive and  Such  surface.  changes  tends  regions  Because  ophthalmologist,  to  to  emphasize  may  these  indicate  regions  i t is desirable  in  that  are  of  they  be  highlighted. Shadows c a s t not  rendered  can  be  in  of  shadows  the  is  and  complicated  by  is  Each  portion  the  on  by  from  the  of  the  images,  original  buffer.  such  that  are  against  using  for  using  the  are  shadows  problem  the cast  perspective  "Shadows c a s t interpret  visual quality  of  shaded  i s obtained  This  i t  by  The  contour  map with  in  corresponding  location  and  of  height  the  the  the in  by  one  ...  and  overlays."  contour  colour  the  associated  upon upon  drawn  the  display  map.  matches that  If  a  that height  by  of is  the a  the  process  point  they  the  scaled  generated  for  1,  into  and  grid  i t checks  with  first  translated  Whenever  point  Chapter  contours  are  dense  contour  in  superposed  the  contours  grid, the  1.12  contours  i s then  finds  sample  and  overlaying  grid.  registers  visible  1.11  field  process.  then  (1981):  Cast  to  scanning-interpolation  point,  surface  another  hard  Figures  visual  elevation  frame  the  thesis.  against  Horn  over  Superposition  surface  a  this  argument  one  another  surface  hidden  The  the  stated  the  in  the  source. to  of  images  solving  light  detract  3. 4 C o n t o u r  surface  of  shape  apparently  one  similar  projection,  shaded  any  calculated  viewpoint  showing  by  at  the  coincide, the  grid  displayed  54  instead  of a grey  This  approach  registration confounded contour  suffers  cannot  map.  To  be  neighbourhoods  of  The  match  from  the  that  matching  straight-line  the contour  i t i s deleted  difficulty  the likelihood  for  in  the  ensured.  improve  searches  searches  from  by t h e a l i a s i n g  process  found,  level.  of c o r r e c t  Once  contour  i n overlapping neighbourhoods  the  matches, the  within  so  i s  in  small  a particular  map  exact  process  segments  candidates map.  an  that  do n o t a r r i v e  match i s different  a t t h e same  candidate. Another contours A  approach  directly  from  noise-free operator  curves  of  addition by  uniform  surface  shaded  surface  approach  3.5  Real-Time  image  surface then  a viewing a  gradient.  grey  model  field  the  original  i n the current  specification  contour  extract,  contours  contours  To a v o i d  of the Light  would  correctly  additional  such  may  the  confuse  data  with  the  confusion,  this  implementation.  Source  and a  lighting  distribution,  i s s y n t h e s i z e d by m a p p i n g  into  pixels.  I f the viewing  a r e dependent an  image  omitted from  surface  Generating  in  extracted  3D  levels  extract  of the surface.  contours,  Although  i s to  connected  an o p e r a t o r  representation.  elements  pixel  visual  i s comparing  Movement  of  Such  these  i s not followed  Given an  width.  model, who  elevation  problem  i s r e q u i r e d t o produce  technician.  ophthalmologist  the contouring  the grid  to the original  the medical  the  to  geometry  on l i g h t i n g  o f t h e same  a n d on  surface  visible  i s fixed, surface  illuminated  55  from  a different  visible If,  surface  instead  stored  of  in  gradients buffer the  direction  gradients, grey  a  frame  i n t o grey  display  form  buffer  then  levels  would  devices  require  these  buffer,  a  lookup  raster-scanned  and  electron  beam  B,  contents  the  beam  contents  used  striking  the map,  image.  Frame  feature in the  3.6).  the  frame  modulate  the  intensity  frame  With  the switch  buffer  and these  new  a r e used values  frame  With the  of  the screen.  of  i n t h e memory  to  surface  between  (see Figure  are  mapping  a mapping  placed  driver  gradients  the  provide  of  has not changed.  table  restore  typically  the  re-computation  unchanging  o f a u s e r - a c c e s s i b l e memory  a t p o s i t i o n A,  values  not  as the viewpoint  levels,  and the e l e c t r o n  switch  does  buffer  are  of  at  the  position  to address  modulate  new  t h e beam  intensity. If table  surface mapping  into  the  identical  gradients  a l l possible  memory to  are stored  map,  i n t h e frame  gradients  then  the  t h e one o b t a i n e d  with  i n t o grey  recovered just  a  buffer,  levels image  frame  and  a  i s loaded should  buffer  of  be grey  levels.  Only  A  restriction  a  small  finite  these  one  grey  N/2  bits  surface component  by t h i s  s e t of gradient  space.  and  posed  must  level.  into  I f each  per gradient  each  exhibiting only  gradient  the space  pixel  component  whose v a r i a t i o n  is finite  pixel  d i r e c t i o n s c a n be e n c o d e d  Furthermore, be p a c k e d  technique  ordinarily  occupies i s a  N bits,  reasonable  i s not d i r e c t i o n a l l y 2 '  h a s two  discrete  in  this  components, occupied  then  biased.  by  assigning  partition  values,  size.  for  With  a  each  quantization  56  Switch  r —  Frame  1  A  Buffer To i  B  Memory Map  Figure  3.6  Frame b u f f e r w i t h a t t a c h e d  memory  map  Intensity Modulator  57  error  the  is  introduced  Since  each  full  gradient  recover  an  into  the  gradient  image  [p  of  component q]  the  restored  is  each The  via a  surface,  memory map  entry  gradients  known.  The  source,  whose  such  as  a  lighting  and  2  ball  joy  N  2 '  values,  directions. be  To  assigned  grey  N 2  a  entries.  reflectance  viewer  is derived  or  to  must  direction  from  stick.  The  a  The  value  function  c o n f i g u r a t i o n c o n s i s t s of  direction  track  .  using the  to  gradients  containing  i s determined  surface  i s quantized  quantized  . levels  image.  are a  memory  R(p,q).  fixed  single,  graphical map  of  distant  input is  and  device  filled  by  N applying  the  reflectance  orientations As  the  source the  new  than  with  the  the  Moving because  the  in  frame  a  as  A shading,  frame  of  be  device  the  the  (with  only  be  calculation  This  before,  of  found  can  surface  not  is typically  256  map  entries  appear  to  with  reflect so  for vary  surface  gradients  represent  the  the  position  a  shaped  much N=8),  in  real  memory  map  information are  serious  shadows  encoded drawback  cast  upon  viewer.  techniques,  and  to  light  source.  irregularly confound  the  map  simulated  only  in Sloan  2  updated  requires  does  memory  the  is  shades  light  but  of  i s manipulated,  Since  surface  cannot  elements,  discussion may  buffer  movement  surface  each  r e f l e c t a n c e map  and  stated  to  quantization.  input  updated,  buffer.  complicated  the  direction.  shadow  surface  because,  the  shadows  about the  and  lighting  is quickly  time  from  graphical  i s moved  smaller it  resulting  function  Brown  including  (1979).  real-time  58  3.5.1  Optimal  Finite quantized  pixel  they  between  quantized  signal.  cumulative  difference  between  A output  level  output  referred  where  distortion  An  aims  to  reduce  the gradient)  function  should  take  frequently  maps  y. ,  measure function  shows D,  frame b u f f e r .  (i.e.  of  into  signal  levels  which  results  occurring  L  where  signal level  level  x  in  signal  the and  account  so  as  when  value  to the  and i t s  must  that  intervals to L the  i = 1,...,L.  interval  The  x^.'s  discrete (x^. ,x'. ^) +  are  often  levels.  for evaluating  the  i s t h e mean-square  h(x) i s the p r o b a b i l i t y density (1960)  be  signal  distortion,  to as decision  quantization  the  i s one t h a t  function  Any s i g n a l  as  A common  Max  Such a  in  components  i s too large.  quantizer levels.  a  the gradient  stored  function  probability density  quantized  are  that  the o r i g i n a l  minimize  is  requires  quantization  discrepancy  the  size  before  effective  the  Quantization  an o p t i m a l  satisfy  two  error  produced  by  d i s t o r t i o n D given  function  quantizer,  conditions:  of the signal  a  by:  x.  which minimizes the  59  =  2x.  -  i  y.  ,  i — 2,...,L  i~ 1  (3.2)  (3.3)  The  first  between y^.  is  the  condition  adjacent the  x  curves Max  at  a  y,  new  from  was  spectra are  the  the  for  then  minimal.  is  reflectance  map.  may  error a  by  in the  by  Because so  r  If  y^  of  p  the  and  for  If  i t i s nonlinear,  c a r e f u l l y minimized  quantization  that  yields  y^  and  X  is  L+J> then  iteratively  frame  buffer,  visible are  the  surface  each  quantized  of  gradients  is  an  image  of  gradients  with  a  this  during an  be  strip.  signal the  is  can  centroid,  r e s u l t i n g "image"  transforming  that  guess  y^. ' s  is  a q  a  x^  the  a  and  recovered  the  resulting  last  over p  obtaining  and  y,  in  q  by  Once  between  the  bounded  y^. , g i v e n  the  i s not  guess  for  x^.'s  h(x)  that  h(x).  known.  3.3.  midway  states  area  levels  e  gradients  quantized.  error  a  If  the =  is  second  procedure  under  centroid  exhibited be  and  L+J  The  surface  obtained  Therefore,  the  However,  the  X  correctly.  intensities  skew  y  and  strip  y^  the  =  quantization  3.2  i s made.  to  of  0,  level  the  a l l successive  1 f  area  and  centroid  and  y,  values  signals  optimally, is  of  encoding of  the x,  y  decision  iterative  Equations  until  For  and  for  guessed  guess  adjusted  x^.  the  an  each  levels,  y  describes  value  centroid  of  =  extremes  a  generated  then  x. , x  levels  signal  made  the  =  that  quantization  component  also  decision the  x  states  transformation quantization.  optimal  gradient  60  encoding  may  The  yield  a  sub-optimal  image  reflectance  maps R ( p , q )  considered  functions [~P in  ~Qi  V  of ],  1 v  image  surface and  gradient  lighting  to  dR,  seems  reasonable  values  where  those  values  "critical" light  R  source The  that  plot is  of shown  points, changes, origin, also of  the  in Figure  and  the  +  R  [-p^  to  the  q]  are,  [~P^  ~<i  in  viewer -q^  of  [p  q]  R(p,q)  quadrant.  major  axis  the  is  about  those  q,  about  However,  these  most  dense.  As  given the  the  of  the  ellipses  the  line  R(p,q)  light  =  0  direction defeats  the  original  real  time  using  the  case  that  the  only  a  critical  by  the  about  or  region  degree  of  [p  the  origin  the  problem  of  3.1  point,  the  reflectance values  contour  direction  moves,  intent  the  Equation  from  and  new  in  A  rotates  source  rotates,  which  lighting  the  the  at  critical  quantization  be  error  and  points  the  may  error  functions  of  as  p  those  figure,  sensitivity To  dR.  function  third  Therefore,  error  are  the  of  in  solve  It  The  finely  errors  are  reflectance  distance  q  varies.  changes  1],  1]«  s  of  and  sensitivity  shading  are  direction  quantization  general,  In  reflectance  lighting  thesis  dq  1  p  3.7.  in  changes.  high  [p  contours  the  dp  P  quantize  values  diffuse  occur  this  q],  gradient  sensitive  direction  iso-brightness  R  maximize  of  critical  =  to  i s most  values  in  where  dR  It  [p  direction  i n t e n s i t y r e s u l t i n g from  proportional  reconstruction.  for  of each  providing table. q]  do  not  61  Figure  3.7  R e f l e c t a n c e map f r o m 9=61°,0=53°  for diffuse surface illuminated [ f r o m Woodham ( 1 9 7 8 ) ]  62  change  appreciably  confined In  to within  this  on  is  distortion, gradient should  3.6  source  for a  given  about  the viewer  dependence  direction  i s not  gradient  viewpoint.  levels,  a  or with  direction.  of  reflectance  investigated.  distortion,  Given  a  relatively  movement  not  sufficient  The image  number  distortion-free  of  image  recovered.  Summary  Reconstructed  visual  orthographically  onto  projection  i s used  expensive  perspective surfaces.  Bui  Phong  Tuong  surface  horizon The image  which  profile The  hidden  the  visible  in  a  may  be  of  and  systematic t o the back  at that  manner of  them  algorithm  i n the " f o c a l " horizon  the  of  model  interest surface  image  by  more  unfamiliar,  highlight rapidly  the  algorithm  intersects The  from  reflectance  the projected from  orthographic  rendering  surface to  An  gained  when  projected  varying to  that a  of  the should  receding-  algorithm.  surface  curves.  profile  front  surface  is  used  are eliminated  hidden  plane  is  are  plane.  little  specular  Portions  surfaces  image  projection  The  features  visible  an  (1975)  ophthalmologist. be  field  because  irregular  not  movement,  the  minimizing  quantization  be  source  however,  light  on  light  a c e r t a i n angle  thesis,  sensitivity focus  with  the  such  takes  with  the  "focuses"  plane focal  planes  on  closest depth.  that  the  surface.  As  surface one  plane  The  focus  plane focal  to the  to  create  at a  to the viewer  focal the  parallel  is  moves plane  time.  becomes shifted from  the  recedes  63  from  the  profiles, Surface  expands, In  an  and  segments  As t h e  image  order image  superposed infer  expansion  do n o t c o n t r i b u t e  displayed.  reduce  the v i s i b l e  and t h i s profile  horizon not  viewer,  surface observe  regions  a  how  elements  and packing  gradients  i s  i s recovered  real memory  time map  by  apparent  into  values  a memory  a  mapping. updating  according  source  obtained. the  t o changes  contents in light  and a r e thus horizon  shape  and t o  contours Another  about  are  way t o  the  surface  of d i f f e r e n t  surface  of v i s i b l e  buffer,  Image i n t e n s i t i e s the  visible  the  surface  surface.  frame  plane.  unfolds.  brightness  by t r a n s f o r m i n g  and  colour-coded  light  surface  image  horizon,  convey  field  new  the currently  the gradients a  with  the  recedes  surface  visual  them  surface  with  plane  By q u a n t i z i n g  coded  below  the o r i g i n a l ,  i s t o move the  in  t o the expanding  focal  shaded,  shape  changes.  falling  more e f f e c t i v e l y  ambiguity,  expands  i s recorded  of the v i s i b l e  to  on  horizon  An  an  image  frame  surface  "image" of  buffer  c a n b e made  intensity contents to vary  of the r e l a t i v e l y source  of  position.  in  small  64  IV.  4.1  Visual  Real charts as  data  record  i t  each  Field  chart  was  the  were  obtained  visual  over  The order,  a  six are  shaded Each  as  inverted  that  should  against  a  plateau  are  is  from  4.1c  years. A  4.1a  to  from  show  the  in  surface and  then  in  chronological  The  corresponding  and  the  same  data  smooth  Figures  0=160°,0=50° 4.6c  eye  image.  4.6a.  in  The  left  The  domain,  assembled  given  to  illuminated  is  to  attached  located  images the  on  the  look  like  understand  A  rising  how  4.1b  to  4.6b.  illuminated  surfaces  from  inverted,  d i r e c t i o n s as  normal to  the x  retina, axis  and  blind  an  upper  to  what  visual  the  a  low  offset  to  limit  on  normal  field  spot,  appear  because  a  peak  should  spots  interpret.  defects  visual  physiological blind  "truncated"  imposes  difficult  and  field.  Pathological are  are  viewer  gradually The  targets  Figures  viewed  four  grey-level  charts.  patients's  triangulated  charts,  renditions  normal  fixation.  Pits  glaucoma  triangulated.  each  in  and  and  continuous-tone,  shaded  important  point.  about  field  non-  surfaces.  The  surface  of  over  Figures  viewed  period  shown  surface  visual  a  field  surface  six  of  visual  6=120°,0=70°. but  a  from  field  digitized  interpolated  displayed  a  Surfaces  were  varies  RESULTS  where as  either  a side  are  also  the  perimeter's  probe  visual are  the the  field  appear center  optic  the  rendered  intensity.  as  finite  as of  nerve  bottomless of  is  manifest  should  at  It  pit  fixation pits. set  Because  of of  Figure  4.1a  Visual  field  chart  1  (March  1978)  Figure  Figure  4.1b  4.1c  Shaded  Inverted  surface  surface  from  data  from  data  in  chart  in  chart  1  1  gure  4.2a  Visual  field  chart  2  (September  1978)  Figure  4.2c  Inverted  surface  from  data  in  chart  2  Figure  4.3a  Visual  field  chart  3  (April  1979)  Figure  4.3c  Inverted  surface  from  data  in chart  3  Figure  4.4a  Visual  field  chart  4  (October  1979)  72  Figure  4.4c  Inverted  surface  from  data  in chart  4  Figure  4.5a  Visual  field  chart  5  (June  1981)  Figure  4.5c  Inverted  surface  from  data  in chart  5  75  Figure  4.6a  Visual  field  chart  6  (December  1981)  Figure  Figure  4.6b  4.6c  Shaded  Inverted  surface  surface  from  data  from  data  in  chart  in  chart  6  6  77  its  known  presence,  in  practice,  the  pits  and often  spots  above  the  appear plain  physiological t o form The to  conveyed defect  of normal  4.2, by  a.  inherent  Fluctuations  occur  fields.  The  periphery pronounced progress cannot two  in of  be  consecutive  diagnosis long to  is  enough  reveal  the  are  fields  to defeat  as  than  glaucomatous  What  surface. Figure  4.1  improvement  i s  may  caused  be  occur  because  as  diseased  and  in in  they  normal  visual  the  field  are  more  fields.  field  spanning  the e f f e c t s of long-term  of the (1984)].  required  of examinations  by  [Drance  condition,  i s  blind  peripheral  the v i s u a l  i n the time-varying  t o 4.6b t h e  from  extensive  in  standing  a n d by a  well  center,  than  pathological  examination  more  by c o m p a r i n g  sequence  any t r e n d  changes  field  4.1b  on e a c h  defect  probed  surfaces,  plateaus,  apparent  fields  examinations. a  inverted  t o improve  This  in perimetric  patient's  evaluated  a  f l u c t u a t i o n s which  diseased  a  appears  Such  i n normal  near  with  fully  i s shallower  In F i g u r e s  central  fluctuations  than  On  p i tvisible  worsen..  in extent.  subjectivity  spots.  vision.  condition  or long-term  i s rarely  as a p i t that  i s merged  shallower  spot  or closed-rimmed  central  and then  diminished  short-term  spot  the large  patient's  Figure  blind  a s mesas,  blind  blind  appears  of p a t h o l o g i c a l  blind  spot  the normal  therefore, charts  for a  from  a  valid  time  period  fluctuations field.  The  and  78  4.2 G r a d i e n t  Quantization  An  experiment  surface  gradient  relatively used r  2  in  = x  2  the  + y .  The t e s t  2  from  4.7,  Figure  control 32  4.8,  image A l l  o f 8,  four  To  illustrate  4.9 w a s  4.10, w h e r e  encoding  for  error  reflectance  varies  produce  The  a  surface  6=160°,0=50°  from  of  ( s i n r ) / r , where  with  and  gradient  respectively. when  The  image  compared with  p a n d q were  synthesized  that  by a  r e s u l t i n g image  quantization  bits  images o f t h e s u r f a c e i n  distortion  transformed  i s shown  image  z =  generated  the s e n s i t i v i t y  in  control  i s viewed The  to  surface.  the function  were  many  each  with  a  the  allotted diffuse  function.  the gradient  0=180°,0=50°  of the  10, a n d 12 b i t s , little  how  sufficient  were  images  gradient,  The  4.9  i n Figure  reflectance  function.  was  surface  and  surface  Figure  are  0=180°,0=70°.  4.9 e x h i b i t s  bits.  to ascertain  information  experiment  representations in  was p e r f o r m e d  d i s t o r t i o n - f r e e image  illuminated Figures  Experiment  i n Figure this  i s most  of reflectance produced  specular  the restored  surface  of the surface 4.11  (Figure  4.12  apparent  in highlighted small  reflectance  not  contains  It  with  image  illuminated  case).  quickly  i s  to errors in  surprising regions  changes  in  from the that where  surface  gradient. The  quantization  algorithm  minimizes  mean-square g r a d i e n t  (1960).  The  probability  distortion  density  linearly  interpolated  from  and  The f u n c t i o n  h(p) derived  q.  used  in  experiment  by t h e method  functions  the corresponding in this  this  of  Max  h ( p ) a n d h ( q ) were histograms  manner  for  p  i s illustrated  Figure  Figure  4.7  4.8  Diffuse  Diffuse  surface  surface  represented  represented  with  with  8-bit  gradients  10-bit  gradients  Figure  4.9  Figure  4.10  Diffuse  Diffuse  surface  surface  represented  represented  with  with  12-bit  64-bit  gradients  gradients  81  Figure  4.11  Specular  surface  represented  with  12-bit  gradients  Figure  4.12  Specular  surface  represented  with  64-bit  gradients  82  Figure  4.13  Graph of h(p) histogram data for  linearly p  interpolated  from  83  in  Figure  4.13.  applications  of  This a  function  three-point  density  function  used  function  i s reproduced  function  h(q)  by  averaging the  in Figure  i s reproduced  was  then  smoothed  operator  quantization 4.14.  in Figure  The 4.15.  to  by  multiple  produce  algorithm.  probability  the This  density  h(p)  o  oo o  o  o  in ©  o"  ©  O  o  —r -4  -10  Figure  4.14  Smoothed  probability density  function  h(p)  85  Figure  4.15  Smoothed  probability density  function  h(q)  86  V.  5.1  CONCLUSIONS  commonly  visual  field  recorded  surface better  human  viewer,  surface  who  display  shaded  a contour conveys  sees  of  from,  superposition surface,  abstract  perimetry  are  the data  as a  Displaying of  the  manifest  and a l s o  image  ambiguity.  data  to  a  by v a r i a t i o n s  in a  for  correct  interpretation.  which,  "plateau"  a  visual  the  p i t into  the  "plain"  a  in  appears  shaded,  as a  large  views,  "mesa",  or closed-rimmed  system  enables  feature. shaded  visual  i s  images  of  Superposing  field  surface  knowledge  still  obscured  the surface plateau,  A peripheral  of  necessary  " p i t " i n the surface,  The bottom  rim i s not closed,  and  the corresponding  i s largely  Inverting  vision.  a s t h e t o p o f t h e mesa. p i t whose  onto  appear  vision.  normal  The  generates from,  to interpret.  normal  of  viewed  depictions,  field  i n a l l but overhead of  zoom  thesis  Nevertheless, a priori  defects  defect  data  difficult  how  A central  contour  onto  normal  surfaces  provides a  remain  this  directions.  to contour  contours  in  field  original  surfaces  reduces  described  visual  superior  original  a  map.  kinetic  the behaviour  arbitrary  of  Although  as  by  the behaviour  system  images  illuminated  visible  gathered  shape.  The  shaded  data  as  shaded  pit  PROPOSALS  Conclusions  Human  the  AND  a  by t h e  transforms  rising  above  o f t h e p i t becomes defect,  i s effectively  appearing  conveyed  by a  87  non-inverted open  A view  end of the r i m r e v e a l s  judicious create  choice  of  a p i t with  reveals  such  surface.  details  a  directed  the walls  zoom  an open  into  and f l o o r  window  rim.  A  over  that  the r i m of the p i t i s opened.  window  i s displayed  drawing  more  detail  shading  in Figure  as a  in Figure than  1.4.  the  The  curvature  the  of the p i t .  The  a central defect surface  can again  The  region  in Figure  shaded  surface  drawing.  Detail  with i s  p a r t i t i o n s the central p i t  surface  The  line  reflectance  and h i g h l i g h t s  1.13  shaded  highlights obtained  model.  from  of the p i t i n t e r i o r .  zoom  line  p i t  non-inverted  The  window  the  a  specular  sensitive  features  that  within 1.14  shows  i s conveyed  by  reflectance  changes  are of  and as a  clearly  surface  to  this  in  interest  surface to  the  ophthalmologist. Wirtschafter the  presentation  visual in  field  et a l . of  visual  into  regions  the r e t i n a .  Field  data  single  measure.  becomes  critical,  be  i n the averaging  lost  with  a  the  2D  array  approach  field. patches  It  that  based  Although  of  a  numbers in  logical  onto  are  subtle  t o model  are aligned  visual  the  a  then may  ophthalmologist The  strength but  field  follow  in  the  of the  visual  surface  the boundaries  boundaries  to  defects  partition  the visual with  layers  reduced  reduction, to  fiber  to the  technique  to evaluate. data  partition  nerve  averaging  leaving  used  the p a r t i t i o n  They  region  an  process,  criterion  i n t e r e s t i n g approach  data.  representing  not  boundaries  an  project  within  of "magic"  seems  whose  partition.  as data  take  field  Selection  lies,  physiologically  (1982)  of  the  by  this shape  88  of to  retinal use  cast  as  the A  its  nerve  boundaries  system  for  provision  for  contour charts  of  of  vary  A  radiance  gradient  Hart  surfaces,  source and  superior  of  for  time  and  zoom, to  with  mobility, creates  traditional  Hartz,  of  a  conveying  with  the  surface  gradient  gradients  word  gradient  surface.  from  components  highly  gradient  when a  shaded  light  data  and  is  of  into  surface  movement formed  vectors  i n t e n s i t y image  recovered  gradient  close  of  to  the  the image  of  by  instead surface  shape a  is  light  encoding of  is  surface recovered  intensities  with  a  map.  be  per  real  as  surface  to  a l .  "image"  An  bundles  a r b i t r a r i n e s s seems  superposition,  drawings  in  pixels  values.  Images  to  shade  which  doubt.  a t t r a c t i v e technique  transforming  memory  et  This  images  field  line  s e l e c t i o n of  mobility,  contour  Wirtschafter  buffer  where  viewer  the  surface  source.  by  to  the  into  synthesizing  human v i s u a l  maps,  Another  frame  promise  inversion,  renditions  bundles,  is arbitrary.  technique's  surface  to  fiber  lengths were  specular  was  of  surface  reflectance  function  error.  simulation so  as  to  images.  was  minimize  to  was  was  was  still  Greater used  the  by  gradients, were  that  images  longer, 12  bits  a  matte  of  discernable  the  gradient  upon  resulted  corresponding quantization  quantizing  mean-square  found  simulate  distortion  because  s e n s i t i v e to performed  8-bit  found  reconstructing  however,  i s more  It  of  optimally,  experiment  for  model  buffer  quantized  conducted.  adequate  the  frame  An  distortion,  inspection  gradients  were  distorted.  Slight  The  a  distortion  surface in  the  89  resulting A  gradient  better  minimize about error.  Quantizing  gradient  purpose  deviation diminish to  the locus  re-quantize  quantization acceptable tighter less  images  light  source  buffer  5.2  of  in  Proposals  light  is  because  made o n  i t would  a  need  gradient yield  directions.  The  movement,  the  images. even  by  frame  restricting  to  reconstruct  buffer  of  shading,  8-bit a  frame  i s required.  Work  f o r conveying  surface  but not implemented  are: stereo  in turn,  to  source  real-time  per pixel  the  the  have  n o t be p o s s i b l e from  new  angular  would,  i t would  that  effective  bits  some  The  light  a  defeats  movement.  lighting  of the author  f o r Further  techniques  These  source  critical source.  and e l i m i n a t e  i s i n the recovered  perform  the Introduction,  system.  gradients  the  It i s possible  within This  finely  to create  buffer  to  reflectance  light  map.  to  direction.  each  movement,  To  a memory  f o r a l l permitted  there  the movement  direction  be s u b - o p t i m a l  12, o r m o r e ,  Other  with  the  i s that  i n t h e frame  d i s t o r t i o n - f r e e images  gradients.  of  i s  by q u a n t i z i n g  approach  source  of c r i t i c a l  i s the opinion  relatively  shading  constraint  distortion It  light  viewing  after  would  the  this  problem  maximize  position  lighting  the  quantization  that  representation  the of  the  each  of r e a l - t i m e  restrict  surface.  the  with  with  after  to  gradients  problem  vary  of the  i n t h e i n t e n s i t y image  critical  The  gradients  to  approach  distortion  those  surface  "image"  pair  generation,  shape  are  i n the current colour-coded  outlined display surface  90  curvature, The  and r e a l - t i m e  current  d i s p l a y program  perform  perspective  generate  two  of  the l e f t  images  produced  procedure  by a that  procedure  that  requires  special  useful  tool  sequence  photography.  into  the  to  viewpoint right.  curvature  could  also  program.  The  at  a point  would,  the  point.  The  surface  point  would  be  colour.  viewpoint  where  a  about  could  lapse  generate  f o r observing of  Frames  interpolated  from  animation how  at a  but the e f f e c t  to  the  to  task  of the  at  hardware,  photography,  simple  from  gradient  curvature  modified  the  be  would  a  surface  demonstrated be t h e t i m e i t  next  frame  in  an  sequence.  a time  show  one  of  the display process  An  a  movement  takes  as  i s then  to the display  surface  curvature  time-lapse  A  be  surface  the radiance  with  animation  easily  the viewpoint  t o convey  surface  surface  It  extension  computes  Real-time  could  viewpoint.  surface:  from  computes  determine  t o map  coded  simple  instead,  changed  o f t h e same  images  of the  projection.  eye, the other  Colour be  movement  charts  surface  two  the surfaces  shape  could  between  sequence  field  also  from  changes  employ  consecutive  corresponding  created varies  visual  to  with  time  time-lapse  charts  could  those  charts.  the in-between  smoothly  recorded  frames  from  be  would  chart  to  chart. In  addition  existing A user  system  part  to  could  the be  above  surface  improvements  to the  made.  of the t r i a n g u l a t i o n  extract  features,  breaklines  procedure from  requires  points  that  a t which  the  visual  91  field  contours  files an  by  keyboard.  automatic  contour  map  surface  is over  a  used  surface. grids, module.  For  the  A  increasing  again  is  would  could  performed  Column a  gaps  succeeding  that  the  image surface applied  column.  be  An  display  this  into  burden, from  a  independently  advantage surfaces  to  the  that  this  are  not  The  experiment  i s such  by  current  estimator  other  of  in  errors  of  of  network.  representable  made  current  twice:  once  modular during  synthesis. profile  than  surface  test a  elevation  hidden  surface the  one  elevation,  using  the  i n t e r p o l a t i o n , and  then  pixel  the  TIN  gaps  a  surface  profile  adaptive planar  patches,  next  unless  in  could  However, searching  addition, columns  representation.  i f the  could  surface  image  be  position adjusted  unassigned  sampling  patches.  In  surface  altogether  the  hidden  scans.  avoided  onto  is  interpolation  the  be  on  by  that  in  these  filling  exactly  Such  of  design  Performing  scanning  one  for  projects  curved  executed  quantization  effect  the  could  represented to  of  sample  sample  from  breaklines  irregular  gradient  eliminate  re-sampling  user  these  image q u a l i t y .  with  surface be  a  image  concurrently procedure  could  scanned  during  to  surfaces  the  disadvantage  surface  enter  surface  interpolation.  displaying  reduce  the  currently  gradient  example,  might  thereby  is  ability  improvements  free  identify  triangulated  in  For  to  and  developed.  and  the  curvature,  order to  synthesis  modularity  surface  In  be  sampling  modelled  high  procedure should  Image  used  exhibit  pixel be  such  in  applied  i t could for  of  an to  not  a be  candidate  92  samples to  within  make  them  of  another  estimate  the  The  directly  specular  is  physical  does  high  law  The  contour  be  longer  shade. of  The the  processes  be  from  problem  i t  a  need  to  nearest  grid  gradient  could  triangular  patch  the  specular  user  interactively  in  law.  the  current  The  specular  surface  model  regions  consistent  with  adopted.  is  exhibits a  from  used  highlights  A  current  contours  model  physical  superposition  closure.  to  no  equation  satisfy  because  The  contour  partitioned  synthesis  point  reflectance  not  improvement.  white  a  pixel  the  curvature.  should  This  image  would  at  compute  surface  desirable  exhibiting  There  from  are  point.  implementation model  to  patches  z.  gradient  in order  the  or  i n t e r p o l a t i o n and  surface  determined  containing  the  in  advantage.  neighbours be  i s performed,  single-valued  Merging has  patches  program  program  illustrated gap.  A  original  does  not  program  that  model,  extraneous  requires  guarantee  in Figure  surface  delete  also  contour  1.12  where  the  would  extract  true  and  then  contours  allow  the  might  be  desirable. A  more  determination the  gradient  functions  were  interpolation approximations should  be  rigorous of  signal  approach level  quantization obtained and upon  experiment.  from  discrete  quantization and  be  taken  probability density  smoothing.  investigated,  should  better  error  histograms effects and  signal  approximations  the  functions  Continuous  The  to  in  density  by  linear  of  these  distortion  sought.  93  The not  be  assertion  was  reconstructed  made  that  from  surface  gradients,  even  experiment  should  be p e r f o r m e d  to either  assertion.  Also,  the tradeoff  between  movement  and  Finally,  for  a  a d i s t o r t i o n - f r e e image  image q u a l i t y  out jagged,  edges  are conspicuous  the  test  experiment.  should  anti-aliasing  smooth  of  limited  occluding  surface  along used  light  be  source verify  freedom  further  techniques surface the outer in  represented  the  by  8-bit  movement.  An  or disprove  this  of  light  source  investigated.  should  edges  could  be e m p l o y e d  in  boundary  to  images.  Such  and c e n t r a l  peak  gradient  quantization  94  BIBLIOGRAPHY  The  following  acronyms  are used  i n the  references:  ACM ASP IEEE IRE SIAM  - A s s o c i a t i o n f o r Computing Machinery - American S o c i e t y of Photogrammetry - I n s t i t u t e of E l e c t r i c a l and E l e c t r o n i c s E n g i n e e r s - I n s t i t u t e of Radio Engineers - S o c i e t y f o r I n d u s t r i a l and A p p l i e d Mathematics  1.  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J . , " P h o t o m e t r i c m e t h o d f o r d e t e r m i n i n g surface orientation from multiple images", Optical Engineering, v o l . 19, n o . 1, J a n . 1 9 8 0 , p p . 1 3 9 - 1 4 4 .  25.  W r i g h t T . J . , "A two-space solution to the hidden line problem for plotting functions of two v a r i a b l e s " , IEEE T r a n s a c t i o n s on Computers, v o l . C-22, n o . 1, J a n . 1973, pp. 28-33.  97  A P P E N D I X A - COMPUTING A DELAUNAY  TRIANGULATION  A 2D Thiessen tessellation partitions a plane into p r o x i m i t y r e g i o n s d e f i n e d about nodes s c a t t e r e d i n t h e plane. Points within a proximity, or Thiessen, region are closer to the region's defining node t h a n t o a n y o t h e r i n t h e node s e t . The boundaries of t h e T h i e s s e n r e g i o n s a r e convex polygons called Voronoi polygons. Nodes are neighbours i f their Voronoi p o l y g o n s s h a r e a common e d g e . The Delaunay triangulation i sthe graph dual of the Voronoi diagram, and i s formed by joining neighbouring nodes with l i n e segments. T h e s e s e g m e n t s become t r i a n g l e edges and t h e nodes become triangle vertices. An algorithm f o r computing a Delaunay t r i a n g u l a t i o n may t h e r e f o r e p r o c e e d by f i n d i n g t h e graph o f n e i g h b o u r i n g nodes i n a Voronoi diagram, and transforming the neighbour links into triangle edges. I t i s t h i s a p p r o a c h t h a t i s t a k e n by F o w l e r (1977). Fowler's a l g o r i t h m g e n e r a t e s a t e s s e l l a t i o n of N nodes from a n e x i s t i n g t e s s e l l a t i o n o f N-1 n o d e s s u c h t h a t t h e i n s e r t i o n o f t h e N ' t h node i s p e r f o r m e d l o c a l l y . This insertion i s nothing more than the construction o f t h e new n o d e ' s T h i e s s e n r e g i o n from p o r t i o n s of nearby r e g i o n s i n t h e e x i s t i n g tessellation. The neighbour l i n k a g e s of these diminished regions a r e the only o n e s a f f e c t e d b y t h e i n s e r t i o n o f t h e new n o d e . The node NEAR, closest to t h e new n o d e N, defines a Thiessen r e g i o n w h i c h c o n t a i n s N. T h e r e g i o n d e f i n e d by node N a l s o c o n t a i n s N. T h e s e two r e g i o n s must intersect, and their Voronoi p o l y g o n s s h a r e an edge. N o d e s NEAR a n d N a r e t h e r e f o r e neighbours. The l i n e c o n t a i n i n g t h e s h a r e d edge i n t e r s e c t s the p o l y g o n s o f t w o n e i g h b o u r s o f n o d e NEAR. Since Voronoi polygons are c o n v e x , a l i n e i n t e r s e c t s a n y p o l y g o n i n a t m o s t two p o i n t s . Of t h e two i n t e r s e c t i n g p o i n t s on e a c h p o l y g o n , t h e one c l o s e r t o t h e m i d p o i n t o f t h e s e g m e n t j o i n i n g N t o NEAR i s a n endpoint of the shared edge. Starting with the endpoint that f a l l s to t h e r i g h t o f t h e d i r e c t e d s e g m e n t f r o m N t o NEAR, the Voronoi polygon of node N may be t r a c e d o u t i n a c l o c k w i s e d i r e c t i o n about N u n t i l i t c l o s e s upon i t s e l f a t t h e e n d p o i n t t o t h e left of the directed segment. The procedure i s outlined i n the following algorithm. (1)  Locate the closest node NEAR to t h e n o d e N, ' a n d s e t CURRENT := NEAR. The closest node i s f o u n d by a w a l k i n g s e a r c h a l o n g t h e l i n k s between n e i g h b o u r i n g nodes i n the existing tessellation. If a candidate node i s n o t t h e c l o s e s t t o N, t h e n o n e o f i t s n e i g h b o u r s i s closer to N than i t i s . The s e a r c h advances t o t h i s node. Starting at an arbitrary node, t h e s e a r c h c o n t i n u e s from node t o node u n t i l one i s found t h a t i s c l o s e r t o N than any of i t s neighbours. T h i s i s t h e c l o s e s t node t o N.  98  (2)  Construct the perpendicular b e t w e e n N a n d CURRENT.  (3)  The b i s e c t o r i n t e r s e c t s t h e polygon of a neighbour o f node CURRENT such that the point of intersection lies to the r i g h t o f t h e d i r e c t e d s e g m e n t f r o m N t o CURRENT. Assign t o NEXT t h e l a b e l o f t h a t neighbour.  (4)  I f NEXT t o 4.  (5)  Stop.  * NEAR,  then  bisector  s e t CURRENT  of the l i n e  := N E X T a n d r e p e a t  segment  steps  2  T h e i n s e r t i o n o f n o d e N c r e a t e s new n e i g h b o u r l i n k a g e s and destroys some e x i s t i n g l i n k s . T h e new n o d e N m u s t b e l i n k e d t o t h o s e e x i s t i n g n o d e s w h o s e new V o r o n o i p o l y g o n s share an edge w i t h N's p o l y g o n . These updates a r e e a s i l y performed d u r i n g t h e process of generating the Voronoi polygon of node N. T h e b i s e c t e d s e g m e n t s d e f i n e new l i n k s b e t w e e n N and the existing nodes. Links corresponding t o o l d edges that a r e c o m p l e t e l y c o n t a i n e d w i t h i n N's p o l y g o n a r e d e l e t e d . Figures A.l a n d A.2 illustrate part of a Thiessen tessellation i n which t h e dashed l i n e s belong t o the Voronoi diagram and t h e s o l i d lines t o t h e Delaunay triangulation. Node 1 i s t h e c l o s e s t n o d e t o N. The p e r p e n d i c u l a r b i s e c t o r of t h e segment from N t o node 1 i s f o l l o w e d u n t i l v e r t e x b on t h e polygon o f node 2 i s reached. Node 2 i s l i n k e d t o N a n d N i s l i n k e d t o n o d e 2. T h e b i s e c t o r o f t h e segment f r o m N t o node 2 is t r a c e d t o v e r t e x c , a t which time nodes 3 and N a r e l i n k e d . The p r o c e s s c o n t i n u e s t h r o u g h v e r t e x d a n d t h e n e, whereupon t h e p a t h r e t u r n s t o t h e p o l y g o n o f n o d e 1. Linkages a r e established b e t w e e n N a n d n o d e 4, a n d b e t w e e n N a n d n o d e 1. The edge yz f r o m t h e o l d V o r o n o i d i a g r a m i s c o n t a i n e d w i t h i n t h e new p o l y g o n of n o d e N, and therefore the link between nodes 1 a n d 3 i s deleted. The a l g o r i t h m s t a r t s w i t h t h r e e n o n - c o l l i n e a r dummy nodes defining a triangular region that c o n t a i n s t h e convex h u l l o f t h e node s e t . A f t e r a l l t h e nodes have been i n s e r t e d , l i n k s to these dummy nodes are stripped from the data structure representing the final triangulation. The T I N d a t a structure records a network of triangle vertices ( t h e nodes of the Voronoi diagram) r a t h e r than a network o f t r i a n g l e s . Instead of each t r i a n g l e being l i n k e d to its three neighbours, each vertex i s l i n k e d t o i t s neighbouring v e r t i c e s such that t h e l i n k s represent t r i a n g l e edges. Peucker et a l . (1978) a r g u e t h a t a n e t w o r k o f t r i a n g l e v e r t i c e s i s more s t o r a g e e f f i c i e n t than a network o f t r i a n g l e s . In a trianglebased data s t r u c t u r e , each t r i a n g l e requires p o i n t e r s t o i t s 3 v e r t i c e s and i t s 3 neighbours f o ra t o t a l of 6. For a data domain containing N points, B o f which a r e on t h e boundary, 6 ( 2 N - B - 2 ) = 12N-6B-12 p o i n t e r s a r e r e q u i r e d i n a d d i t i o n to the  99  Figure  A.1  Figure  A.2  Node  N  Node  introduced  N  into  incorporated  an  existing  into  a new  tessellation  tessellation  100  space occupied by the point coordinates. For a d a t a s t r u c t u r e , o n l y 3 ( 2 N - B - 2 ) + B = 6N-2B-6 p o i n t e r s stored. This represents a savings in storage percent.  vertex-based need to be o f a b o u t 50  The d a t a s t r u c t u r e i s a l i s t o f v e r t i c e s where e a c h vertex in the l i s t p o i n t s t o a number o f o t h e r v e r t i c e s i n t h e l i s t . The l i s t i s a one-dimensional array of records, each containing information about one triangle vertex. A record contains the (x,y,z) c o o r d i n a t e s of the vertex and a l i s t of i t s neighbouring vertices i n the triangulation. Instead of recording a l l vertex neighbours, each record contains a neighbour count and a p o i n t e r to where these neighbours are listed. The n e i g h b o u r s of a l l v e r t i c e s a r e stored i n another l i s t c a l l e d the neighbours list. The n e i g h b o u r s a r e g i v e n by t h e i r i n d i c e s i n t o t h e v e r t e x a r r a y . This data structure, illustrated i n F i g u r e A.3, is further d i s c u s s e d i n Peucker e t a l . (1978).  101  VERTEX LIST  Coordinates  Neighbour Count •  •  NEIGHBOUR LISTS  Neighbour L i s t Pointer  •  •  X  I *I I Z  •  J:  X  J  y  J J Z  A  Neighbours ' of node I M  • 6  N Neighbours ^ of node J  •  Figure  A.3  •  •  N+B-l:  How a t r i a n g l e edge d e f i n e d by v e r t i c e s is represented i n the TIN data structure  I  and  J  102  APPENDIX  B ~ SMOOTH  ( C ) INTERPOLATION  IN  1  The f u n c t i o n v a l u e F a t an arbitrary triangle T may be v e r t i c e s by t h e f o l l o w i n g s t e p s :  TRIANGLES  interior point (x,y) interpolated from data  of an at the  (1)  Transform the point (x,y) i n T to a point (s(x,y),t(x,y)) in a standard t r i a n g l e S with v e r t i c e s a t (0,0), (1,0), and (0,1).  (2)  Perform  (3)  Transform F(x,y) =  the i n t e r p o l a t i o n ( s , t ) back f(s,t).  within  S to obtain  t o i t sp o s i t i o n  f(s,t).  (x,y) i n T  such  that  The arbitrary triangle T with vertices V, = ( x , , y , ) , 2 = (x ,y )r and V = (x ,y ) i s transformed to the standard t r i a n g l e S, s u c h t h a t the interior of T i s mapped to the interior of S. The t r a n s f o r m a t i o n i s a c o n c a t e n a t i o n o f f o u r geometric transformations: translation, rotation, shearing, and scaling. The t r a n s f o r m a t i o n i l l u s t r a t e d i n F i g u r e B.1 m a p p i n g V, t o ( 0 , 0 ) , V t o (1,0), and V t o (0,1) i s :  V  2  2  3  3  2  s  3  (x-x,)(y -y1) U - x , ) (y -y,)  -  (x -x,)(y-y,) U -x,My -y,)  = -(x-x,)(y -y,) (x -x,)(y -y,)  + -  (x -x,)(y-y,) (x -x,)(y -y,)  =  3  2  t  3  2  2  The its  transformation function value  3  changes F(x,y).  3  3  (B.1)  2  2  3  2  the p o s i t i o n of a p o i n t (x,y) but not T h e r e f o r e , f ( s , t ) = F(x,y) and  f(0,0) f(1,0) f(0,1)  Derivatives,  3  = F(V,) = F(V ) = F(V ) 2  3  however, a r e not p r e s e r v e d  and a r e transformed  to:  103  Figure  B.l  Triangle  transformations  104  f  = F  s  f.  = F  t  x  x  s  + F  x. + F  x  t  y y  y  s  1  y. 1  1  where t h e p a r t i a l d e r i v a t i v e s o f x a n d y are c o n s t a n t s . These derivatives may inverse t r a n s f o r m a t i o n by s o l v i n g f o r x t. The inverse of the t r a n s f o r m a t i o n i n  Nielson's triangle i s : f(s,t)  x  =  (1-s-t)x,  + sx  2  + t x  y  =  (1-s-t)y,  + sy  2  + t y  (1980)  = f(s,0)  +  with respect t o s and t be obtained from the and y i n terms of s and E q u a t i o n s B.I i s :  interpolation  + f(0,t) (st/2)  /  J  1  _  /  3  function  - f(0,0)  +  (B.2)  3  over  tg(s) + 2(1-s)  standard  sg(l-t) 2(l-t)  t  q ( w ) - w d - w ) h ( w ) dw w (1-w)  s  z  g(w) = f(w,1-w) - f ( w , 0 )  and  h ( w ) = f (w,1-w) + f (w,1-w) - f ( w , 0 ) - f t  (B.3)  2  where  S  the  - f(0,1-w)  +  f(0,0)  5  t  (0,1-w).  This function interpolates t o the boundary values f(s,0), f(0,t), and f ( s , 1 - s ) , and t o the normals f (s,1-s) + f (s,1-s) on t h e h y p o t e n u s e o f S. s  When f u n c t i o n v a l u e s a n d d e r i v a t i v e s a r e g i v e n o n l y a t t h e triangle vertices, then t h e boundary conditions may be i n t e r p o l a t e d from these d a t a . Cubic Hermite interpolants are used f o r t h e boundary functions, and l i n e a r interpolants a r e u s e d f o r t h e n o r m a l d e r i v a t i v e s on t h e h y p o t e n u s e . The i n t e r p o l a n t i n E q u a t i o n B.3 interpolates to function values on a l l three edges but to n o r m a l s on o n l y o n e , t h e h y p o t e n u s e o f S. T h i s means t h a t i n t h e a r b i t r a r y triangle T, the f u n c t i o n F ( x , y ) i n t e r p o l a t e s t o t h e normals o f o n l y one edge as well. To correct for this, T i s transformed to S three t i m e s , where e a c h t i m e a d i f f e r e n t edge i n T i s mapped to the hypotenuse i n S. A weighted sum of the three resulting i n t e r p o l a n t s i n t e r p o l a t e s t o f u n c t i o n v a l u e s and t o f i r s t - o r d e r d e r i v a t i v e s on a l l t h r e e e d g e s o f T. Introduction of barycentric coordinates b,, b , and b h e l p s t o keep t r a c k of the different transformations. These c o o r d i n a t e s a r e d e f i n e d by: 2  3  105  x = t^x, + b x + y = b,y, + b y + 1 « b, + b + b 2  2  2  2  2  b x b y 3  3  3  3  3  For t h e t r a n s f o r m a t i o n i l l u s t r a t e d i n F i g u r e B.1 where V , V , V a r e mapped t o ( 0 , 0 ) , ( 1 , 0 ) , (0,1), the barycentric coordinates b and b take on the roles of s and t , r e s p e c t i v e l y (See Equations B.2). If the vertices of the triangle T remain numbered consecutively i n a counter-clockwise d i r e c t i o n around t h e t r i a n g l e , t h e n t h e two o t h e r transformations map V^V^V, and V ,V,,V to ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) . U s i n g i n d i c e s i , j , and k, t h e t h r e e t r a n s f o r m a t i o n s c a n be c o n v e n i e n t l y e x p r e s s e d as: f  2  2  3  3  3  2  V.  ==>  ( 0 , 0 )  V  ==>  (1,0)  ==>  (0,1)  where (i,j,k) is a I = {(1,2,3),(2,3,1),(3,1,2)}. written as: F.(x,y) Nielson  shows  that  F(x,y)  =  the  =  member of Each interpolant  f(b  (x,y),b  weighted  babaF^x^y) b b 1  f c  then  set be  (x,y))  sum  + b ^ F ^ x , ^ ) + b b + b b  2  the can  1  3  2  +  b,b F (x,y) 2  3  3  i n t e r p o l a t e s t o F a n d i t s f i r s t - o r d e r d e r i v a t i v e s on t h e entire boundary o f T. R e p l a c i n g boundary f u n c t i o n s with c u b i c Hermite polynomials and interpolating boundary normal derivatives linearly from those at the vertices, a nine-parameter C interpolant i s obtained. This interpolant is given in the following, where the summation i s over a l l t r i p l e s ( i , j , k ) i n the s e t I: 1  F(x,y)  =  Z  [F(V. )[b +  F  k  {  V  i  +  F  ;  ( v  /  )  2 j  (3-2b  [  ) [ b  b  i k b  / y b  ) +  /  +  +  6wb  w b  /  ( 3 b  v  i  i  b  3  b  f  (b^A  +  * /y  +  b  j i k  +  b  A  k  b.A. ^  y " k  '  b  b  y  it )  ) ]  ]  ]  ]  106  where  F)(V.) = (x.-x.) F j  i  j  w = bib b 2  K.. and  3  i  /  x  (V.) + ( y , - y , ) F i  j  (V. ) ,  +  1  1  3  2  2  y  (b b2 b b +b b ),  = dej'+le^'-le*! )  |e. |  i  = U^.-x^)  2  +  2  3  /  (2|e | ), 2  y  (y^.-y^) . 2  i  


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