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Colour image coding using linear transformation and block quantization Seecheran, Cris Anand 1974

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COLOUR IMAGE CODING USING LINEAR TRANSFORMATION AND BLOCK QUANTIZATION by CRIS ANAND SEECHERAN B.Sc.(E.Eng.), U n i v e r s i t y of the West I n d i e s , 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a ABSTRACT A bandwidth compression and redundancy r e d u c t i o n scheme was i n v e s t i g a t e d i n a d i g i t a l c o lour image t r a n s m i s s i o n system employing a l i n e a r t r a n s f o r m a t i o n and b l o c k q u a n t i z a t i o n technique. Colour images were represented by the N.T.S.C.'s Red, Green, Blue and Y, I , Q Re-c e i v e r Primary and Transmission Co-ordinate systems r e s p e c t i v e l y . The Y, I and Q s i g n a l s were assumed to c o n s t i t u t e a homogeneous Gauss-Markov random f i e l d modelled by an exponential c o r r e l a t i o n f u n c t i o n of the form R(x,x',y,y') = exp[-a|x-x'| - $|y-y'|], where a, 8 a r e the c o r r e l a t i o n c o e f f i c i e n t s i n the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s , r e s p e c t i v e l y . L i n e a r transformations employed were the d i s c r e t e F o u r i e r and Hadamard tr a n s f o r m a t i o n s . In the b l o c k coding scheme, b l o c k s i z e s of 8 x 8 and 16 x 16 p i c t u r e elements were used and i n the q u a n t i z a t i o n s t r a t e g y , both optimum uniform and optimum non-uniform q u a n t i z e r s were considered. The F o u r i e r and Hadamard coding schemes were evaluated i n terms of the mean square e r r o r between o r i g i n a l and r e c o n s t r u c t e d images, and by s u b j e c t i v e preference. Based on the t h e o r e t i c a l r e s u l t s , the Hadamard transform was s u p e r i o r to the F o u r i e r transform f o r b i t assignments below approximately 1.0 b i t s per p i c t u r e element ( b i t s / p e l . ) , on each of the Y, I and Q s i g n a l planes. The experimental r e s u l t s showed the Hadamard coded images to be s u p e r i o r to the F o u r i e r coded images from both the mean square e r r o r c r i t e r i o n and from a s u b j e c t i v e e v a l u a t i o n of the processed images. P i c t u r e s coded w i t h b l o c k s i z e s of 16 x 16 p i c t u r e elements were su p e r i o r i i to those coded w i t h 8 x 8 b l o c k s i ^ e s . There was no n o t i c e a b l e d i f f e r -ence i n image q u a l i t y between the optimum uniform or optimum non-uniform q u a n t i z e r . Good q u a l i t y images were obtained f o r an assignment of 2.0 b i t s / p e l . on the Y s i g n a l and 0.375 b i t s / p e l . on the I and Q s i g n a l s . For some Hadamard transformed images, good q u a l i t y reproduc-t i o n s were obtained f o r a t o t a l b i t assignment as low as 1.75 b i t s / p e l . Some experiments employing simple z o n a l low pass f i l t e r i n g schemes were a l s o c a r r i e d out when transform coding the e n t i r e image i n i t s 256 x 256 format. When usin g t h i s scheme, as compared to the b l o c k q u a n t i z a t i o n scheme, b e t t e r q u a l i t y images were obtained f o r the F o u r i e r system f o r e q u i v a l e n t b i t assignments. There was not much d i f f e r e n c e f o r the Hadamard coded images. i i i TABLE OF CONTENTS Page I. INTRODUCTION 1.1 M o t i v a t i o n and Background 1 1.2 Transform Coding System - Block Diagram 3 1.3 Scope of Thesis 3 I I . COLOUR IMAGE REPRESENTATION 2.1 Concepts 6 2.2 Col o r i m e t r y 7 2.2.1 Colour Processes 7 2.2.2 Trich r o m a t i c Theory 7 2.3 Colour Co-ordinate Systems 10 2.4 Choice of Colour Co-ordinate System 12 2.4.1 N.T.S.C. Receiver Primary Colour Co-ord i n a t e System 12 2.4.2 N.T.S.C. Transmission Colour C o - o r d i -nate System 12 I I I . TWO DIMENSIONAL TRANSFORM CODING OF COLOUR IMAGES 3.1 P i c t u r e S t a t i s t i c s 15 3.2 S t a t i s t i c a l Image Model 15 3.3 Li n e a r Transformations 16 3.3.1 D i s c r e t e 2-Dimensional F o u r i e r Transform.. 18 3.3.2 D i s c r e t e 2-Dimensional Hadamard Transform. 19 3.4 Energy D i s t r i b u t i o n i n the Transform Domain 20 VI. QUANTIZING FOR MINIMUM DISTORTION 4.1 Block Coding . . 22 4.2 Quantizing the Transform C o e f f i c i e n t s 23 4.3 Block Q u a n t i z a t i o n 24 i v Page 4.4 Variances of Transform C o e f f i c i e n t s 26 4.5 Qu a n t i z a t i o n E r r o r 29 4.6 A Note on Optimum Quantizers 30 V. EXPERIMENTS AND RESULTS 5.1 E s t i m a t i o n of C o r r e l a t i o n C o e f f i c i e n t s a, 6 35 5.2 Parameters Under I n v e s t i g a t i o n 36 5.2.1 Block S i z e s 36 5.2.2 B i t Assignments 36 5.2.3 Quantization 41 5.3 System Implementation 41 5.4 Performance C r i t e r i a 41 VI. PERFORMANCE ANALYSIS 6.1 I n t r o d u c t i o n 63 6.2 F o u r i e r v s . Hadamard 63 6.3 B i t Assignment Strategy 64 6.4 Edge E f f e c t s 65 6.5 Quantizer Performance 67 6.6 A Comparison Between T h e o r e t i c a l and Experimental R e s u l t s 67 V I I . ADDITIONAL EXPERIMENTS AND RESULTS 7.1 Implementation of a Simple Zonal Coding Scheme •••• 70 7.2 Smoothed F o u r i e r Images 71 7.3 Coding the E n t i r e P i c t u r e 72 V I I I . CONCLUDING REMARKS 8.1 Summary of R e s u l t s 74 8.2 Recommendations f o r Future Research 75 v Page APPENDIX A. Conversion from Receiver Primary to Transmission Co-ordinate S i g n a l s and V i c e Versa 78 APPENDIX B. Brightness Amplitude Histograms of Test P i c t u r e s 79 APPENDIX C. Variances of Transform C o e f f i c i e n t s 84 APPENDIX D. Optimum Quantizers 87 REFERENCES 91 v i LIST OF ILLUSTRATIONS F i g u r e Page 1.1 Block schematic of transform coding system 4 2.1 T r i s t i m u l u s v a l u e s of C L E . S p e c t r a l P r i m a r i e s . (Red = 700 nm, Green = 546.1 nm, Blue = 435.8 nm) .. 9 2.2 Ch r o m a t i c i t y Diagram f o r C.I.E. XYZ Primary System 9 2 4.1 P a r t i t i o n i n g of data arra y i n t o M sub-blocks ...v... 22 4.2 Schematic of b a s i s r e s t r i c t e d t ransformation 23 4.3 Variances of transform c o e f f i c i e n t s ordered according to t h e i r magnitude 27 4.4 Variances of transform c o e f f i c i e n t s ordered according to t h e i r magnitude 28 4.5 Comparison of b i t s per p e l v s . mean square e r r o r f o r F o u r i e r and Hadamard systems based on (4.11) 31 4.6 Comparison of b i t s per p e l v s . mean square e r r o r f o r F o u r i e r and Hadamard systems based on (4.11) 32 4.7 Comparison of b i t s per p e l v s . mean square e r r o r f o r F o u r i e r and Hadamard systems based on (4.11) 33 4.8 Comparison of b i t s per p e l v s . mean square e r r o r f o r F o u r i e r and Hadamard systems based on (4.11) 34 5.1 Measured c o r r e l a t i o n i n each s i g n a l plane and l e a s t squares exponential f i t . Smooth l i n e s are-the f i t t e d curves 37 5.2 Measured c o r r e l a t i o n i n each s i g n a l plane and l e a s t squares exponential f i t . Smooth l i n e s are the f i t t e d curves. 38 5.3 Measured c o r r e l a t i o n i n each s i g n a l plane and l e a s t squares exponential f i t . Smooth l i n e s are the f i t t e d curves. 39 v i i F i g u r e Page 5.4 Measured c o r r e l a t i o n i n each, s i g n a l plane and l e a s t squared exponential f i t . Smooth l i n e s a re the f i t t e d curves 40 5.5 Format of i l l u s t r a t i o n s i n P l a t e s I - V 42 5.6(a) Mean square e r r o r v s . b i t s per p e l . Mean square e r r o r s approximately the same f o r each colour plane ^4 5.6(b) Mean square e r r o r v s . b i t s per p e l . Mean square e r r o r s d i f f e r e n t f o r each colour plane 54 5.7 Mean square e r r o r v s . b i t s per p e l . Y-=3.0, 2.0, 1.0 b i t s / p e l . I,Q = 0.5 b i t s / p e l 5 5 5.8 Mean square e r r o r v s . b i t s per p e l . Y = 3.0, 2.0, 1.0 b i t s / p e l . I,Q = 0.5 b i t s / p e l 5 6 5.9 Mean square e r r o r v s . b i t s per p e l . Y: = 3.0, 2.0, 1.0 b i t s / p e l . I,Q = 0.375 b i t s / p e l 5 7 5.10 Mean square e r r o r v s . b i t s per p e l . Y- = 3.0, 2.0, 1.0 b i t s / p e l . I,Q = 0.375 b i t s / p e l 58 5.11 Mean square e r r o r v s . b i t s per p e l . Y, = 3.0, 2.0, 1.0 b i t s per p e l . I,Q = 0.5 b i t s / p e l 59 5.12 Mean square e r r o r v s . b i t s per p e l . Y = 3.0, 2.0, 1.0 b i t s per p e l . I,Q:;= 0.5 b i t s / p e l 60 5.13 Mean square e r r o r v s . b i t s per pel.' Performance of systems as given by experimantal r e s u l t s r e s u l t s 61 B . l Brightness amplitude histograms of GARDEN p i c t u r e . . . . 80 B.2 Brightness amplitude histograms of HOUSE p i c t u r e 81 J;B';"3 Brightness amplitude histograms, of FACE p i c t u r e 82 B.4 Brightness amplitude histograms of TEST PATTERN 83 v i i i F i g u r e Page D.l Quantizer 87 D.2 Quantizer f o r a symmetric p r o b a b i l i t y d i s t r i b u t i o n ... 87 D.3 Input ranges and corresponding out put l e v e l s f o r optimum non-uniform quantizer 89 i x LIST OF PLATES P l a t e I I I I I I IV V Page 44 46 48 50 52 x ACKNOWLEDGEMENT I wish to thank my s u p e r v i s o r , P r o f e s s o r R. W. Donaldson, f o r h i s encouragement and suggestions during the course of my research. I a l s o wish to thank Mr. A. Soubigou f o r the use of h i s data and f o r the many u s e f u l d i s c u s s i o n s we shared. My thanks a l s o go to Mr. M. Koombes f o r h i s t e c h n i c a l a s s i s t a n c e and to Mrs. A. Semmens f o r t y p i n g the manuscript. Acknowledgement i s a l s o made to the Canadian Commonwealth and Scholarship Committee and to the N a t i o n a l Research C o u n c i l of Canada, under NRC Grant A-3308. F i n a l l y , I wish to express my thanks to a l l my f r i e n d s at U.B.C. who helped to make my stay a pleasant and p r o f i t a b l e one. x i 1. I . INTRODUCTION 1.1 M o t i v a t i o n and Background A major concern i n the design of a d i g i t a l image tr a n s m i s s i o n scheme i s that of minimizing the b i t r a t e through use of v a r i o u s data compression and redundancy r e d u c t i o n techniques. During the l a s t decade, much e f f o r t has been devoted to the e f f i c i e n t coding and tr a n s m i s s i o n of monochromatic images. Many schemes have been examined which e x p l o i t e i t h e r the s t a t i s t i c a l p r o p e r t i e s of the monochromatic image sources or the p s y c h o v i s u a l p r o p e r t i e s of the human v i s u a l system. A l s o , both adaptive and non-adaptive techniques have been t r i e d , and much l i t e r a -t u r e now e x i s t s i n the area of p i c t u r e processing. W i l k i n s and Wintz [33] have published a comprehensive b i b l i o g r a p h y of p i c t u r e coding t e c h -niques . One technique t h a t has re c e i v e d much a t t e n t i o n l a t e l y i s that of image coding by l i n e a r t ransformation and block q u a n t i z a t i o n . This scheme has the two-fold advantage of employing some aspects of both s t a t i s t i c a l and psychovisual coders. The concept of l i n e a r transform-a t i o n and block q u a n t i z a t i o n was f i r s t introduced by Huang and S c h u l t h e i s s [17]. Many researchers have si n c e adopted t h i s technique i n image coding schemes. Wintz [18] has surveyed a l l known a p p l i c a t i o n s of l i n e a r t r a n s f o r m a t i o n and block q u a n t i z a t i o n to monochromatic image coding. In the area of colour image coding, there has not been as much a t t e n t i o n . Gronemann [9] i n v e s t i g a t e d the p o s s i b i l i t y of d i g i t a l l y coding colour images and by employing low pass averaging techniques on 2. c o l o u r images represented by one luminance and two chrominance planes, was a b l e to produce good q u a l i t y images f o r a t o t a l b i t assignment of 5.55 b i t s per p i c t u r e element ( b i t s / p e l ) . Bhushan [10] has i n v e s t i -gated the s u b j e c t i v e e f f e c t s of noise on a r e a l time colour image t r a n s -m i s s i o n scheme employing Red, Green and Blue primary colour s i g n a l s . P r a t t [7] a p p l i e d the technique of l i n e a r t r a n s f o r m a t i o n i n coding colour images represented i n d i f f e r e n t c o l o u r co-ordinate systems. He a l s o i n v e s t i g a t e d block coding of the colour images employing zonal low pass f i l t e r i n g schemes. His r e s u l t s i n d i c a t e d that good q u a l i t y images could be obtained f o r a t o t a l b i t assignment of 3.75 b i t s / p e l . By using a h y b r i d coding scheme employing the Karhunen-Loeve and Hadamard tran s f o r m a t i o n s , P r a t t [7] was able to o b t a i n good q u a l i t y images f o r a t o t a l b i t assignment of 1.75 b i t s / p e l . P r a t t a l s o i n v e s t i g a t e d the e f f e c t s of channel noise on a PCM c o l o u r t r a n s m i s s i o n scheme [34]. Limb et a l [8] performed a r e a l time i n v e s t i g a t i o n i n t o the coding of c o l o u r Picturephone s i g n a l s represented by one luminance and two chromi-nance components usi n g an element d i f f e r e n t i a l q u a n t i z a t i o n scheme. Their r e s u l t s showed that good q u a l i t y images were obtainable f o r a t o t a l assignment of 4.0 b i t s / p e l . In the present study, separate baseband encoding us i n g the technique of l i n e a r t ransformation and b l o c k q u a n t i z a t i o n was performed on colour images represented i n the N a t i o n a l T e l e v i s i o n System Committee (N.T.S.C.) co l o u r t e l e v i s i o n format. The l i n e a r transformations employed were the d i s c r e t e F o u r i e r and Hadamard transformations. 3. 1.2 The Transform Coding System F i g . 1.1 shows a block diagram of the transform coding system. R ( x , y ) , G(x,y) and B(x,y) represent r e s p e c t i v e l y the Red, Green and Blue primary planes of the colour image. Through a mat r i x conversion these three colour s i g n a l s are converted i n t o the Y ( x , y ) , I ( x , y ) and Q(x,y) t r a n s m i s s i o n s i g n a l s . The Block O r g a n i z a t i o n U n i t p a r t i t i o n s each of the t r a n s m i s s i o n s i g n a l s i n t o sub-blocks of s p e c i f i e d dimen-s i o n N, and a l i n e a r t r a n s f o r m a t i o n i s then performed on each sub-block. In the transform domain subsequent q u a n t i z a t i o n and coding i s performed on the c o e f f i c i e n t s to be t r a n s m i t t e d , as s e l e c t e d by the Block Quanti-z e r . Both the s e l e c t i o n and assignment of q u a n t i z a t i o n l e v e l s to i n -d i v i d u a l samples are based on the v a r i a n c e s of the transform c o e f f i -c i e n t s . At the r e c e i v i n g end of the d i g i t a l channel, the decoder r e -covers the t r a n s m i t t e d s i g n a l s and the i n v e r s e l i n e a r t r a n s f o r m a t i o n i s performed. The r e - o r g a n i z a t i o n or reassembling of the sub-blocks form the r e c o n s t r u c t e d t r a n s m i s s i o n s i g n a l s Y ( x , y ) , I ( x , y ) and Q(x,y). The inverse, co-ordinate conversion gives r i s e to the Receiver primary colour s i g n a l s R ( x , y ) , G(x,y) and B(x, y ) . The d i g i t a l channel was assumed to be an e r r o r f r e e b i n a r y symmetric channel. The image trans m i s s i o n system was simulated on a d i g i t a l com-puter and the r e c e i v e d colour images were d i s p l a y e d as t r i p l e exposures on Ektachrome f i l m using a computer c o n t r o l l e d f l y i n g spot scanner. 1.3 Scope of t h e s i s In Chap. 2 a short t r e a t i s e on c o l o r i m e t r y i s given. Reasons B(xy) • COORDINATE CONVERSION Y(x,y) BLOCK ORGANISATION f,(x,y) •LINEAR TRANSFORM Ffav) BLOCK QUANTIZER ENCODER K*,Y) . f2(x,y) F2(u,v) BLOCK ORGANISATION LINEAR TRANSFORM BLOCK QUANTIZER Q(*,Y) . F3(u,v) BLOCK ORGANISATION LINEAR TRANSFORM BLOCK QUANTIZER DIGITAL CHANNEL R(x,y) G(x,y) B(x,y) INVERSE COORDINATE CONVERSION Y(x,y) BLOCK REORGANISATION f,(x,y) LINEAR INVERSE TRANSFORM DECODER Kx,y) f/*.y) F2(u,v) BLOCK REORGANISATION LINEAR INVERSE TRANSFORM 0(x,y) f3(x,y) BLOCK REORGANISATION LINEAR INVERSE TRANSFORM FIG. 1.1 BLOCK SCHEMATIC OF TRANSFORM CODING SYSTEM. 5. f o r employing the p a r t i c u l a r set of primary and tra n s m i s s i o n co-ordina t e systems are a l s o o u t l i n e d . In Chap. 3 a t h e o r e t i c a l model f o r image data i s g i v e n , and the two l i n e a r transformations employed i n the study, the F o u r i e r and Hademard, are discussed. Energy d i s t r i b u t i o n among transform samples i s a l s o discussed and an expression f o r a mean square approximation e r r o r i s given. In Chap. 4 the Block Q u a n t i z a t i o n scheme i s described. Ex-pressions f o r the va r i a n c e s of the transform c o e f f i c i e n t s are given and an equation r e l a t i n g the t o t a l mean square e r r o r , comprising approx-imation;, e r r o r plus q u a n t i z a t i o n e r r o r i s given. Chap. 5 describes the experimental procedures used i n the study. The es t i m a t i o n of the c o r r e l a t i o n c o e f f i c i e n t s i n both h o r i z o n -t a l and v e r t i c a l d i r e c t i o n s are o u t l i n e d . A c o n s i d e r a t i o n of the d i f f -erent parameters under i n v e s t i g a t i o n i n c l u d i n g b l o c k s i z e s , b i t a s s i g n -ment schemes and q u a n t i z e r s , are given. The performance measure used i n the study i s described. Chap. 6 i n c l u d e s a - d i s c u s s i o n of the r e s u l t s obtained f o r the two transform coding schemes. A comparison of experimental and theo-r e t i c a l r e s u l t s i s made and a system e v a l u a t i o n i s done. Advantages and disadvantages of performance measures are discussed. Chap. 7 de s c r i b e s some a d d i t i o n a l experiments performed and r e s u l t s obtained. Aysmoothing a l g o r i t h m f o r the F o u r i e r system i s proposed and a coding scheme a p p l i e d to the e n t i r e p i c t u r e (256 x 256 format) i s considered. Chap. 8 contains a summary of the major r e s u l t s obtained and some recommendations f o r fu t u r e research. 6. I I . COLOUR IMAGE REPRESENTATION 2.1 Concepts The concept of colour may be t r e a t e d on both a p s y c h o l o g i c a l (the aspect of v i s u a l perception) and psychophysical (the c h a r a c t e r i s -t i c s of v i s i b l e r a d i a n t energy) b a s i s [ 2 ] , As defined i n [2, p 229]: Colour i s that c h a r a c t e r i s t i c of a v i s i b l e r a d i a n t energy by which an observer may d i s t i n g u i s h d i f f e r -ences between two s t r u c t u r e - f r e e f i e l d s of view of the same s i z e and shape, caused by d i f f e r e n c e s i n the s p e c t r a l composition of the r a d i a n t energy con-cerned i n the observation. Wintringham [1] has traced the development of colour science l e a d i n g to the three-colour or T r i c h r o m a t i c Theory of Colour V i s i o n f i r s t p o s t u l a t e d by Thomas Young i n 1807. E s s e n t i a l l y t h i s theory s t a t e s that there are three types of colour receptors i n the r e t i n a of the eye and that the whole gamut of colour sensations i s d e r i v e d from v a r i a t i o n s i n the magnitude of the responses i n these three sensors. Much work has been done to determine the p h y s i o l o g i c a l processes that take place i n these three areas and v a r i o u s sets of s p e c t r a l s e n s i t i v i t y curves have been proposed as a b a s i s of t h i s t r i c h r o m a t i c theory of colour p e r c e p t i o n [3. F i g . 2.7 p 45]. The most important evidence supporting the t r i c o l o u r theory i s the f a c t t h a t , over a wide range of c o n d i t i o n s . o f o b s e r v a t i o n , most col o u r s can be matched completely by a d d i t i v e mixtures i n s u i t a b l e pro-p o r t i o n s of three f i x e d primary c o l o u r s . The choice of three primary colours i s not a r b i t r a r y ; any:?primary must not be matched by a mixture of the other two p r i m a r i e s . 7. Much work remains to be done i n the development of a complete theory of colour v i s i o n . However, the theory has proven s a t i s f a c t o r y f o r both c o l o u r photography and colour t e l e v i s i o n . Using t h i s t r i c o l o u r theory, P r a t t [4] developed a s i m p l i f i e d model of colour v i s i o n . 2.2 Colorimetry 2.2.1 Colour Processes Fundamentally, there are two processes which may occur i n the reproduction of any c o l o u r * These are e i t h e r an a d d i t i v e process or a s u b t r a c t i v e process. The a d d i t i v e process i s one i n which colours are produced by the a d d i t i v e combination of coloured l i g h t s . This i s the process which i s employed i n colour t e l e v i s i o n , the coloured l i g h t s being produced by the phosphors i n the cathode ray tubes. The s u b t r a c t i v e process, which i s employed i n c o l o u r photo-traphy or p r i n t i n g , i s based on an e n t i r e l y d i f f e r e n t p r i n c i p l e . In t h i s process, an i n c i d e n t l i g h t i s s u c c e s s i v e l y modified by three l a y e r s of what can e s s e n t i a l l y be termed 'band e l i m i n a t i o n ' m a t e r i a l s . These three l a y e r s are each s e l e c t e d to absorb c e r t a i n wavelengths of l i g h t and r e f l e c t o thers. In a colour photograph, f o r example, one l a y e r r e -f l e c t s the red and green l i g h t and absorbs the b l u e ; one l a y e r r e f l e c t s blue-red and absorbs green and the t h i r d l a y e r r e f l e c t s green and blue and absorbs red. I t i s t h i s s u b t r a c t i o n of the i n c i d e n t l i g h t to which the process owes i t s name. 2.2.2 T r i c h r o m a t i c Theory The science of c o l o r i m e t r y deals w i t h the numerical measure ment and s p e c i f i c a t i o n of c o l o u r . The general r u l e s governing a d d i t i v e 8. colour matching may be found i n [1] . Apart from the broad g e n e r a l i z a -t i o n of the t r i chromat i c theory as s tated i n Sect . 2 .1 , one may further strengthen the g e n e r a l i z a t i o n to s tate that p r o p o r t i o n a l i t y and add-i t i v i t y hold s t r i c t l y over a wide range of observing cond i t i ons . A colour match may be stated i n the form of an equat ion. Consider an a r b i t r a r y spectrum colour 1 C matched by the a d d i t i v e mix-ture of i?K u n i t s of primary R, c7^units of primary G , and u n i t s of primary B. This can be w r i t t e n i n the form C = RjR + c?1Q- + BjB (2.1) I t i s conventional to normalize the matching u n i t s R^t G^ and B^ by the amount of energy of each primary needed to produce a ' r e f e r -ence white ' source. The 'reference white ' source normally employed i s the hypothe t i ca l constant energy source which has un i t energy through-out the v i s i b l e spectrum. Suppose that these un i t matching energy u n i t s are R , G and B , then the quant i t i e s w w w R = ^ (2.2) w Gl c? = (2.3) w Bl and B = (2.4) D W are known as the t r i s t i m u l u s values of the given colour wi th respect to the p a r t i c u l a r set of pr imaries R, G and B. Eqn 2.1 may be r e w r i t t e n as C = R..R + GR + B B (2.5) note for the reference white source, the t r i s t i m u l u s values are a l l u n i t y . F i g . 2.1 shows a t y p i c a l set of t r i s t i m u l u s curves for the F i g . 2.2 C h r o m a t i c i t y Diagram f o r C L E . , XYZ Primary System 10. s p e c t r a l p r i m a r i e s Red (700 nm), Green (546.1 nm) and Blue (435.8 nm). From these curves, the matching u n i t s needed f o r the production of any given colour may be obtained. As noted b e f o r e , the choice of a set of pr i m a r i e s i s not unique and i f the t r i s t i m u l u s values of a colour f o r a p a r t i c u l a r set of p r i m a r i e s ; i s known, a simple co-ordinate conversion can be performed to determine the t r i s t i m u l u s values to another set of p r i m a r i e s . As noted i n F i g . 2.1, some t r i s t i m u l u s values may ber.negative at some wavelengths. The p h y s i c a l s i g n i f i c a n c e of t h i s i s that a colour match may be achieved by adding the primary w i t h the negative t r i s t i m u l u s v a l u e to the colour to be matched and then matching the r e s u l t a n t colour w i t h the remaining p r i m a r i e s . In a p r a c t i c a l system, such negative t r i s t i m u l u s values are n o n r e a l i z a b l e and there remains, t h e r e f o r e , some co l o u r s which cannot be produced i n a p r a c t i c a l c o lour d i s p l a y u n i t . However, by a j u d i c i o u s choice of colour p r i m a r i e s , these 'non-reproducible' c o l o u r s can be l i m i t e d to r a r e l y n a t u r a l l y o c c u r r i n g c o l o u r s . 2.3 Colour Co-ordinate Systems Ass o c i a t e d w i t h each set of t r i s t i m u l u s values R., C7, B (say) f o r any primary system are the c h r o m a t i c i t y co-ordinates r , g, b defined by the r e l a t i o n s h i p s r = —-4 =• (2.6) R + G + B 8 = R +G +B (2'7) b = „ , ? . ^  (2.8) and hence r + g + b = 1 (2.9) I t may be noted from (2.9) that only two of the c h r o m a t i c i t y co-ordinates are independent q u a n t i t i e s . Since f o r the complete s p e c i -f i c a t i o n of a colour three independent q u a n t i t i e s are needed, the chrom-a t i c i t y co-ordinates are not a complete s e t . The t h i r d q u a n t i t y needed i s the luminance or brightness of the c o l o u r . F i g . 2.2 shows a p l o t of two c h r o m a t i c i t y co-ordinates and i s c a l l e d a c h r o m a t i c i t y c h a r t . The chart shows the locus of the chromat-i c i t y co-ordinates f o r the v i s i b l e spectrum. Note that the locus i s e i t h e r s t r a i g h t or convex. This i m p l i e s that the colour a r i s i n g from the mixture of any two wavelengths must l i e e i t h e r on or w i t h i n the l o c u s . The bounded area i n c l u d e s a l l r e a l c o l o u r s p e r c e p t i b l e to the average human eye. The co l o u r s l o c a t e d on the periphery are s p e c t r a l l y pure w i t h the exception of the purples which are combinations of blue and red. The numbers along the periphery represent wavelengths i n nanometres of the s p e c t r a l c o l o u r s . P o i n t C represents the 'reference white' source - standard d a y l i g h t . < There have been many c o l o r i m e t r i c co-ordinate systems proposed, mainly as a b a s i s of s t a n d a r d i z a t i o n . In 1931, the Commission I n t e r -n a t i o n a l e de l ' E c l a i r g e ( C L E . ) proposed a standard primary system w i t h p r i m a r i e s at wavelengths: Red (700 nm), Green (546.1 nm) and Blue 435.8 nm). This primary system i s defined by t r i s t i m u l u s curves as shown i n F i g , 2.1. Because some t r i s t i m u l u s values may sometimes be neg a t i v e , the C L E . developed a 'non-negative t r i s t i m u l u s v a l u e ' system - the X, Y, Z primary system [ 1 ] . Many other systems have been proposed and developed from time to time, the choice of any system l a r g e l y dependent upon the s p e c i f i c a p p l i c a t i o n [1]. 2.4 Choice of Colour Co-ordinate System In the present study the choice of the colour co-ordinate system employed was determined p r i m a r i l y by the system c u r r e n t l y i n use f o r colour t e l e v i s i o n i n North America. This i s the system proposed by the N a t i o n a l T e l e v i s i o n System Committee, (N.T.S.C.) [6]. 2.4.1 N.T.S.C. Receiver Primary Colour Co-ordinate System In t h i s system, the t r i s t i m u l u s values are normalized so that the values are equal when matching the 'reference white'. Since the phosphors employed i n the cathode ray tubes are not pure monochromatic sources of r a d i a t i o n , the gamut of c o l o u r s p r o d u c i b l e i s smaller than that o b tainable from C L E . p r i m a r i e s . The t r i a n g l e i n F i g . 2.2 encloses the area of c o l o u r s capable of being reproduced by the N.T.S.C. system. 2.4.2 N.T.S.C. Transmission Colour Co-ordinate System In t h i s system, the three primary c o l o u r s mentioned i n Sect. 2.3 are transformed i n t o a luminance component Y, and two colour d i f f -erence c h r o m a t i c i t y v a l u e s , I , Q which s p e c i f y the hue and s a t u r a t i o n of the colour image. O v e r a l l , though, the choice of the Y, I , Q t r a n s m i s s i o n sys-tem was governed by the f a c t that i t met q u i t e c l o s e l y the general r e -quirements of the colour transform coding scheme employed. These r e -quirements may be s t a t e d as f o l l o w s : i ) : ' To provide a compatible system w i t h the e x i s t i n g monochrome systems. In t h i s r e s p e c t , the Y s i g n a l component (luminance) 13. alone can be decoded by monochrome r e c e i v e r s , i i ) To provide colour planes that are e s s e n t i a l l y u n c o r r e l a t e d w i t h each other. This property i s e s p e c i a l l y important f o r the transform coding scheme si n c e each colour plane i s coded s e p a r a t e l y . i i i ) A major f a c t o r to be considered i s the bandwidth requirement. A d e s i r a b l e system w i l l be one that r e q u i r e s i d e a l l y no greater bandwidth than i s normally employed f o r the t r a n s -m i s s i o n of monochrome images. In order to implement such a system, however, the luminance s i g n a l w i l l have to be de-graded somewhat s i n c e a d d i t i o n a l b i t s w i l l be needed f o r the chrominance components of the s i g n a l . H o p e f u l l y t h i s de-g r a d a t i o n of the luminance s i g n a l w i l l be overset by the add-i t i o n a l colour i n f o r m a t i o n to provide an o v e r a l l greater sub-j e c t i v e q u a l i t y . Thus, i n the transform coding scheme, chromi-nance s i g n a l s must be such that no a p p r e c i a b l e n o t i c e a b l e de-g r a d a t i o n must be inherent when the s i g n a l s are s e v e r e l y b a n d l i m i t e d . I t has been shown [7], [8] that the I , Q com-ponents do s a t i s f y t h i s requirement and i n f a c t can be coded w i t h a maximum of l . p b i t s / p e l . i v ) Since q u a n t i z a t i o n e r r o r s are inherent i n any d i g i t a l t r a n s -m i s s i o n system, the s i g n a l planes must not be o v e r l y s e n s i -t i v e to these e r r o r s . S i g n a l s a l s o must not be o v e r l y s e n s i -t i v e to channel e r r o r s , v) I t i s important, i n the transform coding scheme to compact the energy of, the s i g n a l i n t o the l e a s t number of t r a n s m i t t e d samples. I t has been shown [7] that Y, I , Q co-ordinate conversion does provide a n e a r l y optimum energy compaction between planes. Yet another advantage of having three separate s i g n a l s f o r coding i s that i t allows f o r f l e x i b i l i t y i n choosing d i f f e r e n t schemes s u i t a b l e to each s i g n a l plane. T h i s f l e x i b i l i t y a l l o w s f o r f u t u r e ex-t e n s i o n of frame-to-frame techniques f o r time v a r y i n g images. This technique of separate baseband encoding has been p r e v i o u s l y i n v e s t i -gated, notably by P r a t t [ 4 ] . Transformation matrices f o r the conversion of Receiver Prim-a r i e s to Transmission Co-ordinates and v i c e versa may be found i n Appendix A. I I I . TWO DIMENSIONAL TRANSFORM SODING OF COLOUR IMAGES 3.1 P i c t u r e S t a t i s t i c s S t a t i s t i c a l p r o p e r t i e s c h a r a c t e r i z i n g p i c t u r e s i g n a l s have been i n v e s t i g a t e d by many re s e a r c h e r s , 111]-[14]. Kretzmier [11] de-termined that the c o r r e l a t i o n between p i c t u r e p o i n t s of a sampled mono-chrome image were of an exponential form, and Franks [12] developed a simple model f o r c h a r a c t e r i z i n g the s t a t i s t i c a l p r o p e r t i e s of a random video s i g n a l . This model c h a r a c t e r i z i n g the a u t o c o r r e l a t i o n f u n c t i o n of the video s i g n a l was expressed as a product of three f a c t o r s - the p o i n t - t o - p o i n t , l i n e - t o - l i n e , and frame-to-frame c o r r e l a t i o n of the s i g n a l . Again an exponential c o r r e l a t i o n f u n c t i o n was used. F r e i and Jaeger [13] proposed the modelling of colour p i c t o r i a l sources by e i t h e r a continuous or d i s c r e t e s t o c h a s t i c process depending on whether or not the p i c t u r e s i g n a l had been sampled and quantized. However, they have shown, through an i n v e s t i g a t i o n of colour images, that s t a t i o n a r i t y i n the s t r i c t sense cannot be assumed but that an assumption of wide sense s t a t i o n a r i t y was q u i t e c o n s i s t e n t w i t h t h e i r s t a t i s t i c a l estimates. Through e x p l o i t a t i o n of many of these s t a t i s t i c a l p r o p e r t i e s of images, a redundancy r e d u c t i o n amy be e f f e c t e d and an o v e r a l l lower b i t r a t e may be achieved f o r any image tr a n s m i s s i o n system. 3.2 S t a t i s t i c a l Image Model Any image can be modelled by s p e c i f y i n g i t s b r i g h t n e s s l e v e l f ( x , y ) at each s p a t i a l co-ordinate ( x , y ) . Based on the r e s u l t s as ob-ta i n e d by ot h e r s , [11]-[12] and under the assumption of wide sense s t a t i o n a r i t y , the image can be modelled as a random Markov f i e l d w i t h an a u t o c o r r e l a t i o n f u n c t i o n of the form [14]: R(Ax,Ay) = E [ f ( x , y ) f ( x ' , y ' ) ] = exp (- a|Ax| - B|Ay|) (3.1) where A x = x - x ' , A y = y - y ' and a,B are the c o r r e l a t i o n c o e f f i c i e n t s i n the x,y d i r e c t i o n s r e s p e c t i v e l y . In the f o l l o w i n g a n a l y s i s the assumption of square images i . e . 0 <^  x,y _< N i s made but the g e n e r a l i z a t i o n to other shapes as w e l l as the i n c l u s i o n of a t h i r d dimension ( i . e . time) i s q u i t e s t r a i g h t forward [15]. An assumption of Gaussian. ;zero mean image data i s a l s o made. Such assumptions, although made f o r an e a s i e r t h e o r e t i c a l e v a l u a t i o n are not completely unfounded as can be seen from the b r i g h t -ness amplitude d i s t r i b u t i o n s of some of the t e s t p i c t u r e s as shown i n Appendix B. 3.3 L i n e a r Transformations As mentioned above, there i s co n s i d e r a b l e s t r u c t u r e inherent i n p i c t u r e data. Source encoding s t r a t e g i e s that seek to match the encoder to the p i c t u r e data s t r u c t u r e w i l l tend to perform w e l l . Such source coding, s t a t i s t i c a l coding, has been i n v e s t i g a t e d p r e v i o u s l y [16]. Another important aspect of coding p i c t u r e data i s t h a t of employing encoders which take the c h a r a c t e r i s t i c s of the human v i s u a l system i n t o account. Such coders are c a l l e d p sychovisual or psycho-p h y s i c a l coders. In transform coding some aspects of both s t a t i s t i c a l and p s y c h o v i s u a l coding are e f f e c t e d . E s s e n t i a l l y t h i s i s accomplished i n a two-sequence o p e r a t i o n . The f i r s t o p e r a tion i s a l i n e a r t r a n s -formation which seeks to transform the set of s t a t i s t i c a l l y dependent p i c t u r e elements to a set of c o e f f i c i e n t s which are s t a t i s t i c a l l y inde-pendent. Such a l i n e a r t r a n s f o r m a t i o n has so f a r not been discovered. However a l i n e a r t ransformation which produces a set of u n c o r r e l a t e d c o e f f i c i e n t s does e x i s t - the Karhunen Loeve or H o t e l l i n g transformation [17]. The second operation seeks to i n d i v i d u a l l y quantize and code each c o e f f i c i e n t i n a manner best s u i t e d to the human v i s u a l system. Conceptually one can v i s u a l i z e the f i r s t o p e r a t i o n , i . e . that of o b t a i n i n g the set of u n c o r r e l a t e d v a r i a b l e s , as that of a ' r o t a t i o n ' of data a r r a y s i n a mul t i d i m e n s i o n a l space [18]. Another i n t e r p r e t a t i o n i s that of c o n s i d e r i n g the data a r r a y as a l i n e a r combination of ortho-normal b a s i s a r r a y s . Consider the o r i g i n a l N x N d i s c r e t e image array f ( x , y ) , x,y = 1,2 ... N, which can be w r i t t e n as a l i n e a r combination of ortho-normal d i s c r e t e b a s i s images as f o l l o w s : N - l N - l f ( x , y ) = I I F(u,v) 9*(x,y,u,v) (3.2) u=0 v=0 where x,y = 1,2 ... N and 9(x,y,u,v) i s a two dimensional b a s i s f u n c t i o n . The a s t e r i s k (*) denotes complex conjugation. Here N N F(u,v) = I I f ( x , y ) 6 (u,v,x,y) (3.3) x=l y=l where u,v = 0, ... N - l Eqn (3.3) describes the tran s f o r m a t i o n which transforms the image f ( x , y ) 2 i n t o the N samples F ( u , v ) , u,v = 0,1 ... N - l . S i m i l a r l y Eqn (3.2) de s c r i b e s the i n v e r s e transformation which recovers the data a r r a y f ( x , y ) from the transform samples and gives the p i c t u r e f ( x , y ) as a weighted sum of the b a s i s p i c t u r e s 0(u,v,x,y). The f a c t o r s which weight the b a s i s a r r a y s are simply the transform c o e f f i c i e n t s as defined by (3.3). 3.3.1 D i s c r e t e 2-Dimensional F o u r i e r Transform The d i s c r e t e two-dimensional F o u r i e r transformation i s ob-tai n e d by choosing b a s i s images of the form e(u,v,x,y) = | exp [-2TTJ if- + f-)] (3.4) In the f o l l o w i n g j = /-T so that N N F(u,v) =±l I f ( x , y ) exp [-2irj ( ^ + ^ ) ] (3.5) x=l y=l and -N-l N - l f (X»Y) = jjf I I F(u»v) exp [+2^j(^- + ^ ) ] (3.6) u=0 v=0 u,v = 0,1 ... N - l x,y = 1,2 ... N I t should be noted that the b a s i s f u n c t i o n i s separable and symmetric and consequently the two dimensional transform can be computed as two s e q u e n t i a l one dimensional transforms [19], [20]. Of note, t oo, i s the f a c t that even though f ( x , y ) i s a r e a l p o s i t i v e function, F(u,v) 2 i s , i n ge n e r a l , complex. Thus, w h i l e the image contains N sample 2 2 p o i n t s , the transformed image contains 2N c o e f f i c i e n t s - N Real and 2 N Imaginary s p a t i a l frequency components. However, the apparent need to transmit twice as many components i s e a s i l y r e s o l v e d because the transform domain e x h i b i t s a property of conjugate symmetry, i . e . , r .. F(u,v) = F*(-u,-v) (3.7) As a r e s u l t of t h i s property i t i s necessary to transmit only one h a l f of the transform plane's samples. Hence the F o u r i e r transform of the 2 image can be c h a r a c t e r i z e d by N c o e f f i c i e n t s . Fast computational algorithms have been developed f o r the d i s c r e t e f o u r i e r transform, the so c a l l e d Fast F o u r i e r transform, which r e q u i r e s i n the order of - 2 2 2.N. l o g 2 N complex operations [19], [20]. 3.3.2 D i s c r e t e 2-Dimensibnal Hadamard Transform The d i s c r e t e two dimensional Hadamard transformation i s ob-t a i n e d by choosing b a s i s images of the form: 6(u,v,x,y) = i ( - l ) F ( x ' y ' U ' v ) (3.8) where n - l p(x,y,u,v) = I (u.;X » v y . ) (3.9) i=0 N = 2 n i . e . n = l o g 2 N (3.10) The symbol €> denotes modulo-2 a d d i t i o n and u_, v^, x^ and y^ are b i n a r y r e p r e s e n t a t i o n s of u, v, x and y r e s p e c t i v e l y , e.g. (u) , „. o l = (u , u - ... u u n ) , , u e(0,l} (3.11) decimal n-± n-z J. u binary l The above expressions give the Hadamard b a s i s a r r a y i n an 'ordered' form [21], From (3.8) i t can be seen that the elements of any b a s i s a r r a y can only be +1 or-1. Hence i t i s p o s s i b l e to order these a r r a y s i n terms of the number of s i g n changes i n a p a r t i c u l a r row. Harmuth [22] has coined the word 'sequency' to designate the number of s i g n changes along a p a r t i c u l a r row. Another r e p r e s e n t a t i o n of the b a s i s images e x i s t s i n which the sequency of each row i s greater than the preceding one. In t h i s case n - l p(x,y,u,v) = I [g.(u)x O g.(v)y ] (3.12) i=0 where g (u) = u .. o n - l g l ( u ) = Un-1 * Un-2 g 2(u) = u n _ 2 • u n _ 3 Sn-1 = u. e u 1 o In a l l cases the summations are performed modulo-2. The two- dimension-a l Hadamard transform can be computed i n e i t h e r n a t u r a l or ordered form w i t h a f a s t computational a l g o r i t h m analogous t o the Fast F o u r i e r t r a n s -2 2 form [21]. This a l g o r i t h m r e q u i r e s i n the order of 2N l o g 2 N operations. 3.4 Energy D i s t r i b u t i o n i n the Transform Domain One important consequence of transforming the p i c t u r e data a r r a y i s th a t c o e f f i c i e n t s of unequal variances are produced. This f e a t u r e , coupled w i t h the f a c t that the c o e f f i c i e n t s are e s s e n t i a l l y u n c o r r e l a t e d , form the b a s i s of the subsequent Block Q u a n t i z a t i o n Scheme. Eqn (3.2) represents an image as the sum of b a s i s images weighted by the transform c o e f f i c i e n t s . Owing to the unequal variances of the transform c o e f f i c i e n t s , i t i s p o s s i b l e to order these b a s i s images by the va r i a n c e of the c o e f f i c i e n t s weighting them so that suc-c e s s i v e terms c o n t r i b u t e p r o p o r t i o n a l l y l e s s to the t o t a l sum. In gener a l , the v a r i a n c e s of some c o e f f i c i e n t s may be so sm a l l that t r u n -2 c a t i o n of the ordered b a s i s images a f t e r the f i r s t n terms (say) may s t i l l r e s u l t i n a c l o s e approximation to the image, i . e . n - l n - l f ( x , y ) - f ( x , y ) = 1 1 F(u,v) <f>*(x,y,u,v) (3.13) u=0 v=0 The mean square approximation e r r o r between the o r i g i n a l image f ( x , y ) and the reconstructed image f ( x , y ) i s given by E = 7 ^ I [f (x,y) - f(x,y)] Z} (3.14) N x=l y=l Use of the orthonormality condition of the basis arrays and Eqns (3.5), (3.6) reduced (3.14) to,the following expression: N-l N-l 4 = ^ 2 I • I ff <u>v> (3.15) N u=n v=n = R(0,0) - ^  J, I 0 < u> v) (3-16) N u=0 v=0 where 2 N N N N a (u,v) = 1 1 I I R(x,x',y,y') cf>(u,v,x,y) <|>*(u,v,x,y) (3.17) x=l x'=l y=l y,'=l and R(x,x',y,yr) i s the autocorrelation function of the image. R(x,x ,y,y') = E[f(x,y),f(x + Ax, y + Ay)] (3.18) = exp (-a | Ax| - 6|.Ayj) (3.19) as given in (3.1). In considering any particular image, R(0,0) may be inter-2 preted as the image energy per unit area and the variance a (u,v) as the energies in the samples F(u,v). Eqn (3.15) states that the mean square approximation error i s given by the sum of the variances of the discarded coefficients. IV QUANTIZING FOR MINIMUM DISTORTION 4.1 Block Coding So f a r the problem of e f f i c i e n t l y coding an o r i g i n a l L x L data a r r a y (say) has been reduced to th a t of coding a smaller set 2 2 n < L of e s s e n t i a l l y u n c o r r e l a t e d c o e f f i c i e n t s . Although t h i s r e -d u c t i o n a f f o r d s s u b s t a n t i a l savings i n the number of elements to be coded, i t i s o f t e n convenient, from a computational p o i n t of view, to f i r s t p a r t i t i o n the o r i g i n a l data a r r a y i n t o sub-blocks of s i z e N < L and to code each sub-block s e p a r a t e l y - the r e c o n s t r u c t e d image being r e a l i z e d by simply reassembling the sub-blocks. F i g . 4.1 shows such p a r t i t i o n i n g of a data a r r a y . (, /_ j, M*l s H-1 5M s 2M V S 2 M F i g . 4.1 P a r t i t i o n i n g of Data Array i n t o M Sub-blocks Such b l o c k coding schemes have been s t u d i e d i n the past [14], [ 2 3 ] , along w i t h the much more s o p h i s t i c a t e d problem of a s s i g n i n g b i t s to the se l e c t e d transform c o e f f i c i e n t s [ 1 7 ] , [24]. 4.2 Quantizing the Transform C o e f f i c i e n t s As mentioned e a r l i e r , a primary m o t i v a t i o n f o r employing t r a n s form coders i s the f a c t t h a t the transform c o e f f i c i e n t s can be e f f i c i e n t l y quantized. This i s a d i r e c t consequence of the f a c t t h a t due to the uncorrelatedness of the c o e f f i c i e n t s , q u a n t i z e r s can be designed to i n d i v i d u a l l y operate on the transform c o e f f i c i e n t s . A l s o , s i n c e the va r i a n c e s of the c o e f f i c i e n t s a r e very d i v e r s e , i t would be i n e f f i c i e n t to use the same quan t i z e r f o r a l l c o e f f i c i e n t s . P e a r l et a l . [ 2 5 ] has r e f e r r e d to such a scheme of l i n e a r t r a n s formation and i n d i v i d u a l q u a n t i z a t i o n as that of a 'basis r e s t r i c t e d t r a n s f o r m a t i o n ' . The Encoder-Decoder c h a i n i s as shown i n F i g . 4.2, where the input v e c t o r f i s represented as a s u p e r p o s i t i o n of b a s i s v e c t o r s . LINEAR TRANSFORM INVERSE F LINEAR TRANSFORM F i g . 4.2 Schematic of Basis R e s t r i c t e d Transformation Since i n a transform coding scheme one i d e a l l y considers the transform c o e f f i c i e n t s to be s t a t i s t i c a l l y independent, one can consider a s e r i e s of N encoders to operate i n d i v i d u a l l y on each of the transform c o e f f i c i e n t s F_^ . The q u a n t i z a t i o n s t r a t e g y c o n s i s t s of the mapping F = Q(F). The scheme i s r e s t r i c t i v e i n the sense that each operator i n v o l v e s only one input F^ and one output F . I t i s t h i s f a c t which allow s f o r the computational s i m p l i c i t y of the scheme. Since the c o e f f i c i e n t s w i t h the l a r g e r v a r i a n c e s c o n t r i b u t e s i g n i f i c a n t l y more to a reconstructed image, i t i s expected that d i s -t o r t i o n due to q u a n t i z a t i o n can be lessened by a s s i g n i n g more qu a n t i z a -t i o n l e v e l s to c o e f f i c i e n t s w i t h the l a r g e r v a r i a n c e s and p r o p o r t i o n a l l y fewer to the c o e f f i c i e n t s w i t h the smaller v a r i a n c e s . Huang and S c h u l t h e i s s [17] and Wintz and Kurtenbach [24] have considered the pro-blem of o p t i m a l l y q u a n t i z i n g the u n c o r r e l a t e d c o e f f i c i e n t s , assumed to be Gaussianly d i s t r i b u t e d , r e l a t i v e to a mean square e r r o r c r i t e r i o n . The Gaussian assumption i s not u n j u s t i f i e d and owes i t s v a l i d i t y to the summing operations i n the l i n e a r t r ansformations. As expected, the c o e f f i c i e n t s w i t h the l a r g e r v a r i a n c e s are coded w i t h a higher number of b i t s . 4.3 Block Q u a n t i z a t i o n The scheme, c o n s i s t i n g of the i n i t i a l block coding of the image data, followed by the assignment of a t o t a l of M b i t s (say) to a set of n samples of unequal v a r i a n c e s , i s u s u a l l y r e f e r r e d to as Block Quanti-z a t i o n . The b i t assignment .algorithm as proposed by Kurtenbach and Wintz [24] , and which was adotped i n the study, i s given below: 1) F i r s t compute the n numbers M = ^ + 2 l o g a, 2- £ I l o g a, 2 (4.1) i = 1, 2 ... n Each M_^(i = 1, 2 ... n) i s rounded to the nearest i n t e g e r , n 2) I f I M. 4 M then a r b i t r a r i l y adjust some of the M.'s i = l according to the f o l l o w i n g r u l e s : n i ) I f M < I M. 1=1 1 take that corresponding to the l a r g e s t i such that > 1 and re p l a c e i t by - 1 n i i ) I f M > 7 M." i - 1 1 take that corresponding to the smallest i such that M. = M and re p l a c e i t by M. + 1 l n x .The above a l g o r i t h m produces b i t v a l u e s , M , such t h a t f o r 2 2 2 2 1 2 - 3 - n M > > M„ > ^ M 1 2 - 3 n A s i m i l a r a l g o r i t h m was proposed by Huang and S c h u l t h e i s s [17], but whereas Kurtenbach and Wintz considered an optimum uniform q u a n t i z e r , Huang and S c h u l t h e i s s considered an optimum non-uniform q u a n t i z e r . Since an optimum uniform quantizer i s much e a s i e r to implement than an optimum non-uniform quantizer and s i n c e the former's performance i s a almost as good as that of the non-uniform quantizer f o r Gaussian da t a , Kurtenbach and Wintz's a l g o r i t h m was used i n the study. A l s o Wood [26], has shown that the optimum uniform quantizer minimizes the entropy of the quantizer output f o r a l a r g e c l a s s of d i s t o r t i o n measures provided the q u a n t i z i n g i s r e l a t i v e l y f i n e . Since proper encoding enables a s i g n a l to be trans m i t t e d at an average b i t r a t e approaching the s i g n a l entropy [27], the choice of the uniform quantizer i s app r o p r i a t e . 4.4 Variances of Transform C o e f f i c i e n t s The va r i a n c e s of both the F o u r i e r and Hadamard transform co-e f f i c i e n t s were needed f o r the e v a l u a t i o n of the r e l a t i v e performances of the transformations as w e l l as i n the Block Q u a n t i z a t i o n process. As defined i n Sect. 3.4 and repeated here f o r convenience, the v a r i a n c e of the transform c o e f f i c i e n t s i s given by: a 2(u,v,) = E[F(u,v), F*(u,v)] = 1 1 I I R(x,x',y,y') 4 (u,v,x,y) <f>*(u,v,x,y) (4.2) N N N N I ] y=l y' where, as before R(x,x',y,y') = e - ° i x " X | ; e " P l y ~ y' ' (4.3) and the b a s i s f u n c t i o n s <j>(u,v,x,y) are as p r e v i o u s l y defined (3.4), (3.8). The s e p a r a b i l i t y of the a u t o c o r r e l a t i o n f u n c t i o n lends e a s i l y to a simple computational procedure and i t i s e a s i l y seen that 2 2 2 a (u,v) = aa (u). 0 ^ (v) (4.4) The exact mathematical d e r i v a t i o n s f o r the var i a n c e s of both transforma-t i o n s w i l l not be presented here but may be found i n Appendix C. The variances of the transform c o e f f i c i e n t s , ordered according to t h e i r magnitude, are shown f o r each of the s i g n a l planes of two p i c t u r e s employed i n the study i n F i g s . 4.3 and 4.4. F i g . 4.4 Variances of transform c o e f f i c i e n t s ordered according to t h e i r magnitude. 29. 4.5 Q u a n t i z a t i o n E r r o r I t has p r e v i o u s l y been shown, Sect. 3.4, that a f a i r l y good approximation of an o r i g i n a l image may be obtained when the low v a r i a n c e transform c o e f f i c i e n t s are di s c a r d e d . T h i s t r u n c a t i o n r e s u l t e d i n a l o s s i n image energy and was c h a r a c t e r i z e d by the mean square a p p r o x i -2 mation e r r o r , e , as defined p r e v i o u s l y (3.14). This was reduced to c l the form G 2 = R(0,0) - l I a 2(u,v) (4.5) a N u=0 v=0 Further d i s t o r t i o n i s introduced during the q u a n t i z a t i o n 2 xs cnaraccerxzea Dy tne quantxzatxon e r r o r the r e l a t i o n process and i s c h a r a c t e r i z e d by the q u a n t i z a t i o n e r r o r e ^ , defined by e 2 = E{ I I [F(u,v) - F ( u , v ) ] 2 (4.6) q N u=l v=l The q u a n t i z a t i o n e r r o r depends on the quantizer s t r u c t u r e , i n c l u d i n g the number of q u a n t i z a t i o n l e v e l s , t r a n s i t i o n l e v e l spacing, the q u a n t i z a t i o n values and the s t a t i s t i c s of the input samples. There has been con-s i d e r a b l e a t t e n t i o n devoted to the design and a n a l y s i s of performance of optimum (Uniform and Non-Uniform) qua n t i z e r s [26], 128], [29]. From published r e s u l t s [17], [24], [14], i t has been shown that the q u a n t i -z a t i o n e r r o r f o r the optimum (minimum mean square e r r o r ) uniform q u a n t i -zer i s 2 2. . -m(u,v)/2 e = a (u,v). 10 (4.7) q where m(u,v) i s the number of b i t s assigned to sample F(u , v ) . Further, Totty and C l a r k [30] have shown that the t o t a l system 2 mean square e r r o r , z , may be decomposed i n t o the sum of three p a r t s , namely 30. 2 2 2 2 e = E + e + E z (4.8) a q c 2 2 2 where e & , are as p r e v i o u s l y defined and i s the channel e r r o r . 2 In the study an e r r o r l e s s channel was assumed i . e . = 0, so that (4.8) reduces to 2 2 2 e = e + e (4.9) a q where e = i E { I I [ f ( x , y ) - f ( x , y ) ] Z } (4.10) N x=l y=l s u b s t i t u t i n g (4.5) and (4.7) i n t o (4.9) we get e 2 = R(0,0) - ^ I I a 2 ( u , v ) [ l - 1 0 ^ ( u » v ) / 2 ] (4.11) N u=l v=l 2 As a measure of performance, values of e were computed f o r v a r i o u s b i t assignments when both the F o u r i e r and Hadamard transformations were employed. This computation was done f o r each s i g n a l plane of each p i c t u r e as i l l u s t r a t e d i n F i g s . 4.5 - 4.8. 4.6 A Note on Optimum Quantizers As mentioned p r e v i o u s l y , a great d e a l of e f f o r t has been devoted to the design and performance e v a l u a t i o n of optimum quantizers [26], [28]. In our study transform samples were quantized to a max-imum of eight b i t s . M ainly f o r comparison purposes, some p i c t u r e s were processed us i n g both the optimum uniform and optimum non-uniform quanti-z e r s . Max [28] published q u a n t i z a t i o n s t r u c t u r e s having a maximum of f i v e b i t s . I t was t h e r e f o r e necessary to compute q u a n t i z a t i o n l e v e l s f o r s i x , seven and eight b i t assignments. A computational a l g o r i t h m f o r the optimum non-uniform quantizer may be found i n Appendix D. 33. 34. V. EXPERIMENTS AND RESULTS 5.1 E s t i m a t i o n of C o r r e l a t i o n C o e f f i c i e n t s a, B As seen e a r l i e r , values of the c o r r e l a t i o n c o e f f i c i e n t s i n both the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s of the p i c t u r e data are needed f o r modelling the image. These values were computed usi n g the r e l a t i o n s h i p ( i n c o n s i d e r i n g the h o r i z o n t a l d i r e c t i o n ) . .. N-T P X ( X , T ) = ± I [ f ( x , y ) - f ( x , y ) ] - [ f ( x , y + x) - f ( x , y ) ] 7 = 1 (5.1) 2 a where x = 1,2 .. . N x = 1, ... 16 2 and a i s the varia n c e of the image data. The values f o r each c o r r e l a t i o n d i s t a n c e ("0 were then averaged along the rows (x) and an exponential l e a s t squares f i t was made to - the p o i n t s . This f i t followed the equation: p x ( t ) = e " a ! T l (5.2) from which a was obtained. S i m i l a r l y the c o r r e l a t i o n c o e f f i c i e n t (3) i n the v e r t i c a l d i r e c t i o n was obtained. In the study, four p i c t u r e s were employed. These were chosen to c h a r a c t e r i z e r e s p e c t i v e l y , p i c t u r e s of h i g h , medium and low d e t a i l . A t e s t p a t t e r n was a l s o considered. These p i c t u r e s chosen were as f o l l o w s FACE - Low d e t a i l HOUSE - Medium d e t a i l GARDEN - High d e t a i l TEST PATTERN. Values of a , f3 were obtained f o r each of the Y, I , Q s i g n a l planes of each p i c t u r e . B i g s . 5.1 - 5.4 show measured and estimated values of the c o r r e l a t i o n f u n c t i o n s . 5.2 Parameters Under C o n s i d e r a t i o n 5.2.1 Block S i z e s Two d i f f e r e n t b l o c k s i z e s were i n v e s t i g a t e d f o r both of the transform coding schemes employed. These were block, s i z e s of 8 x 8 and 16 x 16 p i c t u r e elements. The choice of these b l o c k s i z e s was based on r e s u l t s of previous researchers [14], [7] and indeed, based on the sen-s i t i v i t y of the human v i s u a l system as a f u n c t i o n of s p a t i a l frequency, i t has been shown that a b l o c k s i z e of 16 x 16 may, i n f a c t , be an o p t i -mum choice [23]. 5.2.2 B i t Assignments As mentioned i n Sect. 2.4.2, a major c o n s i d e r a t i o n f o r u s i n g the Y,I,Q t r a n s m i s s i o n co-ordinate system was the f a c t that the chromi-nance components I , Q can be s e v e r e l y band l i m i t e d w h i l e s t i l l r e t a i n -ing a r e l a t i v e l y h i g h s u b j e c t i v e q u a l i t y . For both coding schemes, and f o r each block s i z e considered, b i t assignments of 0.375 and 0.5 b i t s / p e l were i n v e s t i g a t e d f o r the I , Q components. The luminance s i g n a l , Y component, i s more s u s c e p t i b l e t o image degradation f o r small b i t assignments. In order to i n v e s t i g a t e the s e v e r i t y of degradation f o r each of the parameters considered above, b i t assignments of 3.0, 2.0 and 1.0 b i t s / p e l . were i n v e s t i g a t e d f o r the Y s i g n a l . 37. SIGNRL:FACE X HORIZONTAL A VERTICRL RLPHR= 0.047 BETH =0.045 X HORIZONTAL A VERTICRL RLPHR= 0.042 BETR =0.052 DISTANCE DISTANCE 38 39. Q SIGNAL'.GARDEN X HORIZONTAL A VERTICAL ALPHA= 0.229 BETA =0.390 F i g . 5.3 Measured c o r r e l a t i o n i n each s i g n a l plane and l e a s t squares exponen-t i a l f i t . Smooth l i n e s are the f i t t e d curves. DISTANCE 40. Y SIGNAL .-TEST PAT U j c i C C C C o X HORIZONTAL A VERTICAL ALPHA= 0.020 BETA -: 0.020 U J o C C cc a B 12 DISTANCE 1E — l 20 I SIGNAL:TEST PAT X HORIZONTAL A VERTICAL A L P H A S 0.021 BETH =0.016 DISTANCE - r ~ 12 I 16 ee O X HORIZONTAL A VERTICAL ALPHA= 0.040 BETA = 0.021 F i g . 5.4 Measured c o r r e l a t i o n i n each s i g n a l plane and l e a s t squares exponen-t i a l f i t . Smooth l i n e s are the f i t t e d curves. i DISTANCE 12 16 —I 20 41. 5.2.3 Q u a n t i z a t i o n For reasons o u t l i n e d e a r l i e r , Sect. 4.3, f o r each block s i z e and b i t assignment considered, images were processed u s i n g the optimum uniform q u a n t i z e r . In order to evaluate d i f f e r e n c e s i n performance, some images were processed u s i n g the optimum non-uniform q u a n t i z e r . 5.3 System Implementation The o r i g i n a l p i c t u r e s were d i g i t i z e d on a 256 x 256 g r i d through Red (#65), Green (#50-55) and Blue (#50) i n t e r f e r e n c e f i l t e r s (Spectracoat V a r i p a s s f i l t e r s - Optics Inc., U.S.A.). Each p i c t u r e point was quantized to a l e v e l of eight b i t s ; the bri g h t n e s s i n t e n s i -t i e s ranged from 0 - 255. The e n t i r e image coding and tr a n s m i s s i o n system, shown i n F i g . 1.1, was simulated on an IBM Model 370/68 computer u s i n g the MTS (Michigan Terminal System) Operating System. Processed images were d i s p l a y e d using a f l y i n g spot scanner c o n t r o l l e d by a Data General Nova 840 minicomputer. The di s p l a y e d p i c t u r e s were made as t r i p l e exposures through the above mentioned Red, Green and Blue i n t e r f e r e n c e f i l t e r s u sing Kodak Ektachrome EX—120 f i l m . For purposes of i l l u s t r a t i o n P l a t e s I - V show hard copies of the processed images made from the colour s l i d e s . 5.4 Performance C r i t e r i a For the colour t r a n s m i s s i o n scheme employed i n the design of. the coding and q u a n t i z a t i o n schemes, the performance c r i t e r i o n considered was the mean square e r r o r between the o r i g i n a l and reco n s t r u c t e d images. Hence, as one means of comparing the performances of the F o u r i e r and Hadamard coding systems, the mean square e r r o r between the o r i g i n a l and PICTURE #2 PICTURE #3 PICTURE #4 PICTURE #5. PICTURE #6 5.5 Format of illustrations in Plates I - V. PLATE I O r i g i n a l t e s t image R,G,B s i g n a l s each represented by 8 . 0 b i t s / p e l . T o t a l b i t assignment «• 2 4 . 0 b i t s / p e l . O r i g i n a l t e s t image. R,G,B s i g n a l s each represented by 8 . 0 b i t s / p e l . T o t a l b i t assignment = 2 4 . 0 b i t s / p e l . O r i g i n a l t e s t image R,G,B s i g n a l s each represented by 8 . 0 b i t s / p e l . T o t a l b i t assignment = 2 4 . 0 b i t s / p e l . O r i g i n a l Test image R,G,B s i g n a l s each represented by 8 . 0 b i t s / p e l . T o t a l b i t assignment = 2 4 . 0 b i t s / p e l . F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2 b i t s / p e l ; I,Q = 0 . 3 7 5 b i t s / p e l . T o t a l b i t assignment = 2 . 7 5 b i t s / p e l . Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2 . 0 b i t s / p e l ; I,Q = 0 . 3 7 5 b i t s / p e l . T o t a l b i t assignment - 2 . 7 5 b i t s / p e l . 45. PLATE I I 1. F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 2.75 b i t s / p e l . 2. Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 2.75 b i t s / p e l . 3. F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q - 0.375 b i t s / p e l . T o t a l b i t assignment = 2.75 b i t s / p e l . 4. Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q 0.375 b i t s / p e l . T o t a l b i t assignment - 2.75 b i t s / p e l . 5. F o u r i e r Coding: B l o c k s i z e 16 x 16: Non-uniform Quantizer Y = 2.0 b i t s / p e l ; 1,0 = 0.375 b i t s / p e l . T o t a l b i t assignment = 2.75 b i t s / p e l . 6. Hadamard Coding: B l o c k s i z e 16 x 16: Non-uniform Quentizer Y = 2.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 2.75 B i t s / p e l . PLATE I I I F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1-0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q » 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . 49. PLATE IV 1. F o u r i e r Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y = 1.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . 2. Hadamard Coding: B l o c k s i z e 16 x 16: Uniform Quantizer Y =1.0 b i t s / p e l . ; I,Q = 0.375 b i t s / p e l . T o t a l b i t assignment = 1.75 b i t s / p e l . 3. F o u r i e r Coding B l o c k s i z e 8 x 8 : Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q = 0.5 b i t s / p e l . T o t a l b i t assignment - 3.0 b i t s / p e l . 4. Hadamard Coding: B l o c k s i z e 8 x 8 : Uniform Quantizer Y = 2.0 b i t s / p e l . I,Q = 0.5 b i t s / p e l . T o t a l b i t assignment = 3.0 b i t s / p e l . 5. F o u r i e r Coding: B l o c k s i z e 8 x 8 : Uniform Quantizer Y. = 2.0 b i t s / p e l ; I,Q = 0.5 b i t s / p e l . T o t a l b i t assignment = 3.0 b i t s / p e l . 6. Hadamard Coding: B l o c k s i z e 8 x 8 : Uniform Quantizer Y = 2.0 b i t s / p e l ; I,Q = 0.5 b i t s / p e l . T o t a l b i t assignment = 3.0 b i t s / p e l . 51. PLATE V 1. F o u r i e r Coding C i r c u l a r Zonal Coding w i t h 16 x 16 b o l c k s i z e s Y = 3.0 b i t s / p e l ; I,Q = 0.5 b i t s / p e l . T o t a l equivalent b i t assignment = 4.0 b i t s / p e l . 2. Hadamard Coding Hyperbolic Zonal Coding i n 256 x 256 format Y = 2.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l equivalent b i t assignment = 2.75 b i t s / p e l . 3. F o u r i e r Coding C i r c u l a r Zonal Coding i n 256 x 256 format Y = 3.0 b i t s / p e l ; r I,Q = 0.375 b i t s / p e l . T o t a l equivalent b i t assignment = 3.75 b i t s / p e l 4. Hadamard Coding Hyperbolic Zonal Coding i n 256 x 256 format Y = 3.0 b i t s / p e l ; . I,Q = 0.375 b i t s / p e l . T o t a l equivalent b i t assignment— 3.75 b i t s / p e l . 5. F o u r i e r Coding Smoothed F o u r i e r coded image usi n g 16 x 16 b l o c k s i z e Y = 3.0 b i t s / p e l ; I,Q = 0.5 b i t s / p e l . T o t a l b i t assignment = 4.0 b i t s / p e l . 6. Hadamard Coding Hyperbolic Zonal Coding i n 256 x 256 format Y = 3.0 b i t s / p e l ; I,Q = 0.375 b i t s / p e l . T o t a l equivalent b i t assignment = 3.75 b i t s / p e l . reconstructed p i c t u r e s on each of the three colour s i g n a l planes was measured. Since the f i n a l r e constructed images were formed from the Red, Green and Blue s i g n a l planes, i t was decided to measure the mean square e r r o r s i n these planes. The measure adopted was that of a normal-i z e d mean square e r r o r defined as f o l l o w s f o r the Red s i g n a l : N I;N ±~ I I [R(x,y) - R ( x , y ) ] Z E r2 - N _ x f l ^ l _ _ _ _ _ ( 5 3 ) I I R 2(x,y) - [I- l I R ( x , y ) ] 2 N x=l y=l N x=l y=l The mean square e r r o r s f o r the Green and Blue planes are s i m i l a r l y d e f i n e d . The n o r m a l i z i n g f a c t o r , as seen i n the above equation i s simp-l y the va r i a n c e of the s i g n a l plane. This n o r m a l i z a t i o n was done i n order to d e f i n e an o v e r a l l mean square e r r o r as f o l l o w s : 2 2 2 2 £ R 4 + £ G + E B e T = 3 (5.4) Ca l c u l a t e d values showed that there was u s u a l l y a wide v a r i a t i o n i n the values obtained f o r the Blue plane as compared w i t h values obtained i n the Red and Green planes, w i t h the r e s u l t that an o v e r a l l mean square e r r o r sometimes gave a somewhat misl e a d i n g r e s u l t . T y p i c a l cases are as shown i n F i g s . 5.6(a), 5.6(b). For both transform coding schemes employed and under v a r y i n g b i t assignments f o r the two block s i z e s s t u d i e d , graphs of mean square e r r o r v s . number of b i t s / p e l . were p l o t t e d . T y p i c a l graphs are as shown i n F i g s . 5. -5.12. Yet another set of mean square e r r o r s were c a l c u l a t e d to 0.135-0.120-0.105-cn 0.090• a in s |o .07S-•—I cr g z 0.060H 0.045H 0.030 HBUSE;BLBCKS17.E 16X16 Y VARYING 1.0 V1TH 0.5 BPEL FBURIER A RED X GREEN Q BLUE • OYERRLL 1.0 1 I 1 2.0 i—|—i—i—i—i—|—i—i—r* 3.0 4.0 7.o 0.145 0.130-0.1)5-^ g CC U J cc 0.100• in XL. ^0.085-d 63 0.070 H J.055-0.04D FRCEiBLQCK5IZE 16X16 Y VRRYING 1.0 VJTH 0.5 6PEL FOURIER A RED X GREEN • BLUE • OYERRLL i i i i | i i i • | i 1.0 2.0 3.0 4.0 7.o TOIRL BITS PER PICT. ELEMENT TOTAL BITS PER PICT. ELEMENT (a) 00 F i g . 5.6 (a) Mean square e r r o r v s . b i t s per p e l . Mean square e r r o r s approximately the same f o r each colour plane. F i g . 5.6 (b) Mean square e r r o r v s . b i t s per p e l . Mean square e r r o r s d i f f e r e n t f o r each colour plane. RED SIGNALiFRCE A FOURIER X HflDRNfiRD BLOCKSIZE 16X16 O.OIO — I— i — i — i — I — i— I— i— r — j — I— i i i — p -1.0 2 . 0 3 .0 4.0 TOTAL BITS PER PICT. ELEMENT 0 .100 0 . O B 5 H | 0.070 •0.055H S in g 0 .040 0 . 0 2 5 M i — l 0 . 0 1 0 -5 . 0 1.0 GREEN SIGNflLiFRCE A FOURIER X HflDPMfKD BLOCKSIZE 16X16 ' I • • 2 . 0 3 . 0 r — [ — i — r - i -4 . 0 5.0 TOTAL BITS PER PICT. ELEMENT 0 . 1 4 0 T BLUE SIGNAL;FACE A FOURIER X HflOflMARO BLOCKSIZE 16X16 0 .020 | • i—r-1.0 T—|—I—I—r—i—|—i—i—i—r—|—i—i—i—i—| 2 . 0 3 .0 4 . 0 5 . 0 TOTAL BITS PER PICT. ELEMENT F i g . 5.7 Mean square e r r o r v s . b i t s , per p e l . Y = 3.0", 2.0, 0.1 b i t s / p e l . I , Q - 0.5 b i t s / p e l . 1.0 RED SJGNflLsTEST PflT. A FOURIER X HHDRMRRD BLOCKSIZE 16X16 ' I ' 1 ' ' I ' 2.0 3.0 4.0 TOTAL BITS PER PICT. ELEMENT 0.03S 0.030 S 0 . 0 2 5 H 5 0.020 s in So.ois-0.010 i — | o.oos-5.0 1.0 GREEN SIGNPI.:TEST PAT. A FOURIER X HflORMPKO BLOCKSIZE 16X16 1 I " • • ' I 1 2.0 3.0 1 I 1 4.0 TOTAL BITS PER PICT. ELEMENT 7.0 0.040-, BLUE SIGNAL:TEST PAT. A FOURIER X HROAMARD BLOCKSIZE 16X16 F i g . 5.8 Mean square e r r o r v s . b i t s per p e l . Y = 3.0, 2.0, 1.0 b i t s / p e l . I , Q = 0.5 b i t s / p e l . \ — I — i — - i — i — r -2.0 ' I 1 3.0 1 I 1 ' • ' I 4.0 S.O TOTAL BITS PER PICI. ELEMENT RED SIGNRLiHOUSE A FOURIER X HRQUMRKD BLOCKSIZE 16X16 \ \ V . J.OIO - j — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — I 1.0 2.0 3.0 4.0 5.0 TOTRL BITS PER PICT. ELEMENT O.lOO-i 0.055 H S0.070H 50.035-g 0 . 0 4 M J.010 GREEN S1GNRL;H0USE A FOURIER X HRORMRRO BLOCKSIZE 16X16 \ \ \ i — i — i — i — | — i — r 1.0 2.0 IT T — i — i — I — i — i — r -4.0 TOTRL BITS PER PICT. ELEMENT 7.0 BLUE SIGNRL:HOUSE A FOURIER X HRDRHflRD BLOCKSIZE 16X16 F i g . 5.9 Mean square e r r o r v s . • b i t s per p e l . Y = 3.0, 2.0, 1.0 b i t s / p e l . I , Q = 0.375 b i t s / p e l . V — i i — i — i — i — i — i — i — | — i — i — i — i — | — i — i — i — i — I 1.0 -.0 3.il 4.0 5.0 TOTRL BITS PER PICT. ELEMENT RED SJGNfitiGfWOEM A FOURIER X HfiDflMRRD BLOCKSIZE 16X16 1 w w \ \ \ x \ \ \ \ \ \ K \ \ \ \ 1.0 ' ' ' I ' ' ' ' I ' 2.0 3.0 0.160 0.143 go.120 a CC a g 0.100 go.oetH 0.060 r _ T - 1 4.0 TOTAL BITS PER PICT. ELEMENT ' • | 0.040 5.0 1.0 GREEN SIGNflLiGPROEN A FOURIER X HRDftHARO 4 BLOCKSIZE 16X16 V \ \ \ \ \ » \ » \ ' \ \ > \ v \\ \ \ \ \ \ -> I l ' ' — • — I — I — • — • — i , , , 1 2-0 3.0 4.0 5.0 TOTAL BITS PER PICT. ELEMENT BLUE SIGNAL-.GAROEN A FOURIER X HflDRMARD BLOCKSIZE 36X16 F i g . 5.10 Mean square e r r o r v s . b i t s per p e l . Y = 3.0, 2.0, 1.0 b i t s / p e l . I , Q = 0.375 b i t s / p e l . * * N . X 0.560-1.0 2.0 3.0 r—r~> 4.0 TOTAL BITS PER PICI. ELEMENT ^1 5.0 0 . 1 0 0 I i • • 1 . 0 RED SIGNflLiFRCE A FOURIER X HflDflMfKO BLOCKSIZE BXB \ \ ' I ' 1 1 ' I 1 2 . 0 3 . 0 1 I ' 4 . 0 TOTAL BITS PER PICT. ELEMENT 0 . 3 5 0 - 1 0 . 3 2 5 -E 0 . 3 0 0 • 5 0 . 2 7 5 S K 0 . 2 5 0 0 . 2 2 5 H i — | 0 . 2 0 0 - | — i — • — i — i — | — i 5 . 0 1 . 0 2 . 0 GREEN SIGNRL-.FRCE A FOURIER X HRDHHRRD BLOCKSIZE BXB \ \ r — i — i — | — i — r 1 J 3 . 0 ' I ' 1 ' • I 4 . 0 5 . 0 TOTAL BITS PER PICT. ELEMENT BLUE SIGNRL:FRCE A FOURIER X HflORMRRD BLOCKSIZE BXB F i g . 5.11 Mean square e r r o r v s . b i t s per p e l . Y = 3,0, 2.0, 1.0 b i t s / p e l . I , Q = 0.5 b i t s / p e l . \ 0 . 3 0 0 | . i 1.0 2 . 0 3 . 0 1 I ' 4 . 0 TOTAL BITS PER PICT. ELEMENT 5.0 4 \ \ \ \ \ \ * RED SIGNAL:Gf«OEN A FOURIER X HHOAMARO BLOCKSIZE BXB \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ * \ \ 0 . 3 4 0 - . 0.300 g 0.260 (X a to 5 0 .220 g o . i e o 0.143 1 1 1 j 1 1 1 1 1 1 1 ) 1 1 r-1.0 2.0 3.0 4.0 TOTAL BITS PER PICT. ELEMENT 5.0 0.100 1.0 GREEN S1GNAL:GAR0EN A FOURIER X HROHMfiRO BLOCKSIZE BXB \ \ \ \ \ * \ \ -\ \ -\ \ \ \ \ N A \ \ \ \ \ \ 2 - ° 3.0 4.0 s'.O TOTAL BITS PER PICT. ELEMENT BLUE SIGNAL:GARDEN A FOURIER X HADAMARD BLOCKSIZE BXB F i g . 5.12 Mean square e r r o r v s . b i t s per p e l . Y =3.0, 2.0, 1.0 b i t s / p e l I , Q = 0.5 b i t s / p e l . \ \ \ V ' I 1 2.0 1.0 3.0 1 I ' ' 4.0 TOTAL BITS PER PICT. ELEMENT 5 . 0 61. O.OTO-i 0 . 0 H H S a a. • 0 . 0 5 0 -SO 0 3 0 -0 . 0 2 0 H 0 . 0 1 0 -Y SJGNALiFRCE A FOURIER X HHQAMARO BLOCKSIZE 16X16 I .0 1 i 1 2 .0 T-3 . 0 1 1 1 1— 8MS PER P I C T . ELEMENT O.lOO - i 0 .085 £ 0 .070 ft 0 .055 4 . 0 0.025H O.OJO-Y SIGNAL:HOUSE A FOURIER X HROHMARD BLOCKSIZE 16X16 \ \ \ V. . I 1 1 .0 ' I 1 2.0 3 .0 ' ' I 4 . 0 BITS PER PICT. ELEMENT 0 . 1 3 0 -O.llOH §0.090-1 IX. 1 • 0 . 0 7 0 - J K 0 . 0 3 H 0 . 0 3 0 H Y SIGNAL-.GARDEN A FOURIER X HRORMORO BLOCKSIZE 16x16 \ \ \ H \ \ \ \ \ 0.035-1 0.030 cc §0 .025 - ) 5 a. : 0 .020 H 1 0 . 0 1 5 0.010 0 . 0 1 0 - 1 — i — • — i — i — I — i — i — i — i — I— i — i — i — ' — 1 — • — • — < — ' — I 0 . 0 0 5 -1.0 2 . 0 3 .0 4 . 0 BITS PER PICT. ELEMENT Y SIGNAL:TEST PflT. A FOURIER BLOCksi'HE 16X16 V.. ' 1 I 1 .0 2 .0 ' ' I ' 3 . 0 BITS PER P I C ; , ELEMENT 4 . 0 F i g . 5.13 Mean square e r r o r v s . b i t s per p e l . Performance of systems as given by experimental r e s u l t s . 62. f u r t h e r evaluate the. performance of the system. Since the coding s t r a t e g y was based on the Y, I , Q tran s m i s s i o n co-ordinate system, i t was decided to measure the e r r o r i n these s i g n a l planes p r i m a r i l y to compare w i t h t h e o r e t i c a l r e s u l t s . T y p i c a l graphs are shown i n F i g . 5.13. V I . PERFORMANCE ANALYSIS 6.1 I n t r o d u c t i o n As i n any image coding system, the u l t i m a t e measure of perform-ance i s s u b j e c t i v e preference. Because of the tedious task i n v o l v e d i n s u b j e c t i v e l y e v a l u a t i n g processed images, many researchers have sought to formulate o b j e c t i v e performance measures which agree c l o s e l y w i t h sub-j e c t i v e t e s t s [31], [15]. No one o b j e c t i v e performance measure has yet been shown to be completely c o n s i s t e n t w i t h s u b j e c t i v e r a t i n g s ; the o b j e c t i v e measure most commonly adopted, the mean square e r r o r between o r i g i n a l and recon s t r u c t e d images, was chosen i n t h i s study. I t should be mentioned that a mean square e r r o r e v a l u a t i o n was not the only per-formance measure adopted. Throughout the study many parameters were v a r i e d , e s p e c i a l l y i n the b i t assignment s t r a t e g y , on the b a s i s of the author's own s u b j e c t i v e preference. 6.2 F o u r i e r v s . Hadamard As shown i n F i g s . 5.6-5.12, the Hadamard system e x h i b i t e d a smaller mean square e r r o r than the F o u r i e r system f o r the same b i t assignment on each of the three s i g n a l planes considered. Thus, con-s i d e r i n g a l l p i c t u r e s processed under v a r y i n g b i t assignments and f o r the two bl o c k s i z e s s t u d i e d , one can c o l l e c t i v e l y say th a t the Hadamard transform coding system performed b e t t e r than d i d the F o u r i e r system, based on the mean square e r r o r s c a l c u l a t e d between the o r i g i n a l and processed images. T h i s s u p e r i o r i t y of performance, on the p a r t of the Hadamard system, agrees w e l l w i t h a s u b j e c t i v e assessment of the p i c t u r e s as evidenced i n p l a t e s I .V. In most cases, e s p e c i a l l y i n the p i c t u r e s of the FACE and HOUSE, there was always a pronounced d i f f e r e n c e i n favour of the Hadamard system i n the q u a l i t y of the reconstructed images. In both the GARDEN and TEST PATTERN p i c t u r e s , t h i s d i f f e r e n c e was l e s s pronounced. , 6.3 B i t Assignment Strategy As mentioned i n Sect. 4.3, a b i t assignment a l g o r i t h m was used which assigned more b i t s to the h i g h v a r i a n c e samples and p r o p o r t i o n a l l y fewer to those of low v a r i a n c e . For cases where the t o t a l computed b i t n value ( J M.) was l e s s than the proposed val u e (M), the f o l l o w i n g r u l e 1=1 1 f o r a d j u s t i n g some of the assigned v a l u e s was given: n i f M > I M. 1=1 1 take that corresponding to the smallest i such that M. = M and r e p l a c e i t by M. + 1. i n r J I During the course of the experiments i t was found that b e t t e r r e s u l t s were obtained, p u r e l y from a s u b j e c t i v e viewpoint, i f the above r u l e was m o d i f i e d to read: n i f M > I M 1=1 take that corresponding to the smallest i such that < 6 and r e p l a c e i t by 6. I n v a r i a b l y the computed value was l e s s than the proposed val u e so that t h i s modified r u l e had the e f f e c t of reducing the number of samples r e t a i n e d w h i l e i n c r e a s i n g the r e s o l u t i o n of those samples that were r e t a i n e d . Another important aspect of the assignment s t r a t e g y was the f a c t that the d.c. component of the transform samples, F ( 0 , 0 ) , was always quantized to 8 b i t s , r e g a r d l e s s of the v a l u e as assigned by- the b i t assignment a l g o r i t h m . Thus the average v a l u e of each sub-block was always a c c u r a t e l y transmitted thereby h e l p i n g to decrease the 'edge e f f e c t s ' as discussed i n the next s e c t i o n . 6.4 Edge E f f e c t s The obvious drawback of the b l o c k coding scheme, as can be seen i n the processed p i c t u r e s was that the blocked s t r u c t u r e was some-times v i s i b l e i n some areas of the processed images. These sub-block boundaries - the so c a l l e d 'edge e f f e c t s ' - were more pronounced i n the F o u r i e r coded images. In the F o u r i e r r e p r e s e n t a t i o n these e f f e c t s a r i s e mainly because of the edge-to-edge d i f f e r e n c e s i n b r i g h t n e s s l e v e l s of any sub-block s i n c e each s u b - b l o c k . i s considered to be p e r i o d i c . Thus any exge-to-edge d i f f e r e n c e s appear as sharp d i c o n t i n u i t i e s i n the p e r i o d i c f u n c t i o n and ' r i n g i n g ' occurs. Aside from these edge-to-edge d i f f e r e n c e s , any l a r g e t r a n s i s t i o n i n b r i g h t n e s s l e v e l s between a d j a -cent p i c t u r e p o i n t s were a l s o p o o r l y reproduced i n both systems. This behaviour can probably be a t t r i b u t e d to the f i l t e r i n g performed on the image planes, s i n c e p i c t u r e s processed w i t h higher b i t assignments showed l e s s degradation i n such areas. I t i s q u i t e evident, as seen i n the processed images, that the v i s i b i l i t y of the 'edges' i n both systems was dependent on the amount of d e t a i l inherent i n the p i c t u r e s . The e f f e c t was a l s o depend-ent on the b l o c k s i z e used, being more pronounced i n the 8 x 8 b l o c k s i z e . In the p i c t u r e of greatest d e t a i l , GARDEN, the edges were b a r e l y v i s i b l e even f o r v e r y low b i t assignments f o r both the F o u r i e r and Hadamard systems. As mentioned above, though, f o r b l o c k s i z e 8 x 8 at the lower b i t assignments of 1.75 b i t s / p e l . the e f f e c t s became more n o t i c e a b l e . In the HOUSE p i c t u r e , r e p r o d u c t i o n was e x c e p t i o n a l l y good using the Hadamard system even f o r very low b i t assignments of 1.75 b i t s / p e l . w i t h the 16 x 16 b l o c k s i z e . For 8 x 8 b l o c k s i z e s there was a severe degradation of the images even at the r e l a t i v e l y higher a s s i g n -ments. In the F o u r i e r system, f o r the 16 x 16 b l o c k s i z e r e c o n s t r u c t e d images were of f a i r l y good q u a l i t y except that the 'edge e f f e c t s ' gave r i s e to the formation of 'shadows' around the chimneys. Again, as i n ; the Hadamard system, f o r the 8 x 8 b l o c k s i z e , the images were s e v e r e l y degraded. In the FACE p i c t u r e , the behaviour was the same. For the 16 x 16 b l o c k s i z e , good q u a l i t y p i c t u r e s were obtained f o r the Hadamard system f o r b i t assignments as low as 2.75 b i t s / p e l . In the F o u r i e r system, the 'edge e f f e c t s ' were q u i t e n o t i c e a b l e even at the higher b i t assignments. In both systems, though, the blocked s t r u c t u r e was q u i t e v i s i b l e i n the area of the s h i r t c o l l a r . As mentioned p r e v i o u s l y , l a r g e t r a n s i t i o n s at block boundaries present d i f f i c u l t y i n the block coding scheme. Again, f o r both systems, the 8 x 8 b l o c k s i z e images were of poorer q u a l i t y . In the TEST PATTERN the performance of both systems was the most s i m i l a r , at l e a s t from the s u b j e c t i v e viewpoint. For the 16 x 16 b l o c k s i z e , though, the Hadamard system was again s u p e r i o r . Good q u a l i t y p i c t u r e s were obtained f o r b i t assignments as low as 1.75 b i t s / p e l . 6.5 Quantizer Performance No app r e c i a b l e d i f f e r e n c e s were n o t i c e d i n the p i c t u r e s pro-cessed u s i n g e i t h e r the optimum uniform or optimum non-uniform q u a n t i z e r . This was t r u e f o r both the F o u r i e r and Hadamard systems. I t seems, t h e r e f o r e , that the choice of employing the optimum uniform quantizer was a p p r o p r i a t e , more so when one considers the reduced complexity r e -quired f o r i t s implementation. Two of the p i c t u r e s processed us i n g the optimum non-uniform quantizer may be seen i n P l a t e I I . 6.6 A Comparison Between T h e o r e t i c a l and Experimental R e s u l t s In Chap. 4, graphs of mean square e r r o r v s . b i t s / p e l . were p l o t t e d f o r each of the Y, I and Q tra n s m i s s i o n s i g n a l planes based on Eqn (4.11). These are shown i n F i g s . 4.5-4.8. As seen, smaller mean square e r r o r s f o r the Hadamard coded over the F o u r i e r system f o r b i t assignments below 1.0 b i t s / p e l . (approx.) are p r e d i c t e d f o r each of the s i g n a l planes. F i g . 5.12 shows graphs of mean square e r r o r v s . b i t s / p e l . f o r the Y s i g n a l planes as given by experimental r e s u l t s . Because of the separate baseband encoding, a v a r i a t i o n of b i t assignments on the Y s i g n a l d i d not a f f e c t the mean square e r r o r s on the I , Q planes. How-ever, from the two sets of mean square e r r o r values corresponding to assignments of 0.-375 and 0.5 b i t s / p e l . on the I , Q planes, the p r e d i c t e d behaviour as shown i n F i g s . 4.5-4.8 was observed on these two planes. However, i n the Y s i g n a l , even f o r b i t assignments of 3.0 and 2.0 b i t s / p e l . , i t was found that the Hadamard system always gave smaller mean square e r r o r s than d i d the F o u r i e r system. This behaviour was observed on each of the four p i c t u r e s processed. This anomaly i n behaviour between the t h e o r e t i c a l and a c t u a l r e s u l t s i n the Y s i g n a l could probably be a t t r i b u t e d to the t h e o r e t i c a l model adopted f o r the system. I t must be remembered that an assumption that the image c o n s t i t u t e d a Gauss Markov random f i e l d was used f o r modelling the data source. A l s o , the q u a n t i z a t i o n e r r o r , Eqn (4.7), was based on curves obtained using Gaussian data. There i s one aspect of the r e s u l t s obtained which seems to support t h i s anomalous behaviour of the Y s i g n a l w i t h regard to the model used. This i s the f a c t t h a t , from the amplitude histograms p l o t t e d f o r the Y, I and Q s i g n a l s (shown i n Appendix B), the I and Q s i g n a l s do seem to possess more of a Gaussian d i s t r i b u t i o n than the Y s i g n a l which i s more s i m i l a r to a Ra y l e i g h than a Gaussian d i s t r i b u t i o n . An i n t e r e s t i n g comparison may be drawn between the t h e o r e t i -c a l l y obtained performance curves and performance curves obtained by B a i l l i e [15]. B a i l l i e has made a t h e o r e t i c a l i n v e s t i g a t i o n of t r a n s -form p i c t u r e coding systems, under the assumption that the p i c t o r i a l source data c o n s t i t u t e d a homogeneous Gauss Markov random f i e l d . He too has employed a separable a u t o c o r r e l a t i o n f u n c t i o n and has i n v e s t i g a t e d the performance of h i s systems r e l a t i v e to a mean square e r r o r c r i t e r i o n . Considering s e p a r a t e l y , 1-dimensional, 2-dimensional and 3-dimensional coders, he i n v e s t i g a t e d d i s t o r t i o n due to v a r i a t i o n i n b l o c k s i z e s f o r d i f f e r e n t c o r r e l a t i o n c o e f f i c i e n t s . His r e s u l t s i n d i c a t e d t h a t , r e -gardless of d i m e n s i o n a l i t y , the Hadamard transform was s u p e r i o r to the F o u r i e r transform when employing r e l a t i v e l y short b l o c k lengths on c o r r e l a t e d data. At longer block, lengths the reverse was t r u e . Many researchers [14], [25] had p r e v i o u s l y expressed the view that the F o u r i e r transform was always s u p e r i o r to the Hadamard transform - a t l e a s t from a mean square e r r o r c r i t e r i o n . The t h e o r e t i c a l r e s u l t s , as shown i n F i g s . 4.5-4.8, agree w i t h B a i l l i e ' s r e s u l t s . For both b l o c k s i z e s considered ( 8 x 8 and 16 x 16) p i c t u r e elements), i t can be seen that the Hadamard transform d i d out-perform the F o u r i e r transform, a t l e a s t f o r the 2-dimensional coding scheme employed i n the study, f o r b i t assignments below 1.0 b i t s / p e l . (approx.). As o u t l i n e d above, t h i s behaviour was r e f l e c t e d to a c e r -t a i n extent i n the s u b j e c t i v e q u a l i t y of the processed images. V I I . ADDITIONAL EXPERIMENTS AND RESULTS 7.1 Implementation of a Simple Zonal Coding Scheme In the previous chapter i t was mentioned that the 'edge e f f e c t s ' i n the processed images were more v i s i b l e i n the F o u r i e r than i n the Hadamard system. Although these e f f e c t s were p r i m a r i l y due to the b l o c k coding technique, i t was f e l t that q u a n t i z a t i o n e r r o r s , e s p e c i a l l y f o r those samples coded w i t h l e s s than s i x b i t s , f u r t h e r enhanced the e f f e c t . In order to check j u s t how much f u r t h e r degradation was i n t r o d u c e d , a simple zonal f i l t e r i n g technique was used in.which each transform co-e f f i c i e n t was assumed to be coded w i t h eight b i t s . This assumption was made i n order to d e f i n e an equivalent b i t assignment f o r any image pro-cessed us i n g t h i s s i m p l i f i e d coding scheme. Based on the d i s t r i b u t i o n of .the h i g h energy samples i n the F o u r i e r transform domain, a zone was e s t a b l i s h e d which was c i r c u l a r l y symmetric about the o r i g i n . R e t a i n i n g samples w i t h i n such a zone i n the transform domain i s c a l l e d C i r c u l a r Zonal Low Pass f i l t e r i n g . However, some high frequency content of the image i s r e t a i n e d by the f u r t h e r i n c l u s i o n of some samples along the s p a t i a l co-ordinate axes. As men-tioned above, each r e t a i n e d sample was assumed to be coded w i t h the r e s o l u t i o n of e i g h t b i t s and f o r any b i t assignment on the image, a p r o p o r t i o n a l number of samples were r e t a i n e d . The above procedure was done f o r b l o c k s i z e 16 x 16 w i t h a s s i g n -ments of 2.0 and 3.0 b i t s / p e l . on the Y s i g n a l s and 0.5 b i t s / p e l . each on the I,Q s i g n a l s . T y p i c a l r e s u l t s are as shown i n P l a t e IV. Images produced usi n g t h i s scheme showed no n o t i c e a b l e d i f f -erences from thoselmages produced us i n g the b l o c k q u a n t i z a t i o n t e c h n i -que. This f a c t f u r t h e r supported the n o t i o n that the blocked s t r u c t u r e and a s s o c i a t e d 4edge e f f e c t s ' were p r i m a r i l y due to the b l o c k coding of the F o u r i e r images as explained i n Sect. 6 . 4 . 7.2 Smoothed F o u r i e r Images In an attempt to reduce the undesirable'edge e f f e c t s ' i n the F o u r i e r coded images, a l i n e a r i n t e r p o l a t i o n scheme was implemented based on the s e n s i t i v i t y of the human v i s u a l system to b r i g h t n e s s changes. Much research has been done i n i n v e s t i g a t i n g the c o n t r a s t s e n s i t i v i t y of the eye. One p a r t i c u l a r measure, the Weber f r a c t i o n , r e l a t e s a j u s t n o t i c e a b l e b r i g h t n e s s d i f f e r e n c e AB, over a b r i g h t n e s s background B + AB. The Weber f r a c t i o n AB/B, i s found to be n e a r l y constant at about two percent over a wide range of b r i g h t n e s s [ 1 6 ] . Using t h i s f a c t , a smoothing a l g o r i t h m was developed. The b r i g h t n e s s l e v e l s of adjacent p o i n t s i n the sub-block boundaries of the processed F o u r i e r images were examined and the abso-l u t e percentage d i f f e r e n c e between the l e v e l s , computed. For a per-centage d i f f e r e n c e of l e s s than two percent, the l e v e l s were l e f t un-changed . Based on experimental r e s u l t s obtained during the study, e i t h e r a 2-point or 4-point i n t e r p o l a t i o n scheme was used depending on whether the percentage d i f f e r e n c e was between two and s i x percent or greater than s i x percent, r e s p e c t i v e l y . In the i n t e r p o l a t i o n scheme, the brightness l e v e l s of a d j o i n i n g p i c t u r e p o i n t s were adjusted such that they each e x h i b i t e d an equal change i n b r i g h t n e s s i n t e n s i f i c a t i o n s . In the 2-rpoint scheme o n l y the two adjacent p o i n t s at a sub-block boundary were adjusted whereas i n the 4-point scheme, two p o i n t s on e i t h e r s i d e of the sub-block boundary were adjusted. The p i c t u r e s whose 'edges' were removed showed a very g r a i n y e f f e c t due to the averaging done on the p o i n t s . A t y p i c a l case i s as shown i n P l a t e v . In any image t r a n s m i s s i o n scheme, post processing at the r e c e i v e r i s u n d e s i r a b l e . The above mentioned smoothing of the F o u r i e r images was not done as p a r t of the coding system but r a t h e r was done i n order to compare the q u a l i t y of the smoothed images to the p i c t u r e s obtained i n the Hadamard system. From a s u b j e c t i v e v i e w p o i n t , the Hadamard images remained s u p e r i o r . 7.3 Coding the E n t i r e P i c t u r e So f a r i t has been shown that although the b l o c k coding scheme o f f e r s many advantages, among them being i t s computational s i m p l i c i t y , a major concern i s the i n t r o d u c t i o n of the 'edges' i n the processed images, e s p e c i a l l y i n the F o u r i e r system. For comparison purposes, some p i c t u r e s were processed u s i n g the e n t i r e p i c t u r e frame 256 x 256 format. I t would not be f e a s i b l e from a computational standpoint to employ the same pro-cedure as was used i n the block coding system. Therefore, zonal f i l t e r -i n g techniques, as o u t l i n e d f o r the F o u r i e r system i n Sect. 7.1 were used. Since the high energy samples are not as c l u s t e r e d around the o r i g i n i n the transform domain f o r the Hadamard system as i n the F o u r i e r system, a d i f f e r e n t zone was e s t a b l i s h e d . In t h i s case, the h i g h energy samples are d i s t r i b u t e d more along the co-ordinate axes but w i t h a higher d e n s i t y around the o r i g i n . E s t a b l i s h i n g a zone i n which these h i g h energy samples are r e t a i n e d i s c a l l e d Hyperbolic Low Pass f i l t e r i n g . P i c t u r e s were coded w i t h an equivalent of 2.75 and 3.75 b i t s / p e l . As expected, there were no severe d i f f e r e n c e s i n the q u a l i t y of the Hadamard coded images f o r the same b i t assignment using the two schemes. There was some improvement i n the F o u r i e r coded images s i n c e the major source of impairment, the 'edges', were not v i s i b l e . I t seems, th e r e -f o r e , that transform coding of the e n t i r e p i c t u r e o f f e r s a b e t t e r q u a l i t y r eproduction f o r the F o u r i e r system. 74. V I I I . CONCLUDING REMARKS 8.1 Summary of R e s u l t s The Hadamard coding system, from the p o i n t of view of produc-ing a smaller mean square e r r o r between the o r i g i n a l and rec o n s t r u c t e d images, performed b e t t e r than d i d the F o u r i e r coding system. The Hadamard coded images were a l s o of a higher q u a l i t y , based on a s u b j e c t -i v e assessment, than the F o u r i e r coded images. In p a r t i c u l a r : i ) In both the Y, I , Q and R, G, B planes, i t was found that the Hadamard system gave a smaller mean square e r r o r between the o r i g i n a l and reconstructed images, than the F o u r i e r system. i i ) Images coded w i t h b l o c k s i z e s of 16 x 16 p i c t u r e elements were superi o r to those coded w i t h 8 x 8 b l o c k s i z e s f o r the same b i t a s s i g n -ment on each of the s i g n a l planes. This was based on both the mean square e r r o r c r i t e r i o n and a l s o from the s u b j e c t i v e q u a l i t y of the pro-cessed images. i i i ) I t was found that the Y s i g n a l component of the image could be coded usi n g 2.0 b i t s / p e l . and s t i l l r e t a i n a r e l a t i v e l y h i g h image q u a l i t y . This was mainly true f o r the Hadamard coded images. For the F o u r i e r system, because of the 'edge e f f e c t s ' produced as a r e s u l t of the bl o c k coding, a comparable b i t assignment on the Y s i g n a l was dependent on the amount of d e t a i l inherent i n the coded image. For both the Hadamard and F o u r i e r systems, i t was p o s s i b l e to code both the I , Q s i g n a l s w i t h as few as 0.375 b i t s / p e l . i v ) There was no appreciable d i f f e r e n c e i n image q u a l i t y when e i t h e r the optimum uniform or optimum non-uniform quantizer was used. v) Spurious e f f e c t s , i . e . the 'edges' i n the F o u r i e r coded images can be e l i m i n a t e d through transform coding the e n t i r e image i n the 256 x 256 format. Good q u a l i t y images, e s p e c i a l l y - i n the Hadamard system,, were obtained f o r a t o t a l b i t assignment of 3.75 b i t s / p e l . v i ) Based p u r e l y on a t h e o r e t i c a l b a s i s , i t was found that the Hadamard transform was s u p e r i o r to the F o u r i e r transform when c o n s i d e r -ing a mean square e r r o r c r i t e r i o n , f o r b i t assignments below approx-imately 1.0 b i t s / p e l . In summary then, i t i s r e a l i z e d that a s u b s t a n t i a l bandwidth r e d u c t i o n can be achieved through the coding system employing a l i n e a r t r ansformation and block q u a n t i z a t i o n technique. I t was seen that the q u a l i t y of the reconstructed images depended not only on the type of l i n e a r transformation but a l s o on the b l o c k s i z e s used. Good q u a l i t y c o lour images were obtained f o r a t o t a l b i t assignment of 2.75 b i t s / p e l . Indeed, i n some cases, depending on the transformation employed and on the d e t a i l inherent i n the image, i t was p o s s i b l e to o b t a i n good q u a l i t y r e p r o d u c t i o n f o r b i t assignments as low as 1.75 b i t s / p e l . 8.2 Recommendations f o r Future Research 1) I t i s recognized that i n the b l o c k coding scheme i n v e s t i g a t e d , average s t a t i s t i c s of the e n t i r e image were used f o r each sub-block. Of course the s i g n a l s t a t i s t i c s (mean, v a r i a n c e , etc.) w i t h i n each sub-block w i l l vary from these average s t a t i s t i c s , some more than others. An adaptive scheme, t h e r e f o r e , i n which the model i s based only on the s t a t i s t i c s of the considered sub-block should r e s u l t i n a b e t t e r per-formance. Such an adaptive b l o c k coding scheme, however, w i l l be very time consuming when one considers that there are three s i g n a l planes comprising the colour image. Such an adaptive technique could be per-formed s o l e l y on the Y s i g n a l , which contains most of the image energy and the non adaptive scheme performed on the I , Q s i g n a l s which possess a very small percentage of the image energy. 2) In the colour image coding system s t u d i e d , some aspects of both s t a t i s t i c a l and psyc h o v i s u a l coding were incorporated i n the system. One f u r t h e r p s y c h o v i s u a l aspect whick may be included i s that of the c o n t r a s t s e n s i t i v i t y of the eye. I t has been shown that the s e n s i t i v i t y f u n c t i o n i s of a l o g a r i t h m i c nature [16], [32] and t a k i n g the l o g a r i t h m of the image br i g h t n e s s before processing and exponent-i a t i n g at the r e c e i v e r compensates f o r the eye's c o n t r a s t s e n s i t i v i t y . I t i s hoped that higher q u a l i t y images, e s p e c i a l l y f o r the F o u r i e r system, would be achieved. 3) In the t h e o r e t i c a l modelling of the images, an exponential a u t o c o r r e l a t i o n f u n c t i o n was used. Although t h i s model performed s a t i s f a c t o r i l y , e s p e c i a l l y because i t a f f o r d e d computational s i m p l i c i t y , some d e f i c i e n c i e s i n a c c u r a t e l y c h a r a c t e r i z i n g the images were inherent P o s s i b l y higher order s t a t i s t i c s are needed as pointed out by F r e i and Jaeger [13], to f u l l y c h a r a c t e r i z e an image but t h i s would f u r t h e r e n t a i l a higher degree of complexity. In e s t a b l i s h i n g some s o r t of t r a d e - o f f then, some i n v e s t i g a t i o n i n t o more r e a l i s t i c models can be made. 4) A major c o n s i d e r a t i o n i n image coding systems i s i t s perform-ance e v a l u a t i o n . Many researchers have t r i e d to e s t a b l i s h meaningful mathematical measures, i . e . ones r e f l e c t i n g image q u a l i t y or s e v e r i t y of degradation. Wilder [31] has done much research i n t h i s area and has shown that p o i n t - t o - p o i n t e r r o r measures i n c l u d i n g mean square e r r o r between o r i g i n a l and processed images are not always s u b j e c t i v e -l y r e l e v a n t . Further he has advocated the use of measures which i n v o l v e area p r o p e r t i e s of v i s i o n . Indeed, some measures have been developed which take the aspect of a human v i s u a l model i n t o account [32], [15]. However, a l l of these measures have been i n v e s t i g a t e d only f o r mono-chromatic imagery. Colour p i c t u r e s present f u r t h e r d i m e n s i o n a l i t y to the problem. Work remains to be done, t h e r e f o r e , i n i n v e s t i g a t i n g performance measures f o r colour image coding schemes. 5) In the study, an e r r o r f r e e b i n a r y symmetric channel was assumed. I t w i l l be u s e f u l to i n v e s t i g a t e the e f f e c t of channel n o i s e on the colour coding system. 6) F i n a l l y , a most important extension of the work i s i t s u l t i m a t e a p p l i c a t i o n to interframe coding. The separate baseband encoding i s very favourable to any frame-to-frame techniques and i t i s hoped that w i t h the advent of high memory d e v i c e s , such frame-to-frame encoding may become f e a s i b l e . APPENDIX A Conversion from Receiver Primary to Transmission Co-ordinate S i g n a l s and V i c e Versa C±) Receiver Primary System to Transmission Co-ordinate System: RGB -> YIQ Y I Q 0.299 0.587 0.114 0.596 -0,274 -0.322 0.211 -0.523 -0.312 R G B ( i i ) Transmission Co-ordinate System to Receiver Primary System: YIQ -> RGB R G B 1.000 0.956 0.621 1.000 -0.272 -0.647 1.000 -1.106 -1.703 Y I Q 79. i APPENDIX B Brightness Amplitude Histograms of Test P i c t u r e s Brightness amplitude histograms f o r each of the p i c t u r e s employed i n the study are shown i n the f o l l o w i n g graphs. F i g . B . l Brightness amplitude histograms of GARDEN p i c t u r e . 11 I O o ' z U i QL ZD o £2 a LU K g I 1 25.0 50.0 Y SIGNAL'.FACE - 1 — 75.0 0.0 100.0 125.0 AMPLITUDE LEVEL 150.0 175.0 I 1 1 200.0 225.0 250.0 a &2 I SIGNAL:FACE 1 1 j= -30.0 -15.0 0.0 15.0 30.0 45.0 AMPLITUDE LEVEL 60.0 75.0 SO.O -1 1 105.0 120.0 O o ' O o o-0 SIGNAL:FACE -30.0 -15.0 0.0 15.0 10~0 <5~o AMPLITUDE LEVEL 90.0 -1 1 105.0 120.0 F i g . B.3 Brightness amplitude histograms of FACE p i c t u r e , Q SIGNAL:TEST PflT. r n U J - ; F i g . B.4 Brightness amplitude histogram of TEST PATTERN. APPENDIX C Variances of Transform C o e f f i c i e n t s As mentioned i n Sect. 4 . 4 , the s e p a r a b i l i t y of the f u n c t i o n s i n the expression used f o r the determination of the v a r i a n c e s of the transform c o e f f i c i e n t s g r e a t l y f a c i l i t a t e s computations. As defined i n Sect. 3 . 4 , the v a r i a n c e s of the transform co-e f f i c i e n t s were given by: 2 N N N N cr (u,v) = I I 1 1 R(x,x' ,y,y' )<(>(",v,x,y)<!> (u,v,x',y*) N N N N 1 , i D, - • I I I I e - a l X " X I. e - P l ^ W,v,x,y)**(u,v,x' x=l x'=l y=l y'=l ( c 2 ) ( i ) Hadamard C o e f f i c i e n t s The b a s i s f u n c t i o n i s of the form n - l I- [g (u)x ® g (v)y ] 1, i = 0 1 1 1 1 <!>(u,v,x,y) = f (-D." " ( C 3 ) where fl> denotes Modulo-2 a d d i t i o n and g^(u), g ^ ( v ) , x^ and y^ are as p r e v i o u s l y given. I t i s e a s i l y seen that X=l X -1 y=l y =1 so that 2 2 2 er (u,v) = er a C u ) . c r g (y) (C.6) ( i i ) F o u r i e r C o e f f i c i e n t s The b a s i s f u n c t i o n i s of the form K u , v , x , y ) = | exp [-2TTj ( ^ + ffi J ( c > ? ) where j = /-T and u,v r= o, ... N - l x,y = 1 , ... N Again, i t may e a s i l y be shown th a t a a 2 ( u ) - i f f e ^ l ^ ' l . e x p t ^ x ' - x ) ] (C.8) x=l x'=l a B 2 ( v ) = I l l ^l7~rl- e x p [ ^ i ( y ' - y ) ] (C.9) y=l y'=l Now the F o u r i e r c o e f f i c i e n t s a r e , i n g e n e r a l , complex. This means that the var i a n c e s of the r e a l and imaginary p a r t s have to be computed s e p a r a t e l y . For a complex random v a r i a b l e , the v a r i a n c e of the random v a r i a b l e i s given by the sum of the v a r i a n c e of the r e a l p a r t and the v a r i a n c e of the imaginary p a r t . For the 2-dimensional F o u r i e r t r a n s f o r m a t i o n , w i t h the b a s i s f u n c t i o n as given by (C.7) the var i a n c e s of the r e a l and imaginary p a r t s are as given below: Variance of Real P a r t : 2 1 a (u,v) = 2-[C^(u,a). Cf(v,g) - S~(u,a). S~(v, B) + C + ( u , a ) . C +(v,g) - S + ( u , a ) . S +(v ,B)] ( C I O ) 8.6. V a r i a n c e of Imaginary P a r t :c~cu, - C + ( u , a ) . C +(v,6) + S + ( u , a ) . S+(v,-6')] ( C l l ) 2 1 - - - -cr (u,v) = ylC (u a). C Cv,£) - S (u,a). S (v,&) where Note that C ^ u . a ) - ' ^ f f e - a l X - X ' l . Cos ^ ( x ± x ' ) (C.12) N Z x=l x'=l N i N N , , , „ S ( u , a ) - V I I e - a l X - x I . S i n ^ x i x ' ) N x=l x'=l N S (u,a) = 0 so that ( C I O ) , ( C l l ) are f u r t h e r s i m p l i f i e d . ( C 1 4 ) APPENDIX D Optimum Quantizers The problem of mi n i m i z i n g the mean square q u a n t i z a t i o n e r r o r has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n the past [26], [ 2 8 ] , [29], being f i r s t considered by Max [28], when the 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 , P x ( X ) , of the input s i g n a l was known and the number of q u a n t i z a t i o n l e v e l s , N, of the quan t i z e r was assumed to be f i x e d . Max a l s o made the assumption that the t r a n s m i s s i o n channel was e r r o r f r e e . I n the design of optimum q u a n t i z e r s , one may e i t h e r consider an optimum non-uniform qu a n t i z e r (non-uniform q u a n t i z e r l e v e l spacing) or an optimum uniform quantizer (uniform q u a n t i z e r l e v e l s p e c i n g ) . In the f o l l o w i n g , the number of q u a n t i z a t i o n l e v e l s N = 2™, where m i s the number of b i t s assigned to the sample to be quantized. Note that i n t h i s case, N i s always even. The q u a n t i z e r operates i n the f o l l o w i n g f a s h i o n : QUANTIZER F i g . D.l Quantizer F i g . D.2 Quantizer f o r a symmetric p r o b a b i l i t y d i s t r i b u t i o n . For a continuous, s i g n a l , x, of known p r o b a b i l i t y d e n s i t y , the quantizer maps x i n t o one of a f i n i t e set of r a t i o n a l numbers governed by the f o l l o w i n g r u l e : I f u < x < u ^ + 1 => y = v , ( j = 1, . .. N) Optimum Uniform Quantizer For a symmetric 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 and the number of q u a n t i z a t i o n l e v e l s even, from [29], v.. = (j r , j = 1, ... N (D.l) u. = ( i - 1 - |) r , i = 2, ... N (D.2) r i s the t r a n s i t i o n l e v e l spacing and may be computed from ([29] - Eqn (13)) Optimum Non-Uniform Quantizer For t h i s q u a n t i z e r , the equations r e l a t i n g the optimum i n t e r v a l s are (from [28], [29]): / V 1 J Xp (X)dX u. x v. = — , j = 1, ... N (D.3) 2 , j + l J p (X)dX u. x 3 Eqn (D.3) may be r e w r i t t e n as: u / J + J - ( X - v,)p (X)dX = 0, j= 1, ... N (D.5) u. 2 3 (D.3) and (D.5) form the b a s i s of the computational procedure. Note t h a t , from (D.5), v. i s the c e n t r o i d of the area p (X) between u. 3 x 3 and u. , J + l . I n the transform coding scheme, the transform samples to be quantized were assumed to be G a u s s i a n l y d i s t r i b u t e d . Since the number of q u a n t i z a t i o n l e v e l s was even and s i n c e the assumed p r o b a b i l i t y den-s i t y f u n c t i o n was symmetric, i t was only necessary to compute one-half of the q u a n t i z a t i o n l e v e l s . J p (X) \ \ 1 V i v X N N F i g . D.3 Input ranges and corresponding output l e v e l s f o r optimum non-uniform q u a n t i z e r . The computational procedure, as proposed by Max, was as f o l l o w s : i ) Set u1 = 0 i i ) Choose a s u i t a b l e v^ i i i ) Compute the area of P x 0 0 between u^, v ^ i v ) F i n d such t h a t (D.5) i s s a t i s f i e d w i t h i n some l i m i t of accuracy, v) F i n d v 2 u s i n g (D .4) Repeat from ( i i i ) s u c c e s s i v e l y c a l c u l a t i n g the u^'s and v ^ ' s . vi.) I f i s the c e n t r o i d between and °°, then v^ was p r o p e r l y chosen. (A choice of X = 12 i s s u f f i c i e n t l y l a r g e to be considered as »). Otherwise modify v^ and repeat from ( i i i ) u n t i l ( v i ) i s s a t i s f i e d . Max has published the q u a n t i z a t i o n i n t e r v a l s f o r values of N'up to 32. I t was necessary, t h e r e f o r e , to compute values of M = 6,7,8 (N = 64, 128, 256). F a i r l y accurate s t a r t i n g values of v^ were obtained from [29]. REFERENCES [I] W.T. Wintringham, "Colour T e l e v i s i o n and Co l o r i m e t r y " , Proc. of I.R.E. V o l 39, No 10, Oct. 1951, pp 1135-1172. [2] G. Wyszecki and W.S. S t i l e s , Colour Science, John Wiley and Sons, New York (1967). [3] W.D. Wright, The Measurement of Colour, Adam H i g l e r L t d . , London (1969), 4 t h Ed. [4] W. P r a t t , D i g i t a l Colour Image Coding and Transmission, USCEE Report 403, Univ. of Southern C a l i f o r n i a , June 1971. [5] R.W.G. Hunt, The Reproduction of Colour, John Wiley and Sons, New York (1967). [6] D.G. F i n k , e d i t o r , T e l e v i s i o n Engineering Handbook, McGraw H i l l , New York (1957). [7] W. P r a t t , " S p a t i a l Transform Coding of Colour Images", IEEE Trans. Comm. Tech. Vol-Com-19, No 6, Dec. 1971, pp 980-992. [8] J . Limb, C. Rubenstein, and K. Walsh, " D i g i t a l Coding of Colour Picturephone S i g n a l s by E l e m e n t - D i f f e r e n t i a l Q u a n t i z a t i o n " , IEEE Trans. Comm. Tech. Vol-Com-19, No 6, Dec. 1971, pp 992-1005. [9] U.F. Gronemann, Coding Colour P i c t u r e s , Tech. Report 422, M.I.T., June 1964. [10] A. Bushan, " E f f i c i e n t Transmission and Coding of Colour P i c t u r e s " , P i c t u r e Bandwidth Compression Symposium, M.I.T., A p r i l 1969. [ I I ] E.R. Kretzmer, " S t a t i s t i c s of T e l e v i s i o n S i g n a l s " . B e l l Syst. Tech. J o u r n a l , J u l y 1952. [12] L.E. Franks, "A Model f o r the Random Video Process", B e l l Syst. Tech. J o u r n a l , A p r i l 1966. [13] W. F r e i and P.A. Jaeger, "Some Basic Considerations f o r the Source Coding of Colour P i c t u r e s " , Communications Conference, S e a t t l e , 1973, pp 48.26-48.29. [14] A. H a b i b i and P.A. Wintz, "Image Coding by L i n e a r Transformation and Block Q u a n t i z a t i o n " , IEEE Trans. Comm. Tech. V o l Com-19, No 1, Feb. 1971. [15] S. B a i l l i e , A t h e o r e t i c a l study of transform p i c t u r e coding systems and an i n v e s t i g a t i o n of a new measure of d i s t o r t i o n i n processed images, M.A.Sc. t h e s i s , U.B.C., 1973. [16] W.F. Schr e i b e r , ' P i c t u r e Coding', Proc. IEEE ( S p e c i a l Issue on Redundancy Reduction), V o l 55, March 1969, pp 320-330. [17] J . Huang and P. S c h u l t h e i s s , "Block Q u a n t i z a t i o n of C o r r e l a t e d Gaussian Random V a r i a b l e s " , IEEE Trans. Comm. Syst. V o l CS-11, Sept. 1963, pp 289-296. [18] P. Wintz, "Transform P i c t u r e Coding", Proc. IEEE, V o l 60, No 7, J u l y 1972. [19] G.D. Bergland, "A Guided Tour of the Fast F o u r i e r Transform", IEEE Spectrum, V o l 6, J u l y 1969. [20] E.O. Brigham and R.E. Morrow, "The Fast F o u r i e r Transform", IEEE Spectrum, V o l 4, Dec. 1967. [21] W. P r a t t , J . Kane and H. Andrews, "Hadamard Transform Image Coding", Proc. IEEE V o l 57, No 1, Jan. 1969. [22] H.F. Harmuth, "A Generalized Concept of Frequency and Some A p p l i -c a t i o n s " , IEEE Trans. I n f o . Theory, V o l IT-14, No 3, May 1968, pp 375-382. [23] G.B. Anderson and T. Huang, "Piecewise F o u r i e r Transformation f o r P i c t u r e Bandwidth Compression", IEEE Trans. Comm. Tech. V o l Com-19, No 2, A p r i l 1971. [24] P.O. Wintz and A.J. Kurtenbach, "Waveform E r r o r C o n t r o l i n PCM Telemetry", IEEE Trans. I n f o . Theory, Sept. 1968, pp 650-661. [25] J . P e a r l , H. Andrews and W. P r a t t , "Performance Measures f o r Trans-form Data Coding", IEEE Trans, on Comm., June 1972. [26] R. Wood, "On Optimum Q u a n t i z a t i o n " , IEEE Trans. I n f o . Theory, V o l IT-15, No 2, March 1969, pp 248-252. [27] R. G a l l a g e r , Information Theory and R e l i a b l e Communication, John Wiley and Sons, 1968. [28] J . Max, "Quantizing f o r Minimum D i s t o r t i o n " , IEEE Trans. I n f o . Theory, V o l IT-6, March 1960. [29] A.J. Kurtenbach, "Quantizing f o r Noisy Channels", IEEE Trans. Comm. Tech. V o l Com-17, A p r i l 1969, pp 271-302. [30] R.E. Totty and G.C. C l a r k , "Reconstruction E r r o r i n Waveform Trans-m i s s i o n " , IEEE Trans. I n f o . Theory, (Correspondence), V o l IT-13, A p r i l 1967, pp 336-338. [31] W.C. Wi l d e r , " S u b j e c t i v e l y Relevant E r r o r C r i t e r i a f o r P i c t o r i a l Data P r o c e s s i n g " , TR-EE 72-34, Purdue U n i v e r s i t y , December 1972. [32] T. Stockham, "Image processing i n the context of a V i s u a l Model", IEEE Proc. V o l 60, No 7, July 1972, pp 828-842. [33] L.C. W i l k i n s and P. Wintz, " B i b l i o g r a p h y on Data Compression, P i c t u r e P r o p e r t i e s and P i c t u r e Coding", IEEE Trans. I n f o . Theory, V o l IT-17, March. 1971, pp 180-197. [34] W. P r a t t , "Binary Symmetric Channel E r r o r E f f e c t s on PCM Colour Image Transmission", IEEE Trans. I n f o . Theory, V o l IT-18, No 5 Sept. 1972. 

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