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A special purpose computer to simulate a visual receptor network for pattern recognition studies Connor, Denis John 1965

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A SPECIAL PURPOSE COMPUTER TO SIMULATE A VISUAL RECEPTOR NETWORK FOR PATTERN RECOGNITION STUDIES by DENIS JOHN CONNOR B.A.Sc, University of B r i t i s h Columbia, 1963 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1965 In presenting th i s thes i s in p a r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, 1 agree that the L ibrary sha l l make i t f r ee l y a va i l ab l e fo r reference and study. I fur ther agree that per-mission for extensive copying o f t h i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by his representa t i ves 4 It i s understood that copying or p u b l i -cat ion of th i s thes i s for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date <Ct-k_^_fl-J / ( y i ^ L S 11 A B S T R A C T A set of equations i s introduced which describes the inh i b i t o r y interaction between the receptors i n the eye of the horse—shoe crab, Limulus. The terms primary i n h i b i t i o n f i e l d , and i n h i b i t i o n f i e l d are defined, and the results of a d i g i t a l computer study of their interdependence with respect to size and to the type of k. . and t. . functions employed i , 0 i j O are given* It i s concluded that a large receptor array can be simulated by solving the set of equations for the output of the central receptor i n much smaller sub-arrays* If the sub-arrays are chosen so as to cover the large array, then the resulting set of central receptor outputs provides a good approximation to the output of the large array. A device which simulates a large receptor array by repetitive solution of the set of equations for 9x9 sub-arrays i s described. Various switching arrangements which reduce the complexity of the device are detailed. The c i r c u i t diagrams for a l l the required elements are given. In order to evaluate the possible u t i l i t y of the device as a tool i n pattern recognition studies, the output from the device for certain simple patterns was simulated on the d i g i t a l computer. The results of this study are given, and some possible applications to the f i e l d of pattern recognition are mentioned. It i s pointed out that the design of the device makes i t a v e r s a t i l e instrument for simulating various types of receptor arrays. TABLE OF CONTENTS Page L i s t of Illustrations.....................•»•••»•.•••.• v L i s t of Tables. v i i Acknowledgement*.............................•••»•.* *.• v i i i L i s t of Special Terms.....**......•••.••.••••***•••.... ix 1* Introduction.•.»••••••.••••••»•••*..««**•*•***....• 1 1.1 The Application of Bi o l o g i c a l Studies to the F i e l d of E l e c t r i c a l Engineering..*........ 1 1.2 Functional and Structural Models of Bio l o g i c a l Systems .•••.•••••«*••«*•*...•• 1 1.3 The Bi o l o g i c a l Basis for the Proposed S i m u l a t o r . • . . . . . . . . . . . . . . . . . . . . a * * . . . . . * . * * . . . 2 1*4 J u s t i f i c a t i o n for the Simulation from an Engineering Point of View..................... 5 2. A D i g i t a l Computer Study of the Output from a Uniformly Illuminated Receptor Array.•••••*••«••••• 7 2*1 The Main Problem i n the Simulation of Bio l o g i c a l Systems ••••••«•*.*••.. 7 2*2 An Ideal Receptor Array* Primary Inhibition F i e l d , and Inhibiti o n F i e l d . . . . . . . . . . . . . . . . . . . 7 2.3 Simulation of Square Receptor Arrays.*,***.....• 9 3. Design of the Simulator............................ 16 3.1 Preliminary Considerations and the Block Diagram.........................••****•»*••... • 16 3.2 The Computation Process! Iteration Cycle, Computation Cycle.••..........•••••»»«»«.••»•. 18 3.3 The Input Scanning Unit. 22 3*4 The Output Scanning Unit *••••..... 24 3.5 The Timing Unit. •.••••.•.••.•...*».***«*.•••.... 25 3*6 The Operation of the Routing Gates* the Multiplying Units, and the Threshold Units.... 27 V, Page 4. Pattern Recognition Studies. .........»,••*.****•••. • 36 4.1 The Ef f e c t of an "Edge" i n the 9x9 Array*..... 36 4.2 The Effe c t of a Right Angle Corner of Illumination Intensity..............*••••....• 39 4.3 The Ef f e c t of a Variable Angle Illumination Intensity Corner.....••*......••••*»****•••..• 42 4.4 Projected Applications for the Simulator*..... 44 5. Conclusion*.••....•.........*..•••«*•*••«*•****••• • 46 Appendix I C i r c u i t s and System Operation*.»**»•***..•. 48 AI*1 The Ring Counter........*.•...*.•***•***••••• 48 AI.2 The Memory Element..............»•»*..«••...• 52 AI«3 Routing Gate, Multiplying Unit, and Threshold Unit * ... 54 AI*4 Staircase Generators............•*.•*••»..... 57 Al*5 Miscellaneous Circuits.........».••*•»»•••.•• 61 Appendix II Modifications to the Multiplying and Threshold Scheme Given i n Section 3*6..... 63 References............................................ 66 LIST OF ILLUSTRATIONS Figure Page 2-1 Portion of a Receptor Array. ...»•..•.•»••••••.»•.• • 8 2-2 x (m) for Equations ( 2 - 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . 11 c 2- 3 x (ra) for Equations ( 2 - 3 ) . . . . . . . . . . . . . . . . * . . . . . . . . 13 G 3- 1 An Array of Receptor Inputs..*.................•.. 16 3-2 Simulator Block Diagram..........•••.•*•••«*•....• 18 3-3 Block Diagram of Computation Unit..........»••... • 20 3-4 Input Unit Scanning Pattern as Viewed from the P h o t o — m u l t i p l i e r • • • . • . • . , . . . . . • . • . • • » • » » » » « » • • • . . • 23 3—5 Timebase Block Diagram.............. e . . . . • . o . . . . . . 24 3-6 Block Diagram of the Timing Unit . . . o . . . . . . . « < > . . . . • 25 3-7 Multiplying Configuration for x ?^............... 28 (n) 3-8 Multiplying Configuration for x^ 29 3-9 Memory Element to M u l t i p l i e r Interconnections..... 31 3- 10 Threshold Units and Outputs... 34 4- 1 a) Edge Orientation and Central Receptor Locus................................ . . . . . . . . . 37 b) Var i a t i o n i n x as Central Receptor Moves Away from Edge................................ 37 4-2 a) Edge Orientation and Central Receptor L o c u s . . » . . . . . . . . . . . . . . . . . . . . . . . . . . • • . » • • » . . . • • 38 b) Variation i n X q as Central Receptor Moves Away from Edge•«.».».»»•.».»..•.o».». .«»». . • • • 38 4-3 a) Corner Orientation and Central Receptor Locus* <»..«.. • * • • . . . . . . . . . * . . . . . . « • • • • • . . • • . . . . 40 b) Var i a t i o n i n x as Central Receptor Moves Away from the c Vertex..... 40 4-4 a) Corner Orientation and Central Receptor L o c u s . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . » » « » . « • • . . « 41 b) Variation i n x as Central Receptor Moves Away from the Vertex. .......••».«•••,«««.*••• 41 Figure Page 4-5 a) Variable Angle Corner Orientation*•«*«*.•*•.... 43 b) Variation i n x with Variation i n ^ . ».«••••«... 43 c AI-1 MECL Flip~]?lop MC352G C i r c u i t . . . . ....... 48 AI-2 Ring Counter Circuit..................»•••...»..•• 50 AI-3 Synchronizing Gate Circuit........•••»»...••••.... 51 AI-4 Memory Element Circuit...*.........•••»••»•••.•.•• 53 AI-5 Memory Element Row Output Circuit..•»..»••*...••.. 54 AI-6 Input Voltage vs. Output Voltage for the Memory E l e m e n t * . a * . . . t . . . . . . . . . . « » . . . . « . . • . 55 Al—7 Routing Gate C i r c u i t * « . . . . . . . . * . . . • 56 AI-8 a) Multiplying Unit Circuit........••.»•••••••••• 58 b) Threshold Unit Circuit•••••••••>»«><>•••.• • • 58 AI-9 Gated Potential Divider Staircase Generator C i r c u i t . . . . ....••...> . 4 ...... 59 AI-10 F l i p — F l o p Staircase Generator C i r c u i t * * . . * ^ . . . * • • 60 AI-11 Integrator-Inverter C i r c u i t . . . . . . . . . . , . . * • • . . . . . . 62 ix LIST OF TABLES Table Page 2-1 Figures of Merit, K , for Equations (2-2). 12 2- 2 Figures of Merit, K ., for Equations (2-3)......... 14 3- 1 Relationship between Clock, Counters, and Units They Control,........... o...................... 26 3-2 Routing Gate Connections between M u l t i p l i e r Row Inputs and Memory Element Row Outputs.»••*••••...» 32 3-3 Simultaneous Gating Sequences for Gate Set #1 and Gate Set $3•«.........«......,*..*•••...«..... 32 AII-1 Expanded Routing Gate Connections between M u l t i p l i e r Row Inputs and Memory Element Row Outputs.•.......................••.•••.•••*».... 65 AII-2 Simultaneous Gating Sequences for the Expanded Gate Sets #1 and #3.................... 65 X LIST OP SPECIAL TERMS F i r s t Defined i n Section Figure of Merit, K 2.2 Primary Inhibit i o n F i e l d 2.2 In h i b i t i o n F i e l d 2.2 Iteration Cycle 3.2 Computation Cycle 3.2 Lattice Scan Unit 3.3 Lattice Movement Unit 3.3 Gate 3.6 Routing Gate 3.6 ACKNOWLEDGEMENT The author wishes to thank the supervising professor, Dr. M. P» Beddoes, for his help and i n s p i r a t i o n during the course of thi s study. The author i s also indebted to Dr. Z. A. Melzak f o r his encouragement. Grateful acknowledgement i s given to the Mr» and Mrs. P. A. Woodward Foundation for an assistantship awarded i n 1963, to the B r i t i s h Columbia Telephone Company for a scholarship i n 1964, and to the National Research Council for an assistant-ship i n 1965* The work described i n this thesis was supported by a grant from the National Research Council. 1 A SPECIAL PURPOSE COMPUTER TO SIMULATE A VISUAL RECEPTOR NETWORK FOR PATTERN RECOGNITION STUDIES •1.: -INTRODUCTION 1.1 The Application of B i o l o g i c a l Studies to the F i e l d of E l e c t r i c a l Engineering. B i o l o g i c a l systems possess many properties which could be usefully applied i n the f i e l d of e l e c t r i c a l engineering. They are able to take i n data i n fiv e d i s t i n c t forms (corre-sponding to the senses), analyze i t to obtain the relevant i n -formation, and carry out the appropriate response. In recent years the e l e c t r i c a l engineer has had to deal to an increasing extent with analogous operations, namely, sensing devices, data processing, and effector mechanisms. The methods employed by nature can be the i n s p i r a t i o n for devising new techniques to carry out these operations. B i o l o g i c a l simulators provide useful tools i n the search for such techniques. The present study i s concerned with the v i s u a l processes of a certain b i o l o g i c a l system. The design of a special purpose computer for simulating this simple v i s u a l system i s presented. The possible u t i l i t y of the computer i n pattern recognition i s pointed out and examples are given. Curiosity and u t i l i t y are equal motives for the work. 1.2 Functional and Structural Models of B i o l o g i c a l Systems. A b i o l o g i c a l simulator can be either a "functional" or a " s t r u c t u r a l " model of the b i o l o g i c a l system. The former i s a physical analogue of the system, while the l a t t e r can be any 2 "black-box", the response function of which i s roughly the same as that of the b i o l o g i c a l system. The structural model i s constructed on the basis of the physiological studies of the system and duplicates certain a c t i v i t i e s of the individual units so as to simulate the overall a c t i v i t y of the system. The functional model on the other hand i s concerned only with the input-output transformation of the system, and i s con-structed on the basis of an ad hoc mathematical model. If the intention i s to use the model i n physiological studies of the system, then a structural model i s indicated. If the aim i s to study the possible applications of the input-output transformation to some sp e c i f i c problem, then a less exact, i . e . functional, model i s s u f f i c i e n t . The simulator to be discussed i n the thesis i s to be used i n pattern recognition studies and w i l l be a functional model, 1,3 The B i o l o g i c a l Basis for the Proposed Simulator. A t r a d i t i o n a l picture of a b i o l o g i c a l nervous system might be one i n which the brain receives data concerning the environment from the sense organs, analyzes i t to extract the relevant information, and then sends orders to the appropriate response^organs. In recent years, physiological evidence has been accumulating which indicates that this picture i s i n many ways misleading. It now appears that the environmental data undergoes a good deal of processing before any information reaches the brain. In addition, not a l l responses are neces-s a r i l y i n i t i a t e d by the brain, and even those that are, are not 3 under s p e c i f i c subsequent control. The p o s s i b i l i t y that some information processing takes place i n the sensory nerve system was f i r s t recognized by Ernst Mach^^, who suggested that reciprocal r e t i n a l i n h i b i t i o n would (2) accentuate contours i n the visu a l f i e l d . Be"ke"sy pointed out the possible role of mutual i n h i b i t i o n as a "sharpening" mechanism i n the auditory system. These speculations have since been borne out by direc t physiological observation of the a c t i v i t y of neural elements. A simple i n h i b i t o r y interaction has been found to exist i n the l a t e r a l eye of the horse-shoe crab, Limulus. A detailed study of thi s process was made by Hartline* R a t l i f f , and their (3) collaborators. The l a t e r a l eye of Limulus i s a coarsely faceted compound eye containing approximately 1000 ommatidia, each of which appears to function as a single receptor unit, excited only by l i g h t entering i t s own corneal facet. Each such receptor unit when so excited, discharges trains of impulses i n one and only one optic nerve f i b e r . These receptor units are not independent i n their action: each one may be inhibited by i t s near neighbors, and i n turn may i n h i b i t them. The anato-mical basis for this interaction i s a plexus of fine nerve fi b e r s an extensive three-dimensional network of f i b e r bundles immediately behind the layer of receptor units. t h Consider a group of n receptors. If the i receptor i s illuminated with l i g h t of intensity 1^, i t s optic nerve f i b e r discharges at some frequency e^, which varies i n an approximately linear fashion with log 1^. If a group of recep-th tors containing the i one i s illuminated with l i g h t of inten-th s i t y I^» the output frequency of the i receptor i s x^, less than e ^ provided the discharge frequencies of the other recep-tors are above the i r respective i n h i b i t i o n thresholds. The th reduction i n f i r i n g frequency of the i receptor i s propor-ti o n a l to the output frequencies of the i n h i b i t i n g receptors less the thresholds, and not to the l e v e l of input stimulus. A l l this i s summed up i n the following set of equations; n x. e. - 5~ k - i max(0,x.-t. .) i = l , . . . , n . . . ( l - l ) th where k. . i s the c o e f f i c i e n t of i n h i b i t i o n of the j receptor th on the i receptor, and t. . i s the corresponding threshold. 1 »J Since x., e., and t. . are pulse-rates or frequencies, they x l l , 3 cannot take on negative values. In addition since the interaction i s s t r i c t l y i n h i b i t o r y , the inequality k. . ^ 0 must hold. The magnitudes of k. . and t. . are, on the average, dependent on 1 » J 1 » J the distance separating the two receptors, k. ^ decreases and t. . increases as the separation becomes greater, 1»3 The set of equations ( l - l ) provide a mathematical de-s c r i p t i o n for the i n h i b i t o r y interaction between the receptors making up the eye of Limulus. They form a non—linear, algebraic set and require an i t e r a t i v e method for solution. They are the basis of the functional model to be described i n this t h e s i s : the problem i s to design a special-purpose computer which for an arbitrary input pattern of e^'s w i l l determine the x.'s given the k. . and t. . functions. 5 1.4 J u s t i f i c a t i o n for the Simulation from an Engineering Point of View. Prom an academic point of view the study and simulation of a v i s u a l receptor network may be an interesting problem. From an engineering point of view, what p r a c t i c a l p o s s i b i l i t i e s can be cited? The fact that a good deal of information processing takes place i n the sensory nervous system i s evident from physio-l o g i c a l data. Although there are approximately 130 m i l l i o n receptors i n the human retina, there are only one m i l l i o n neural pathways i n the optic nerve system. Indeed, the results of a number of workers (Hartline, K u f f l e r ^ ^ , and Barlow^ have shown that the outputs from the r e t i n a l mosaics of diverse types of creatures exhibit a marked refinement of the input information. As a general rule the processing of the information emphasizes dis c o n t i n u i t i e s (in a space or time sense) i n the information. Hartline, i n his studies of Limulus. found that the transformation given i n the set of equations ( l - l ) caused a d e f i n i t e enhancement of vis u a l contours. Barlow was able to show that certain receptors i n the eye of the frog acted as "moving—edge" detectors, reacting vigorously to a change i n the in t e n s i t y of impinging l i g h t , but remaining r e l a t i v e l y dormant under conditions of constant illumination. A method of information processing which emphasizes discontinuities would appear to have useful applications i n the f i e l d of pattern recognition* For example, the type of transformation given i n 6 the set of equations ( l - l ) provides a method for the "edging" of patterns so as to increase the pattern-to-background contrast. 7 2. A DIGITAL COMPUTER STUDY OP THE OUTPUT PROM A UNIFORMLY ILLUMINATED RECEPTOR ARRAY This chapter begins with a discussion of the main problem i n any simulation of a b i o l o g i c a l system* namely* the large number of sub-units which are involved. The concepts of primary i n h i b i t i o n f i e l d , and i n h i b i t i o n f i e l d are introduced, and the results of a study of their interdependence are given. 2.1 The Main Problem i n the Simulation of B i o l o g i c a l Systems. In any simulation of a b i o l o g i c a l system, the main problem i s to overcome the necessity of considering the very large number of individual units which make up the system. In terms of a v i s u a l receptor network, one would hope to be able to simulate the output of a given receptor by considering only the influence of a small number of receptors i n i t s immediate neighborhood. The work of Hartline on Limulus provides some j u s t i f i c a t i o n for this approach, i n that the influence exerted by one receptor on another was found to decrease quite rapidly with increasing separation. This fact implies that i n the solu-t i o n of the set of equations ( l - l ) for the value of some par t i c u l a r x^, we can neglect the effect of most of the receptors which make up the eye of Limulus. 2.2 An Ideal Receptor Array: Primary Inhibition F i e l d , and Inhibit i o n F i e l d . In order to f a c i l i t a t e the subsequent discussion, l e t us assume that the receptor inputs i n the eye of Limulus are arranged i n a square l a t t i c e - l i k e array, the central portion of 8 which i s shown i n P r g. 2-1, Central Receptor J u Boundary of m-square array u u = unit distance F i g . 2-1. Portion of Receptor Array. Assume that the overall array has mQ receptors to a side, and consider the sub—array having m receptors (m being an odd number) to a side as shown i n F i g . 2-1• Also* assume that the influence exerted by one receptor on another (as determined by the k. . and t. . functions) i s dependent only on the distance separating them. Let the illumination of the receptor f i e l d be such that e^ i s constant and non-zero within the boundary of the m—square array, and zero outside. The output, x , of the c 9 central receptor i n this array i s a function of m. For example, for m = 1, x (m) = e.; whereas for m > 1, x (m)< e.. Thus, i t C 1 C 1 i s possible to define a figure of merit, x (m) - x (m ) c c o x (m ) c o which gives a measure of the difference between the value of the central receptor output when the m-square array alone i s illuminated, and the value when the whole array i s illuminated. If we specify the k. . and t. . functions for the set of equations ( l - l ) , we can make the following d e f i n i t i o n s : 1, The primary i n h i b i t i o n f i e l d of a receptor i s the smallest m—square array which contains a l l the receptors that are related to the central receptor by non-zero k. .. 11J 2. The i n h i b i t i o n f i e l d of the central receptor i s the smallest m—square array which contains a l l the concentric square arrays for which 0.1, 2,3 Simulation of Square Receptor Arrays, The solution of the set of equations ( l — l ) on the d i g i t a l computer for various square receptor arrays had a two-fold purpose. F i r s t , a rough measure of the size of the i n h i b i t i o n f i e l d for a specified size of primary i n h i b i t i o n f i e l d was needed. Second, some information was required about the rate of convergence of the i t e r a t i v e method of solution for equations ( l - l ) given below. Before proceeding, we require the statement of a theorem (6) f i r s t proved by Melzak '; The set of equations ( l - l ) has a 10 unique solution for an arbitrary set of e.*s and t. .'s i f and | i 1»J only i f n_ k. .k. . < 1 i = l , . . . , n ...(2-1) This theorem determines the maximum inhib i t o r y i nteraction between a receptor and the receptors which influence i t . As a f i r s t approximation i n the solution of the set of equations ( l - l ) , the x^'s on the right side of the equations are replaced by the e.'s and the equations are solved for the x^, i = l , . . . , n . This set of x^'s constitute a second approx-imation and i s used i n the right-hand side of the set of equa-tions ( l - l ) to obtain a t h i r d approximation. This process i s repeated as often as necessary. A 5 x 5 direc t i n h i b i t i o n f i e l d was chosen for the studies of the set of equations ( l - l ) . The value of x as a function of m was computed separately for three levels of illumination intensity (equivalent to e^ values of 45, 55 and 65 pulses/sec. (p.p.s.)), and for two types of k. . and t. . 1tJ 1»1 functions. The f i r s t k. . and t. . functions were chosen to give a maximum amount of in h i b i t o r y i nteraction between the receptors. Thus k. . = I 0.51-0.17r 0<rc3 1 , 0 I 0 r=0, r>3 ( 1 ...(2-2) t. . ={0.51-0.17r 0<r^3 1 , J { 0 r=0,r>3 where r_ i s the separation between the i ^ * 1 and j ^ * 1 receptors, 11 and i s measured i n terms of the unit distance shown i n Pig. 2-1. Ve solve equation (2-l) for this p a r t i c u l a r k. . function, and fin d : 25 0£i k. .k. . 0.998 i = 1,...,25 Hence, a unique solution of the set of equations ( l - l ) i s obtainable. The results obtained for the three levels of illumination under the influence of the above k. . and t. . functions are shown i n P i g . 2-2. x (m) c v ' (p.p.s.) 25 20 15 10 = 4f p-p-s. et = ^ 5" p . p . s . . ei - p.p.s. _ _ _ xc(~„\ c- - <.s-p p . s . 1 3 5 7 9 11 13 15 m Pig. 2-2. x (m) for Equations (2-2). c The values of x c(m), for m< mQ were obtained using the IBM 7040 d i g i t a l computer. The values of x c(m Q) were obtained under the assumption that i f whole array i s uniformly illuminated, a l l the receptors have the same output. The figures of merit, K m, for the various cases are shown i n Table 2-1. 12 e^=45p.p.s. ei=55p*p,s. e^=65p.p.s. m=5 0.087 0.150 0.46 m=7 0.39 0.44 0.46 Km=9 0.20 0.21 0.23 Km=ll 0.008 0.007 0.00 Km=13 0.06 - -Km=15 0.008 - -Table 2-1. Figures of Merit, Kffi, for Equations (2-2), The second k. . and t. . functions were chosen to be a 1 > J 1»3 better quantitative agreement with the experimental findings of Hartline; k. . i>3 i»3 0,3-O.lr 1 0 1 0.3-O.lr V.0 0< r ss3 r=0, r >3 0 < r sc3 r=0, r > 3 ...(2-3) The solution of equation (2-1) for this case gives: 25 .k. . = 0.348 i = 1,...,25 1^ 1 x>3 3»i and hence the set of equations ( l - l ) can be solved. The k. . and t. . functions of both equations (2-2) and 113 1 > 3 (2-3) are i n qualitative agreement with the findings of Hartline since the k. . are decreasing functions and the t. . 1» 3 1»3 13 are increasing functions of the separation, r . The maximum k. . i n equations (2-2) for r=l i s 0.34. The maximum inhib i t o r y constant detected by Hartline was 0.2, and this i s i n agreement with the maximum k. . i n equations (2—3). 1 » J The results obtained from the set of equations ( l - l ) and the k. . and t. .functions of equation (2—3) for the three levels of illumination are shown i n F i g . 2-3. x c(m) F i g . 2-3. * c(m) for Equations (2-3). The figures of merit, , for the various cases are shown i n Table 2-2. It i s evident from Table 2-1 that for the k. . and 113 t. . functions of equations (2-2), K <0*1 for the 11x11 array. Thus, for the 5x5 primary i n h i b i t i o n f i e l d , the i n h i b i t i o n f i e l d of the central receptor i s an 11x11 array. This result corresponds to the maximum inh i b i t o r y i n t e r a c t i o n . For the 14 = 45p.p.s. = 55p.p.s. = 65p.p.s. 0.15 0.18 0.20 Km=7 0.13 0.14 0.15 Km=9 0.03 0.03 0,02 Km=ll 0.01 0.01 0.02 Km=13 0.01 0.009 0.01 Km=15 0.005 - — Table 2-2. Figures of Merit, K m, for Equations (2-3). more r e a l i s t i c type of k. . and t. . functions of equations 1 » 3 1 » 3 (2-3), Km<0.1 for the 9x9 array. Hence i t would appear that as long as one keeps the inh i b i t o r y interaction between the receptors well below the maximum as defined by the Melzak theorem, the output, x c» for the central receptor i n a 9x9 array provides a good approximation to the output of the same receptor when the whole array i s illuminated. In conclusion, to obtain a good simulation of the output from a large receptor array, i t i s s u f f i c i e n t to break the array up into much smaller, overlapping square arrays whose size i s dependent on the size of the primary i n h i b i t i o n f i e l d and on the type of k. . function considered. The set of 113 equations ( l — l ) i s solved for the output of the central receptor i n each of these arrays. The resulting set of central receptor values constitutes the simulation of the output of the receptor array* I t gives a good approximation to the set of individual 15 receptor outputs which would be obtained i f one solved the set of equations ( l - l ) f or the whole array. 16 3. DESIGN OF THE SIMULATOR This chapter deals with the design of a device to simulate a v i s u a l receptor network: the design consists of considerations leading to an overall block diagram. The operations of the units are described. 3.1 Preliminary Considerations and the Block Diagram. Consider the portion of a large but f i n i t e array of receptor inputs shown i n F i g . 3-1, and assume that the f i n i t e array i s illuminated by some pattern of l i g h t . .1 .2 #3 .4 ,5 .6 .7 ,8 # 9 .10 .11 .12 .13 J.4 ^5 ,16 ±1 .18 .19 .20 .21 .22 .23 .24 «25 £6 •27 • 28 .29 .30 .31 W32 .33 «4 .35 .36 .37 .38 .39 .40 .41 .42 ,43 .44 .45 46 47 48 49 .50 51 52 53 .54 .55 .56 .57 .58 .59 .60 £1 £2 .63 .64 .65 .66 .67 ^8 ^9 JO Jl ,72 .73 74 75 76 ' 77 78 79 80 .81 • • • • • • • • • F i g . 3—1. An Array of Receptor Inputs. The pattern w i l l give r i s e to a unique set of outputs from the receptors as determined by the set of equations ( l - l ) . The 17 proposed method of simulation operates on the receptor inputs* 81 at a time, i n a 9x9 square array. The arrays cover the complete f i e l d and overlap one another: the f e a t r a l points describe the path of the f a m i l i a r t e l e v i s i o n raster with a v e r t i c a l l i n e scan. For any one 9x9 array, the operation on the inputs i s described by the set of equations: 81 x. == e. - } k. . max(0,x. - t. .) i = 1*,..,81 ...(3-1) This set of equations i s solved by an i t e r a t i v e procedure to be described i n section 3-2 with a view to obtaining the value at the: center, x = x.,« The computer studies f o r constant e. "* C 41 1 have shown that the output, gives a good approximation to the output pf the receptor i n the same position i n the large but f i n i t e array. In our simulator the k. . and t, . functions are dependent only on the geometrical r e l a t i o n between the receptors, and are not dependent on signal amplitude or anything else. (in simulating a Limulus-type receptor network* the geometrical r e l a t i o n i s especially simple; see equations (2-?2) and (2-3).) The geometrical dependence of k. . i s i l l u s t r a t e d i n the following e q u a l i t i e s : k l , l l = k2,12 5 k l , 1 2 = k2*13 ' k l , 1 3 = k2,14 k l * 1 4 = k2,15 5 5 k l , 1 7 = k 2 , l 8 The basic block diagram of the simulator i s shown i n F i g . 3-2* 18 r i 1 Input Scanning Unit Computation Unit Output Scanning Unit — . ..... ..^ p. F i g , 3-2. Simulator Block Diagram, The input scanning unit obtains the e^ signals from a photo-I transparency which simulates the l i g h t pattern impinging on a receptor network. The computation unit solves the sets of equations ( 3 — l ) . The output scanning unit displays the re s u l t s . Each of these units i s dealt with i n d i v i d u a l l y i n the following sections. 3.2 The Computation Process: Iteration Cycle, Computation Cycle, The set of equations (3-1) can be rewritten i n the following form* 81 x i = e i ~ max(0,ki^^x^ - k ^ t ^ . . ) i = 1,**.,81 ...(3-2) 19 This form of the equations simplifies the switching requirements necessary f o r their solution. Thus, since the individual k. .t. . products are constant while the k. .x. product varies depending 1 » J J on the input e., i t i s simpler to multiply the variable x. by 3 J k. . and then to subtract the constant k. .t, . than to subtract a constant t. . from the variable and then to multiply the resultant variable by k. ., 1 > J We avoid using the 81 p a r a l l e l feedback paths of the set of equations (3-2) by solving a number of rather simple equations i n time sequence, 81 memory elements are used i n this process to store converging approximations to the x^ values. We obtain a " f i r s t approximation" to the x^ss x^"^ using the apparatus of Pig, 3—3» By sequential scanning using a staircase waveform the values of e^ are obtained. The f i r s t term* x ^ ^ = e-^ , follows from the fac t that we start with zero information i n a l l the memory elements and there i s consequently no feedback. The information i s stored i n memory element #1, While the scanning spot i s producing the next signal, e 2, the information i n memory element #1 is' sampled and the term max(0,k2 -^ y^  -k_ , t 0 .) i s produced. (Fig. 3-3 explains why y. i s used i n / \ place of ») We subtract this term from e 2 and store the difference i n memory element #2. The input from "summing junction" #2 to the memory elements moves i n synchronism with the sequential scanning spot used to obtain the e^ values. Thus, we have: x 2 ( l ) = e2 ~ m a x ( 0 ' k 2 , l y l - k 2 , l t 2 i l ) y r ^ Memory 1 \^ % Element #1 ~ l e. 1 Summing Junction #2 (n) Memory Y HElement #i Memory 8^1 Element#8l 81 y max(0,k. .y. - k. .t. .) "Routing" Gates Multiplying Units k i , j y j k. .t Threshold Units I J Integrating- I max(0,k. .y.-k. .t. .) 1 » J J -1-»J X » J Inverting Unit Maximum Operator k. .y.-k. .t. . ^ Summing Junction #1 Note: 1, Portion within dashed line i s discussed i n Section 3.6. lemory e i f j < i i f j > i Fi g . 3-3. Block Diagram of Computation Unit. 2. For simplicity of representation the memo lement outputs are given as y. t where 21 stored i n memory element #2. This value and the previously stored value x ^ ^ ^ are now operated on i n time sequence by the feedback loop to give us the two terms: max(0,k^ 1 y 1 - k^ ^ t ^ ^ ) and max(0,k.j ^  ~ k 3 , 2^3 2) Thus, we obtain: x = e- - >^ max(0,k^ .y. - k„ .t„ .) J -> j =1,2 » J J -> » J ->» J m = e - 3 max(0,k .y . - k . t .) m m ' m,j Jj m,j m,j y When m •= 81 we have concluded the f i r s t cycle, which we c a l l an i t e r a t i o n cycle, and we have 81 values of x^^^ stored i n the 81 memory elements. (2) In a second cycle of operations we obtain x^ . The cycle of events i s an exact re p e t i t i o n with the difference that we are starting the cycle with information stored i n a l l the memory elements and with feedback operating a l l the time. The convergence of the solution requires a maximum of 15 cycles, as determined by the computer simulation* The whole process i s termed a computation cycle. The section of the computation unit which operates on the values stored i n the memory elements to produce the k. .y. and the -k. .t. . terms i s described i n section 3-6. 22 3.3 The Input Scanning Unit. The purpose of the input scanning unit i s to di r e c t the correct e^ signals to "summing junction" #2 of the computation unit (Pig, 3-3). It must sample the e^'s i n order of ascending i corresponding to the space pattern or l a t t i c e i n Pig. 3-1 once every i t e r a t i o n cycle. At the end of a computation cycle i t must s h i f t the square array of points to a new position and repeat the above operation. The input scanning unit consists of a timebase, an oscilloscope, and a photo-multiplier. A transparency of the pattern to be studied i s placed against the face of the oscilloscope. The timebase moves the scanning spot sequentially to points of the face of the oscilloscope i n the pattern shown i n F i g . 3—1. At the end of each computation cycle i t s h i f t s the square array of light-points so that a complete analysis of the pattern i s obtained. The operation as viewed from the photo—multiplier i s shown diagramatically i n F i g . 3-4. A 3x3 square array i s used for sim p l i c i t y of i l l u s t r a t i o n . The photo—multiplier monitors the intensity af l i g h t transmitted through the transparency. The output from the photo-multiplier i s the e^ signal for the particular point of the pattern. The timebase consists of the four units shown i n F i g . 3-5. "Horizontal l a t t i c e scan" and " v e r t i c a l l a t t i c e scan" units generate the square array of dots; "horizontal l a t t i c e movement" and " v e r t i c a l l a t t i c e movement" units s h i f t the array at the end of each computation cycle. A " l a t t i c e scan" unit i s a gated potential divider 1. Computation Cycle #1 2. Computation Cycle #2 3» Computation Cycle #n o- possible light—point position for specified computation cycle. Pig* 3—4, Input Unit Scanning Pattern as Viewed from the Photo-multiplier* to 24 V e r t i c a l Lattice Scan Unit V e r t i c a l Lattice Movement J Unit To V e r t i c a l 'Oscilloscope Control Horizontal La t t i c e Scan Unit Horizontal Lattice Movement Unit To Horizontal Iscilloscope Control F i g . 3-5. Timebase Block Diagram* giving a staircase output. A " l a t t i c e movement" unit i s a f l i p - f l o p staircase generator i n which the number of steps per cycle can be varied. In conjunction with a v a r i a t i o n i n the size of the steps from the " l a t t i c e scan" units, varying the number of steps from the " l a t t i c e movement" units enables one t o obtain any desired resolution. For a given pattern analysis, the resolution i s constant over the whole pattern. The c i r c u i t s for these units are given i n Appendix I. 3.4 The Output Scanning Unit. At the end of each computation cycle the value determined for the central receptor, x.,, i s displayed. The intensity 25 control of the display unit, an oscilloscope, i s connected by a tran s i s t o r gate to the output from memory element #41 at the end of each computation cycle. The scanning spot for the output unit moves i n synchronism with the scanning spot of the input unit. 3.5 The Timing Unit. The block diagram of the timing unit i s given i n Fi g . 3-6. Clock Counter #1 Counter #3 Counter #4. Synchronizing Gate Counter #2 Fi g * 3-6. Block Diagram of the Timing Unit. The clock and the counters are related to the units they control i n Table 3-1. It can be b r i e f l y summarized? the clock 26 Clock Counter #1 Counter #2 Counter #3 Counter #4 Input Scanning Unit V e r t i c a l Lattice Scan Unit V e r t i c a l Lattice Movement Unit Horizontal Lattice Scan Unit Horizontal Lattice Movement Unit Controlled by V e r t i c a l Lattice Movement Unit Computation Unit Memory Element a) Input Gates b) Output Gates c) Set—zero Gates Multiplying Units Threshold Units Routing Gates Integrator Gate Summing Junction #2 Output Scanning Unit Memory Element #41 Output Gate Timing Unit Synchronizing Gate v " Counter #1 Counter #2 Counter #3 Counter #4 I — Table 3—1* Relationship between Clock, Counters, and Units They Control. 27 controls counters #1 and #2; counters #1 and #2 control the (n) individual steps i n the computation of an x^ '$ counter #3 (n) determines which x^ i s being computed; counter #4 controls the number of i t e r a t i o n cycles i n a computation cycle. In addition, counters #1 and #3 keep counter #2 synchronized. A detailed description of the interaction of the timing unit with the computation unit, the input scanning unit* and the output scanning unit i s given i n Appendix I, along with the c i r c u i t diagrams for the various units. 3.6 The Operation of the Routing Gates, the Multiplying Units, and the Threshold Units. This section deals with the portion of the computation unit which generates the k. .y. and -k. .t. . terms. In the solution of the set of equations (3-2), (8l) values of k. 1 » J and (81) values of k. .t. . are required. By taking advantage 1 i J 1 11 of the recurrence of pa r t i c u l a r values of k. . and t. . (as indicated i n section 3 . l ) , i t i s possible to generate a l l the k. . values using a set of 81 m u l t i p l i e r s , and a l l the k. .t. . values using a set of 81 threshold units. To i l l u s t r a t e how this can be accomplished, consider, for the sake of simplicity a 3x3 square array of receptors whose interaction i s described by the set of equations? 9 x± ~ e± ~ X^ki»a*X0 1 = 1 ' * 9 * * 9 ..*(3-3) j=l Note that the threshold values, k. .t. ., have been set to zero. The case of non-zero thresholds w i l l be dealt with l a t e r . 28 In order to obtain a clear picture of how the recurrence property of the k. . values can be used to our advantage, l e t us for the moment ignore the "routing" gates and the memory element output gates shown i n F i g . 3-3. Consider a set of nine (n) mu l t i p l i e r s and assume we are attempting to calculate x^ v '. If the outputs, y., from the nine memory elements are connected d i r e c t l y to the multipliers as shown i n F i g . 3-^ 7, and i f the potentiometer of the j mu l t i p l i e r , j = 1,..»,9, i s set to multiply by k_ ., then the multipliers have the outputs shown. M, X 5*3 k 5_4_y3 y, 6 > k5,6 X ^ 6 M 8 M. = M u l t i p l i e r #i L5,9 X 5^959 M, y. = Output from memory element #j F i g . 3-7. Multiplying Configuration for X,^ (n) 29 If the outputs from the multipliers are summed, and the result i s subtracted from e,. the value of x,. (n) i s obtained. Can the same mu l t i p l i e r potentiometer settings used i n (n) (n) the above computation of ' be used to calculate any x^ '? Consider the arrangement of memory element output-to-multiplier input connections shown i n Pig. 3-8 for the computation of (n) x, . The mu l t i p l i e r outputs are as shown. 5»1 J2 X 5,2 k 5_^3 0 "5,3 X X 5,4 ~k 5_t4^5 ^6 X 5,5 ^ 6 L5,6 0 X o X 5,7 k 5*£ y8 X 5,8 k J I ^ 9 Pig. 3-8. Multiplying Configuration f o r X ^ U ^ » Due to the geometrical dependence of the k. . function, the following equalities holds = k, k 5 , l ~ k6,2 L5,2 " ^6,3 k5,4 ~ k6,5 k5,5 = k6,6 k5,7 _ k6,8 k — k K5,8 ~ K6,9 30 Hence, i f the outputs from the multipliers are summed and the (n) result i s subtracted from e^, the value of x^ ' i s obtained, (n) S i m i l i a r l y , any x^ v 1 can be calculated using the same nine potentiometer settings. However, we are s t i l l l e f t with the p r a c t i c a l problem of how to obtain the necessary rearrangements of the memory element output-to-multiplier input connections (n) for the sequential computation of the x^ , i t = 1,..,,9. The solution of this problem requires the following modifications to the method given aboves 1. A number of gates i s used. Each gate i s composed of one or more electronic switches to open or close signal paths as determined by timing pulses, 2. Each memory element has an output gate, (See gate set #1 i n F i g . 3-9.) 3. Gates #1,2,3; #4 ,5,6; #7,8,9 of gate set #1 connect to three common points (called memory element row outputs) as shown i n F i g . 3-9. 4. The multipliers #1,2,3$ #4,5,6; #7.8*9 have common input points (called m u l t i p l i e r row inputs) as shown i n F i g . 3-9. 5. The multi p l i e r s connect through gate set #3 to a common output point. 6. The outputs from gate set #1 connect to the multi-p l i e r row inputs through three "routing" gates (gate set #2) as indicated i n Table 3-2, and as shown i n F i g , 3-9. 7. The gates i n set #1 are always activated i n the order 1-2-3-4-5-6-7-8-9. Memory Elements J . I i y 4 y5-8 y8 Gate Set #1 Memory Element Row Output #1 Memory Element Row Output #2 Memory Element 'T >• Row Output #3 Gate Set #2 ("Routing" Gates) M u l t i p l i e r Row Input #1 M u l t i p l i e r Row Input #2 M u l t i p l i e r Row Input #3 Mu l t i p l i e r s x x X ^ 4 x 3 * 4 x X 3^1 X X 3*3 X Gate Set #3 Output F i g . 3-9. Memory Element to M u l t i p l i e r Interconnections. UJ 32 Memory Element Output Row Number Mu l t i p l i e r Row Routing Gate #1 #2 #3 #1 - 1 2 #2 1 2 3 #3 2 3 — Table 3-2. Routing Gate Connections between M u l t i p l i e r Row Inputs and Memory Element Row Outputs. Gate Number of Gate Set #3 To Compute: Gate Number of , Gate Set #1 (n) X l (n) X2 (n) x 3 (n) x4 (n) x 5 (n) x6 (n) x7 (n) X8 (n) x9 1 5 4 3 2 1 9 8 7 6 2 6 5 4 3 2 1 9 8 7 3 7 6 5 4 3 2 1 9 8 4 8 7 6 5 4 3 2 1 9 5 9 8 7 6 5 4 3 2 1 6 1 9 8 7 6 5 4 3 2 7 2 1 9 8 7 6 5 4 3 8 3 2 1 9 8 7 6 5 4 9 4 3 2 1 9 8 7 6 5 Table 3-3. Simultaneous Gating Sequences for Gate Set #1 and Gate Set #3. 33 8, The gates i n set #3 are activated i n sequence but the starting point can be any of the nine positions depending on which x ^ ^ n ^ i s to be computed as indicated i n Table 3-3. The effect of these modifications w i l l now be i l l u s t r a t e d by a study of the action of the various gates during the (n) computation of x^ . Since the sixth receptor l i e s i n receptor row #2, gate #2 of set #2 i s closed and connects the memory elements to the multipliers as shown i n Pig. 3—9. Gate #1 of set #1 and gate #9 of set #3 are closed simultaneously, as indicated i n Table 3-3. No resultant signal appears at the output due to the action of the gates i n set #2. The simul-taneous closure of gate #2 of set #1 and of gate #1 of set #3 causes the signal k^ ^y^ = k^ ^y2 ^° a P P e a r a"t "kke output. Similarly as we proceed through the rest of the gating sequence (n) for x^ « the k^ ^y. signals are obtained i n time sequence at the output* In practice we must introduce thresholds into the set of equations (3-3) as follows: 9 x. - e - > max(0,k. .x. - k. .t. .) i — 1.....9 1 p i x'3 3 ^ x»3 ...(3-4) Consider a set of nine threshold units having gated outputs a l l of which connect to a common point. Arrange the threshold units as shown i n F i g . 3-11, and set the potentiometers so that gating the various units w i l l cause the outputs shown to appear at the common output point. Due to the geometrical dependence of the k. .t. . function, i t i s possible to generate a l l the k. .'t. . values using this set of nine threshold units* keeping the 34 potentiometers at the settings specified above. The operation of the threshold units follows that given for the mu l t i p l i e r s , and w i l l not be repeated. ^ 2 * 5 , 2 ~k5.3*5,3 k c j t - A - 5^4 5,4 " k5,7 t5,7 -k c T l8 " k5,9 +5,9 >• T ± - Threshold Unit #i. Fi g * 3-10. Threshold Units and Outputs* The introduction of the thresholds as given i n the set of equations (3-4), does not a l t e r the multiplying scheme given (n) for the computation of the x^ ' i n the set of equations (3—3), The common output point from the multipliers i s connected to one of the inputs of "summing junction" #1 (Fig* 3—3), The common output point from the threshold units i s connected to the other input point* Due to the fact that the k. . and the k. . t. . i n the set of equations (3-4) have i d e n t i c a l subscripts] 1»3 1»3 the gating sequences for the threshold units and the multipliers 35 are the same* Hence, the numbers i n the body of Table 3-2 refer to both the mu l t i p l i e r and the threshold output gates. The introduction of the threshold units causes certain unwanted signals to appear at the output from "summing junction" #1, As mentioned above, the synchronous closure of gate #1 of set #1 and gate #9 of set #3 during the computation of x ^ n ^ gives no resultant signal at the common mu l t i p l i e r output due to the action of the gates i n set #2, However, the simultaneous gating of the output of threshold unit #9 causes —kj. gtj. ^ to appear at the common threshold output and hence at the output from "summing junction" #1. Since the signal must now proceed through the maximum operator (Pig. 3-3) which sets a l l negative signals to zero, we can ignore this otherwise anomalous action. We have shown that i t i s possible to generate the k. . 1 » J values and the k. .t. . values necessary for the solution of the set of equations (3-4) using a set of nine mul t i p l i e r s and a set of nine threshold units. The set of equations describes the interactions between receptors i n a 3x3 array. We want to solve the set of equations (3-2) which describes the interactions between receptors i n a 9x9 array. We can obtain a method f o r doing this by simply expanding the results obtained for the 3x3 array. The necessary modifications are given i n Appendix I I . 36 4. PATTERN RECOGNITION STUDIES As mentioned i n section 1-2, the main reason for designing and building a simulator based on the set of equations (3—l) was to test the usefulness of i>h.is transformation i n the f i e l d of pattern recognition. In this chapter the results of some d i g i t a l computer studies of certain simple patterns are detailed, and some speculations on the use of the device are made. In the following sections, the levels of illumination i n t e n s i t y are specified i n terms of the equivalent value of e^. Thus, an illumination l e v e l of intensity 1^  w i l l be referred to as an illumination level of e^ pulses/sec* In addition a l l the studies carried out used the k. . and t. . functions of equations (2—3). 4.1 The E f f e c t of an "Edge" i n the 9x9 Square Array. The f i r s t pattern to be studied vas a simple step i n illumination i n t e n s i t y from a le v e l of e^=0 pulses/sec. to a lev e l of e^s=45 pulses/sec* The edge was f i r s t oriented so that i t was p a r a l l e l to one of the sides of the 9x9 square array, as shown i n Pig. 4—1(a). The central receptor locus i s shown. The va r i a t i o n i n the output, x c, of the central receptor as a function of the distance from the edge i s shown i n F i g . 4-1(b). Next the edge was oriented so that i t was p a r a l l e l to one of the diagonals of the array, as shown i n F i g . 4—2(a)* The central receptor locus i s shown. The v a r i a t i o n i n x i s given i n c F i g . 4-2(b). In both, cases the output of the central receptor undergoes a series of fl u c t a t i o n s above and below the uniform illumination 3 7 Central Receptor Locus F i g . 4-1 (a). Edge Orientation and Central Receptor Locus. 3 0 4-Central Receptor Output, X Q . (p.p.s.) 2 5 1 2 0 1 5 -1 0 -5 * 0 •I r-Central Receptor 6 7 to Edge Distance F i g . 4-1 (b). Vari a t i o n i n x as Central Receptor c Moves Away from Edge. 38 F i g * 4-2 (a). Central Receptor Locus Edge Orientation and Central Receptor Locus. ^ Central Receptor Output, x c» (p.p.s, ) 30--25-20-15-10-0 —t-1 —*-2 -t r- ~i r-Central Receptor 8 to Edge Distance F i g . 4-2 (b) Vari a t i o n i n x c as Central Receptor Moves Away from Edge. 39 output of 18*4 pulses/sec*, and climbs to a large peak when the central receptor i s adjacent to the unilluminated edge. In the f i r s t case the peak output i s 25.3 pulses/sec*, and i n the second case i t i s 27.3 pulses/sec. The flu c t a t i o n s near the illumination edge correspond to the well-known Mach's bands, which were f i r s t detected i n studies of the human vi s u a l process. 4.2 The E f f e c t of a Right Angle Corner of Illumination Intensity. The second pattern to be studied was a right angle illumination intensity corner. Two cases are considered. In the f i r s t case the edges are p a r a l l e l to the sides of the 9x9 square array, as shown i n F i g . 4-3(a). In the second case the edges are p a r a l l e l to the diagonals as shown i n F i g . 4-4(a). The v a r i a t i o n i n the output of the central receptor, x , for the c two cases i s shown i n F i g . 4-3(b) and F i g . 4—4(b)* As i n the f i r s t example* X q fluctuates about the uniform illumination output of 18.4 p u l s e s / s e c , r i s i n g to a peak when the central receptor i s adjacent to the vertex. In F i g * 4—3(b), the peak i s 31.9 p u l s e s / s e c , while i n F i g . 4—4(b) i t i s 34.0 pulses/sec. The example i n this section and the one i n the preceding section show the marked contour enhancement or "edging" that can be obtained using the transformation contained i n the set of equations ( 3 — l ) * The 9x9 square array with the k . and t. . i , 3 i»J functions of equations (2-3) produces a central receptor output which i s p a r t i a l l y dependent on the r e l a t i v e orientation of the pattern and the array. No steps have been taken to minimize thi s e f f e c t * The difference between the peak outputs for the di f f e r e n t orientations i n the two examples i s around 2 pulses/sec* 40 e. « 0 p*p*s = 45 p*p.s. Central Receptor Locus Pig* 4-3 (a)* Corner Orientantion and Central Receptor Locus* 35 1 Central Receptor Outputj H>* , (p.p.s.) 30 .. 25 -20 .. 15 10 5 0 Central Receptor to Vertex -+-4 - t -5 — t -6 Distance Pig* 4-3 (b). Variation i n x as Central Receptor Moves Away from Vertex. 41 Pig. 4-4 (a). Corner Orientation and Central Receptor Locus. Central Receptor Output, x , (p.p.s.) 35.. 30.. 25 . 10-5.. Central Receptor to Vertex >-0 1 2 3 4 5 6 7 8 Distance F i g . 4-4 (b). Variation i n x as Central Receptor Moves Away cfrom Vertex. Pig, 4-5 (a). Variable Angle Corner Orientation Central Receptor Output, x , (p.p.s.) C 0° 50° 100° 150° 200° 250° 300° 350° a F i g . 4-5 (b). Var i a t i o n i n x c with Var i a t i o n i n a. 44 4.4 Projected Applications for the Simulator* The preceding sections have dealt with the type of output to be expected from the simulator* Some indications have been given as to how t h i s output might be useful when applied to pattern recognition studies. A further indic a t i o n i s contained i n the work done by (7) V, K. Taylor on an automatic analog apparatus for pattern recognition* The "detail f i l t e r " described i n his work consists of a 3x3 array of l i g h t sensors interconnected to give the set of outputs described by the following equations* 9 x i = k l e i " k2 XI X3 1 " 1 , # " » 9 j=l k, 0, 0^ k 2 <C 1 This set of equations i s similar to the set of equations (3—l) which determine the output of the proposed simulator, i n that the i n h i b i t o r y i n t e r a c t i o n i s related to the output l e v e l s , not to the l e v e l of input stimulus. However* the lack of threshold l e v e l s and of varied i n h i b i t o r y constants leads to a less sophisticated type of pattern transformation* The output variations exhibited by the "d e t a i l f i l t e r " i n scanning across an illumination edge or corner have the same form as for the solutions for the set of equations (3—l) given i n sections 4.1 and 4*2 Taylor used the outputs from the "d e t a i l f i l t e r " to provide a contour enhancement or "edging" of the pattern signals before they proceeded into the actual recognition stages of his apparatus. Our approach w i l l be to optimize the k. . and 45 t. . functions used i n the set of equations (3—l) with respect to the recognition of given classes of patterns* This more sophisticated pre-processing of the patterns should lead to a reduction i n the complexity of the f i n a l recognition apparatus. In broadest terms, then, the simulator w i l l be used to obtain optimum k. . and t. . functions so as to simplify any given recognition apparatus. This purpose was constantly kept i n mind i n designing the simulator. For example, the use of potentiometers i n the multiplying and threshold units provides a means of r e a d i l y varying the k. . and t. . functions. Also, the 81 receptor positions do not have to remain fixed i n a 9x9 square array* The l a t t i c e scan units (Fig, AI—9) can be expanded to give p r a c t i c a l l y any desired l a t t i c e array of 81 points from the input scanning u n i t . The "routing" gates (gate set #2, F i g , 3—9) can be rewired to obtain the solution of the r e s u l t i n g set of equations. The d e s i r a b i l i t y of being able to simulate non-square receptor arrays i s indicated by certain studies carried out by (8) Hubel and Wiesel on the v i s u a l processes of the cat. They found that the receptive f i e l d i n the r e t i n a of the cat for a single c o r t i c a l unit i n the brain was often e l l i p s o i d a l i n shape* and hence the reaction of the receptive f i e l d to an illumination edge was dependent on the orientation of the edge. Such a strongly orientation dependent reaction would be very useful i n detecting preferred directions or "granularity" i n a pattern* 46 5. CONCLUSION The f e a s i b i l i t y of simulating a large v i s u a l receptor network of the type described by Hartline, et a l , has been studied* For a 5x5 primary i n h i b i t i o n f i e l d and a " r e a l i s t i c " l e v e l of i n h i b i t o r y i n t e r a c t i o n between the receptors, i t was shown that a 9x9 array contained the i n h i b i t i o n f i e l d of the central receptor. Hence, i n order to obtain a good simulation of the output from a large receptor array, i t i s s u f f i c i e n t to break i t up into much smaller, overlapping square—arrays and to solve the sets of equations (3—l) for the output of the central receptor* The set of such outputs gives a good approximation to the outputs from the overall receptor array. The design of a simulator based on the above findings has been presented i n i t s f i n a l form. By making use of a number of sets of gates, we found that we could generate a l l the k. . and k. .t. . values necessary for the solution of the set i»3 1*3 1*3 J of equations (3—2) using only 81 m u l t i p l i e r s and 81 threshold units* The c i r c u i t diagrams for the various elements making up the system are given. The construction of the simulator can now proceed* D i g i t a l computer studies of the type of output to be expected from the simulator indicate that i t w i l l be a useful tool i n the study of pattern recognition problems* In p a r t i c u l a r , the method used to generate the k. . and t. . functions lends i»3 1#3 i t s e l f to the optimization of these functions with respect to the recognition of any given class of patterns. Also, the 47 design of the simulator allows one to obtain asymmetrical receptor arrays from the basic symmetrical square array configuration* The author believes that the proposal contained i n this thesis provides the f i r s t p r a c t i c a l means for simulation of a large receptor network. 48 APPENDIX I A l . C i r c u i t s and System Operation. This appendix i s devoted to a description of the various c i r c u i t s employed i n the simulator, and to an outline of the interactions between these c i r c u i t s which leads to successful system operation. AI.l The Ring Counter. The basic units of the ring counter are MC 352G integrated-circuit f l i p - f l o p s produced by Motorola Inc. The c i r c u i t diagram for the ten terminal device i s given i n F i g . AI-1. The numbers on the diagram refer to the various terminals. VCC ° v o l t s  V E E ~ ~ 5 v o l t s F i g . AI-1. MECL Fl i p - F l o p MC352G- C i r c u i t . 49 For the supply voltages shown, the OFF-ON voltage levels from terminal #5 are -0.75v and -1.55v respectively. Terminal #4 outputs the opposite voltage l e v e l s . Positive-going voltage pulses into terminals #9 or #10 turn #5 ON; into #6 or #7 they turn #5 OFF. The c i r c u i t diagram for the ring counter i s given i n F i g . AI-2. The numbers on the various stages refer to the terminals indicated i n Pig. AI-1. The c i r c u i t i s designed to be "sure—starting" and "se l f - c o r r e c t i n g . " The l i n e marked A insures that i f the reset stage i s ON, then stage #1 i s ON and the rest of the stages are OFF. The line marked B insures that i f stages #2 to #n are OFF, then the reset stage i s ON. A positive—going, O.lusec, 1.2-1.8v pulse into the point marked C turns the ON stage OFF and the resulting positive-going pulse couples through an emitter follower, a capacitor and a diode to turn the succeeding stage ON. The output lines from the various stages are marked by the l e t t e r D. This c i r c u i t i s designed to work at a lOOKc clock pulse rate, and has been successfully tested up to 500Kc. Counter #1 i s driven d i r e c t l y by a lOOKc clock. Counter #2 i s driven i n d i r e c t l y by the clock through the synchronizing gate for which the c i r c u i t diagram i s given i n F i g . Al—3. The "or"-gate inputs marked by the A^, i = 1,...,13 connect to the 81 to 93 stages respectively of counter #1. Activation of the gate by one of these stages prevents the succeeding clock pulse from reaching counter #2. Hence, counter #2 remains i n the same state, while counter #1 proceeds through states #82 to #93 and back to state #1. Then as counter #1 49 For the supply voltages shown, the OFF-ON voltage levels from terminal #5 are -0.75v and -1.55v respectively. Terminal #4 outputs the opposite voltage l e v e l s . Positive-going voltage pulses into terminals #9 or #10 turn #5 ON; into #6 or #7 they turn #5 OFF. The c i r c u i t diagram for the ring counter i s given i n F i g . AI—2. The numbers on the various stages refer to the terminals indicated i n F i g . AI-1. The c i r c u i t i s designed to be "sure—starting" and "self - c o r r e c t i n g . " The l i n e marked A insures that i f the reset stage i s ON, then stage #1 i s ON and the rest of the stages are OFF. The line marked B insures that i f stages #2 to #n are OFF, then the reset stage i s ON. A positive—going, O.lusec, 1.2—1.8v pulse into the point marked C turns the ON stage OFF and the resulting positive-going pulse couples through an emitter follower, a capacitor and a diode to turn the succeeding stage ON, The output lines from the various stages are marked by the l e t t e r D. This c i r c u i t i s designed to work at a lOOKc clock pulse rate, and has been successfully tested up to 500Kc, Counter #1 i s driven d i r e c t l y by a lOOKc clock. Counter #2 i s driven i n d i r e c t l y by the clock through the synchronizing gate for which the c i r c u i t diagram i s given i n F i g . AI—3. The "or"-gate inputs marked by the A^, i = 1,...,13 S "b I ' d connect to the 81 to 93 stages respectively of counter #1. Activation of the gate by one of these stages prevents the succeeding clock pulse from reaching counter #2. Hence, counter #2 remains i n the same state, while counter #1 proceeds through states #82 to #93 and back to state #1. Then as counter #1 Reset Stage MC 352 G _4 7- 9 m I O K / 1 , N. B Stage #1 MC 352 G 5 Z ML IOK fa.ic •^>|—1 3<*K Stage #2 MC 352 G 5 7 10 Stage #n MC 352 G 5 .7 12. A l l transistors—TI415 A l l diodes - 1N191 Pig. AI-2. Ring Counter C i r c u i t , 51 s h i f t s to state #2, counter #2 switches to i t s next state. Thus, counter #2 drops behind one state with respect to the f i r s t 81 states of counter #1 every time counter #1 cycles. A, H4-toon o.«>f IK t (.OK Prom Clock 5-v To Counter #2 Pulse Input 1 l13 IOK B, A l l t ransistors - TI415 A l l diodes - 1N191 tOOK< O.iy-f I OtC -5V To output " O r l i n e from 4. of counter #2 Pig, AI-3. Synchronizing Gate C i r c u i t , The "and M-gate inputs marked B^ and connect to the output of the "or"-gate shown and to stage #81 of counter #3. This gate i s activated at the end of every i t e r a t i o n cycle and i n conjunction with the inverter shown insures that counter #2 i s i n the correct state for the start of the next i t e r a t i o n cycle. The driving pulses for counter #3 come from a diode-52 capacitor coupled inverter operated by stage #93 of counter #1. Thus, counter #3 advances one stage every time counter #1 cycles. AI.2 The Memory Elements th ' The c i r c u i t diagram for the i memory element, i = 1,...,81, i s given i n F i g . AI-4. The point A i s common with the same point on the other 80 memory elements. It connects to "summing junction" #2 through a diode "or"-gate activated by stages #82 to #92 of counter #1. The point C connects to a s *t "fch gate composed of the outputs from the 51 to the 80 stages of counter #1 i n a diode "or"-gate configuration which feeds into th a diode "and"—gate along with the output from the i stage of counter #3. Activation of this "and" gate causes the voltage on the 0.47uf. Mylar capacitor marked M to be reduced to zero. The th point marked B connects to the output from the i stage of counter #3. Activation of this "read-in" gate i n conjunction with the activation of the gate from "summing junction" #2 causes a new voltage to be stored on capacitor M. t h The point marked D connects to the output from the i stage of counter #1. Activation of t h i s gate causes the voltage stored across capacitor M to be read out at the point E, which connects to the point A of the memory element row output shown i n F i g . AI-5. To obtain a measure of the accuracy of the memory element, a plot was made of the input versus output voltage lev e l s as shown i n F i g , AI-6. From the plot i t i s evident that the output voltage i s from l»6v to 2.5v less than the input voltage. If a 2.0 v o l t referencing voltage level i s provided at the output A Input 3.5K< M IK IK - looKl H 0.002 ^ i-f E M f 9 / Output B Read—in Point Set-to-zero Point D Read—out Point A l l transistors - TI415 A l l diodes except those shown — 1N456 Fi g . AI-4. Memory Element C i r c u i t . u i 54 A TT4IS A 3.3 k B /OO-fl-i F i g . AI-5. Memory Element Row Output C i r c u i t . from "summing junction" #2 as shown i n F i g . Al—6, the output from the memory element reproduces the input signal l e v e l to within — 0.5v at the extreme ends of the 0-16v range with the error decreasing to zero at 9v. For a lOOKc clock rate, the read-in time for the memory element i s 0*11 msec, the set-to-zero time i s 0*30 msec, the read-out time i s 10 usee, and the storage time i s 75 msec. The overall performance of the memory element i s adequate with regard to the operation of the simulator. AI.3 Routing Gate, Multiplying Unit, and Threshold Unit. The "routing" gate c i r c u i t diagram i s shown i n F i g . AI-7. The "or"—gate inputs A-^  to kg connect to the appropriate stages of counter #3, e.g. either stages #1 to #9, #10 to #18,..,, or #72 to #81 respectively. The points B to B connect to the point Pig. Al—6. Input Voltage vs. Output Voltage for the Memory Element. U l U l A J 1<--Kr-I O K B m ' r > B A l l transistors - TI415 A l l diodes - 1N191 B .ooa^ f 4. K . 6 0 1 ^ t - r Pig* AI-7. Routing Gate C i r c u i t * 57 marked B i n Pig. AI-5 of the memory element row outputs i n accordance with the groupings given i n Table AII?2» Similarly, the points marked C to C, connect to the m u l t i p l i e r rows i n accordance with the same table, as explained i n section 3-6. The c i r c u i t diagrams for a multiplying unit and a threshold unit are given i n F i g . Al-8. The point A-^  on the multiplying unit i s common with the rest of the units i n the row and connects to the appropriate outputs from the routing gates as specified above. The points B 1 and on the multiplying unit and the threshold unit respectively connect to the same output from the appropriate stage of counter #2* as follows: The i * * 1 multiplying or threshold unit as specified i n Appendix II "til connects to the (i+40) counter stage i f i ss 41 and to the ( i - 4 l ) s ^ stage i f i > 41. The 2K potentiometers are set so that i f y. v o l t s i s input to the multiplying unit, a gating pulse w i l l cause the output of k. .y. vo l t s and -k. .t. . v o l t s into the two terminals of "summing junction" #1. AI.4 Staircase Generators. Two d i f f e r e n t types of staircase generators are used i n the simulator. The gated potential divider type shown i n F i g . AI-9 i s used i n the "horizontal" and " v e r t i c a l l a t t i c e scan" u n i t s . The f l i p - f l o p type shown i n F i g . AI—10 i s used i n the "horizontal" and " v e r t i c a l l a t t i c e movement" units, and i n counter #4. The operation of both types of unit i s quite straightforward* Consider the outputs from the 81 stages of counter #3 TI-+15 IK . 0 0 1 / 4. 1 — 1 < B, Pig, AI-8 (a). Multiplying Unit C i r c u i t , F i g . AI-8 (b). Threshold Unit C i r c u i t . 59 5Z. 10 n 4C0 K .ooj. =F A" I O K 9 Output •-4.LK 4 t 0 K 4= * -JWvV— — f O K A l l transistors - TI415 A l l diodes - 1N191 / O K =1= o.iy/ 1 i A, F i g . AI-9* Gated Potential Divider Staircase v • "Generator C i r c u i t * to be arranged sequentially i n a 9x9 square—array* If the members of each rov are connected to form a diode "or"—gate, and the members of each column are connected i n the same fashion, then the row—gates and column-gates that are obtained can be used i n conjunction with the gated potential divider as " v e r t i c a l " arid "horizontal l a t t i c e scan" units. If the outputs from row-gates #1 to #9 are connected to the points A 1 to Ag respectively, a " v e r t i c a l l a t t i c e scan" unit i s obtained. Similarly connecting the outputs from column-gates #1 to #9 to points A-^  to A^ gives a "horizontal l a t t i c e scan", unit . Flip-Plop #1 Set 0 Set-Reset Flip-Flop #2 O.I ;*-f Set 0 Set Resei 19 K 1K _ v v w < -IK TI<rlS" c IOK + 2«fv Output F i g . AI-10. Flip-Flop Staircase Generator C i r c u i t . o 61 The f l i p - f l o p staircase generator shown i n Pig. AI-10 operates i n the following fashion. The eight f l i p - f l o p s and the r e s i s t i v e network constitute a digital-to-analog converter. Input pulses at A are counted by the f l i p - f l o p array and the analog voltage i s output at B. Eventually the voltage at C becomes s l i g h t l y positive causing the double-inverter c i r c u i t to emit a positive pulse resetting the f l i p - f l o p array. Variation of the 20K potentiometer setting causes a va r i a t i o n in the number of steps i n the staircase output at B. Counter #4 consists of the f l i p - f l o p staircase generator and an "and"-gate connected to the f l i p - f l o p s i n such a fashion as to be activated after the correct number of i t e r a t i o n cycles. The potentiometer i s set so that counter #4 recycles on the pulse after the activation of the "and"-gate. The input pulses to counter #4 come from tljie 81 stage of counter #3 through an inverter. I The " v e r t i c a l " and "horizontal l a t t i c e movement" units are f l i p - f l o p staircase generators. The recycling pulses of counter #4 are also the input pulses to the " v e r t i c a l l a t t i c e movement" unit, and i t s recycling pulses are the input pulses to the "horizontal l a t t i c e movement" unit. AI.5 Miscellaneous C i r c u i t s . The output gate on memory element #41 i s i d e n t i c a l with the transistor-diode switch shown on the output of the memory element i n Pig. AI-4. The switch i s activated by an "and"-gate consisting of the inverted output from the "and"-gate on counter 62 #4 and the output from stage #42 of counter #3* The integrator-inverter c i r c u i t diagram i s given i n F i g . AI-11. -/5V _[ F i g . AI-11. Integrator-Inverter C i r c u i t . The point A connects to the output from "summing—junction" #1 through the grounded diode maximum-operator. The point B connects to one of the inputs of "summing—junction" #2. The point C connects to stage #93 of counter #1. A negative pulse into C causes the voltage across capacitor C^ to be set to zero, This completes the description of the c i r c u i t s and of the operation of the system. 63 APPENDIX II AH. Modifications to the Multiplying and Threshold Scheme Given i n Section 3.6. In order to solve the set of equations (3—2), the following modifications to the method given i n section 3.6 for the solution of the set of equations (3-4) are necessary: 1. Gate set #1 has 81 gates. 2* Gates #1 to 9; #10 to 18; #19 to 27,* #28 to 36; #37 to 45j #46 to 54; #55 to 63; #64 to 72; #73 to 81 of set #1 connect to nine common points respectively. 3. M u l t i p l i e r s #1 to 9; #10 to 18; .••; #72 to 81 have common input points. 4* The memory element row outputs connect to the multi-p l i e r row inputs through the nine gates of gate set #2 as i n d i -cated i n Table A l l - l . 5. The 81 gates of set #3 connect the multi p l i e r s to a common output point. 6. The output gates of the 81 threshold units connect to a common point. 7* The gates of set #1 are always activated i n the order 1-2-3-4-....-80-81. 8. The gates i n set #3 are activated i n a sequence (n) which depends on which x^ ' i s to be computed as shown i n Table A l l — 2 , The gating sequence can also be determined from the formula: G 3 = Gr - i + 41 + 8ld 64 where = gate number i n set #3 = gate number i n set #1 i = subscript of x. d =0,-1 such that G^ takes on only-integral values from 1 to 81 th 9* The potentiometer i n the j mu l t i p l i e r i s set to multiply by k 4 1 ., j = l , . . . , 8 l . th 10* The potentiometer i n the j threshold unit i s set to give the output -k., .t., .. 4 ± , j 4 i , 3 If these modifications are carried out the resu l t i n g system w i l l solve the set of equations (3-2) i n the same manner as the o r i g i n a l system solved the set of equations (3-4). Memory Element Output Row Number Mu l t i p l i e r Row Routing Gate #1 #2 #3 #4 #5 #6 #7 #8 #9 #1 - - - - 1 2 3 4 5 #2 - - - 1 2 3 4 5 6 #3 - - 1 2 3 4 5 6 7 #4 - 1 2 3 4 5 6 7 8 #5 1 2 3 4 5 6 7 8 9 #6 2 3 4 5 6 7 8 9 -#7 3 4 5 6 7 8 9 - — #8 4 5 6 7 8 9 - - -#9 5 6 7 8 9 - - - -Table AII-1* Expanded Routing Gate Connections between M u l t i p l i e r Row Inputs and Memory Element Row Outputs. Gate Number of Gate Set #3 Gate Number of Gate Set #1 To Compute: (n) X l 2 (n) X3 (n) X4 (n) X5 x80 (n) X81 1 41 40 39 38 37 43 42 2 42 41 40 39 38 • 44 43 3 43 42 41 40 39 • 45 44 4 44 43 42 41 40 • 46 45 5 45 44 43 42 41 • 47 46 • • • • • • • • • • • * • • • • « • 81 40 39 38 37 36 • 42 41 Table AII-2. Simultaneous Gating Sequences for the Expanded Gate Sets #1 and #3. 66 REFERENCES 1. Mach, E., "Uber die ¥irkung der raumlichen Vertheilung des Lichtreizes auf die Netzhaut—I", Sitzber. Akad.  Viss . Vien Math, naturv. K l . I I t Vol. 52, pp. 303-322, 1865. 2. Be'kesy, G, von, "Zur Theorie des Horensj Die Schvingungsform der Basilar-membran", Physik. Z.« V o l . 29, pp. 793 -810, 1928. 3. Hartline, H. K., R a t l i f f , F., and M i l l e r , W.H., "Inhibitory Interaction i n the Retina and i t s Significance i n Vision", Nervous Inhibition, edited by E. Florey, Pergamon Press, N.T., 1961. 4. Barlow* H*B., "Summation and Inhibit i o n i n the Frog's Retina", J . Physiol.. V o l . 119, pp. 69-88, January, 1953. 5. K u f f l e r , S.W., "Discharge Patterns and Functional Organiza-ti o n of Mammalian Retina", J . Neurophysiol., Vol. 16, pp. 37-68, January, 1953. 6. Melzak, Z.A., "On a Uniqueness Theorem and i t s Application to a Neuro-physiological Control Mechanism", Information and Control, V o l . 5, pp. 163-172, 1962. 7. Taylor, W.K., "Pattern Recognition by Means of Automatic Analog Equipment", Proc. I.E.E., Vol. 106, Pt. B, March, 1959. 8. Hubel, D.H., and Wiesel, T.N., "Receptive Fields of Single Neurones i n the Cat's Striate Cortex", J . Physiol., Vol. 148, No. 3, pp. 574-591, 1959. 

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