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Lateral inhibition and the area operator in visual pattern processing Connor, Denis John 1969-12-31

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LATERAL INHIBITION AND THE AREA OPERATOR IN VISUAL PATTERN PROCESSING by DENIS JOHN CONNOR M.A.Sc B.A.Sc • 5 • > U n i v e r s i t y o f B.C U n i v e r s i t y o f B.C 1963 1965 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f E l e c t r i c a l E n g i n e e r i n g We ac c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d R e s e a r c h S u p e r v i s o r Members o f the Committee Head o f the Department Members o f the Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA June, 1969 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 the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I a g r e e t h a t the 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 and Study. I f u r t h e r agree 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 of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department 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 not 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 . The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Department of ABSTRACT The s t a t i c i n t e r a c t i o n of the r e c e p t o r nerves i n the l a t e r a l eye o f the horsesoe c r a b , L i m u l u s , • i s c a l l e d l a t e r a l i n h i b i t i o n . I t i s d e s c r i b e d by the H a r t l i n e e q u a t i o n s . A s i m u l a t o r has been b u i l t t o s t u d y l a t e r a l i n h i b i t i o n w i t h a v i e w t o a p p l y i n g i t i n a p r e - p r o c e s s o r f o r a v i s u a l p a t t e r n r e c o g n i t i o n system. The a c t i v i t y i n a l a t e r a l i n h i b i t o r y r e c e p t o r network i s maximal i n r e g i o n s of non-uniform i l l u m i n a t i o n . This' enhancement of i n t e n s i t y c o n t o u r s has been e x t e n s i v e l y s t u d i e d f o r the case of b l a c k and w h i t e p a t t e r n s . I t i s shown t h a t t h e l e v e l of a c t i v i t y near a. b l a c k - w h i t e boundary p r o v i d e s a measure of i t s l o c a l g e o m e t r i p r o p e r i t e s . However, the l e v e l of a c t i v i t y i s dependent on the boundary o r i e n t a t i o n . A number of methods f o r r e d u c i n g t h i s o r i e n t a t i o n dependence a r e e x p l o r e d . The a c t i v i t y i n a l a t e r a l i n h i b i t o r y network a d j a c e n t t o a boundary can be modelled by an a r e a o p e r a t o r . I t i s shown t h a t the v a l u e o f t h i s o p e r a t o r a l o n g an i n t e n s i t y boundary p r o v i d e s a d e s c r i p t i o n of t h e boundary t h a t i s r e l a t e d t o i t s i n t r i n s i c d e s c r i p t i o n — c u r v a t u r e as a f u n c t i o n of a r c l e n g t h . S i n c e the o p e r a t o r i s maximal on an i n t e n s i t y boundary, t h i s d e s c r i p t i o n has been c a l l e d the r i d g e f u n c t i o n f o r the boundary. A r i d g e f u n c t i o n can a l s o be o b t a i n e d u s i n g a l a t e r a l i n h i b i t o r y , network. The p r o p e r t i e s of t h i s f u n c t i o n a r e d i s c u s s e d . I t i s shown how r i d g e f u n c t i o n s might be i n c o r p o r a t e d i n t o a p a t t e r n r e c o g n i t i o n a l g o r i t h m . A n o v e l method f o r d e t e c t i n g t h e b i l a t e r a l and r o t a t i o n a l symmetries i n a p a t t e r n i s d e s c r i b e d . TABLE OF CONTENTS Page LIST OP ILLUSTRATIONS v LIST OF TABLES X V ACKNOWLEDGEMENT . . x v i 1. ANIMAL AND HUMAN VISUAL SYSTEMS 1 1.1 I n t r o d u c t i o n 1 1.2 The V i s u a l Systems of V a r i o u s A n i m a l s 1 (1) The Fr o g 1 (2) The Cat 2 (3) Other Animals - 5 1.3 The Human V i s u a l System 6 1.4 L a t e r a l I n h i b i t i o n i n Limulus 8 (1) A Model f o r t h e L a t e r a l Eye of Limulus 8 (2) The L a t e r a l I n h i b i t i o n S i m u l a t o r 11 1.5 A Summary of the T h e s i s 12 2. LATERAL INHIBITION, 14 2.1 The I n i t i a l S t u d i e s of L a t e r a l I n h i b i t i o n . 14 2.2 Methods f o r Reducing O r i e n t a t i o n Dependence 24 3. EXPERIMENTAL STUDIES ON LATERAL INHIBITION .. 29 3.1 I n t r o d u c t i o n 29 3.2 The Li m u l u s k. . (d) F u n c t i o n 32 3.3 .The Un i f o r m k. . (d.) F u n c t i o n . . . 41 3.4 The I n v e r s e k. .(d) F u n c t i o n 53 3.5 Peak Re c e p t o r A c t i v i t y and C i r c u l a r P a t t e r n s . . . . 61 3.6 Re c e p t o r A c t i v i t y Near D i s k s and Wedges 70 3.7 "Feed-Forward" I n h i b i t o r y I n t e r a c t i o n 79 i i i Page 4. LATERAL INHIBITION AND THE AREA OPERATOR 84 4.1 I n t r o d u c t i o n 84 4.2 D i s k A r e a Operator 84 4.3 Peak Response t o Wedges 86 4.4 Peak Response t o D i s k s . 92 5. THE AREA OPERATOR AND CURVATURE 103 5.1 I n t r o d u c t i o n 103 5.2 G e n e r a l Form f o r the Area Operator 103 5.3 Curves i n a Plane 110 5.4 Shape D e s c r i p t i o n U s i n g Area O p e r a t o r s - 112 6. SECONDARY PROCESSING OP RECEPTOR ACTIVITY FUNCTIONS.. 129 6.1 I n t r o d u c t i o n ' 129 6.2 Boundary P r e d i c t i o n U s i n g Peak R e c e p t o r A c t i v i t y 130 6.3 Concave-Convex F i g u r e s 135 6.4 R i d g e - P o i n t O p e r a t o r s . . 137 6.5 The "Ridge-Runner" A l g o r i t h m 143 7. THE RIDGE FUNCTION AND APPLICATIONS 147 7.1 I n t r o d u c t i o n 147 7.2 The R e c e p t o r A c t i v i t y Ridge F u n c t i o n ' 147 7.3 Three Types o f Ridge F u n c t i o n s 154 7.4 The Ridge F u n c t i o n and P a t t e r n R e c o g n i t i o n 157 8. CONCLUSIONS -. 171 REFERENCES 173 i v LIST OF ILLUSTRATIONS F i g u r e Page 1.2.1 The V i s u a l System of t h e Cat 3 1.3.1 R e l a t i v e Frequency o f P o i n t Placement f o r P r e s e r v i n g Shape 8 1.3.2 The P o i n t s o f Maximum C u r v a t u r e on a Cat 9 1.4.1 (a) R e c e p t o r F i r i n g a t Rate e^ 10 (b) R e c e p t o r A c t i v i t y Reduced t o Rate x^ 10 2.1.1 The K „ ( d ) F u n c t i o n Used i n the I n i t i a l S t u d i e s . . . 15 2.1.2 (a) R e c e p t o r A r r a y Superimposed on a B l a c k and White Square 16 (b) R e c e p t o r A c t i v i t y a l o n g AA' 16 (c) R e c eptor A c t i v i t y a l o n g BB' 16 2.1.3 (a) R e c e p t o r A c t i v i t y A l o n g D i a g o n a l of the Square 17 (b) R e c e p t o r s w i t h A c t i v i t y i n t h e Range 132 x ± 200 p.p.s 17 2.1.4 (a) A Set o f R e c e p t o r s and a W h i t e - B l a c k Edge 18 (b) V a r i a t i o n i n the A c t i v i t y o f r-, as the Edge Moves Away 7 18 2.1.5 (a) A White Square a t 45° t o t h e R e c e p t o r Rows.... 19 (b) R e c e p t o r A c t i v i t y a l o n g AA' 19 (c) R e c e p t o r A c t i v i t y a l o n g BB' 19 2.1.6 A Wedge at. an Angle. 0 w i t h r e s p e c t t o the Rec e p t o r Rows " 20 2.1.7 Peak R e c e p t o r A c t i v i t y as a F u n c t i o n o f Wedge Angle f o r V a r i o u s O r i e n t a t i o n s 21 2.1.8 A 100° Wedge i n V a r i o u s O r i e n t a t i o n s 23 (a) 0 = 0° 23 (b) 0 = 15° 23 (c) 0 =. 30° 23 (d) 0 = 4.5° 23 (e) The k. . f o r the C e n t r a l R e c eptor 23 v F i g u r e Page 2.2.1 (a) Edge Response f o r a 9x9 A r r a y w i t h a 5x5 F i e l d o f D i r e c t I n h i b i t i o n 26 (b) Edge Response f o r a 5x5 A r r a y w i t h a 5x5 F i e l d o f D i r e c t I n h i b i t i o n 26 2.2.2 The Rounded 9x9 A r r a y 27 2.2.3 A Rounded 9x9 A r r a y o f R e c e p t o r s w i t h a F i n i t e F i e l d o f View 28 3.1.1 A T y p i c a l Computer P r i n t o u t of Peak A c t i v i t i e s a t V a r i o u s O r i e n t a t i o n s 30 3.1.2 The Peak A c t i v i t y Envelope f o r the Set of Graphs i n F i g . 2.1.7 31 3.2.1 The l i m u l u s k. .(d) F u n c t i o n ' 33 l j 3.2.2 Peak A c t i v i t y Envelope f o r Limulus k..(d) w i t h D = O.Ou and x Q = 27 p.p.s 7 ? 34 3.2.3 Peak A c t i v i t y Envelope f o r Limulus k. .(d) w i t h D = 0.5u and X q = 27 p.p.s Ti 1 34 3.2.4 Peak A c t i v i t y Envelope f o r Limulus k..(d) w i t h D = l.Ou and X q = 27 p.p.s 7? 35 3.2.5 Peak A c t i v i t y Envelope f o r Limulus k. .(d) w i t h D = 1.5u and x Q = 27 p.p.s 7? 35 3.2.6 Peak A c t i v i t y Envelope f o r Limulus k..(d) w i t h D = 2.0u and X q = 27 p.p.s .7? 36 •3.2.7 Peak A c t i v i t y Envelope f o r Limulus k..(d) w i t h D = 2.5u and X q = 27 p.p.s 7 i 1 36 3.2.8 Average Peak A c t i v i t y Spread a g a i n s t D f o r L i m u l u s k. . (d) 37 ±2 3.2.9 Change i n Median Peak A c t i v i t y w i t h I n c r e a s i n g D f o r V a r i o u s Wedge A n g l e s . . . . . . . 38 3.2.10 P o s i t i o n o f Peak A c t i v i t y f o r Two D i f f e r e n t V a l u e s o f D 39 3.2.11 Peak A c t i v i t y Envelope f o r Limulus k..(d) w i t h x Q = 50 p.p.s. and D = 1.5u 7? 40 3.2.12 Peak A c t i v i t y E n v e l o p e . f o r Limulus k..(d) w i t h x Q =.- 75 p.p.s. and D = 1.5u 7? 40 v i F i g u r e 3.3.1 The U n i f o r m k. .(d) F u n c t i o n 3.3.2 Peak A c t i v i t y Envelope f o r Un i f o r m k..(d) w i t h D = O.Ou and x = 0 p.p.s 7? o r r 3.3.3 Peak A c t i v i t y Envelope f o r Un i f o r m k..(d) w i t h D = 0.5u and X Q = 0 p.p.s 3.3.4 Peak A c t i v i t y Envelope f o r U n i f o r m k. .(d) w i t h D = 1.Ou and x = 0 p.p.s, o r r 3.3.5 Peak A c t i v i t y Envelope f o r Un i f o r m k. .(d) w i t h D = 1.5u and X Q = 0'p.p.s, 1 3 3.3.6 Peak A c t i v i t y Envelope f o r U n i f o r m k. .(d) w i t h D = 2.0u and x = 0 p.p.s, o r r 3.3.7 Peak A c t i v i t y E nvlope f o r U n i f o r m k..(d) w i t h D = 2.5u and x Q = 0 p.p.s, 3-D 3.3.8 Change i n Median Peak A c t i v i t y w i t h I n c r e a s i n g D f o r V a r i o u s Wedge A n g l e s 3.3.9 Average Peak A c t i v i t y Spread a g a i n s t D f o r U n i f o r m k. . ( d ; w i t h x = 0 p.p.s 3.3.10 Peak A c t i v i t y Envelope f o r Un i f o r m k. .(d) w i t h x Q = 25 p.p.s. and D = 1.5u. I D 3.3.11 Peak A c t i v i t y Envelope f o r U n i f o r m k. .(d) w i t h xQ = 50 p.p.s. and D = 1.5u 7? 3.3.12 The Hexagonal R e c e p t o r A r r a y . . . 3.3.13 Peak A c t i v i t y Envelope f o r U n i f o r m k i j ( d ) w i t h 3 D = O.Ou, X Q = 0. p.p.s. , and an Hexagonal A r r a y . . . .3.14 Peak A c t i v i t y Envelope f o r U n i f o r m k^-(d) w i t h D = 0.5u, x = 0 p.p.s., and an Hexagonal A r r a y . . . . 3.3.15 Peak A c t i v i t y Envelope f o r U n i f o r m kj_^(d) w i t h D = l.Ou, X q = 0 p.p.s., and an Hexagonal A r r a y . . . 3.3-16 Peak A c t i v i t y Envelope f o r U n i f o r m k^-j(d) w i t h D = 1.5u, x Q = 0 p.p.s., and an Hexagonal A r r a y . . . 3.3.17 Peak A c t i v i t y Envelope f o r U n i f o r m k^-j (d); w i t h D = 2.0u, X q = 0 p.p.s., and an Hexagonal A r r a y . . . 3.3.18 Peak A c t i v i t y Envelope f o r U n i f o r m k^-(d) w i t h D = 2.5u, X q = 0 p.p.s., and an Hexagonal A r r a y . . . 3.4.1 The I n v e r s e k ^ ( d ) F u n c t i o n 3.4.2 Peak A c t i v i t y Envelope f o r I n v e r s e k..(d) w i t h D = O.Ou and X q ' = 0 p.p.s............ i. . v i i F i g u r e Page 3.4.3 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d). w i t h D - 0.5u and x = 0 p.p.s ? 55 o 3.4.4 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h D = l.Ou and x = 0 p.p.s ? 56 o 3.4.5 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h D = 1.5u and x = 0 p.p.s j1 56 o 3.4.6 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h D = 2.0u and X Q = 0 p . p . s , . U , 57 3.4.7 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h D = 2.5u and X Q = 0 p.p.s .V 57 3.4.8 Change i n Median Peak A c t i v i t y w i t h I n c r e a s i n g D f o r "Various Wedge Ang l e s 58 3.4.9 Average Peak A c t i v i t y Spread a g a i n s t D f o r I n v e r s e k ^ ( d j w i t h X Q = 0 p.p.s 59 3.4.10 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h x Q = 25 p.p.s. and D = 2.0u .Y 60 3.4.11 Peak A c t i v i t y Envelope f o r I n v e r s e k..(d) w i t h x Q = 50 p.p.s. and D = 2.Ou i1 60 3.5.1 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the • Lim u l u s k. . ( d ) , D = 1.5u, and x = 2 7 p.p.s 62 1 - ) > > 0 3.5.2 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the Un i f o r m k i - j ( d ) . D = !-5u," and X Q = 0 p.p.s 64 3.5.3 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the I n v e r s e k. . ( d ) , D = 2.0u, and x = 0 p.p.s 65 3.5.4 E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r Li m u l u s k. . ( d ) , D = 1.5u and x = 27 p.p.s 7? 66 o 3.5.5 E q u a l Peak A c t i v i t y Wedge and D i s k t o S c a l e w i t h 9x9 A r r a y , L i m u l u s C o n f i g u r a t i o n . . . . ' 67 3.5.6 E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r the U n i f o r m k. .(d), D = 1.5u, and x = 0 p.p.s. 7? ' 68 o 3.5.7 E q u a l Peak A c t i v i t y Wedge and D i s k t o S c a l e w i t h 9x9 A r r a y , U n i f o r m C o n f i g u r a t i o n 68 3.5.8 E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r I n v e r s e k. . ( d ) , D = 2.0u, and x = 0 p.p.s. 69 v i i i F i g u r e ' Page 3 . 5 . 9 E q u a l Peak A c t i v i t y . Wedge and D i s k t o S c a l e w i t h 9x9 A r r a y , I n v e r s e C o n f i g u r a t i o n 69 3.6.1 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r L i m u l u s C o n f i g u r a t i o n 71 3.6.2 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r U n i f o r m C o n f i g u r a t i o n 72 3.6 . 3 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r I n v e r s e C o n f i g u r a t i o n 74 3.6.4 (a) A Large White D i s k and t h e 9x9 A r r a y 75 (b) A S m a l l White D i s k and the 9x9 A r r a y 75 3.6.5 (a) Contour P l o t of the Receptor A c t i v i t y F u n c t i o n f o r a 100° Wedge, Limulus C o n f i g u r a t i o n 76 (b) An I s o m e t r i c View of the 100° Wedge, R e c e p t o r A c t i v i t y F u n c t i o n , L i m u l u s C o n f i g u r a t i o n . 76 3.6.6 (a) Contour P l o t of the Rec e p t o r A c t i v i t y F u n c t i o n f o r a 100° Wedge, U n i f o r m C o n f i g u r a t i o n . 77 (b) An I s o m e t r i c View of the 100° Wedge, Rec e p t o r • A c t i v i t y F u n c t i o n , U n i f o r m C o n f i g u r a t i o n 77 3.6.7 (a) Contour P l o t o f the Receptor A c t i v i t y F u n c t i o n f o r a 100° Wedge, I n v e r s e C o n f i g u r a t i o n 78 (b) An. I s o m e t r i c View of the 100° Wedge, Re c e p t o r A c t i v i t y .Function, I n v e r s e C o n f i g u r a t i o n . . . . 78 3.7.1 "Feed-Back" and "Feed-Forward" R e c e p t o r I n t e r a c t i o n 80 3.7.2 "Feed-Back" and "Feed-Forward" Peak A c t i v i t y E n v e l o p e s f o r Wedges; U n i f o r m k. . ( d ) , D = 1.5u, and x - 0 p.p.s 7 ? 81 o r r 3.7 . 3 "Feed-Back" and "Feed-Forward" Peak A c t i v i t y Envelopes f o r D i s k s ; U n i f o r m k ^ ( d ) , D = 1.5u, and X q = 0 p.p.s. 82 4 . 3-1 The Ope r a t o r D i s k and a Wedge.... 86 4 . 3.2 (V ) as a F u n c t i o n of the Wedge An g l e 87 y op max to 0 4 . 3 - 3 L i m u l u s k. .(d) Peak Response Curve and (V ) t J s —. i j op max L i n e f o r Wedges 88 4 .3.4 U n i f o r m k. .(d) Response Curve and (V ) L i n e op max f o r Wedges 90 i x F i g u r e Page 4.3.5 I n v e r s e k. .(d) Response Curve and (V ) L i n e -e i,7 3 i . l op max n~. f o r Wedges. 91 4.4.1 P a t t e r n D i s k w i t h the O p e r a t o r D i s k i n the (V ) - n • 1 • op max r**? P o s i t i o n 93 4.4.2 U n i f o r m k. .(d) Response Curve and (V ) L i n e 1.2 op max f o r D i s k s . 95 4.4.3 L i m u l u s k ^ ( d ) Response Curves f o r D i s k s as a F u n c t i o n of the Area of I n t e r s e c t i o n ". 97 4.4.4 L i m u l u s k^_.(d) Response Curves f o r D i s k s as a F u n c t i o n o f the Weighted Area of I n t e r s e c t i o n . . . 99 4.4.5 I n v e r s e k ^ ( d ) Response Curves f o r D i s k s as a F u n c t i o n of the A r e a o f I n t e r s e c t i o n 100 4.4.6 I n v e r s e k ^ ( d ) Response Curves f o r D i s k s as a F u n c t i o n of the Weighted Area of I n t e r s e c t i o n . . . 101 5.2.1 G e n e r a l Geometry of the Area Operator . 104 5.2.2 A T h r e e - L e v e l I(x,y) F u n c t i o n 107 5.2.3 (a) O p e r a t o r Response a l o n g AA' . . 108 (b) O p e r a t o r Response a l o n g BB' 108 (c) O p e r a t o r Response a l o n g CC'.. ' 108 (d) Operator Response a l o n g DD' 108 5.4.1 A Mesa I n t e n s i t y F u n c t i o n , I ( x , y ) 113 5.4.2 The R e l a t i o n between N and N a t two P o i n t s on a Curve ? 114 5.4.3 An I ( x , y ) Mesa F u n c t i o n w i t h a Convex M^ R e g i o n . 117 5.4.4 (a) A S m a l l Convex M 2 R e g i o n 119 (b) A Large Convex M^ Region 119 5.4 .5 The Operator D i s k on C w i t h a Convex Region.. 121 5.4.6 (a) The Operator D i s k a t ( x Q , y ) 122 (b) The D i s k Moved a D i s t a n c e An a l o n g N 122 3? 5.4.7 (a) A D i s k a t Two P o i n t s on a Curve w i t h D e c r e a s i n g C u r v a t u r e 126 x F i g u r e Page 5.4.7 (To) The D i s k s Superimposed.. 126 5.4.8 The D i s k Operator i n t h e Neighborhood of a S e p a r a b l e F e a t u r e 127 6.1.1 (a) I s o m e t r i c o f the R e c e p t o r A c t i v i t y F u n c t i o n f o r a White P o l y g o n 130 (b) I s o m e t r i c of the R e c e p t o r A c t i v i t y F u n c t i o n f o r a White D i s k 130 6.2.1 (a) The P o l y g o n a l e±(x,y) F u n c t i o n 131 (b) The Contoured x ^ ( x , y ) F u n c t i o n 131 (c) The L o c a l Maxima of the x ^ X j y ) F u n c t i o n . . . . 131 (d) The Expanded L o c a l Maxima 131 6.2.2 (a) S e a r c h P a t t e r n f o r L o c a t i n g One S i d e o f the Wedge 133 (b) T e s t i n g f o r the Second Side 133 (c) P a r t of the P o l y g o n Boundary 133 (d) - A Change i n t h e D i r e c t i o n o f the Boundary... 133 (e) The Complete P o l y g o n Boundary 133 6.3.1 (a) A B l a c k P o l y g o n w i t h a Convex V e r t e x 136 (b) R e c e p t o r A c t i v i t y F u n c t i o n f o r the B l a c k P o l y g o n , 136 6.3.2 (a) I s o m e t r i c of the x J ( x , y ) F u n c t i o n f o r the B l a c k P o l y g o n 137 (b) I s o m e t r i c View of the Combined A c t i v i t y F u n c t i o n 138 6.4.1 I s o m e t r i c of an A c t i v i t y F u n c t i o n w i t h Superimposed Ridge L i n e 138 6.4.2 (a) The H o r i z o n t a l Ridge P o i n t O p e r a t o r 139 (b) The D i a g o n a l Ridge P o i n t O p e r a t o r 139 (c) The Combined Ridge P o i n t Operator 139 6.4.3 (a) Contoured A c t i v i t y F u n c t i o n f o r P o l y g o n .140 (b) Ridge P o i n t s D e t e c t e d by H o r i z o n t a l O p e r a t o r . . 140 (c) Ridge P o i n t s D e t e c t e d by D i a g o n a l O p e r a t o r . . . . 140 x i F i g u r e Page 6.4.3 (d) Ridge P o i n t s D e t e c t e d by Combined O p e r a t o r . . . . 140 6.4.4 (a) Contoured R e c e p t o r A c t i v i t y .Function f o r a B l a c k "S" 141 (b) Ridge P o i n t s D e t e c t e d by H o r i z o n t a l O p e r a t o r . . 141 (c) Ridge P o i n t s D e t e c t e d by D i a g o n a l O p e r a t o r . . . . 141 (d) Ridge P o i n t s D e t e c t e d by Combined O p e r a t o r . . . . 141 6.4.5 (a) Combined Receptor A c t i v i t y F u n c t i o n f o r an "E" i n Contour 143 (b) Ridge P o i n t s D e t e c t e d by H o r i z o n t a l O p e r a t o r . . 143 (c) Ridge P o i n t s D e t e c t e d by D i a g o n a l O p e r a t o r . . . . 143 (d) Ridge P o i n t s D e t e c t e d by Combined O p e r a t o r . . . . 143 6.5.1 (a) R i d g e - L i n e Traced t h r o u g h R i d g e - P o i n t s o f F i g . 6.4.5(h) 146 (b) R i d g e - L i n e Traced t h r o u g h R i d g e - P o i n t s o f F i g . 6.4.5(d) ' 146 (c) R i d g e - L i n e f o r P o i n t s i n F i g . 6.4.3(d) 146 (d) R i d g e - L i n e f o r P o i n t s i n F i g . 6 .4 .4(c) 146 7.2.1 The R i d g e - L i n e P a t h and Ridge F u n c t i o n f o r a B l a c k "G" 148 7.2.2 The R i d g e - L i n e and Ridge F u n c t i o n f o r . an "M" 150 '7.2.3 (a) The R i d g e - L i n e s and Ridge F u n c t i o n s f o r an "H" and a 1 1Z" 151 (b) The R i d g e - L i n e s and Ridge F u n c t i o n s f o r an "M" and a "W" 151 7.2.4 (a) The Ridge Funct i o n f o r a L e t t e r w i t h One A x i s of Symmetry. • 152 (b) The Ridge F u n c t i o n f o r a L e t t e r w i t h Two Axes of Symmetry.. 152 7.2.5 Two L e t t e r s w i t h a Common Sequence o f F e a t u r e s . . . . 153 7.2.6 The Two Ridge F u n c t i o n s f o r an "0" 153 7.3.1 (a) I s o m e t r i c o f x ± ( x , y ) f o r a White "H" 155' (b) I s o m e t r i c of x ! ( x , y ) f o r a White "H" 155 x i i F i g u r e . Page 7.3.1 (c) I s o m e t r i c of X ^ x ^ ) f o r a White "H" 155 7.3.2 (a) The x ± ( x , y ) Ridge F u n c t i o n f o r a White "H" . 156 (b) The x | ( x , y ) Ridge F u n c t i o n I56 (c) The X i ( x , y ) Ridge F u n c t i o n 156 (d) Transformed X^(x,y) Ridge F u n c t i o n 156 7.4.1 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r "K" , White on B l a c k 158 7.4.2 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples o f the L e t t e r "X", White on B l a c k . . 159 7.4.3 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r "H", White on B l a c k 160 7.4.4 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples o f the L e t t e r "C", White on B l a c k 161 7.4.5 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples o f the L e t t e r "U" , White on B l a c k 162 7.4.6 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r " J " , White cn B l a c k . . . . 163 7.4.7' (a) The L e t t e r "X" has Second Order R o t a t i o n a l Symmetry and Two Axes o f Symmetry. 166 (b) The L e t t e r "N" has Second Order R o t a t i o n a l Symmetry ' 166 (c) The S w a s t i k a has F o u r t h Order R o t a t i o n a l Symmetry 166 x i i i LIST OF TABLES T a b l e Page 2.4.1 Sum of k^ . V a l u e s a t V a r i o u s O r i e n t a t i o n s 24 3.2.1 L i m u l u s k ^ ( d ) A m b i g u i t y V a l u e s f o r V a r i o u s R e c e p t i v e ' F i e l d S i z e s 33 3.2.2 A m b i g u i t y V a l u e s f o r V a r i o u s T h r e s h o l d s w i t h the L i m u l u s k. .(d) F u n c t i o n 41 -L-) 3.3.1 A m b i g u i t y V a l u e s f o r a U n i f o r m k..(d) a t V a r i o u s V a l u e s o f D 7? 42 3.3.2 A m b i g u i t y V a l u e s f o r V a r i o u s T h r e s h o l d s w i t h the U n i f o r m k. .(d) F u n c t i o n "49 3.3-3 A m b i g u i t y V a l u e s f o r V a r i o u s D, U n i f o r m k..(d) and an Hexagonal A r r a y 7^ ...... . 53 3.4.1 A m b i g u i t y V a l u e s f o r the I n v e r s e k..(d) a t V a r i o u s V a l u e s o f D 7? . . 58 3.4.2 A m b i g u i t y V a l u e s f o r V a r i o u s T h r e s h o l d s w i t h the I n v e r s e k. .(d) F u n c t i o n 61 x i v ACKNOWLEDGEMENT I w i s h t o acknowledge the a s s i s t a n c e r e c e i v e d from a number of my c o l l e a g u e s . Dr. John MacDonald and Dr. M.P. Beddoes s u p e r v i s e d the t h e s i s . Dr. MacDonald p r o v i d e d en couragement and i n s p i r a t i o n throughout the work. Dr. Beddoes was most generous w i t h h i s a s s i s t a n c e i n p o l i s h i n g the f i n a l v e r s i o n of the t h e s i s . Dr. A l e c Melzak p r o v i d e d enthusiasm and a d v i c e , p a r t i c u l a r l y w i t h the t h e o r y i n c h a p t e r 5- L a c h l a n Brown, Stan Semrau and John Bennett gave a g r e a t d e a l of h e l p w i t h computer programming. Robert Epp p o i n t e d out the e l e g a n t p r o o f f o r Theorem 5-4.8. Tony Leugner drew the f i g u r e s which grace the t h e s i s . Bev Harasymchuk q u i c k l y and c h e e r f u l l y typed i t . F i n a n c i a l l y , t he c o n t r i b u t i o n s of the N a t i o n a l R e s e a r c h C o u n c i l , the M e d i c a l R e s e a r c h C o u n c i l , and the U n i v e r s i t y o f B r i t i s h Columbia a r e g r a t e f u l l y acknowledged. T h i s t h e s i s c o u l d not have been .completed w i t h o u t the l o v e and u n d e r s t a n d i n g o f my w i f e , D e l i a . xv 1 1. ANIMAL AND HUMAN VISUAL SYSTEMS 1.1 I n t r o d u c t i o n I n t h i s c h a p t e r we g i v e a b r i e f r e v i e w of p r e s e n t knowledge of the p r o c e s s i n g p r o p e r t i e s of a n i m a l v i s u a l systems. T h i s r e v i e w p r o v i d e s some p e r s p e c t i v e f o r the remainder of the t h e s i s i n which we d e a l w i t h d e t a i l e d s t u d i e s of the m a t h e m a t i c a l model of. the l a t e r a l eye of L i m u l u s , the horse-shoe c r a b . 1.2 The V i s u a l Systems of V a r i o u s Animals (1) The P r o g (1 2) The work of L e t t v i n , e t a l ' ' has shown t h a t much p r o c e s s i n g of v i s u a l i n f o r m a t i o n t a k e s p l a c e b e f o r e any s i g n a l s a r r i v e a t the b r a i n . They r e c o r d e d the s i g n a l s ( v a r i a b l e f r e  quency p u l s e t r a i n s ) t h a t a r o s e i n s i n g l e nerve f i b e r s of the o p t i c t r a c t (the bundle of nerve f i b e r s l e a d i n g from the r e t i n a t o the b r a i n ) i n response t o v a r i o u s s t i m u l i i n the v i s u a l f i e l d . They^"^ d e s c r i b e f o u r main o p e r a t i o n s on the image i n the f r o g ' s eye, each of which i s t r a n s m i t t e d by a p a r t i c u l a r group of f i b e r s . The f i r s t group of f i b e r s p r o v i d e s u s t a i n e d c o n t r a s t  d e t e c t i o n . Each f i b e r i s a s s o c i a t e d w i t h an i l l u m i n a t i o n edge (or c o n t r a s t ) i n a p a r t i c u l a r r e g i o n of the r e t i n a , i t s r e c e p t i v e f i e l d . The second group of f i b e r s are the net c o n v e x i t y d e t e c t o r s . They f u n c t i o n q u i t e e f f e c t i v e l y as "bug d e t e c t o r s " . They s i g n a l the presence of s m a l l dark o b j e c t s or s h a r p l y c u r v i n g convex c o r n e r s i n t h e i r r e s p e c t i v e r e c e p t i v e f i e l d s . I f the o b j e c t i s moving the response i s more i n t e n s e than when i t i s s t a t i o n a r y . A l s o , the g r e a t e r the c u r v a t u r e the more v i g o u r o u s i s the r e s p o n s e . 2 The l a s t two groups of f i b e r s encode the o c c u r r e n c e of change i n the v i s u a l f i e l d . The moving-edge d e t e c t o r responds t o any d i s t i n g u i s h a b l e edge moving through i t s r e c e p t i v e f i e l d . A f a s t moving edge causes a more v i g o u r o u s response t h a n a slow moving one. The net dimming d e t e c t o r s show a response t o a sudden decrease i n i l l u m i n a t i o n w i t h i n t h e i r r a t h e r l a r g e r e c e p t i v e f i e l d s . Thus i f a l a r g e dark o b j e c t moves i n t o the f r o g ' s f i e l d of view and s t o p s , these u n i t s s i g n a l i t s p o s i t i o n . The f o u r groups of f i b e r s d e s c r i b e d above are i n t e r m i x e d i n the o p t i c t r a c t . When they r e a c h the tectum of the f r o g t hey s e p a r a t e and t e r m i n a t e i n f o u r d i s t i n c t l a y e r s . Each l a y e r e x h i b i t s a c o n t i n u o u s map of the r e t i n a i n terms of i t s p a r t i c u l a r o p e r a t i o n ; a l l f o u r maps are i n r e g i s t r a t i o n . (2) I n a l a t e r paper^ L e t t v i n , e t a l d e s c r i b e two types of t e c t a l neurons t h a t r e c e i v e i n f o r m a t i o n from the o p t i c nerve f i b e r s . One type d e t e c t s n o v e l t y i n v i s u a l e v e n t s , the o t h e r , c o n t i n u i t y i n time of i n t e r e s t i n g o b j e c t s i n the f i e l d of v i s i o n . (2) The Cat A schematic, of the c a t v i s u a l system i s shown i n F i g . 1.2.1. An. image f a l l i n g on the r e t i n a causes s i g n a l s t o course back through the o p t i c t r a c t . t o the o p t i c chiasm. About h a l f the f i b e r s from one eye c r o s s over and i n t e r m i x w i t h h a l f the f i b e r s from the o t h e r . The s i g n a l s proceed t o the l a t e r a l g e n i c u l a t e b o d i e s where the f i r s t s y n a p t i c j u n c t i o n beyond the r e t i n a o c c u r s . From here they c o n t i n u e on to the v i s u a l a r e a s of the c o r t e x . K u f f l e r ^ ^ and Barlow, e_t a l ^ ^ i n t h e i r s t u d i e s of the r e s p o n s e s of c a t g a n g l i o n c e l l s t o s m a l l s p o t s o f l i g h t showed t h a t t h e r e c e p t i v e f i e l d s of these c e l l s were, b a s i c a l l y c i r c u l a r . They Eye R e t i n a O p t i c T r a c t O p t i c Chiasm L a t e r a l G e n i c u l a t e Body S t r i a t e C o r t e x • F i g . 1.2.1 The V i s u a l System of the Cat c o n s i s t e d of a m u t u a l l y a n t a g o n i s t i c c e n t r a l d i s k and s u r r o u n d i n g a n n u l u s . I n some c e l l s whenever a s m a l l spot of l i g h t f o c u s s e d on th e c e n t r a l d i s k was t u r n e d on, the c e l l would f i r e . I f the spot was then moved t o the a n n u l u s , the c e l l f i r e d whenever the spot was t u r n e d o f f . These c e l l s were s a i d t o have ON c e n t e r s and OFF s u r r o u n d s , and were c a l l e d ON-OFF c e l l s . An a p p r o x i m a t e l y e q u a l number of OFF-QN c e l l s whose response p a t t e r n was the r e v e r s e of the ON-OFF c e l l s were found. ( R o d i e c k ^ ^ has s i n c e shown t h a t t h e s e response p a t t e r n s a r e mediated by two s e p a r a t e a g e n c i e s , one 4 of which i s e f f e c t i v e over the whole r e c e p t i v e f i e l d , the o t h e r b e i n g e f f e c t i v e o n l y i n the c e n t r a l d i s k . ) I n b o t h types of c e l l s i f two s p o t s o f l i g h t were used, one on the surr o u n d and one on the d i s k , the e f f e c t s tended t o c a n c e l . ' 6) Hubel and W i e s e l v ' i n t h e i r s t u d i e s of c a t l a t e r a l g e n i  c u l a t e c e l l s found t h a t t h e i r r e s p o n s e s were q u a l i t a t i v e l y s i m i l a r t o those i n the r e t i n a l g a n g l i o n . However, the p e r i p h e r y of the r e c e p t i v e f i e l d of the g e n i c u l a t e c e l l had g r e a t l y enhanced c a p a c i t y t o c a n c e l the e f f e c t s of the c e n t e r * much g r e a t e r t h a n was the case f o r g a n g l i o n c e l l s . Thus, the l a t e r a l g e n i c u l a t e c e l l s a r e even more s p e c i a l i z e d t h a n g a n g l i o n c e l l s i n r e s p o n d i n g t o s p a t i a l d i f f e r e n c e s or d i s c o n t i n u i t i e s i n r e t i n a l i l l u m i n a t i o n . A t the l e v e l of the c o r t e x Hubel and W i e s e l v ' ' ; found a r a d i c a l change i n c e l l r e s p o n s e . I n the c a t t h e r e were no c o r  t i c a l c e l l s w i t h c o n c e n t r i c r e c e p t i v e f i e l d s . Three main t y p e s of c e l l s , i n terms of response p a t t e r n s , were found. They were c a l l e d s i m p l e , complex, and hypercomplex c e l l s . The s i m p l e and complex c e l l s respond m a x i m a l l y t o l i n e s t i m u l i - such shapes as s l i t s , d ark b a r s , and edges ( s t r a i g h t - l i n e b o u n d a r i e s between l i g h t and dark r e g i o n s ) . I n b o t h cases the s t i m u l i must be o r i e n t e d c o r r e c t l y . The d i f f e r e n c e between the two t y p e s of c e l l s i s i n p o s i t i o n and movement response. The s i m p l e c e l l s w i l l respond t o l i n e s t i m u l i o n l y i n a s m a l l a r e a , whereas complex c e l l s are not so d i s c r i m i n a t i n g . I n a d d i t i o n , complex c e l l s respond w i t h s u s t a i n e d f i r i n g t o moving l i n e s . I n some cases t h i s response i s d i r e c t i o n a l l y s e n s i t i v e . Thus, a complex c e l l might respond v i g o u r o u s l y t o a slow downward movement of a h o r i z o n t a l b a r , but g i v e o n l y a weak response t o an upward 5 movement, and no response t o a v e r t i c a l bar moved h o r i z o n t a l l y . (9) I n d i s c u s s i n g the hypercomplex c e l l s Hubel and W i e s e l mention two subgroups. A l o w e r o r d e r hypercomplex c e l l responds e i t h e r to a s l i t , an edge, or a dark b a r , but the l e n g t h of the s t i m u l u s must be l i m i t e d ("stopped") i n one or both d i r e c t i o n s . Thus the optimum s t i m u l u s i n some cases was a 90° c o r n e r l y i n g w i t h i n the " a c t i v a t i n g " r e g i o n of the r e t i n a w i t h one edge l y i n g on the boundary between t h a t and the " a n t a g o n i s t i c " r e g i o n . Movement of the s t i m u l u s p a r a l l e l t o t h i s boundary would cause v i g o u r o u s response i n one d i r e c t i o n w i t h none when the movement was r e v e r s e d . I n a n o t h e r case the b e s t s t i m u l u s was a p r o p e r l y p o s i t i o n e d and o r i e n t e d double 90° c o r n e r or "tongue". A h i g h e r o r d e r complex c e l l r e q u i r e d i n g e n e r a l the same s t i m u l u s shapes as the l o w e r o r d e r c e l l , but was l e s s s p e c i f i c as t o b o t h o r i e n t a t i o n and p o s i t i o n . Thus, the s t i m u l u s c o u l d be o r i e n t e d i n one of two d i r e c t i o n s 90° a p a r t , and would evoke a u n i f o r m l y v i g o u r o u s response over i t s e n t i r e r e c e p t i v e f i e l d . (3) Other Animals The v i s u a l pathways of a number of o t h e r c r e a t u r e s have been s t u d i e d by v a r i o u s w o r k e r s . The work of Hubel and W i e s e l on t h e monkey o p t i c n e r v e , l a t e r a l g e n i c u l a t e b o d y ^ " ^ , and (12) s t r i a t e c o r t e x i s the most complete. The response p r o p e r t i e s of the c e l l s i n each case were s i m i l a r t o those of the c a t , except t h a t a t each s t a g e i n the v i s u a l system of the monkey t h e r e were a few c e l l s t h a t responded i n a s p e c i f i c way t o c o l o u r e d s t i m u l i . F o r example, i n the s t r i a t e c o r t e x they found t h a t most of the c e l l s c o u l d be c a t e g o r i z e d as s i m p l e , complex, or l o w e r o r d e r hypercomplex, w i t h response p r o p e r t i e s v e r y s i m i l a r to those d e s c r i b e d i n the c a t . On the average, however, the r e c e p t i v e f i e l d s were s m a l l e r and t h e r e was a g r e a t e r s e n s i t i v i t y t o changes i n s t i m u l u s o r i e n t a t i o n . A s m a l l p r o p o r t i o n of the c e l l s were c o l o u r coded. Barlow, e t a l i n t h e i r s t u d i e s of s i n g l e c e l l r e s  ponses i n the r e t i n a of the r a b b i t found t h a t s e p a r a t e c l a s s e s . of g a n g l i o n c e l l s a b s t r a c t e d the d i r e c t i o n and speed of moving edges, as w e l l as l o c a l i z e d dimming and b r i g h t e n i n g . Maturana and F r e n k ^ ^ found c e l l s w i t h s i m i l a r l y s e l e c t i v e response i n the r e t i n a l g a n g l i o n of the p i g e o n . One c l a s s would respond t o an edge i n any o r i e n t a t i o n i f i t was moving i n a p a r t i c u l a r d i r e c t i o n . Movement i n the r e v e r s e d i r e c t i o n gave no response.. Another c l a s s of c e l l responded t o v e r t i c a l movement of h o r i z o n  t a l edges. Other workers have c a r r i e d out s i m i l a r s t u d i e s on the (is) (16) v i s u a l systems of the ground s q u i r r e l , the g o l d f i s h , (17) and the octopus . 1.3 The Human V i s u a l System P r a c t i c a l l y a l l c o n c r e t e knowledge of the p r o c e s s i n g p r o p e r t i e s of the human v i s u a l system has been o b t a i n e d by e x p e r i m e n t a l p s y c h o l o g i s t s . Of n e c e s s i t y , they d e a l m a i n l y w i t h the o v e r a l l response of t h i s system. Consequently, i t i s v e r y d i f f i c u l t t o d e r i v e from t h e i r work any d e t a i l e d i n f o r m a  t i o n about the f u n c t i o n of the v a r i o u s s t a g e s i n the human v i s u a l system. I n f o r m a t i o n of t h i s n a t u r e must be d e r i v e d by i n f e r e n c e from the work done on a n i m a l s . N e v e r t h e l e s s , the work of some of the e x p e r i m e n t a l p s y c h o l o g i s t s does p r o v i d e c l u e s as t o how the human v i s u a l system 7 p r o c e s s e s images f a l l i n g on the r e t i n a . The work of one p s y c h o i  ds) o g i s t i n p a r t i c u l a r , F r e d A t t n e a v e , had a good d e a l of i n f l u e n c e on the work d e s c r i b e d i n t h i s t h e s i s . He was a b l e t o show t h a t f o r any g i v e n f i g u r e the p o i n t s on i t s c o n t o u r s h a v i n g maximum c u r v a t u r e were the most c r i t i c a l from the p o i n t o'f view of e i t h e r r e c o g n i t i o n or r e p r e s e n t a t i o n of the f i g u r e . I n h i s (18) own words , "An experiment r e l e v a n t t o the p r i n c i p l e t h a t i n f o r m a t i o n i s c o n c e n t r a t e d a t p o i n t s where a co n t o u r changes d i r e c t i o n most r a p i d l y may be summarized b r i e f l y . E i g h t y s u b j e c t s were i n s t r u c t e d t o draw, f o r each of 16 o u t l i n e shapes, a p a t t e r n of 10 d o t s which would resemble the shape as c l o s e l y as p o s s i b l e , and t h e n t o i n d i c a t e on the o r i g i n a l o u t l i n e the e x a c t p l a c e s w hich the d o t s r e p r e s e n t e d . A good example of the r e s u l t s i s shown i n ( F i g . 1.3.1): r a d i a l b a r s i n d i c a t e the r e l a t i v e f r e q u e n c y w i t h w h i c h d o t s were p l a c e d on each of the segments i n t o which the c o n t o u r was d i v i d e d f o r s c o r i n g purposes. I t i s c l e a r t h a t s u b j e c t s show a g r e a t d e a l of agreement i n t h e i r a b s t r a c t i o n s of p o i n t s b e s t r e p r e s e n t i n g the shape, and most of these p o i n t s a re t a k e n from r e g i o n s where the co n t o u r i s most d i f f e r e n t from a s t r a i g h t l i n e . T h i s c o n c l u s i o n i s v e r i f i e d by d e t a i l e d compari sons of dot f r e q u e n c i e s w i t h measured c u r v a t u r e s on b o t h the f i g u r e shown and o t h e r s . "Common o b j e c t s may be r e p r e s e n t e d w i t h g r e a t economy, and f a i r l y s t r i k i n g f i d e l i t y , by c o p y i n g the p o i n t s a t which t h e i r c o n t o u r s change d i r e c t i o n m a x i m a l l y , and then c o n n e c t i n g these p o i n t s a p p r o p r i a t e l y w i t h a s t r a i g h t edge. ( F i g . 1.3.2) was drawn by a p p l y i n g t h i s t e c h n i q u e as m e c h a n i c a l l y as p o s s i b l e , t o a r e a l s l e e p i n g c a t . The i n f o r m a t i o n a l c o n t e n t o f a drawing l i k e 8 P i g . 1.3.1 R e l a t i v e Frequency of P o i n t Placement f o r f 18) - P r e s e r v i n g Shape ( A t t n e a v e ^ ') t h i s may be c o n s i d e r e d t o c o n s i s t of two components: one d e s c r i b i n g the p o s i t i o n s o f the p o i n t s , the o t h e r i n d i c a t i n g which p o i n t s are connected t o which o t h e r s . " The sense of the above paragraph has been s t a t e d more (19) c o n c i s e l y by MacDonald v , "The c r i t i c a l f e a t u r e s f o r the r e c o g n i t i o n of a shape are the p o i n t s of maximum c o n t o u r c u r  v a t u r e , t h e i r l o c a t i o n , and t h e i r c o n n e c t i v i t y . " 1.4 l a t e r a l I n h i b i t i o n i n L i m u l u s ( l ) A Model f o r the L a t e r a l Eye of L i m u l u s A l l of the work on v i s u a l systems d i s c u s s e d p r e v i o u s l y was b a s i c a l l y q u a l i t a t i v e . I n c o n t r a s t to t h i s , H a r t l i n e and (20 21) R a t l i f f ' ' have developed a m a t h e m a t i c a l model t o d e s c r i b e the s t e a d y s t a t e i n t e r a c t i o n between r e c e p t o r s i n the l a t e r a l eye of t h e horseshoe c r a b , L i m u l u s . T h i s model i s the main b a s i s f o r the work d e s c r i b e d i n t h i s t h e s i s . 9 F i g . 1.3-2 The P o i n t s of Maximum C u r v a t u r e on a Cat ( A t t n e a v e ^ 1 8 ^ ) The l a t e r a l eye of L i m u l u s i s a c o a r s e l y f a c e t t e d compound eye c o n t a i n i n g a p p r o x i m a t e l y 1000 .ommatidia, each of which appears t o f u n c t i o n as a s i n g l e r e c e p t o r u n i t e x c i t e d o n l y by l i g h t e n t e r i n g i t s own c o r n e a l f a c e t . Each such r e c e p t o r u n i t when so e x c i t e d , d i s c h a r g e s t r a i n s of i m p u l s e s i n one and o n l y one o p t i c nerve f i b e r . These r e c e p t o r u n i t s are not independent i n t h e i r a c t i o n : each one may be i n h i b i t e d by i t s near n e i g h b o r s , and i n t u r n may i n h i b i t them. C o n s i d e r a group of n r e c e p t o r s , two of which are shown i n F i g . 1.4-1. I f the i ^ * 1 r e c e p t o r a l o n e i s i l l u m i n a t e d w i t h l i g h t of i n t e n s i t y 1^, as shown i n F i g . 1 . 4 . 1 ( a ) , i t s o p t i c nerve f i b e r d i s c h a r g e s a t some f r e q u e n c y e^, which v a r i e s i n an a p p r o x i m a t e l y l i n e a r f a s h i o n w i t h l o g 1^. I f a group of r e c e p t o r s c o n t a i n i n g the i ^ * 1 one i s i l l u m i n a t e d w i t h l i g h t of i n t e n s i t y 1^, as i n F i g . 1.4.1(b), the output f r e q u e n c y of the i ^ * 1 r e c e p t o r 10 li site of pulse generat ion F i g . 1.4.1(a) R e c e p t o r F i r i n g a t Rate e^ Ii I i F i g . 1.4.1(b) R e c e p t o r A c t i v i t y Reduced t o Rate x^ becomes x ^ , l e s s t h a n e^, p r o v i d e d the d i s c h a r g e f r e q u e n c i e s o f a t l e a s t some of the o t h e r r e c e p t o r s a re above t h e i r r e s p e c t i v e i n h i b i t i o n t h r e s h o l d s . The r e d u c t i o n i n f i r i n g f r e q u e n c y t h of t h e i r e c e p t o r i s p r o p o r t i o n a l t o the output f r e q u e n c i e s of the i n h i b i t i n g r e c e p t o r s l e s s the t h r e s h o l d s . I t i s not p r o p o r  t i o n a l t o the l e v e l of i n p u t s t i m u l u s . A l l t h i s i s summed up i n the f o l l o w i n g s e t of e q u a t i o n s : 11 n x. = e. - / k. . max (0,x.-t. .) i = l , . ... ,n 1.4.1 1 1 JT-L 1D D i j t h 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 r e c e p t o r ID t h on t h e i r e c e p t o r , and t ^ i s the c o r r e s p o n d i n g t h r e s h o l d . S i n c e x., e., and t . . are p u l s e - r a t e s or f r e q u e n c i e s , they cannot t a k e on n e g a t i v e v a l u e s . I n a d d i t i o n , s i n c e the i n t e r  a c t i o n i s s t r i c t l y i n h i b i t o r y , the i n e q u a l i t y k.-^O must h o l d . H a r t l i n e and R a t l i f f i n t h e i r s t u d i e s on L i m u l u s found t h a t the magnitudes of L ^ and t ^ a r e , on the average, dependent on the d i s t a n c e s e p a r a t i n g the two r e c e p t o r s : k ^ d e c r e a s e s and t ^ i n c r e a s e s as the s e p a r a t i o n becomes g r e a t e r . The s e t of e q u a t i o n s 1.4.1 p r o v i d e a m a t h e m a t i c a l d e s c r i p  t i o n f o r the i n h i b i t o r y i n t e r a c t i o n between the r e c e p t o r s making up the l a t e r a l eye of L i m u l u s . They form a n o n - l i n e a r , a l g e b r a i c s e t and r e q u i r e an i t e r a t i v e method f o r s o l u t i o n . (2) The L a t e r a l I n h i b i t i o n S i m u l a t o r (22) I n a paper by Beddoes, et a l v ' some f e a s i b i l i t y s t u d i e s were g i v e n f o r a d e v i c e t h a t would s i m u l a t e a r e c e p t o r network whose response was governed by the s e t of l a t e r a l i n h i b i t i o n e q u a t i o n s 1.4.1. I t was shown t h a t i n o r d e r t o s i m u l a t e the a c t i  v i t y of a l a r g e number of r e c e p t o r s , i t i s s u f f i c i e n t t o break the network up i n t o much s m a l l e r , o v e r l a p p i n g sub-networks. The s e t of e q u a t i o n s 1.4-1 can then be s o l v e d f o r the x^ v a l u e of the c e n t r a l r e c e p t o r i n each of these sub-networks. The s e t of such v a l u e s or a c t i v i t i e s g i v e s a good a p p r o x i m a t i o n t o the a c t i v i t y of the o v e r a l l r e c e p t o r network. On the b a s i s of t h e s e f e a s i b i l i t y s t u d i e s a 9x9 sub-network or a r r a y of r e c e p t o r s was chosen as the most r e a s o n a b l e compromise between computation time and s i m u l a t i o n a c c u r a c y . Hence the s e t of e q u a t i o n s used i n the s i m u l a t o r has n = 81 or l e s s . I n a d d i t i o n the s e t of e q u a t i o n s 1.4.1 are r e w r i t t e n a s : 81 x. = e. 5 max(0,k. .x.-k. . t . .) i = l , . . . , 8 1 1.4.2 1 1 j t i ! J J 1 3 ID The d e s i g n of the s i m u l a t o r i s ' d e a l t w i t h i n the M.A.Sc. (21) t h e s i s of the a u t h o r . The c o n s t r u c t i o n of the s i m u l a t o r was begun i n the summer of 1965, and was completed a y e a r l a t e r . V a r i o u s problems w i t h t h e o r i g i n a l c i r c u i t d e s i g n s were encountered and overcome. As t h e r e s e a r c h p r o g r e s s e d a d d i t i o n a l f e a t u r e s were added t o the b a s i c s i m u l a t o r - i n some i n s t a n c e s t o f a c i l i t a t e the s t u d y , and i n o t h e r s t o answer q u e s t i o n s t h a t a r o s e . These m o d i f i c a t i o n s w i l l be mentioned a t the a p p r o p r i a t e p l a c e s i n t h i s t h e s i s . 1.5 A Summary of the T h e s i s The a c t i v i t y i n a l a t e r a l i n h i b i t o r y network i s g r e a t e s t i n r e g i o n s of non-uniform i l l u m i n a t i o n . T h i s s u g g e s t s the p o s s i b l e use of l a t e r a l i n h i b i t i o n as a p r e - p r o c e s s o r i n a v i s u a l p a t t e r n r e c o g n i t i o n machine a l o n g the l i n e s d e s c r i b e d by T a y l o r ^ 4 ) ^ The enhancement of i n t e n s i t y c o n t o u r s by l a t e r a l i n h i b i t i o n i s e x t e n s i v e l y s t u d i e d i n the case of b l a c k and w h i t e p a t t e r n s . The r e c e p t o r a c t i v i t y i n a l a t e r a l i n h i b i t o r y network near a b l a c k - w h i t e boundary p r o v i d e s a measure, of the l o c a l g e o m e t r i c p r o p e r t i e s of t h a t boundary. P r e l i m i n a r y s t u d i e s i n d i c a t e d , however, t h a t this measure was h i g h l y dependent on the r e l a t i v e o r i e n t a t i o n of the boundary and the r e c e p t o r network. V a r i o u s methods f o r r e d u c i n g t h i s o r i e n t a t i o n dependence have been e x p l o r e d . The concept of an a r e a o p e r a t o r i s i n t r o d u c e d . I t i s shown e x p e r i m e n t a l l y t h a t the peak a c t i v i t y i n a l a t e r a l i n h i b i t o r y network can be mod e l l e d by the a r e a o p e r a t o r . T h i s f a c t l e a d s i n t o a t h e o r e t i c a l s t u d y of the r e l a t i o n s h i p between the a r e a o p e r a t o r and the c u r v a t u r e of a boundary. The f u n c  t i o n o b t a i n e d by t r a c i n g the a r e a o p e r a t o r a l o n g an i n t e n s i t y boundary has maxima and minima near the p o i n t s of maximum and minimum c u r v a t u r e , of the boundary. An e q u i v a l e n t r i d g e f u n c t i o n can be o b t a i n e d u s i n g a l a t e r a l i n h i b i t o r y network. The p r o p e r t i e s of t h i s r i d g e f u n c t i o n a r e such t h a t i t can be used f o r p a t t e r n r e c o g n i t i o n . 14 2. LATERAL INHIBITION 2•1 The I n i t i a l S t u d i e s of L a t e r a l I n h i b i t i o n I n s t u d y i n g the s e t of e q u a t i o n s 1.4.2 we a r e i n t e r e s t e d i n d e t e r m i n i n g how a p a t t e r n of i l l u m i n a t i o n , r e p r e s e n t e d by the e . ( x , y ) , i s t r a n s f o r m e d under the a c t i o n of v a r i o u s k. . and k . . t . . v a l u e s i n t o a p a t t e r n of r e c e p t o r a c t i v i t y , x . ( x , y ) . I n the s t u d i e s d e t a i l e d i n t h i s t h e s i s the k.. are f u n c t i o n s o n l y of the d i s t a n c e d s e p a r a t i n g the i ^ * 1 and j ^ * 1 r e c e p t o r s . The ^ i j ^ i j a r e a s s u m e d e Q . u a l a n ( i are kept c o n s t a n t i n any g i v e n study or experiment. T h i s c o n s t a n t may change from experiment t o experiment. The f i r s t k . f u n c t i o n s t u d i e d was the same one used i n " (22) the f e a s i b i l i t y s t u d i e s mentioned above: k (d) = f ° ' 3 ~ ° - l d 0 < d < 3u 2..1.1 0 d=0, d > 3u The graph of t h i s f u n c t i o n i s g i v e n i n P i g . 2.1.1. The k ^ t ^ v a l u e was chosen t o be z e r o . T h i s v a l u e m i n i m i z e s the c e n t r a l r e c e p t o r output i n the case where a l l the r e c e p t o r s are e q u a l l y i l l u m i n a t e d . ( H e r e a f t e r , t h i s w i l l be r e f e r r e d t o as t h e con d i t i o n of u n i f o r m i l l u m i n a t i o n . ) The f i r s t p a t t e r n p r o c e s s e d was a w h i t e square on a b l a c k background, shown s c h e m a t i c a l l y i n F i g . 2 . 1 . 2 ( a ) . An a r r a y of r e c e p t o r s has been superimposed on the square. I n F i g . 2.1.2(b) and (c) the r e c e p t o r a c t i v i t y , x^ a l o n g the l i n e s AA' and BB' i s shown. Note the a c t i v i t y peaks t h a t o c c u r near the boundary of the square. The a c t i v i t y i n the u n i f o r m l y i l l u m i n a t e d c e n t e r of the square i s 131 p u l s e s / s e c , whereas a d j a c e n t to' an edge i t 15 DISTANCE, d F i g . 2.1.1 The k ^ ( d ) F u n c t i o n Used i n the I n i t i a l " S t u d i e s i s 161 p.p.s., and i n a c o r n e r , 189 p.p.s. T h i s phenomenon i s c a l l e d d i f f e r e n t i a l c o n t o u r enhancement, or c o n t o u r enhance ment f o r s h o r t . I t a r i s e s i n the f o l l o w i n g f a s h i o n . C o n s i d e r the t h r e e r e c e p t o r s marked r-^, and r ^ i n F i g . 2 . 1 . 2 ( a ) . The r e c e p t o r r-^ i s surrounded by a c t i v e , i l l  uminated r e c e p t o r s a l l c f which a c t t o reduce i t s a c t i v i t y . However, about h a l f of the r e c e p t o r s t h a t a re capable of i n  h i b i t i n g r ^ are i n darkness and hence i n a c t i v e . Thus, r ^ r e c e i v e s l e s s i n h i b i t i o n and has a g r e a t e r a c t i v i t y than . S i m i l a r l y , t h r e e q u a r t e r s of the r e c e p t o r s t h a t d i r e c t l y i n f l u e n c e r ^ a r e i n d a r k n e s s , and c o n s e q u e n t l y i t s a c t i v i t y i s even g r e a t e r than t h a t of . An i n t e r e s t i n g phenomenon occ u r s i n the r e c e p t o r a c t i v i t y f u n c t i o n a l o n g the d i a g o n a l of the square. T h i s f u n c t i o n i s 16 • « • c A • • o • 0 c • \ © c o v 6 o e v o c e v o e vo o o \ o o o \ o o o \ o o e>\o -o—o o g o c o o o—o—o— o e o o e e o e e o o e o o o o e e e o o e o © o o o © o o o o « o o o o o e c * © © © -o—o— -e o- • • • o • e o o o o e • • o e o © • • e o e » o e • c o o o o o o e o c c e e e o o c c o o o © c © — O — f i ' _o o DARK \ e o o \ 0 e o o o LIGHT 9 \ 0 O . 0 \ © © O \ 0 O © \ O C 0 \ D P i g . 2. ACT! • © © \ c o o \ o o • \ o © e \ © o o 1 . 2(a) R e c e p t o r A r r a y Superimposed on a B l a c k and White Square VITY (p.p.s.) . ' ACTIVITY (p.p.s.) I 131 161 B -189 B F i g . 2 . 1 . 2(b) R e c e p t o r A c t i v i t y a l o n g AA' F i g . 2 . 1 . 2 ( c ) Re c e p t o r A c t i v i t y a l o n g BB' shown i n F i g . 2 . 1 . 3 ( a ) . Note the secondary peak i n a c t i v i t y j u s t a d j a c e n t t o the wide p l a t e a u of u n i f o r m a c t i v i t y . Another way of d e m o n s t r a t i n g t h i s e f f e c t i s shown i n F i g . 2 . 1 . 3(b) where a l l the r e c e p t o r s h a v i n g an a c t i v i t y i n the range from 132 p.p.s. t o 200 p.p.s. are d i s p l a y e d . Note the f o u r groups of r e c e p t o r s a d j a c e n t t o the c o r n e r s of the square. These a l o n g w i t h the peak and the d e p r e s s i o n i n r e c e p t o r a c t i v i t y 17 v . Jj F i g . 2.1.3(a) R e c e p t o r A c t i v i t y (b) R e c e p t o r s with. A c t i v i t y A l o n g D i a g o n a l o f the Square i n the Range 132< x^< 200p.p.e a d j a c e n t t o the boundary of the square are analogous t o the / o~\ 25) well-known Mach bands ' . T h e i r presence p r o v i d e d a n i c e c o n f i r m a t i o n t h a t the s i m u l a t o r was f u n c t i o n i n g c o r r e c t l y . The i n i t i a l peak i n a c t i v i t y a d j a c e n t t o the boundary has been e x p l a i n e d above. The d e p r e s s i o n and the secondary peak have a s i m i l a r e x p l a n a t i o n except t h a t they a r e h i g h e r o r d e r e f f e c t s caused by an i n d i r e c t i n t e r a c t i o n between r e c e p t o r s C o n s i d e r the s i t u a t i o n shewn i n F i g . 2.1.4(a). Assume t h a t o n l y the k. . r e l a t i n g a d j a c e n t r e c e p t o r s are non-zero. When the edge moves as shown, the a c t i v i t y of r-, , x^, w i l l vary, i n the f a s h i o n i n d i c a t e d i n F i g . 2.1.4(b). As r 2 i s exposed, i t s a c t i v i t y , x 2 > i n c r e a s e s by A x 2 > c a u s i n g a decrease i n x-^  of Ax-j^  = - k 1 2 A x2 As r ^ i s exposed, i t s a c t i v i t y i n c r e a s e s by Ax^, c a u s i n g a decrease i n x 2 of A x 0 = - k 0 ^ Ax^ 2 23 3 18 f / WHITE EDGE MOVEMENT r3 r« rs BLACK F i g . 2.1.4(a) A Set of R e c e p t o r s and a W h i t e - B l a c k Edge Xi DISTANCE FROM r, TO EDGE F i g . 2.1.4(h) V a r i a t i o n i n the A c t i v i t y of as the Edge Moves Away which i n t u r n causes a change i n x ^ of A x ~ ' k^ 2 "^^ "2 2 ^" 2 3 ^ That i s , the i n c r e a s e i n x^ i n d i r e c t l y causes an i n c r e a s e i n X-^. Thus, the secondary peak i n the a c t i v i t y f u n c t i o n shown i n F i g . 2.1.3(h) a r i s e s v i a a second o r d e r or i n d i r e c t i n t e r  a c t i o n between r e c e p t o r s . I n our f e a s i b i l i t y s t u d i e s f o r the s i m u l a t o r we found t h a t c o n t o u r enhancement always o c c u r r e d but t h a t the magnitude of the enhancement was. dependent on the o r i e n t a t i o n of the r e c e p t o r a r r a y w i t h r e s p e c t t o the boundary.. T h i s l e d us t o perform the f o l l o w i n g experiment w i t h the s i m u l a t o r . We r e  o r i e n t e d the square as shown i n F i g . 2.1.5(a). and r e c o r d e d the 19 15x15 ARRAY F i g . 2.1.5(a) A White Square a t 45° t o the Rece p t o r Rows ACTIVITY (pp.s.) 131 ACT! VITY (p.p.s) 169 F i g . 2.1.5(b) Re c e p t o r A c t i v i t y a l o n g AA' F i g . 2 .1.5(c) Rec e p t o r A c t i v i t y a l o n g BB' r e c e p t o r a c t i v i t y a l o n g the l i n e s AA' and BB'. These r e s u l t s a re shown i n F i g . 2.1.5(b) and ( c ) . . Note t h a t t h e shape of the a c t i v i t y f u n c t i o n s i n t h i s case i s the same as i n F i g . 2.1.2(b) and ( c ) . However, t h e magnitude of the a c t i v i t y has changed. A l o n g an edge the a c t i v i t y i s now 169 p.p.s. whereas b e f o r e i t was 161 p.p.s.; i n a c o r n e r the a c t i v i t y i s 194 p.p.s. as op posed t o 189 p.p.s. p r e v i o u s l y . The a c t i v i t y under u n i f o r m 20 CENTRAL ROW OF 9x9 ARRAY BISECTOR F i g . 2.1.6 A Wedge a t an Angle 0 w i t h r e s p e c t t o the Re c e p t o r Rows i l l u m i n a t i o n i s of course the same. (22) I n Beddoes, e_t a l v ' i t was i n d i c a t e d t h a t the magnitude of the r e c e p t o r a c t i v i t y might be used f o r measuring a n g l e s t o an a c c u r a c y of + 2 0 ° . The problem of the dependence of the magnitude of a c t i v i t y on the o r i e n t a t i o n of the a r r a y c a s t s some doubt on t h i s p o s s i b i l i t y . I n o r d e r t o t e s t i t we c a r r i e d out the f o l l o w i n g s t u d y . C o n s i d e r a wedge of angl e a whose b i s e c t o r makes an a n g l e 0 w i t h r e s p e c t t o the c e n t r a l row of the 9x9 square a r r a y shown i n F i g . 2.1.6. Move the 9x9 a r r a y i n s m a l l s t e p s so t h a t the c e n t r a l r e c e p t o r c o v e r s the a r e a near the v e r t e x of the wedge. Record the peak r e c e p t o r a c t i v i t y . Now r e o r i e n t the wedge w i t h r e s p e c t t o the .- 9x9 a r r a y and r e p e a t t h i s p r o c e s s . Do t h i s f o r o r i e n t a t i o n s of 0 = 0°, 1 5 ° , 3 0 ° , and 4 5 ° f o r wedges whose angl e a = 10°, 20°,..., 180°. The r e s u l t s of t h i s study are g i v e n i n t h e graphs shown i n F i g . 2.1.7. The v a r i a t i o n i n peak r e c e p t o r a c t i v i t y w i t h 240-\ 220A to Q. 200A O JSQ. Lu 160- 140 20' T T 40* 60" 80" 100" WEDGE ANGLE ad. 720 740 76*0 180 F i g . 2.1.7 Peak R e c e p t o r A c t i v i t y as a F u n c t i o n of Wedge Ang l e f o r V a r i o u s O r i e n t a t i o n s wedge angl e a i s shown f o r each o r i e n t a t i o n . The h e a v i e r l i n e s form the peak a c t i v i t y envelope f o r t h i s p a r t i c u l a r k. .(d) f u n c - t i o n . T h i s envelope c o n s i s t s of two l i n e s , one f o l l o w i n g the max imum peak r e c e p t o r a c t i v i t y a t each v a l u e of a f o r the v a r i o u s o r i e n t a t i o n s , and the o t h e r f o l l o w i n g the minimum a c t i v i t y . I t i s e v i d e n t from the graph t h a t the peak r e c e p t o r o u t  put f o r t h i s k. .(d) f u n c t i o n p r o v i d e s a v e r y poor measure of the a n g l e a . G i v e n a peak r e c e p t o r o u t p u t , the u n c e r t a i n t y i n the a n g l e a t o which i t c o r r e s p o n d s would be about + 40°. Thus, one s h o u l d be a b l e t o d i f f e r e n t i a t e a s t r a i g h t edge from a r i g h t - a n g l e , but not much more. The r e a s o n f o r t h i s o r i e n t a t i o n dependence problem i s r e a d i l y apparent i f we examine a p a r t i c u l a r case. I n F i g . 2.1.8(a) t o (d) we have s k e t c h e d a 100° wedge i n f o u r d i f f e r e n t o r i e n t a t i o n s w i t h r e s p e c t t o the 5x5 f i e l d over which the r e c e p t o r s i n t e r a c t d i r e c t l y w i t h the c e n t r a l r e c e p t o r . The wedge has been p o s i t i o n e d so t h a t the c e n t r a l r e c e p t o r i s i l l u m i n a t e d but the number of i n t e r  a c t i n g r e c e p t o r s i s a minimum. T h i s i s the p o s i t i o n f o r which the c e n t r a l r e c e p t o r w i l l have peak a c t i v i t y . I n F i g . 2.1.8(e) we have noted b e s i d e each r e c e p t o r i n the 5x5 a r r a y the v a l u e of the k ^ ( d ) f u n c t i o n . T h i s v a l u e determines the amount of i n h i b i  t i o n e x e r t e d by t h a t r e c e p t o r on the c e n t r a l r e c e p t o r . Hence, a t l e a s t t o a f i r s t a p p r o x i m a t i o n , the sum of the k^^(d) v a l u e s of the i l l u m i n a t e d r e c e p t o r s p r o v i d e s a measure of the t o t a l i n h i b i t i o n e x e r t e d on the c e n t r a l r e c e p t o r . The l a r g e r t h i s sum, the l o w e r the c e n t r a l r e c e p t o r a c t i v i t y . I n Table 2.1.1 we g i v e t h i s sum f o r the v a r i o u s o r i e n t a t i o n s . (a) <!>= 0 (b) f = 15° .02 .08 .10 .08 .02 .08 .16 .20 .16 .08 .10 .20 O'.O .20 .10 .08 .16 .20 .16 .08 .02 .08 .10 .08 .02 (e) F i g . 2.1.8(a) t o (d) A 100° Wedge i n V a r i o u s O r i e n t a t i o n s , 0° t o 45° (e) The k ^ j f o r the C e n t r a l R e c e p t o r 24 O r i e n t a t i o n Sum of k. . of I l l u m i n a t e d . .Angle, 0 R e c e p t o r s 0° 0.82 15° 0.72 30° 0.64 45° 0.98 Table 2.1.1 Sum o f k^ . V a l u e s a t V a r i o u s O r i e n t a t i o n s On the b a s i s of t h e s e sums then, we would p r e d i c t t h a t f o r a 100°--wedge the maximum a c t i v i t y o c c u r s a t an o r i e n t a t i o n of 30°, and the minimum a t an o r i e n t a t i o n of 45°. I f we check t h i s p r e d i c t i o n w i t h the r e s u l t s f o r a = 100° i n the graph g i v e n i n P i g . 2.1.7>we see t h a t sequence of o r i e n t a t i o n s from h i g h e s t t o lowe a c t i v i t y i s 30°, 15°, 0°, 45°. T h i s i s the same as the sequence of o r i e n t a t i o n s i n o r d e r of i n c r e a s i n g v a l u e of the k.. sum. Why does t h i s k.. sum change as we v a r y the o r i e n t a t i o n of the a r r a y ? I t i s o b v i o u s l y due t o the d i s c r e t e n a t u r e of the a r r a y . Por t h i s r e a s o n , ' t h e o r i e n t a t i o n dependence of the r e c e p  t o r a c t i v i t y can never be t o t a l l y e l i m i n a t e d . However, i t can be s u b s t a n t i a l l y reduced by a number of t e c h n i q u e s which are d i s c u s s e d i n the f o l l o w i n g s e c t i o n . 2.2 Methods f o r Reducing O r i e n t a t i o n Dependence The most obvious method f o r r e d u c i n g the o r i e n t a t i o n dependence of the response i s t o i n c r e a s e the number of r e c e p t o r s t h a t i n t e r a c t d i r e c t l y w i t h the c e n t r a l r e c e p t o r . I n d o i n g t h i s , however, we reduce the a c c u r a c y of our a p p r o x i m a t i o n t o an a c t u a l l a t e r a l i n h i b i t o r y r e c e p t o r network. As was shown i n the f e a s i b i l i t ; s t u d y o n e can approximate the response of t h i s type of network by s o l v i n g the s e t of e q u a t i o n s 1.4-1 f o r a s u b f i e l d of a much l a r g e r a r r a y . T h i s s u b f i e l d , however, must be l a r g e r than the d i r e c t i n h i b i t i o n f i e l d , i . e . the f i e l d over which k. .(d) i s non-zero. I n the case of a 5x5•• d i r e c t f i e l d i t was shown t h a t the response of the c e n t r a l r e c e p t o r i n a u n i f o r m l y i l l u m i n a t e d 9x9 a r r a y was w i t h i n t h r e e p e r c e n t of 'the response f o r a r e c e p t o r i n a u n i f o r m l y i l l u m i n a t e d i n f i n i t e a r r a y . Use of a 5x5 a r r a y w i t h a 5x5 d i r e c t f i e l d l e d t o a 15$ e r r o r i n the r e s ponse. I n a d d i t i o n the response i s q u a l i t a t i v e l y a l t e r e d due t o the e l i m i n  a t i o n of i n d i r e c t i n t e r a c t i o n s . I n F i g . 2.2.1(a) we show the response f u n c t i o n f o r an edge u s i n g a 9x9 f i e l d w i t h a 5x5 d i r e c t f i e l d . Note the o s c i l l a t i o n s i n r e c e p t o r output as we proceed away from the edge. I n F i g . 2.2.1(b) we show the c o r r e s  ponding f u n c t i o n u s i n g a 5x5t f i e l d w i t h a 5x5 d i r e c t f i e l d . The o s c i l l a t i o n s have d i s a p p e a r e d . A s i m i l a r e f f e c t o c c u r s when we extend the range of non-zero k: .(d) t o i n c l u d e the whole a r e a of the 9x9 a r r a y . I n d o i n g t h i s we thus l o s e some f i d e l i t y i n our s i m u l a t i o n of l a t e r a l i n h i b i t i o n . T h i s l o s s o f the h i g h e r o r d e r e f f e c t s i s a c c e p t a b l e s i n c e we are s t u d y i n g the p r o c e s s w i t h a view t o a p p l i c a t i o n , r a t h e r than e l u c i d a t i o n , of the mechanism. Hence one of the methods t h a t w i l l be used to reduce the o r i e n t a t i o n dependence w i l l be t o i n c r e a s e the number of r e c e p t o r s i n the 9x9 a r r a y t h a t i n t e r a c t d i r e c t l y w i t h the c e n t r a l r e c e p t o r . I n c o n j u n c t i o n w i t h t h i s method another approach i n v o l v e s the use of k ^ ( d ) f u n c t i o n s d i f f e r e n t from t h a t encountered i n L i m u l u s . Two types of f u n c t i o n s have been s t u d i e d . In the f i r s t t y p e , the k..(d) f u n c t i o n i s c o n s t a n t f o r a c e r t a i n range of d, and z e r o o u t s i d e t h a t range. I n the second t y p e , the k..(d) 26 F i g . 2.2.1(a) Edge Response f o r a 9x9 a r r a y w i t h a 5x5 F i e l d of D i r e c t I n h i b i t i o n F i g . 2.2.1(b) Edge Response f o r a 5x5 A r r a y w i t h a 5x5 F i e l d of D i r e c t I n h i b i t i o n f u n c t i o n i s a l i n e a r l y i n c r e a s i n g f u n c t i o n of d out t o a c e r t a i n v a l u e of d and z e r o beyond. Another obvious method f o r r e d u c i n g the o r i e n t a t i o n dependence i n v o l v e s r o u n d i n g o f f the 9x9 square a r r a y as shown i n F i g . 2.2.2. By d r o p p i n g the r e c e p t o r s from the c o r n e r s we make the d i s t r i b u t i o n of r e c e p t o r s about the c e n t r a l r e c e p t o r more i s o t r o p i c . S i m i l a r l y , m o d i f i c a t i o n of the s i m u l a t o r a r r a y to g i v e an hexagonal d i s t r i b u t i o n of r e c e p t o r s about the c e n t r a l r e c e p t o r 27 e • • • • • • CENTRAL RECEPTOR F i g . 2.2.2 The Rounded 9 x 9 A r r a y s h o u l d l e a d t o a r e d u c t i o n i n o r i e n t a t i o n dependence. T h i s i s because the hexagonal a r r a y i s s e l f - c o n g r u e n t under 60° r o t a t i o n s as opposed t o 90° r o t a t i o n s f o r the square a r r a y . The f i n a l method used t o reduce o r i e n t a t i o n dependence was suggested by an e x a m i n a t i o n of the anatomy of the l a t e r a l eye of L i m u l u s . I n a compound eye of t h i s s o r t each r e c e p t o r has i t s own l e n s . Thus l i g h t can e n t e r each r e c e p t o r from a l a r g e a n g u l a r c r o s s - s e c t i o n of the v i s u a l f i e l d , w i t h a d j a c e n t r e c e p t o r s s a m p l i n g (26) o v e r l a p p i n g a r e a s of t h i s f i e l d . ( R e i c h a r d t and M a c G i n i t i e have shown t h a t t h i s type of s a m p l i n g . o f the image space does not n e c e s s a r i l y l e a d t o a ' l o s s of i n f o r m a t i o n . ) I n o t h e r words t h e n , the f i e l d of view of each r e c e p t o r i s a n y t h i n g but the p o i n t f i e l d of view used i n the experiments d e s c r i b e d i n s e c t i o n 2.1. These f a c t s l e d us t o modify the s i m u l a t o r so t h a t each r e c e p t o r sampled a f i n i t e c i r c u l a r area, of t h e v i s u a l space. A number of schemes were t r i e d which i n v o l v e d i n t e n s i t y c o n t r o l of the CRT i n the f l y i n g spot scanner. I n no case were we a b l e t o o b t a i n the e i g h t b i t a c c u r a c y t h a t we wanted i n the i n t e n s i t y measurement. F o r t h i s r e a s o n we d e c i d e d t o l i m i t o u r s e l v e s t o a 28 D = 1.5u — CENTRAL RECEPTOR F i g . 2.2.3 -A- Rounded 9x9 A r r a y of Rec e p t o r s w i t h a F i n i t e F i e l d of View s t u d y of b l a c k and wh i t e p a t t e r n s . I n t h i s s i t u a t i o n we s i m p l y scan over a s m a l l c i r c u l a r a r e a i n the neighborhood of the r e c e p t o r p o s i t i o n and measure the w h i t e " a r e a encountered. The r a d i u s of the c i r c u l a r a r e a i s v a r i a b l e , and hence we can s i m u l a t e r e c e p t o r s w i t h d i f f e r e n t s i z e d f i e l d s of view. A rounded 9x9 a r r a y of r e c e p t o r s w i t h f i n i t e f i e l d s of view i s shown i n F i g . 2.2.3. I n t h i s case the d i a m e t e r , D, of the f i e l d s of view i s 1.5u, where u i s the minimum i n t e r - r e c e p t o r d i s t a n c e . 3. EXPERIMENTAL STUDIES ON LATERAL INHIBITION 3.1 I n t r o d u c t i o n I n t h i s c h a p t e r we p r e s e n t the r e s u l t s ( i n the form of a l a r g e number of graphs) of s t u d i e s done on t h r e e d i f f e r e n t t y p e s of k..(d) f u n c t i o n s . I n these s t u d i e s we v a r y a number of p a r a - meters - s i z e of r e c e p t i v e f i e l d of view, t h r e s h o l d l e v e l , r e c e p t o r arrangement - i n o r d e r t o determine the e f f e c t on the r e c e p t o r o u t  put. The t e s t p a t t e r n s c o n s i s t of wedges, and c i r c l e s o r a r c s of c i r c l e s . We are i n t e r e s t e d i n d e t e r m i n i n g which s i m u l a t o r con f i g u r a t i o n w i l l g i v e the b e s t o r i e n t a t i o n - i n d e p e n d e n t enhancement of v i s u a l c o n t o u r s . Por example, we w i s h t o determine which c o n f i g u r a t i o n w i l l enable us t o r e l a t e peak r e c e p t o r a c t i v i t y t o wedge ang l e w i t h minimum a m b i g u i t y . The e x p e r i m e n t a l procedure s t a r t s w i t h the i n s e r t i o n of a p a t t e r n , a wedge f o r example, i n t o the i n p u t s e c t i o n of the s i m u l a t o r . With an i n i t i a l o r i e n t a t i o n of z e r o degrees between the a r r a y rows and the b i s e c t o r of t h e wedge a n g l e , the r e c e p t o r a c t i v i t y a t a l a r g e number of p o i n t s i n the neighborhood of the v e r t e x i s computed and punched onto paper t a p e . T h i s p r o c e s s i s r e p e a t e d f o r o r i e n t a t i o n s of 15°, 30° and 45°• The paper tape i s then p r o c e s s e d on a d i g i t a l computer to determine the peak r e c e p t o r a c t i v i t y f OX* QcL ch o r i e n t a t i o n . A t y p i c a l computer p r i n t o u t i s shown i n F i g . 3.1.1. The f i r s t number i n each group i s the peak r e c e p t o r a c t i v i t y f o r the wedge a t the d e s i g n a t e d o r i e n t a t i o n . (The o t h e r number p a i r s a re the x and y c o o r d i n a t e s of the p o i n t s a t which t h a t a c t i v i t y o c c u r r e d . ) The maximum peak r e c e p t o r a c t i v i t y f o r t h i s wedge has been c i r c l e d t w i c e and the minimum 30 o c •= 100° 7 7 6 = 0° 0 r 15* (j) = 45 o 4 1 4 0 5 1 5 2 5 0 6 1 6 2 6 P i g . 3-1-1 A T y p i c a l Computer P r i n t o u t of Peak A c t i v i t i e s a t V a r i o u s O r i e n t a t i o n s once. These two numbers, a l o n g w i t h the c o r r e s p o n d i n g p a i r s f o r the o t h e r wedges, are p l o t t e d , g i v i n g the type of graph shown i n P i g . 3-1-2. T h i s graph forms the peak a c t i v i t y envelope f o r the s e t of wedges under the a c t i o n of the p a r t i c u l a r s i m u l a t o r c o n f i g u r a  t i o n . T h i s graph i s the envelope shown i n F i g . 2.1.7- I t can be used t o o b t a i n a measure of the a m b i g u i t y t h a t a r i s e s i n r e l a t i n g peak a c t i v i t y t o wedge a n g l e . The worst case a m b i g u i t y , 9 , can be used as a measure of the o r i e n t a t i o n de- amax pendence a s s o c i a t e d w i t h a p a r t i c u l a r s i m u l a t o r c o n f i g u r a t i o n . I n the case of F i g . 3-1-2 the a m b i g u i t y measure would g i v e 0 _ 84°. amax ~ The graphs i n the f o l l o w i n g s e c t i o n s a r e extended t o i n c l u d e wedge a n g l e s g r e a t e r than 180°. The r e c e p t o r a c t i v i t y f u n c  t i o n f o r t h i s type of wedge has a s a d d l e p o i n t near the.wedge 240-i WEDGE ANGLE cxL F i g . 3.1.2 The Peak A c t i v i t y E n v e l o p e f o r the Set of Graphs i n F i g . 2.1.7 v e r t e x . The a c t i v i t y a t the s a d d l e p o i n t i s the peak a c t i v i t y a s s o c i a t e d w i t h the wedge a n g l e . 3.2 The L i m u l u s k..(d) F u n c t i o n I n c h a p t e r 2 some s t u d i e s were d e s c r i b e d t h a t used a k ^ ( d ) f u n c t i o n t h a t was non-zero over the 5x5 c e n t r a l p o r t i o n of the 9x9 a r r a y . The f i r s t k. .(d) f u n c t i o n d e a l t w i t h i n t h i s 1 J c h a p t e r has the same form, but has been extended t o g i v e non z e r o k ^ j over the 9x9 a r r a y . The f u n c t i o n i s f 0.3-0.05d 0< d $ 6u k ± . ( d ) = J 3-2.1 ^ 0 d=0, d > 6u and has the form shown i n F i g . 3.2.1. The t h r e s h o l d or k . . t . . v a l u e used i n most of the s t u d i e w i t h t h i s k..(d) f u n c t i o n was chosen t o g i v e the minimum non- n e g a t i v e r e c e p t o r a c t i v i t y l e v e l , x , f o r u n i f o r m i l l u m i n a t i o n . F o r t h i s k^.(d) f u n c t i o n the r e q u i s i t e k ^ . t ^ . v a l u e gave an a c t i v i l e v e l of x Q = 27 p.p.s I n the f i r s t s e t of experiments the d i a m e t e r of the r e c e p t i v e f i e l d of v i e w , D, was v a r i e d from D = O.Ou t o D = 2.5u i n steps of 0.5u. The r e s u l t s from t h i s s e t of experiments are g i v e n i n F i g s . 3-2.2 t o 3>2.7. The 0 v a l u e s i n v a r i o u s ranges a of wedge a n g l e are marked on the graphs. These v a l u e s a l o n g w i t h 9 f o r the s i x r e c e p t i v e f i e l d s i z e s are brought t o g e t h e r i n max Table 3-2.1. 33 *// (d) 1 2 3 4 5 6u DISTANCE F i g . 3-2.1 The Lim u l u s k^.(d) F u n c t i o n Receptor F i e l d S i z e , D 9 a max 0 f o r a l 0°<a<90° 9 f o r 90°< ou<180° 0 f o r 3 180°$ a*270° 0 f o r a 4 270°< a<360'° O.Ou 26° 26° 22° 22° 18° 0 . 5 u 3 3 ° 3 3 ° 2 3 ° 2 3 ° 18° l.Ou 5 3 ° 5 3 ° 2 0 ° 22° 1 2 ° 1 . 5 u 4 3 ° 4 3 ° 1 5 ° 17° 1 3 ° 2.0u 60° 60° 1 3 ° 1 0 ° 1 2 ° 2 . 5 u 60° 60° 1 0 ° 8° 3 0 ° Table 3-2.1 Lim u l u s k^^(d) A m b i g u i t y V a l u e s f o r V a r i o u s R e c e p t i v e F i e l d S i z e s The f i r s t p o i n t worth n o t i n g i s the expected decrease i n a m b i g u i t y between the r e s u l t s shown i n F i g . 3 - 1 - 2 and those i n F i g . 3 - 2 . 2 . Expanding the d i r e c t i n h i b i t i o n f i e l d t o i n c l u d e a l l the r e c e p t o r s i n the rounded 9x9 a r r a y s h o u l d , and does, l e a d t o a b e t t e r d i s c r i m i n a t i o n of wedge angl e as a f u n c t i o n of peak r e c e p  t o r a c t i v i t y . I n a d d i t i o n , by r o u n d i n g o f f the 9x9 a r r a y the 34 ti WEDGE ANGLE F i g . 3-2.3 Peak A c t i v i t y E n v e l o p e f o r L i r n u l u s k ^ ( d ) w i t h D = 0.5u and x Q = 27 p . p . s . F i g . 3-2.4 Peak A c t i v i t y Envelope f o r L i m u l u s k..(d) w i t h D = l.Ou and x = 27 p.p.s. w i t h D = 1.5u and x = 27 p.p.s. ?240 ti 11 \200 *^ o o 120 lu 0. * 0 - 0 " F i g . 5 0 " ' o • 120 » e  1 180 Ba, 0 a , 6cr 4 /2 240 300 360 WEDGE ANGLE e£ 3-2.6 Peak A c t i v i t y Envelope f o r L i m u l u s k ^ ( d ) w i t h D = 2.0u and x = 27 p.p.s. to * 160 A o g/20 Bi o 2 0. 40 6a, • fla, • 8at- Oaf 10. 30* F i g -i 1 r j - r 60 T _7- 1 r wo" ;«o WEDGE ANGLE oC i r——i 1 1 I 240 300 360* 3.2.7 Peak A c t i v i t y Envelope f o r L i m u l u s k ^ ( d ) w i t h D = 2.5u and x o 27 p.p.s UJ 0 " — i 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 u RECEPTIVE FIELD SIZE, D F i g . 3.2.8 Average Peak A c t i v i t y Spread a g a i n s t D f o r Lim u l u s k.-(d) d i s t r i b u t i o n of r e c e p t o r s i s made more i s o t r o p i c . T h i s a l s o reduces the o r i e n t a t i o n dependence. I t i s e v i d e n t t h a t the use of a non-zero r e c e p t i v e f i e l d of view l e a d s t o a g e n e r a l r e d u c t i o n i n a m b i g u i t y f o r wedge a n g l e s g r e a t e r than 90°. F o r a n g l e s l e s s than t h i s v a l u e the g e o m e t r i c a l i n t e r a c t i o n between the c i r c u l a r r e c e p t i v e f i e l d s and the wedge causes a " f o l d i n g o ver" of the peak response c u r v e s . The decrease i n a m b i g u i t y f o r a n g l e s g r e a t e r than 90° comes about as a r e s u l t of two a n t a g o n i s t i c e f f e c t s . For any g i v e n wedge, i n c r e a s i n g the s i z e of the f i e l d of view l e a d s t o a decrease i n the o r i e n t a t i o n dependent spread of peak a c t i v i t y . I n F i g . 3-2.8 the average peak a c t i v i t y spread i n each of the graphs i n F i g s . 3.2.2 to 3-2.7 i s p l o t t e d a g a i n s t the d i a m e t e r of the r e c e p t i v e f i e l d of view. The decrease i n the average spread -with i n c r e a s i n g D i s ci. 200- o 1 5 0 2 100 0. 5: 50 oc= 190 GC=280 oc = 350 — I — 7.5 — i — 2.0 2.5 u O.O 0.5 1.0 RECEPTIVE FIELD SIZE, D F i g . 3.2.9 Change i n Median Peak A c t i v i t y w i t h I n c r e a s i n g D f o r V a r i o u s Wedge Ang l e s d r a m a t i c . However, the l a r g e r f i e l d of view a l s o causes a decrease i n the median peak a c t i v i t y a s s o c i a t e d w i t h any g i v e n wedge. T h i s tends t o o f f s e t the decrease i n o r i e n t a t i o n dependence caused by the decrease i n peak a c t i v i t y s p r e a d . I n F i g . 3.2.9 the change i n median peak a c t i v i t y as a f u n c t i o n of r e c e p t i v e f i e l d s i z e i s p l o t t e d f o r wedges of 10°, 100°, 190°, 280°, and 350°. The cause of t h i s change i s i l l u s t r a t e d i n F i g . 3-2.10 f o r the case of a 100° wedge. I n F i g . 3-2.10(a) and (b) the i l l u m i n a t i o n p a t  t e r n f o r peak a c t i v i t y i s shown f i r s t f o r p o i n t r e c e p t o r s , D = O.Ou, th e n f o r r e c e p t o r s w i t h a D = 2.5u. ( I n (b) o n l y the r e c e p t i v e f i e l d of view f o r the c e n t r a l r e c e p t o r i s shown.) The c r o s s e s 39 ©—CENTRAL RECEPTOR CENTER (a) D = 0.0 u (b) 0 = 2.5 u F i g . 3'2.10 P o s i t i o n of Peak A c t i v i t y f o r Two D i f f e r e n t V a l u e s of D i n d i c a t e r e c e p t o r s t h a t r e c e i v e no i l l u m i n a t i o n i n (a) and y e t i n (b) have o n e - h a l f or more of t h e i r r e c e p t i v e f i e l d s i l l u m i n a t e d . Thus as p o i n t r e c e p t o r s they do not i n h i b i t the c e n t r a l r e c e p t o r . As t h e i r r e c e p t i v e f i e l d i n c r e a s e s , however, they b e g i n t o i n f l u e n c e the c e n t r a l r e c e p t o r , r e d u c i n g i t s a c t i v i t y . C o n s e q u e n t l y , the peak a c t i v i t y f o r D = 2.5u i s much l e s s than f o r D = O.Ou. The n e x t s e t of experiments u s i n g the L i m u l u s k ^ ( d ) f u n c t i o n i n v o l v e d v a r i a t i o n of the t h r e s h o l d l e v e l . The r e c e p t i v e f i e l d s i z e was s e t a t D = 1.5u and k ^ t ^ was i n c r e a s e d u n t i l the u n i f o r m i l l u m i n a t i o n a c t i v i t y was 50 p.p.s. The experiment de scribed p r e v i o u s l y was r e p e a t e d . The r e s u l t s a re g i v e n i n F i g . 3.2.11. T h i s procedure was r e p e a t e d f o r a k ^ t ^ v a l u e g i v i n g a u n i f o r m i l l u m i n a t i o n a c t i v i t y of 75 p.p.s. These r e s u l t s a re g i v e n i n F i g . 3-2.12. These graphs s h o u l d be compared w i t h the graph i n F i g . 3.2.5. A l t h o u g h f o r a wedge of 350° the d i f f e r e n c e i n ' the median peak a c t i v i t y f o r s u c c e s s i v e i n c r e a s e s i n k. . t . . i s about 25 p.p.s., f o r a wedge of 10° the d i f f e r e n c e i s o n l y about 8 p.p.s. C240 e *> $200 a ^ 160 o § 120 t~ ft. Ui o Ui a: eo 0 a , - 0 a 4 - 0a» - Oaf 52 7 - 1 1 L~T~R o 60 — I a » 1 1 0 I ! 1— r - ! 1 1—T" 120 180 24 0 300 WEDGE ANGLE oi. T 1 360 F i g . 3.2.11 Peak A c t i v i t y Envelope f o r Li m u l u s k^Cd) w i t h x Q = 50 p.p.s. and D = 1.5u C240 *> in •5 200 a * 160 o "* o 120 K . ft. Ui ft: 0O Ui «• 4-0 0a, 0a, 0a. 5 0 " 20 0 a * — - 20 7? ~j~rr 180 — 1 r 2 <0° 1 o 1 300 360 WEDGE ANGLE oc F i g . 3-2,12 Peak A c t i v i t y Envelope f o r Limulus k.^(d) w i t h X Q = 75 p.p.s. and D - 1.5u 41 The a m b i g u i t y measures from the t h r e e graphs are g i v e n i n Table 3.2.2. There i s a g e n e r a l t r e n d i n .these r e s u l t s i n d i c a t i n g t h a t an i n c r e a s e i n the t h r e s h o l d l e v e l causes an i n c r e a s e i n U n i f o r m 111. A c t i v i t y 9 a . max 9 f o r a l 0°<a<90° 9 0 f o r 2 90°<: a*180° 9 f o r 180°< a<270° 9 f o r a 4 270°< a<360° 27 p.p.s. 43° 43° 15° 1.7° 13° 50 p.p.s. 52° 52°. 12° 18° 18° 75 p.p.s. 50° 50° 15° 20° 20° Table 3«2.2 A m b i g u i t y V a l u e s f o r V a r i o u s T h r e s h o l d s w i t h the L i m u lus k^^(d) F u n c t i o n a m b i g u i t y . T h i s i s expected s i n c e the t h r e s h o l d l e v e l can be i n c r e a s e d u n t i l no i n t e r a c t i o n t a k e s p l a c e . In t h i s case t h e r e would be no graded enhancement of c o n t o u r s . Consequently, t h e r e would be t o t a l a m b i g u i t y i f one attempted t o r e l a t e r e c e p t o r a c t i v i t y and wedge a n g l e . 3.3 The U n i f o r m k.-(d) F u n c t i o n . The u n i f o r m k..(d) f u n c t i o n i s c o n s t a n t over a l l the r e c e p t o r s i n the rounded 9x9 a r r a y . The f u n c t i o n i s 0.125 0 <d .^ 4.5u 3-3.1 L. 0 d=0, d > 4-5u and has the form shown i n F i g . 3-3.1. In the i n i t i a l experiments the t h r e s h o l d l e v e l was s e t t o g i v e an a c t i v i t y l e v e l of z e r o p u l s e s / s e c . under u n i f o r m i l l u m i n a t i o n c o n d i t i o n s . The f i r s t s e t of experiments i n v o l v e d the wedges as t e s t p a t t e r n s . The d i a m e t e r , D, of the r e c e p t i v e f i e l d of view was v a r i e d between s u c c e s s i v e e x p e r i m e n t s . The r e s u l t s are g i v e n 42 h,j(d) T 1 r 1 2 3 4 DISTANCE, d F i g . .The U n i f o r m k..(d) F u n c t i o n 1 J < R e c e p t i v e F i e l d S i z e , D 9 a max 9 f o r a l 9 ^2 f o r 9 f o r a 3 9 f o r a 4 10°<oc 90° 90°^ ail80° 180°< oc<270° 270°$ a<360° O.Ou 23° 17° 14° ' 23° 14° 0.5u 20° 1 3 ° 16° 20° 14° / 1. Ou 17° 17° 10° 11° 10° 1.5u 11° . 11° 8° 9° 10° 2.0u 22° 22° 7° 9° 11° 2.5u 35° 35° 4° 5° 10° Table 3«3-l A m b i g u i t y V a l u e s f o r a U n i f o r m k i ^ ( d ) a t V a r i o u s V a l u e s of D. i n F i g s . 3.3.2 t o 3-3>7. The v a r i o u s . 9 v a l u e s are noted on the a f i g u r e s , and a l o n g w i t h 9 are comp i l e d i n Table 3-3-1. 1 max Note the s t e a d y decrease between one f i g u r e and the nex t of the d i s t a n c e s e p a r a t i n g the two s i d e s of the peak a c t i v i t y envelope T h i s i s r e f l e c t e d i n a s t e a d y decrease of the 9 v a l u e s a t l e a s t a up t o D = 1.5u. Above t h i s v a l u e the " f o l d i n g o v e r " of the envelope a t s m a l l wedge a n g l e s causes an i n c r e a s e i n the 9 v a l u e which i s o & 1 the peak a m b i g u i t y i n the 10° t o 90 range of wedge a n g l e s . WEDGE ANGLE F i g . 3«3-2 P e a k A c t i v i t y E n v e l o p e f o r U n i f o r m k _ ( d ) w i t h D = 0 . O u a n d x = 0 p . p . s . 1 - — r r — — r — — , 1 1——1 1 1 „ 1 1 • 0 ' " 1 o 1 1 1 o 0° 60 120 ISO 240 300 360 WEDGE ANGLE oc F i g . 3.3.3 P e a k A c t i v i t y E n v e l o p e f o r U n i f o r m k ^ ( d ) w i t h D = 0.5u a n d x = 0 p . p . s . 4 4 1 1 1 1 1 1 i 1 1 r——i 1 r——i 1 r——i 1 1 0 ° 60° 120° 180 240 300 360 WEDGE ANGLE oc P i g . 3-3.5 P e a k A c t i v i t y E n v e l o p e f o r U n i f o r m k^Cd) w i t h D = 1.5u and x = 0 p . p . s . WEDGE ANGLE F i g . 3-3,6 Peak A c t i v i t y Envelope f o r U n i f o r m k ^ ( d ) w i t h D = 2.0u and x = 0 p.p.s. w i t h D = 2.5u and x = 0 p.p.s. 46 UJ CL Q Uj 5: 200 - 150 100 - 50 - oc=10 oc= 100 oc*280 oc=350 T T 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 u RECEPTIVE FIELD SIZE, D P i g . 3.3-8 Change i n Median Peak A c t i v i t y w i t h I n c r e a s i n g D f o r V a r i o u s Wedge Angles As noted i n the p r e v i o u s s e c t i o n , t w o c o u n t e r v a i l i n g i n f l u e n c e s a r e a t work I n the r e d u c t i o n of a m b i g u i t y . I n c r e a s i n g the s i z e of the f i e l d of view l e a d s i n g e n e r a l t o a decrease i n median peak a c t i v i t y a t any g i v e n wedge a n g l e . T h i s r e d u c t i o n i n the dynamic range o f peak r e c e p t o r a c t i v i t y i s e v i d e n t i n the graphs of P i g . 3-3-8. I t i s more than o f f s e t by the decrease i n the spread o f peak a c t i v i t y caused by changes i n o r i e n t a t i o n . The average v a l u e of t h i s spread i s p l o t t e d i n P i g . 3-3.9 as a f u n c  t i o n o f D. Note t h a t -the s h a r p e s t decrease i n t h i s average o c c u r s between D = O.Ou and D = 1.5u. The o b j e c t of the next two experiments was t o t e s t the e f f e c t on the a c t i v i t y c u r v e s of a v a r i a t i o n i n t h e t h r e s h o l d . I n 47 ft: ^ 0 I r i 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 u RECEPTIVE FIELD SIZE, D F i g . 3.3.9 Average Peak A c t i v i t y Spread a g a i n s t D f o r U n i f o r m k..(d) w i t h x^ = 0 p.p.s. the f i r s t experiment the k . . t . . v a l u e was s e t t o g i v e a u n i f o r m i l l u m i n a t i o n a c t i v i t y of'25 p.p.s., and the second, an a c t i v i t y o f 50 p.p.s. I n b o t h cases D was s e t a t 1.5u. The r e s u l t s a r e p l o t t e d i n P i g s . 3-3-10 and 3.3-H and s h o u l d be compared w i t h the graph i n . P i g . 3-3-5. By r a i s i n g the t h r e s h o l d the a c t i v i t y a s s o c  i a t e d w i t h a g i v e n wedge a n g l e i s i n c r e a s e d . The i n c r e a s e , however, i s g r e a t e s t f o r l a r g e wedge a n g l e s and de c r e a s e s m o n o t o n i c a l l y w i t h d e c r e a s i n g a n g l e . I n Table 3-3.2 the v a r i o u s a m b i g u i t y measures a re com p i l e d . There i s no e v i d e n t change i n the measured a m b i g u i t i e s w i t h i n the range of t h r e s h o l d covered by these e x p e r i m e n t s . In s e c t i o n 2.2 i t was suggested t h a t a n o t h e r p o s s i b l e method f o r r e d u c i n g a m b i g u i t y or o r i e n t a t i o n dependence would be 48 «240 Q.2O0 O o 720 0. lu O Ui * eo Uj 40 H 8°, 8a1 8 a, 8ai 10° 10° ' o 1 60 120" T T 180° 240" WEDGE ANGLE 0 6 300 1 1 , 360 F i g . 3.3.10 Peak A c t i v i t y E n v e l o p e f o r U n i f o r m k^Cd) w i t h x = 25 p.p.s. and D = 1.5u 1240 i <0 tl "5 200 Q. C 160 o a. Ul o UJ ce 80 Ui 0 . 40 8a, • 8at 8a, 8a4 10" 8° 10° 10. 7 ? 120 180 240 WEDGE ANGLE c*s 300 360 F i g . 3 . 3 . I I Peak A c t i v i t y E n v e l o p e f o r U n i f o r m k ^ d ) w i t h = 50 p . p . s . and D = 1.5u x. < 49 CENTRAL RECEPTOR F i g . 3-3.12 The Hexagonal R e c e p t o r A r r a y U n i f o r m 111. A c t i v i t y 9 a max 9 f o r a l „o n~o 0 <<x<90 "•" --"1 9 f o r 90°< asl80° — 9 o f o r 5 180 °s a<270° \9 f o r - a 4 270°$a<360° 0 p.p.s. 11° 11° 8° 9° 10° 25 p.p.s. 10° 10° 8° 10° 10° 50 p.p.s. 10° 10° 8° 10° 10° Table 3.3-2 A m b i g u i t y V a l u e s f o r V a r i o u s ' T h r e s h o l d s w i t h the Uni f o r m k..(d) F u n c t i o n the use of an hexagonal a r r a y of r e c e p t o r s . I n l i n e w i t h t h i s , the s c a n n i n g s e c t i o n of the s i m u l a t o r was m o d i f i e d t o g i v e the hexagonal a r r a y shown i n F i g . 3.3.12. Note t h a t t h i s a r r a y c o n t a i n s 61 r e c e p  t o r s as opposed t o the 69 r e c e p t o r s making up the rounded 9x9 square a r r a y . A l t h o u g h experiments w i t h the hexagonal a r r a y were c a r r i e d out f o r each of the k..(d) f u n c t i o n s , o n l y the r e s u l t s o b t a i n e d w i t h the u n i f o r m ^ ^ ( d ) w i l l be p r e s e n t e d . The wedge experiments w i t h D as a parameter were re p e a t e d u s i n g the hexagonal a r r a y . The t h r e s h o l d was s e t to g i v e z e r o a c t i v i t y under u n i f o r m i l l u m i n a t i o n . 240 n 200 160 120 •] 80 4 40 50 6a. 20 30c 20' 6a. Ba3 —\ r 60° — i r 120° i r 180" WEDGE ANGLE o4 —i r 240° ~i 1— 300e ~i 1 360' F i g . 3.3.13 Peak A c t i v i t y Envelope f o r U n i f o r m k^Cd) w i t h D = O.Ou,xQ = 0 p.p.s., and an Hexagonal A r r a y 240 -> 200 160 120 80 40 8 a, 0 a, 20* ; s c 20° 16' ' o ' r 60 -r-7-1 r 120 —1——1 1 r 180 240- 300 360 WEDGE ANGLE °C F i g . 3.3.14 Peak A c t i v i t y Envelope f o r Uniform. kj-Cd) w i t h D = 0.5u, x 0 p.p.s., and an Hexagonal A r r a y <o 240 Z.200 -\ o o 120 -\ » -0. UJ o UJ cc eo H Uj °- 40 A 60 6a, U 8a2 14 8a3 20 ea^- 10" 120° 180° 240' WEDGE ANGLE oc 300 360 F i g . 3.3.15 Peak A c t i v i t y Envelope f o r U n i f o r m k^Cd) w i t h D = l.Ou, xQ = 0 p.p. and an Hexagonal A r r a y 8al 8a4 2 8 \ 10 •"\ 20 60 120 180" 24 0 WEDGE ANGLE oc 360 F i g . 3 . 3 . I 6 Peak A c t i v i t y Envelope f o r Uniform k i_ j(d) w i t h D - 1 . 5 u , x z 0 p.p.s., and an Hexagonal A r r a y 52 o ^240 o. 2 0 0 C 160 o "<t ct O 120 cu lu o lu cc 5C lu 0. SO 4 40 Ba, Ba, 8 a, Ba. -30 -10° -10' -10' — i r 60° — i r 120° I 1 1 180 240 WEDGE ANGLE — i r 300° 360 F i g . 3-3.17 Peak A c t i v i t y Envelope f o r U n i f o r m k ± . ( d ) w i t h D = 2.0u, X Q = 0 p.p.s., and an Hexagonal A r r a y 0 * 240 < t> ^200 Z 160 k. o %120 cu Ul o lu cc 80 lu cu 40 0a, 0a3 0ai 0a< -40' 10 10° 1 1 1 1 1 1 1 1 1 1 60° 120° 180° 240" WEDGE ANGLE 00 -1 r 300 360 F i g . 3-3.18 Peak A c t i v i t y Envelope f o r Uniform k. .(d) w i t h D = 2.5u, x Q = 0 p.p.s., and an Hexagonal A r r a y 53 The r e s u l t s from t h e s e experiments are p r e s e n t e d i n F i g . 3-3-13 to F i g . 3.3-18. The a m b i g u i t y measures are noted on the f i g u r e s and are c o m p i l e d i n Table 3-3-3. Rec e p t o r F i e l d S i z e D G a max 9 f o r a l 9 f o r ^2 9 f o r 9 f o r & 4 10° a 90° 90° a 180° 180° a 270° 270° a 350° O.Ou 30° 24° 20° 0 30 20° 0.5 20° 20° 16° 20° 16° 1.0 20° 14° 14° 20° 10° 1.5 .28° 28° 10° 10° 20° 2.0 30° 30° 10° 10°, 10° 2.5 40° 40° 8° 10° 10° Table 3>3.3 A m b i g u i t y V a l u e s f o r V a r i o u s D , U n i f o r m k^ . (d), and an Hexagonal A r r a y I n no case i s the maximum a m b i g u i t y f o r the hexagonal a r r a y l e s s t h a n the -corresponding v a l u e f o r the square a r r a y . I t may be t h a t the a c t i v i t y of an hexagonal a r r a y i s i n h e r e n t l y more o r i e n t a t i o n dependent. I n t h i s p a r t i c u l a r case i t i s more l i k e l y t h a t the g r e a t e r a m b i g u i t y r e s u l t s from the use of fewer r e c e p t o r s . I n any event, i n our p a r t i c u l a r s i t u a t i o n the r e c e p t o r response a s s o c i a t e d w i t h the rounded 9x9 square a r r a y was always l e s s o r i e n t a t i o n dependent t h a n t h a t a s s o c i a t e d w i t h the hexagonal a r r a y . 3.4 The I n v e r s e k^.(d) F u n c t i o n J- The k..(d) f u n c t i o n s t u d i e d i s an i n c r e a s i n g f u n c t i o n of J t h t h the d i s t a n c e , d, between the i and j r e c e p t o r s . As such, i t 54 kij(d) A .20- .15 - .05- .10- T 1 1 1 1 / 2 3 4 5 u DISTANCE, d F i g . 3'4.1 The I n v e r s e k. .(d) F u n c t i o n i s the o p p o s i t e or i n v e r s e of the L i m u l u s k ^ ( d ) f u n c t i o n . I t has the form shown i n F i g . 3-4.1 and i s s p e c i f i e d as l e v e l f o r u n i f o r m i l l u m i n a t i o n of z e r o p u l s e s per second. As i n the p r e v i o u s s e c t i o n s the f i r s t e xperiments employed the wedges as t e s t p a t t e r n s . S u c c e s s i v e experiments d i f f e r e d o n l y i n the s i z e . o f the r e c e p t i v e f i e l d , D. The r e s u l t s from these experiments a re g i v e n i n F i g s . 3«4.2 t o 3-4.7. The 9 v a l u e s a i n the v a r i o u s ranges of wedge a n g l e a r e noted on the f i g u r e s and compiled i n Table 3-4-1- There i s a st e a d y decrease i n a m b i g u i t y , a t l e a s t i n the 10° t o 270° range, out t o D=2.0u. At D=2.5u the peak a c t i v i t y envelope f l a t t e n s out a t the s m a l l wedge a n g l e s , c a u s i n g a sharp i n c r e a s e i n a m b i g u i t y i n the 10° t o 90° range. From these r e s u l t s the b e s t v a l u e f o r D i n terms of minimum a m b i g u i t y i s D=2.0u. In F i g . 3-4.8 the change, i n the median peak a c t i v i t y a t v a r i o u s wedge a n g l e s i s p l o t t e d as a f u n c t i o n o f D. There i s .a 0 d=0, d > 4 - 5 u The k ^ . t ^ . v a l u e used i n the i n i t i a l e xperiments gave an a c t i v i t y WEDGE ANGLE oC F i g . 3.4-2 Peak A c t i v i t y Envelope f o r I n v e r s e k^Cd) w i t h D = O.Ou and x = 0 p.p.s. WEDGE ANGLE at F i g . 3-4-3 Peak A c t i v i t y Envelope f o r I n v e r s e k . ^ d ) w i t h D — 0-5u and x = 0 p.p.s. .240-, WEDGE ANGLE F i g . 3.-4.4 Peak A c t i v i t y Envelope f o r I n v e r s e k ^ d ) w i t h D = 1.Ou and x = 0 p.p.s. WEDGE ANGLE 0 6 F i g . 3.4.5 Peak A c t i v i t y Envelope f o r I n v e r s e k^Cd) w i t h D = 1.5u and x - 0 p.p.'s. F i g . 3-4.7 Peak A c t i v i t y Envelope f o r I n v e r s e k. .(d) w i t h D = 2.5u and x = 0 p.p.s. 58 - ci. ol 200-s- - 150 - o UJ 100 - Q 50 - Lu _ <^=10 oczlOO = 190 = 280 oc = 350 0.0 0.5 1.0 1.5 2.0 RECEPTIVE FIELD SIZE, D F i g . 3-4.8 Change i n Median Peak A c t i v i t y w i t h . I n c r e a s i n g D f o r V a r i o u s Wedge Angles I 1.5 2.5 u R e c e p t i v e P i e l d S i z e D 9 amax 9 f o r a l 10°<a<90° 9 f o r a 2 90°<a<180° 9 f o r 180°$a<270° 9 „ a^ f o r 270°$a<360° O.Ou 22° 19° 22° 22° 20° 0.5u 21° 21° 21° 16° 14° l.Ou 17° 8° 18° 16° 17° 1.5u 20° 8° 14° 16° 20° 2.0u 17° 10° 10° 10° 17° 2.5u 20° 20° 9° 10° 16° Table 3 -4-1 A m b i g u i t y V a l u e s f o r the I n v e r s e k ^ ( d ) a t V a r i o u s V a l u e s of D marked decrease i n the median a c t i v i t y w i t h i n c r e a s i n g D f o r wedge a n g l e s a t l e a s t up t o 190°. At 280° t h e r e i s v i r t u a l l y no change, w h i l e a t 350° the a c t i v i t y i n c r e a s e s w i t h i n c r e a s i n g r e c e p t i v e f i e l d 59 £ S -i o Lu 6- 5 i -\ Lu C£ a. to Lu 2 A e> Lu ^ 0 0.0 — I 0.5 2.5 u LO J.S 2.0 RECEPTIVE FIELD SIZE, D F i g . 3»4.9 Average Peak A c t i v i t y Spread a g a i n s t D f o r I n v e r s e k. .(d) w i t h x = 0 p.p.s. l j o ^ ^ s i z e . I n F i g . 3.4.9 the change i n the average v a l u e of the peak a c t i v i t y s pread ( i . e . the average gap between the curv e s i n F i g s . 3.4.2 t o 3.4 .7) i s p l o t t e d w i t h r e s p e c t t o r e c e p t i v e f i e l d s i z e . The curve drops s h a r p l y between D = O.Ou and D = l.Ou and then f l a t t e n s out beyond t h i s v a l u e . I n the next two experiments the t h r e s h o l d l e v e l was v a r i e d . The u n i f o r m i l l u m i n a t i o n a c t i v i t y l e v e l s f o r the two k . . t . . s e t t i n g s were 25 p.p.s. and 50 p.p.s. The r e c e p t i v e f i e l d s i z e was D = 1.5u. In Table 3-4.2 the a m b i g u i t y measures from the graphs i n F i g s . 3.4-5, 3.4.10 and 3-4-11 a r e c o m p i l e d . I t i s e v i d e n t t h a t t h e r e i s no s i g n i f i c a n t change i n a m b i g u i t y w i t h i n the e x p e r i m e n t a l range. — i 1 1 1- 1 1 1 1 1 1 1 r — — i 1 r — - i i i 0 ° 60° 120° 180° 240° 300° 360 WEDGE ANGLE oC F i g . 3-4.11 Peak A c t i v i t y Envelope f o r I n v e r s e k ^ ( d ) w i t h x 0 = 50 p.p.s. and D = 2.0u 61 U n i form 111 L e v e l 9 9 f o r max 1 10°$as90 9 f o r 90°s a$180° 9 f o r 180%-a « 270° 9 f o r 270°< cx.<350° 0 p.p.s 25 p.p.s 50 p.p.s 20 20' 19' 8' 8 8' o 14' 14 C 15' 16' 17' 15 ( 20 20 19' o Table 3-4.2 A m b i g u i t y V a l u e s f o r V a r i o u s T h r e s h o l d s w i t h the I n v e r s e k^^(d) F u n c t i o n 3.5 Peak Receptor A c t i v i t y and C i r c u l a r P a t t e r n s I n t h i s s e c t i o n we d e a l w i t h a s e t of experiments i n which c i r c u l a r f i g u r e s or d i s k s were used as t e s t p a t t e r n s . We have two purposes i n mind. F i r s t , we w i s h t o o b t a i n some i d e a of the o r i e n t a t i o n dependence problem as i t r e l a t e s t o c i r c l e s . Second, we want t o r e l a t e the peak a c t i v i t y a s s o c i a t e d w i t h c i r c l e s and wedges. Each of the d i f f e r e n t k ^ ( d ) f u n c t i o n s was used. The st u d y was, however, l i m i t e d t o those p a r t i c u l a r s i m u l a t o r c o n f i g u r a  t i o n s t h a t had. g i v e n the.minimum a m b i g u i t y i n r e l a t i n g peak a c t i v i t y and wedge a n g l e . The experiments employed b o t h b l a c k on w h i t e , and whi t e on b l a c k d i s k s . The e x p e r i m e n t a l t e c h n i q u e i n v o l v e d the d e t e r m i n a  t i o n of the peak r e c e p t o r a c t i v i t y n e a r the boundary of these d i s k s f o r v a r i o u s o r i e n t a t i o n s of t h e rounded 9x9 a r r a y . The f i r s t experiment i n t h i s s e t made use of the L i m u l u s k ^ ( d ) f u n c t i o n w i t h a r e c e p t i v e f i e l d of view of D = 1.5u, and a k. . t . . s e t t o g i v e an a c t i v i t y of 27 p.p.s. under u n i f o r m «J tJ i l l u m i n a t i o n . The r e s u l t s a re shown g r a p h i c a l l y i n F i g . 3.5.1. The a b s c i s s a of t h i s graph i s the i n v e r s e r a d i u s , l / R , or c u r v a t u r e , T-240 200 WHITE ON BLACK DISKS 4- —T- .25 BLACK ON WHITE DfSKS 4- 40 —T— .75 1.0 u-i 1.0 u-1 —i— .75 .50 .25 .50 INVERSE RADIUS, ± F i g . 3 . 5 . 1 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the L i m u l u s k ^ ^ ( d ) , D = 1 . 5 u , and X Q = 27 p.p.s. 63 of the d i s k s . Use of the i n v e r s e r a d i u s a l l o w s us t o p l o t the r e s u l t s f o r b o t h b l a c k on w h i t e and w h i t e on b l a c k d i s k s as a c o n t i n u o u s envelope. The f i g u r e t h a t s e p a r a t e s t h e s e two types of d i s k s i s the c i r c l e of i n f i n i t e r a d i u s , the s t r a i g h t edge.. The envelope has a steeper, s l o p e f o r w h i t e d i s k s t h a n f o r b l a c k ones. T h i s i n d i c a t e s t h a t the L i m u l u s k..(d) f u n c t i o n g i v e s — i J . a more s e n s i t i v e response t o changes i n the c u r v a t u r e of w h i t e d i s k s . The maximum spread i n the peak a c t i v i t y (due t o o r i e n t a t i o n dependence) oc c u r s w i t h the s t r a i g h t edge. T h i s was a l s o the case when wedges were used as shown i n F i g . 3.2.5. The average spread i n the peak a c t i v i t y i n F i g . 3-5.1 i s 3-8 p.p.s. w i t h a minimum of 1 p.p.s. and a maximum of 8 p.p.s. The same experiment"as above was r e p e a t e d u s i n g the u n i f o r m k..(d) f u n c t i o n , w i t h D = l - 5 u , and u n i f o r m i l l u m i n a t i o n a c t i v i t y of z e r o p.p.s. The r e s u l t s are p l o t t e d i n F i g . 3.5-2. As i n the p r e v i o u s case the peak a c t i v i t y i s more s e n s i t i v e t o change i n the c u r v a t u r e of the w h i t e d i s k s than of the b l a c k . The spread i n the peak a c t i v i t y i s f a i r l y u n i f o r m a l o n g the c u r v e . The average s p r e a d i s 2.7 p.p.s. w i t h a minimum of z e r o and a maximum of 4.O p . p .s. The experiment was r e p e a t e d a t h i r d time u s i n g the i n v e r s e k. .(d) f u n c t i on w i t h D = 2.0u'.and u n i f o r m i l l u m i n a t i o n a c t i v i t y of z e r o p.p.s. The r e s u l t s a r e p l o t t e d i n F i g . 3-5.3. Note t h a t f o r w h i t e d i s k s w i t h a c u r v a t u r e g r e a t e r than 0.5u the peak a c t i v i t y i s p r a c t i c a l l y c o n s t a n t , and the peak a c t i v i t y spread i s z e r o . T h i s phenomenon i s e x p l a i n e d i n s e c t i o n 3-6. The average v a l u e f o r the spread i s 3-6 p.p.s. w i t h a minimum of z e r o and a maximum of 7.0 p.p.s. ? sec  to" CD " 10 a" - K. - 2 0 0 ACTI  - 750 /RECEPTOR  - 720 BLACK ON WHITE DISKS WHITE ON BLACK DISKS i 1 1 J PEAK  i i i i - 40 1 1 1 1 1 — - - i 7.0 u-t .75 . 5 0 . 2 5 0 .25 .50 .75 7.0 u - ' INVERSE RADIUS. F i g . 3 - 5 . 2 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the U n i f o r m k ^ d ) , D = 1 . 5 u , and x n = 0 p.p.s. I 1 1 T 1 1 1 1 ~I 1.0 u-i .75 .50 .25 0 .25 • .50 .75 1.0 u-' INVERSE RADIUS, £ R F i g . 3-5.3 Peak A c t i v i t y Envelope f o r D i s k s U s i n g the I n v e r s e k ^ C d ) , D = 2.0u, and x Q = 0 p.p.s. . V J l 66 F i g . 3 - 5 - 4 E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r the L i m u l u s C o n f i g u r a t i o n The r e s u l t s g i v e n i n these t h r e e graphs i n d i c a t e t h a t the p a r t i c u l a r s i m u l a t o r c o n f i g u r a t i o n employing the u n i f o r m k^.(d) f u n c t i o n g i v e s the l e a s t o r i e n t a t i o n dependent measure of the c u r  v a t u r e of a d i s k . T h i s i s i n agreement w i t h the r e s u l t s o b t a i n e d i n the p r e c e d i n g t h r e e s e c t i o n s where t h i s same c o n f i g u r a t i o n gave the minimum a m b i g u i t y i n a s s o c i a t i n g peak response and wedge a n g l e . . By combining the r e s u l t s g i v e n i n t h i s s e c t i o n w i t h the e q u i v a l e n t r e s u l t s f o r wedges i n the p r e c e d i n g s e c t i o n s one can o b t a i n an e q u a l a c t i v i t y curve r e l a t i n g wedges and d i s k s . I n F i g . 3 - 5 - 4 a curve has been p l o t t e d showing t h i s r e l a t i o n s h i p f o r the L i m u l u s k..(d) c o n f i g u r a t i o n . T h i s c u r v e ' i n d i c a t e s , f o r example, t h a t a 100° wedge and a w h i t e disk" w i t h a c u r v a t u r e of 0-55U-"'" have e q u a l median peak a c t i v i t i e s . I n F i g . 3 . 5 - 5 these two p a t t e r n s are superimposed i n s c a l e w i t h the rounded 9x9 a r r a y o f r e c e p t o r s . The p a t t e r n s are i n the p o s i t i o n f o r which the c e n t r a l 67 F i g . 3-5.5 E q u a l Peak A c t i v i t y Wedge and D i s k t o S c a l e w i t h 9x9 A r r a y , L i m u l u s C o n f i g u r a t i o n r e c e p t o r of the a r r a y has peak a c t i v i t y . I n F i g . 3-5-6 the wedge-disk e q u a l a c t i v i t y curve f o r the u n i f o r m k. .(d) c o n f i g u r a t i o n i s g i v e n . I n F i g . 3-5.7 the 1 0 0 ° wedge and the e q u i v a l e n t a c t i v i t y w h i t e d i s k a re shown r e l a t i v e t o the 9x9 a r r a y . I n F i g . 3-5.8 and 3.5-9 the e q u i - a c t i v i t y curve and the p a t t e r n comparison f o r the i n v e r s e k ^ ( d ) c o n f i g u r a  t i o n a r e g i v e n . Note t h a t i n t h i s case the peak a c t i v i t y i n response t o the w h i t e d i s k o c c u r s when i t i s c o n c e n t r i c w i t h the c e n t r a l r e c e p t o r . I n the two p r e c e d i n g cases the peak a c t i v i t y o c c u r r e d when the edge of the d i s k was tangent t o the edge of the c e n t r a l r e c e p t o r r e c e p t i v e f i e l d . I n a l l c a s e s , the peak a c t i v i t y f o r the 1 0 0 ° wedge occ u r s when both s i d e s of the wedge are tangent t o the r e c e p t i v e f i e l d of the c e n t r a l r e c e p t o r . Because of the p e c u l i a r i t i e s noted above, a more thorough 68 l.On J.O J F i g . 3 . 5 . 6 E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r the U n i f o r m C o n f i g u r a t i o n F i g . 3 . 5 . 7 E q u a l Peak A c t i v i t y Wedge and D i s k t o S c a l e w i t h 9x9 A r r a y , U n i f o r m C o n f i g u r a t i o n 6 9 .75- 1 .50- .25- Uj Cfc :D 0 - .25-Cc :D o .50- .75 - 1.0- P i g . 3 - 5 . 8 WHITE DISKS 180' WEDGE' 240 ANGLE 360 BLACK DISKS E q u a l Median Peak A c t i v i t y Curve R e l a t i n g D i s k s and Wedges f o r I n v e r s e C o n f i g u r a t i o n = 100 R = 3.33 u D = 2.0 u x—CENTER OF CENTRAL RECEPTOR P i g . 3 - 5 . 9 E q u a l Peak A c t i v i t y Wedge and D i s k t o S c a l e v;ith 9 x 9 A r r a y , I n v e r s e C o n f i g u r a t i o n 70 study was made of the o v e r a l l a c t i v i t y a s s o c i a t e d w i t h b o t h d i s k s and wedges. The r e s u l t s of t h i s study a re g i v e n i n the next s e c t i o n . 3.6 Re c e p t o r A c t i v i t y near D i s k s and Wedges There i s a t w o f o l d purpose i n t h i s s e c t i o n . F i r s t , t o e x p l a i n some apparent anomalies i n the r e s u l t s g i v e n i n the p r e c e d i n g s e c t i o n . Second, t o attempt t o pass on t o the r e a d e r some i n t u i t i v e f e e l i n g f o r the form of the r e c e p t o r a c t i v i t y f u n c t i o n s near w h i t e / b l a c k b o u n d a r i e s . We d e a l f i r s t w i t h ' the r e c e p t o r a c t i v i t y f u n c t i o n s a l o n g the d i a m e t e r s of a s e t of w h i t e d i s k s . We then examine i n d e t a i l the r e c e p t o r a c t i v i t y f u n c t i o n s near the v e r t e x . o f a 100° wedge. The t h r e e d i f f e r e n t s i m u l a t o r c o n f i g u r a t i o n s d e a l t w i t h i n the p r e c e d i n g s e c t i o n a r e used i n these s t u d i e s . I n F i g . 3-6.1 the r e c e p t o r a c t i v i t y f u n c t i o n s a l o n g the d i a m e t e r s of a number of w h i t e d i s k s a re g i v e n . These r e s u l t s a r e f o r the L i m u l u s k_^(d) c o n f i g u r a t i o n . Note t h a t i n a l l cases the- p o i n t of peak r e c e p t o r a c t i v i t y i s i m m e d i a t e l y a d j a c e n t t o the edge of the d i s k . That t h i s i s not the case f o r a l l s i m u l a t o r c o n f i g u r a t i o n s i s e v i d e n t from the next two f i g u r e s . I n F i g s . 3.6.2 and 3.6.3 r e c e p t o r a c t i v i t y f u n c t i o n s a l o n g the d i a m e t e r s of the wh i t e d i s k s a re shown. They were produced by the u n i f o r m k^^(d) and the i n v e r s e k.,(d) s i m u l a t o r c o n f i g u r a t i o n s r e s p e c t i v e l y . The d i s k r a d i i a r e as i n d i c a t e d . Note t h a t i n F i g . 3-6.2 the r e c e p t o r a c t i v i t y f u n c t i o n f o r d i s k s w i t h a r a d i u s l e s s t h a n about 3u i s e s s e n t i a l l y c o n s t a n t t h rough the d i s k i n t e r i o r . F o r d i s k s w i t h r a d i i g r e a t e r than t h i s v a l u e the r e c e p t o r a c t i v i t y peaks a t the boundary of the d i s k and then d e c r e a s e s toward the c e n t e r . F i g . 3.6.1 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r L i m u l u s C o n f i g u r a t i o n -r250 4- 150 Uj o Lu 200 R = 1.34 u -R = 2u -R = 2.67 u •R = 3,34 u •R = 4 u -R = 4.67u -R= 5.. T 2 u 2 1 0 1 DISTANCE FROM CENTER OF DISK 5u F i g . 3-6.2 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r U n i f o r m C o n f i g u r a t i o n I n the case of the i n v e r s e k ^ ( d ) c o n f i g u r a t i o n shown i n F i g . . 3-6.3» the r e c e p t o r a c t i v i t y f u n c t i o n s have t h e i r maxima a t the d i s k boundary i f the d i s k r a d i u s i s g r e a t e r than 4.0u. However, f o r d i s k s w i t h a r a d i u s l e s s t h a n or e q u a l t o t h i s v a l u e , the peak i n r e c e p t o r a c t i v i t y o c c u r s a t the c e n t e r of the d i s k . The above r e s u l t s show t h a t as l o n g as the d i s k i s l a r g e r t h a n the 9x9 a r r a y , the peak i n a c t i v i t y o c c u r s n e a r the edge. A l l of the r e c e p t o r s i n F i g . 3.6.4(a) are i l l u m i n a t e d and con s e q u e n t l y the c e n t r a l r e c e p t o r a c t i v i t y i s low. As the d i s k moves some of the r e c e p t o r s e n t e r the b l a c k r e g i o n , c a u s i n g the a c t i v i t y of the c e n t r a l r e c e p t o r t o i n c r e a s e . T h i s r i s e i n a c t i v i t y must c o n t i n u e u n t i l the c e n t r a l r e c e p t o r i s j u s t a d j a c e n t t o the edge. The magnitude of the a c t i v i t y peak a t t h i s p o i n t i s r e l a t e d t o the r a d i u s of the d i s k . I f the d i s k , or any o t h e r p a t t e r n , i s s m a l l e r than the 9x9 a r r a y , the p o s i t i o n of the a c t i v i t y peak i s dependent on the form of the ^..(d) f u n c t i o n . I f the s m a l l d i s k i n Fig.- 3.6.4(b) moves as shown, the more c e n t r a l r e c e p t o r s on the l e f t s i d e a re obscured w h i l e the more p e r i p h e r a l r e c e p t o r s on the r i g h t s i d e are i l l u m i n a t e d . S i n c e f o r t h e L i m u l u s k ^ ( d ) f u n c t i o n the more p e r i  p h e r a l r e c e p t o r s a re l e s s i n h i b i t o r y , the peak r e c e p t o r a c t i v i t y s t i l l o c c u r s a t the edge. C o n v e r s e l y , i t must oc c u r a t the d i s k c e n t e r when the i n v e r s e k. .(d) f u n c t i o n i s employed. I t f o l l o w s 1 J t h e n t h a t - t h e a c t i v i t y f o r the u n i f o r m k. .(d) f u n c t i o n i s r e l a t i v e l y c o n s t a n t i n the i n t e r i o r of the s m a l l d i s k . Note, however, t h a t i n a l l cases the magnitude of the a c t i v i t y peak i s s t i l l dependent on the r a d i u s of the d i s k . In p a r a l l e l w i t h the above s t u d i e s some work was done 5u 4 3 2 1 0 1 2 3 4 5u DISTANCE FROM CENTER OF DISK F i g . 3.6.3 R e c e p t o r A c t i v i t y F u n c t i o n s a l o n g D i s k Diameters f o r I n v e r s e C o n f i g u r a t i o n 75 LOCUS OF DISK MOVEMENT F i g . 3 ' 6 . 4 ( a ) A Large White D i s k and the 9^9 A r r a y LOCUS OF DISK MOVEMENT F i g . 3.6.4(b) A S m a l l White D i s k and the. 9x9 A r r a y on the r e c e p t o r a c t i v i t y f u n c t i o n s f o r wedge shaped i l l u m i n a t i o n p a t t e r m s . A 100° wedge was p r o c e s s e d u s i n g t h e . t h r e e d i f f e r e n t k..(d) s i m u l a t o r c o n f i g u r a t i o n s . The r e s u l t s , w e r e f e d i n t o an IBM 7044 in''order t o o b t a i n c o n t o u r and i s o m e t r i c p l o t s of the r e c e p t o r a c t i v i t y f u n c t i o n s . The p l o t s f o r the L i m u l u s k . ^ d ) , the u n i f o r m k ^ ( d ) , and the i n v e r s e k^^(d) a re p r e s e n t e d i n F i g s . 3-6.5 to 3-6.7 r e s p e c t i v e l y . The r e c e p t o r a c t i v i t y f u n c t i o n s produced by the t h r e e d i f f e r e n t k — ( d ) c o n f i g u r a t i o n s a r e b a s i c a l l y s i m i l a r . They a l l F i g . 3«6.5(b) An I s o m e t r i c View of the 100 Wedge Rec e p t o r A c t i v i t y F u n c t i o n , L i m ulus C o n f i g u r a t i o n F i g . 3.6.6(b) I s o m e t r i c View of the 100° Wedge Receptor A c t i v i t y F u n c t i o n , U n i f o r m C o n f i g u r a t i o n F i g . 3-6.7(b) I s o m e t r i c View of the 100 Wedge Receptor A c t i v i t y F u n c t i o n , I n v e r s e C o n f i g u r a t i o n e x h i b i t d i f f e r e n t i a l c o n t o u r enhancement, h a v i n g r i d g e s running- p a r a l l e l t o the s i d e s of the wedge, which t h e n mount t o a peak a t the wedge v e r t e x . The a c t u a l shape of the peak i s i n each case q u i t e d i f f e r e n t . I t i s determined by the form of the k ^ ( d ) f u n c t i o n . Por the Li m u l u s k ^ ( d ) , the c l o s e r one approaches the peak, the s t e e p e r the s l o p e of the a c t i v i t y ' f u n c t i o n . T h i s r e f l e c t s the f a c t t h a t f o r a Li m u l u s k^.(d) the nearby r e c e p t o r s a r e the most i n h i b i t o r y . S i m i l a r l y , i n F i g . 3-6.6 i t i s e v i d e n t t h a t the s l o p e of the a c t i v i t y f u n c t i o n i s c o n s t a n t a l o n g any g i v e n l i n e i n the i l l u m i n a t e d r e g i o n near the peak. T h i s r e f l e c t s the c o n s t a n t n a t u r e of the u n i f o r m k^.(d) f u n c t i o n . F i n a l l y , i n F i g . 3-6.7 the form of the i n v e r s e k ^ ( d ) f u n c t i o n i s r e f l e c t e d i n the shape of the r e c e p t o r a c t i v i t y f u n c t i o n . The s l o p e of the f u n c t i o n i n the i l l u m i n a t e d r e g i o n near the peak i n c r e a s e s as one moves away from the peak. T h i s i s because the more d i s t a n t r e c e p t o r s have a g r e a t e r i n h i b i t o r y e f f e c t . I t s h o u l d be obvious from the above d i s c u s s i o n of r e c e p  t o r a c t i v i t y f u n c t i o n s t h a t one can make q u a l i t a t i v e p r e d i c t i o n s about t h e i r shape g i v e n the i l l u m i n a t i o n p a t t e r n and the k ^ ( d ) f u n c t i o n . 3-7 "Feed-Forward" I n h i b i t o r y I n t e r a c t i o n I n t h i s s e c t i o n an a l t e r n a t i v e scheme f o r co n t o u r enhance ment i s t r e a t e d i n a p r e l i m i n a r y f a s h i o n . T h i s method i s o b t a i n e d by a m o d i f i c a t i o n of the H a r t l i n e e q u a t i o n s ( e q u a t i o n s 1.4-1) i n the f o l l o w i n g f a s h i o n : n y. = e. - Y ~ k. . max(0, e . - t . .) i = l , . . . , n . 3-7-1 80 U X site of pulse generati on Feed Back Interaction Feed Forward Interaction F i g . 3.7.1 "Feed-Back" and "Feed-Forward" R e c e p t o r I n t e r a c t i o n The v a l u e o f the r e c e p t o r s t i m u l u s , e., has been s u b s t i t u t e d f o r the r e c e p t o r a c t i v i t y x., i n the r i g h t - h a n d s i d e of the e q u a t i o n s . The i n h i b i t o r y i n t e r a c t i o n g i v i n g r i s e t o the new r e c e p t o r a c t i v i t y , y^, i s e l i c i t e d by a " f e e d - f o r w a r d " mechanism as opposed t o the feed-back mechanism o f the H a r t l i n e e q u a t i o n s . The d i f f e r e n c e i s shown s c h e m a t i c a l l y i n F i g . 3.7.1. . (The s e t of e q u a t i o n s 3.7.1 (27^ i s v e r y s i m i l a r t o a r e t i n a l n e u r a l network proposed by F r y , (21) and i s a f i r s t a p p r o x i m a t i o n t o the H a r t l i n e e q u a t i o n s .) I n i t i a l l y , our main i n t e r e s t was t o see whether the o r i e n t a t i o n dependence of the r e c e p t o r a c t i v i t y determined by t h i s s e t o f e q u a t i o n s was markedly d i f f e r e n t from t h a t determined by the H a r t l i n e e q u a t i o n s . A f t e r m o d i f i c a t i o n of the s i m u l a t o r so t h a t the s e t o f equat i o n s 3.7.1 c o u l d be s o l v e d , the minimum a m b i g u i t y c o n f i g u r a  t i o n was s e t up. T h i s was the u n i f o r m k^.(d) f u n c t i o n w i t h D - 1.5u and z e r o a c t i v i t y f o r u n i f o r m i l l u m i n a t i o n . Two o r i e n t a  t i o n dependence experiments were done, one u s i n g t h e s e t of wedges, 0*5 IV) H j CD O P i 4 0 e_i. t) ra I bd P o pb (D II CD CD I £ o s: P P p, p. D Hd II CD P O PT o crt- H - <! H - ct- 13 <J CD H O t i CD ra o s: CD p . 0*} 0 ra o C3 s C) CO O to O H O o PEAK RECEPTOR ACTIVITY (p.p.s.) CD O CD Q3 CD I a . j CD CD CD CO r\> — CQ q> CD CD Q Q Q Q CD C> CD O o o o o to CD o 18 -r 240 to a 5s. WHITE ON BLACK DISKS BLACK ON WHITE DISKS FEED- FORWARD FEED-BACK 1,0 u-i .75 .50 —i 25 0 .25 INVERSE RADIUS, 1 R .50 .75 1,0 u-i F i g . 3.7.3 "Feed-Back" and "Feed-Forward" Peak A c t i v i t y E n v e l o p e s f o r D i s k s ; U n i f o r m k ^ d ) , D = 1.5u,. and x Q = 0 p.p.s. co 85 the o t h e r the s e t of d i s k s . The r e s u l t s are g i v e n i n F i g s . 3.7.2 .and 3-7.3 r e s p e c t i v e l y . F o r purposes of comparison the e q u i v a l e n t r e s u l t s f o r the l a t e r a l i n h i b i t i o n case ( H a r t l i n e e q u a t i o n s ) are p l o t t e d as d o t t e d l i n e s a l o n g w i t h the above. The most s t r i k i n g f e a t u r e about these two s e t s of r e s u l t s i s t h e i r s i m i l a r i t y . Even the a m b i g u i t y measures noted on F i g . 3-7.2 are w i t h i n one or two degrees of each o t h e r . T h i s s i m i l a r i t y caused us t o c a r r y out a more d e t a i l e d comparison between the l a t e r a l i n h i b i t i o n e q u a t i o n s and the " f e e d - f o r w a r d " s e t of e q u a t i o n s . The o b j e c t was t o d i s c o v e r the u n d e r l y i n g common mechanism, i f any. These i n v e s t i g a t i o n s are d e a l t w i t h i n the n e x t c h a p t e r . : 4. LATERAL INHIBITION AND THE AREA OPERATOR 4.1 I n t r o d u c t i o n • I n c h a p t e r 3 we p r e s e n t e d the r e s u l t s from a l a r g e number of experiments on a v a r i e t y of l a t e r a l i n h i b i t o r y networks. The v a r i a b l e s were the form of the k. .(d) f u n c t i o n , the s i z e of the 1 J r e c e p t i v e f i e l d of view, the t h r e s h o l d l e v e l , and the geo m e t r i c arrangement of the r e c e p t o r s . I n a d d i t i o n , the r e s u l t s from two experiments w i t h a f e e d - f o r w a r d r e c e p t o r network were p r e s e n t e d . " f e e d - f o r w a r d " network i s deve l o p e d . The r e s u l t s t h a t would be o b t a i n e d i n p r o c e s s i n g the t e s t p a t t e r n s u s i n g t h i s model are compared w i t h the r e s u l t s o b t a i n e d u s i n g the l a t e r a l i n h i b i t o r y network and the " f e e d - f o r w a r d " network. Some c o n c l u s i o n s about the common mechanism operant i n thes e two networks a re g i v e n . 4.2 The D i s k A r e a Operator • The model i s a- s i m p l e weighted a r e a o p e r a t o r , c o n s i s t i n g of a d i s k of u n i t r a d i u s , and a measurement p o i n t a t the c e n t e r of the d i s k . The w e i g h t i n g f u n c t i o n i s a r a d i a l l y dependent f u n c  t i o n , k ( r ) , which s a t i s f i e s the c o n d i t i o n I n t h i s c h a p t e r a c o n t i n u o u s model f o r the d i s c r e t e 1 4.2.1 0 and hence 1 4.2.2 D i s k 0 0 the a r e a of a u n i t r a d i u s d i s k . 85 The o p e r a t o r p r o c e s s e s w h i t e and b l a c k p a t t e r n s as f o l l o w s . I f the c e n t e r of the d i s k i s i n a b l a c k a r e a , the o p e r a t o r has a v a l u e of z e r o . I f the c e n t e r of the d i s k i s i n a w h i t e a r e a , the o p e r a t o r has a v a l u e of the a r e a of the d i s k , %, l e s s the i n t e g r a l of the w e i g h t i n g f u n c t i o n over the a r e a of the d i s k t h a t i s w h i t e . L e t us c a l l the v a l u e of the o p e r a t o r V , and the v a l u e of the c e n t e r of the d i s k V . V has the op c op l i m i t s 0 £ V < it, and V has the v a l u e of z e r o i n a b l a c k a r e a , op ' c ' and one i n a w h i t e a r e a . M a t h e m a t i c a l l y , the o p e r a t o r has the f o r m I A V Q p = YG\n - J k ( r ) d A \ 4.2.3 w where A i s the a r e a of t h e ' d i s k t h a t i s w h i t e . I t has the same w l i m i t s as V op I t i s obvious t h a t when the d i s k i s t o t a l l y w i t h i n a w h i t e or b l a c k a r e a V i s z e r o . Only when the d i s k i s i n t e r  s e c t i n g a w h i t e - b l a c k boundary w i t h the c e n t e r of the d i s k i n the w h i t e a r e a w i l l V be non-zero. I f the c e n t e r of the d i s k i s i n the w h i t e a r e a , and i s i m m e d i a t e l y a d j a c e n t t o the boundary, the a r e a of the d i s k t h a t i s w h i t e , A , i s a minimum, and hence V V Q p i s a maximum. Under the s e c o n d i t i o n s , e q u a t i o n 4-2.3 becomes ( Vop)max = * - • / k ^ d A ' . 4.2.4. A wm Note t h a t the peak i n r e c e p t o r a c t i v i t y f o r b o t h a l a t e r a l i n h i b i t o r y or a " f e e d - f o r w a r d " network o c c u r s when the r e c e p t o r i s i m m e d i a t e l y a d j a c e n t t o the w h i t e - b l a c k boundary. Thus the (V ) and the ^ op max peak r e c e p t o r a c t i v i t y c o n d i t i o n s are e q u i v a l e n t . BLACK WHITE Awm OPERATOR DISK F i g . 4.3-1 The Operator D i s k and a Wedge 4 - 3 Peak Responses t o Wedges L e t us f i r s t d e v e l o p the (V ) e q u a t i o n f o r a wedge r op max ^ to of a n g l e a . We can then compare the (V ) v e r s u s wedge a n g l e D * op max & & curve t o t h a t o b t a i n e d u s i n g the d i s c r e t e r e c e p t o r networks. I n F i g . 4-3-1 the o p e r a t o r d i s k i s shown i n the (V ) op'max p o s i t i o n w i t h r e s p e c t t o a wedge of angl e a . We have k ( r ) d A 0 -a/2 k ( r ) r d r d 9 = | 4-3.1 m and hence f o r a wedge of angl e a (V ) op max a. 2 4.3-2 f o r a l l k ( r ) t h a t s a t i s f y e q u a t i o n 4.2.1. G r a p h i c a l l y . (V ) r J ' op max v e r s u s wedge a n g l e i s as shown i n F i g . 4-3.2. The v a r i a t i o n i n peak response f o r the o p e r a t o r can now be compared t o t h a t of the l a t e r a l i n h i b i t o r y network over the (VoP)max A F i g . 4.3.2 ( V J as a F u n c t i o n of the Wedge Angle CJ JL/ I I lcl-A. range of wedge a n g l e s . I n o r d e r t o check t h a t the o p e r a t o r i s the c o n t i n u o u s a n a l o g o f the d i s c r e t e " f e e d - f o r w a r d " network the r e s u l t s of experiments c a r r i e d out on t h i s t y p e of network w i l l a l s o be i n c l u d e d . F o r b o t h the l a t e r a l i n h i b i t o r y and the " f e e d  f o r w a r d " cases, the wedges had an o r i e n t a t i o n of 15° w i t h r e s p e c t t o the a r r a y . T h i s o r i e n t a t i o n was chosen i n o r d e r t o a v o i d h a v i n g t h e a x i s o f symmetry of the wedge c o i n c i d e w i t h one o f the axes of symmetry of the 9x9.array of r e c e p t o r s . The f i r s t graph i s shown i n F i g , 4.3.3. These e x p e r i  mental r e s u l t s were o b t a i n e d u s i n g the Limulus k..(d) f u n c t i o n and p o i n t r e c e p t o r s . The c r o s s e s i n d i c a t e the e x p e r i m e n t a l p o i n t s o b t a i n e d u s i n g t h e l a t e r a l i n h i b i t o r y network. The p o i n t s i n d i c a t e the r e s u l t s u s i n g the " f e e d - f o r w a r d " network. The s t e p - l i k e b e h a v i o u r of the e x p e r i m e n t a l f u n c t i o n s i s due t o the d i s c r e t e n a t u r e of the r e c e p t o r a r r a y . The s o l i d l i n e i s the s h i f t e d and s c a l e d (V ) l i n e of F i g . 4.3.2. The s h i f t i n g i s n e c e s s a r y s i n c e under u n i f o r m i l l u m i n a t i o n , r e c e p t o r s w i t h the Limulus k^^.(d) f u n c t i o n have non-zero a c t i v i t y . Note t h a t the ( v r i r , ) r T 1 o Y l i n e i s i n good agreement 00 CO w i t h the " f e e d - f o r w a r d " e x p e r i m e n t a l r e s u l t s . The dashed curve forms an a p p r o x i m a t i o n t o the l a t e r a l i n h i b i t i o n e x p e r i m e n t a l p o i n t s . Note the l o n g l i n e a r r e g i o n i n t h i s response curve s t r e t c h i n g from A = 0.14ft t o A = 0.8ft. * to w w m m T h i s l i n e a r i t y i n d i c a t e s t h a t w i t h i n t h i s r e g i o n the change i n peak response of the l a t e r a l i n h i b i t i o n network i s d i r e c t l y p r o p o r  t i o n a l t o the change i n wedge a n g l e , and hence t o the change i n the a r e a of i l l u m i n a t i o n of the r e c e p t i v e f i e l d . The s l o p e of t h i s l i n e a r p o r t i o n i s -61.6 p . p . s . / u n i t a r e a . The s l o p e of the (V ) l i n e i s -68.5 p . p . s . / u n i t a r e a , op max ' The second s e t of e x p e r i m e n t a l r e s u l t s a re g i v e n i n P i g . 4.3.4'. Here the u n i f o r m k ^ ( d ) f u n c t i o n and p o i n t r e c e p t o r s were used. The (V ) l i n e i s i n good agreement w i t h the op max a ° " f e e d - f o r w a r d " e x p e r i m e n t a l r e s u l t s . The l a t e r a l i n h i b i t i o n r e s  ponse i s a g a i n l i n e a r between A = 0.14ft and A = 0.80ft. The * & W W . m m l i n e a r p o r t i o n has a s l o p e of -75.0 p . p . s / u n i t a r e a , w h i l e the (V ^J™,^ l i n e has a s l o p e of -77.0 p . p . s . / u n i t a r e a , op max Por the l a s t s e t of experiments the i n v e r s e k..(d) f u n c - t i o n w i t h p o i n t r e c e p t o r s was used. The graphs are g i v e n i n P i g . 4.3-5. As b e f o r e the s o l i d (V ) l i n e forms a good a p p r o x i -0 op max b r * mation to the " f e e d - f o r w a r d " e x p e r i m e n t a l p o i n t s . The dashed curve i s l i n e a r between A = 0.1ft and A = 0.86ft. The s l o p e i s w w * • m m -77-0 p.p.s./degree which i s the same as t h a t of the (V ) l i n e . r / o op max Prom t h i s s e t of experiments we can deduce two t h i n g s . F i r s t , the (V ) m r . v o p e r a t o r i s ind e e d the c o n t i n u o u s a n a l o g of op max the d i s c r e t e " f e e d - f o r w a r d " network. Second, the l a t e r a l i n h i b i t o r y network behaves l i k e an a r e a o p e r a t o r under c e r t a i n c o n d i t i o n s . The c o n d i t i o n s a r e as f o l l o w s : L e t M be the r a t i o of the weighted F i g . 4.3.4 U n i f o r m k..(d) Response Curve and (V ) L i n e f o r Wedges l . l on'max « i c u . g c o 250-i vo H 92 a r e a of i l l u m i n a t i o n , of the 9x9 a r r a y t o the t o t a l weighted a r e a . Then f o r M between the l i m i t s 0.14 5 M < 0.80.,the l a t e r a l i n h i b i t o r y network f u n c t i o n s i n the same way as the a r e a o p e r a t o r . That i s , the peak response i s a l i n e a r f u n c t i o n of the weighted a r e a o f i l l u m i n a t i o n o f the r e c e p t i v e f i e l d . T h i s second d e d u c t i o n i s t e n t a t i v e s i n c e i t i s based o n l y on a study of the peak response to wedges. I n o r d e r t o t e s t i t f u r t h e r some a d d i t i o n a l experiments are c a r r i e d out u s i n g d i s k s as t e s t p a t t e r n s . 4.4 Peak Responses t o D i s k s I n d e a l i n g w i t h the wedges i t was found t h a t an a n a l y t i c e x p r e s s i o n c o u l d be o b t a i n e d g i v i n g (V ) as a f u n c t i o n o f the op max a n g l e a. T h i s e x p r e s s i o n was independent of k ( r ) , the r a d i a l w e i g h t i n g f u n c t i o n . F o r d i s k s the e q u i v a l e n t e x p r e s s i o n would g i v e ("V „) as a f u n c t i o n of the r a d i u s o f the d i s k . However, op max. o n l y i f k ( r ) i s a c o n s t a n t can. one f i n d such an e x p r e s s i o n . I f k ( r ) i s a c o n s t a n t , i n o r d e r t o s a t i s f y e q u a t i o n 4.2.1 one must have k ( r ) - 1. 4.4.1 Hence, e q u a t i o n 4.2.4 can be r e w r i t t e n as (V ). = TC - A . 4.4.2 v op max w - f t . * m I n o r d e r t o o b t a i n ( ^ 0 p ) m a x a s a f u n c t i o n of the d i s k r a d i u s , we r e q u i r e an e x p r e s s i o n f o r A w m I n F i g . 4.4.1 a w h i t e d i s k of r a d i u s r on a b l a c k back ground i s shown a l o n g w i t h the o p e r a t o r d i s k i n the (V ) ° -f ^ op max p o s i t i o n . The w h i t e a r e a , A , o f the o p e r a t o r d i s k i s g i v e n m 93 PATTERN DISK OPERATOR DISK F i g . 4 - 4 . 1 P a t t e r n D i s k w i t h the Operator D i s k i n the (V ) P o s i t i o n op max 1 A = 0 + Q r 2 - ( r 2 - \)2 4 . 4 . I wm 4 where (fi , 0 and r are as i n d i c a t e d i n F i g . 4 . 4 . 1 . I t i s e v i d e n t t h a t 0 = at - 2 0 4 . 4 . 2 and by the law o f c o s i n e s cos 0 = | ^ 4 . 4 . 3 and hence 0 = c o s " 1 ^ ) = f - s i r f 1 ^ ) 4 . 4 . 4 Thus, by s u b s t i t u t i o n of. e q u a t i o n s 4 - 4 . 2 and 4 - 4 . 4 i n t o e q u a t i o n 4 . 4 . 1 we o b t a i n A = f + ( 2 r 2 - l ) s i r T 1 ^ ) - ( r 2 - \ ) 2 4 . 4 - 5 m 94 f o r a l l r ^  • I n the case where r < i t can be e a s i l y shown t h a t p A = i t r . 4.4.6 w m I n a d d i t i o n , f o r a b l a c k d i s k on a w h i t e background i t can be shown t h a t f o r r ^ ^, 1 A, = f - ( 2 r 2 - l ) s i n " 1 ^ ) + ( r 2 - \ ) 2 . 4-4-7 and f o r , r < ^ A 2 4-4-8 w = it - i t r . m We can now w r i t e the e x p r e s s i o n s f o r o b t a i n i n g (V ) * . fa op max f o r b o t h w h i t e on b l a c k and b l a c k on w h i t e d i s k s of any r a d i u s . F o r w h i t e on b l a c k d i s k s , . (V ) op max 2 ( 2 r 2 - l ) s i n - 1 ( | r ) - ( r 2 - | ) 2 j i f r >e \ i t ( l - r 2 ) ' i f r < |. 4.4.9 4.4.10 Fo r b l a c k on w h i t e d i s k s , V i t r 2 • i f r < I The v a l u e s o b t a i n e d from these t h e o r e t i c a l e x p r e s s i o n s can now be compared w i t h some e x p e r i m e n t a l r e s u l t s o b t a i n e d u s i n g the u n i f o r m k ^ ( d ) f u n c t i o n and p o i n t r e c e p t o r s . The t e s t p a t t e r n s were b l a c k on w h i t e d i s k s , w h i t e on b l a c k d i s k s , and a s t r a i g h t edge. I n Fig.. 4.4-2 the peak responses t o these p a t t e r n s are g i v e n as a f u n c t i o n of the a r e a of i n t e r s e c t i o n , A , between the w m w h i t e p a r t of the p a t t e r n and the o p e r a t o r d i s k . The e x p e r i m e n t a l 250 n 200A ISOA 1004 50 A max LATERAL INHIBITION ,3TT AREA OTC .In .2JT ,4TC .5 TV ,6n >8n .9TC OF INTERSECTION. Aw, m F i g . 4.4.2 U n i f o r m k. .(d) Response' Curve and (V ) L i n e f o r D i s k s i j op insix r e s u l t s u s i n g the " f e e d - f o r w a r d " network are p l o t t e d as p o i n t s ; the r e s u l t s u s i n g the l a t e r a l i n h i b i t o r y network, as c r o s s e s . The (V ) c u r v e , which f o r u n i f o r m w e i g h t i n g i s l i n e a r w i t h v op max ' . to & r e s p e c t t o A , i s the s o l i d l i n e . I t forms a good a p p r o x i m a t i o n wm t o the " f e e d - f o r w a r d " e x p e r i m e n t a l r e s u l t s . The dashed curve approximates the e x p e r i m e n t a l r e s u l t s o b t a i n e d from the l a t e r a l i n h i b i t o r y network. I t has a l o n g l i n e a r r e g i o n s t r e t c h i n g from A - 0.14^ t o A = 0.8ft. The s l o p e of t h i s p o r t i o n i s - 77.8 p.p. wm wm u n i t a r e a , whereas the s l o p e of the (V ) l i n e i s -77.0 p.p.s./ ' ^ op max * * ' u n i t a r e a . I n . o t h e r words, between the l i m i t s 0.14 £ M <0.80 the l a t e r a l i n h i b i t o r y network behaves i n the same way as the o p e r a t o r . Namely, the v a r i a t i o n i n the peak response i s d i r e c t l y p r o p o r t i o n a l t o the change i n the weighted area, of i n t e r s e c t i o n . I n order 1 t o demonstrate t h a t t h i s h o l d s t r u e f o r non u n i f o r m w e i g h t i n g f u n c t i o n s some f u r t h e r experiments w i t h the d i s k s were c a r r i e d out. I n F i g . 4.4-3 the response curves f o r the L i m u l u s k..(d) f u n c t i o n are g i v e n . Note t h a t these r e s u l t s are p l o t t e d a g a i n s t the a r e a of i n t e r s e c t i o n , not the weighted a r e a of i n t e r s e c t i o n . F o r t h i s r e a s o n the c u r v e s a p p r o x i m a t i n g the e x p e r i m e n t a l r e s u l t s are n o n - l i n e a r . The dashed s t r a i g h t l i n e i s the ( V 0 p ) m a x l i n e t h a t would be o b t a i n e d i f the a r e a w e i g h t i n g f u n c t i o n were c o n s t a n t . I t s e r v e s as a r e f e r e n c e f o r the " f e e d  f o r w a r d " response c u r v e , shown as a s o l i d l i n e . S i n c e the L i m u l u s k ^ ( d ) f u n c t i o n w e i g h t s the c e n t e r of the r e c e p t i v e f i e l d more h e a v i l y than the p e r i p h e r y , the p o r t i o n of the " f e e d - f o r w a r d " curve c o r r e s p o n d i n g t o the w h i t e d i s k s (0 <A < 0.5ft) l i e s wm below t h i s r e f e r e n c e l i n e . The curve c r o s s e s the l i n e a t the F i g . 4.4.3 L i m u l u s ^ ( d ) Response Curves f o r D i s k s as a F u n c t i o n o f " t h e A r e a of I n t e r s e c t i o n 98 s t r a i g h t edge response p o i n t and then l i e s above i t f o r the b l a c k d i s k response p o i n t s ( 0 . 5 f t * A $ ft). T h i s i s as expected. Note t h a t the l a t e r a l i n h i b i t i o n response c u r v e . ( t h e s i n g l y dashed curve) has the.same form as the " f e e d - f o r w a r d " c u r v e . I n P i g . 4-4-4 these r e s u l t s are r e p l o t t e d as a f u n c t i o n of the weighted a r e a of i n t e r s e c t i o n , . The. s o l i d l i n e i n d i c a t e s m the (V ) response f u n c t i o n . The " f e e d - f o r w a r d " e x p e r i m e n t a l v op max * * p o i n t s are i n r e a s o n a b l e agreement w i t h t h i s l i n e . The dashed c u r v e , on the o t h e r hand, p r o v i d e s a good a p p r o x i m a t i o n t o the l a t e r a l i n h i b i t i o n p o i n t s . As i n the case of the wedges, shown i n P i g . 4-3-3j t h i s response curve has a l o n g l i n e a r p o r t i o n between 0.14ft^W^ < 0.8%. The s l o p e of t h i s p o r t i o n i s -65-8 p.p.s./ u n i t m a r e a . T h i s compares w i t h a s l o p e of -61.6 p . p . s . / u n i t a r e a f o r t h e l i n e a r p o r t i o n of the wedge response c u r v e . The s l o p e of the (V ) l i n e i s -68.5 p.p.s. x op max The t h i r d s e t of experiments w i t h the d i s k p a t t e r n s was done w i t h an i n v e r s e k. .(d) f u n c t i o n and p o i n t r e c e p t o r s . The r e s u l t s are p l o t t e d i n P i g . 4-4-5 as a f u n c t i o n of the a r e a of i n t e r s e c t i o n . Note t h a t the e x c u r s i o n s of the " f e e d - f o r w a r d " response curve ( s o l i d l i n e ) w i t h r e s p e c t to the r e f e r e n c e l i n e (dashed s t r a i g h t l i n e ) are the o p p o s i t e of those i n F i g . 4.4.3. T h i s i s because the i n v e r s e ^ ^ ( d ) f u n c t i o n w e i g h t s the p e r i p h e r y of the r e c e p t i v e f i e l d more h e a v i l y than the c e n t e r . Note a l s o t h a t the l a t e r a l i n h i b i t i o n response curve (dashed curve) has b a s i c a l l y the same form as t h e ' f e e d - f o r w a r d " c u r v e . I n F i g . 4-4-6 the e x p e r i m e n t a l r e s u l t s are r e p l o t t e d as a f u n c t i o n of the weighted a r e a of i n t e r s e c t i o n . The(V ) 0 op max •2S0-I 0 1 1 1 r 1 1 1 1 1 1 1 On .In ,2n .3 n .4n ,5n ,6n ,7n .8 n ,9n In WEIGHTED AREA OF INTERSECTION, WAm F i g . 4.4.4 Lim u l u s k. .(d) Response Curves f o r D i s k s as a F u n c t i o n of the Weighted Are a of I n t e r s e c t i o n 250 F i g . 4.4.5 I n v e r s e k. .(d) Response Curves f o r D i s k s as a F u n c t i o n of the A r e a of I n t e r s e c t i o n 250-i F i g . 4.4.6 I n v e r s e k. .(d) Response Curves f o r D i s k s as a F u n c t i o n o f the Weighted J Area of I n t e r s e c t i o n o and the l a t e r a l i n h i b i t i o n response f u n c t i o n s are as i n d i c a t e d . The " f e e d - f o r w a r d " e x p e r i m e n t a l p o i n t s are i n good agreement w i t h the (V ) l i n e . Note t h a t the l a t e r a l i n h i b i t i o n response v op max r curve has the u s u a l l i n e a r r e g i o n between 0.1JC<W^ ^ 0.9ft. The m s l o p e of t h i s p o r t i o n i s -76.7 p . p . s . / u n i t a r e a which compares w i t h a s l o p e of -77.0 p . p . s . / u n i t a r e a f o r the wedge response curve shown i n F i g . 4-3-5- The s l o p e of the (V ) l i n e i s to * K op'max -77.0 p . p . s . / u n i t a r e a . The e x p e r i m e n t a l r e s u l t s f o r non-uniform w e i g h t i n g f u n c t i o n s are i n agreement w i t h those o b t a i n e d u s i n g a u n i f o r m w e i g h t i n g f u n c t i o n . That i s , f o r b o t h d i s k s and wedges.the l a t e r a l i n h i b i t o r y network behaves i n the same way as the a r e a o p e r a t o r p r o v i d e d the weighted a r e a r a t i o , M, i s between the l i m i t 0.14 $0.8. I n o t h e r words, i n t h i s range the v a r i a t i o n i n the peak response from a l a t e r a l i n h i b i t o r y network i s d i r e c t l y p r o  p o r t i o n a l t o the change i n the weighted a r e a of i l l u m i n a t i o n of the r e c e p t i v e f i e l d , t h e ' w e i g h t i n g b e i n g determined by the k ^ ( d ) f u n c t i o n . 103 5. THE AREA OPERATOR AND CURVATURE 5.1 I n t r o d u c t i o n I n view of the c o n c l u s i o n t h a t the l a t e r a l i n h i b i t o r y network f u n c t i o n s as an a r e a o p e r a t o r , we propose t o examine such o p e r a t o r s i n more d e t a i l . F i r s t , we de v e l o p a more g e n e r a l f o r m u l a t i o n f o r a r e a o p e r a t o r s t h a n t h a t g i v e n i n s e c t i o n 4.2. We g i v e a s h o r t s e c t i o n on the t h e o r y of cur v e s i n a p l a n e , and t h e n go on t o de v e l o p some of the t h e o r y a s s o c i a t e d w i t h a p a r t i c u  l a r o p e r a t o r . I n t h i s development we t r y t o show the r e l a t i o n s h i p between the a r e a o p e r a t o r response and the c u r v a t u r e of- i l l u m i n a  t i o n b o u n d a r i e s . Examples of the use of the e i g h t theorems developed a r e g i v e n i n s e c t i o n 7.2. 5.2 G e n e r a l Form f o r the Are a Operator I n d e v e l o p i n g the f o l l o w i n g f o r m u l a t i o n f o r the a r e a o p e r a t o r we were guided by two o b j e c t i v e s . F i r s t , we wished to o b t a i n a f o r m u l a t i o n t h a t would cover n o n - c i r c u l a r o p e r a t o r s h a v i n g n o n - p o i n t c e n t e r s . Second, and more d i f f i c u l t t o a c h i e v e , we wanted an o p e r a t o r t h a t would be independent of the i n t e n s i t y of i l l u m i n a t i o n , and y e t be capable of h a n d l i n g grey l e v e l s . These o b j e c t i v e s a r e r o o t e d i n the body of m a t e r i a l p r e s e n t e d i n c h a p t e r 1. F o r example, the work of Hubel and W i e s e l on the v i s u a l a r e a s of the c a t s t r i a t e c o r t e x demonstrated t h a t the c i r c u l a r r e c e p t i v e f i e l d s ( c e n t r a l d i s k and a n n u l a r surround) of the g a n g l i o n and l a t e r a l g e n i c u l a t e had g i v e n way t o more complex, l i n e a r l y c o n f i g u r e d r e c e p t i v e f i e l d s . The a r e a o p e r a t o r f o r m u l a t i o n s h o u l d p e r m i t t h i s type of geometry. I n a d d i t i o n , s i n c e the n e u r o p h y s i o l o g i c a l d a t a 104 F i g . 5.2.1 G e n e r a l Geometry of the Are a Operator i n d i c a t e s t h a t the response of neurons t o i l l u m i n a t i o n p a t t e r n s i s dependent on l o c a l d i f f e r e n c e s r a t h e r than a b s o l u t e i n t e n s i t i e s , the a r e a o p e r a t o r s h o u l d have the same p r o p e r t y . I n F i g . 5.2.1 a g e n e r a l r e p r e s e n t a t i o n of an a r e a o p e r a t o r i s g i v e n c o n s i s t i n g of a c e n t r a l r e g i o n , A , and a s u r r o u n d i n g a r e a , A . A s s o c i a t e d w i t h t h i s a r e a o p e r a t o r i s a c o o r d i n a t e system (x',y') and a w e i g h t i n g f u n c t i o n k ( x ' , y ' ) such t h a t k ( x ' ,y 1 )dA = i t . 5.2.1 A <J A s c (The w e i g h t i n g f u n c t i o n i s made t o operat e over both A and A s c i n l i n e w i t h the f i n d i n g s of Rodieck • > Assume t h a t t h e r e i s some i l l u m i n a t i o n f u n c t i o n , l ( x , y ) , w h ich i s t o be pr o c e s s e d by the a r e a o p e r a t o r . L e t the o r i g i n i n the x'y' r e f e r e n c e frame be a t ( x Q , y ) i n the xy frame. Then the v a l u e of the a r e a o p e r a t o r a t (x ,y ) i s d e f i n e d t o be V, (x ,y ) = JC 1 o J o S . C k(x',y») R 1 ( x , y ) dA 5.2.2 105 where R-^Cx.y) = max(0, I(x,y) - I~(x,y), 5.2.3 I(x,y) - lTx,y7 iT^ yJ = j~ S I(x,y)dA 5.2.4 c A c (x',y') = (x-xQ, y-yQ), 5.2.5 under the assumption t h a t l i m R-^x.y)• = 1. 5-2.6 I -^1 I f the p o i n t ( x 0 , y o ) ranges over a l l ( x , y ) , one o b t a i n s the f u n c t i o n V-^(x,y). Note t h a t the r a t i o R-^(x,y) can o n l y take on the v a l u e s zero or one. T h i s means t h a t V-^(x,y) i s independent of the a b s o l u t e magnitude of I ( x , y ) , and depends o n l y on the r e l a t i v e magnitudes of l(x,y) and lTx,y"7. I n o r d e r t o demonstrate the p r o p e r t i e s of the a r e a o p e r a t o r assume t h a t i t has the d i s k geometry used irj the a r e a o p e r a t o r of c h a p t e r 4. That i s , A^ i s a p o i n t a t the c e n t e r of a u n i t r a d i u s d i s k , the remainder of the d i s k b e i n g A^. A l s o , assume t h a t k ( x ' , y ' ) i s dependent o n l y on the d i s t a n c e from A . I t i s e a s i l y shown t h a t i n t h i s case V^(x,y) responds t o b l a c k - w h i t e p a t t e r n s i n the same way as the o p e r a t o r , V Qp> d e f i n e d i n s e c t i o n 4.2. I f A c i s i n a bla.ck r e g i o n a t the p o i n t (x-^,y^), then T(x2>y-]_) - 0, and hence I(x-^,y-^) = 0. A l l the p o i n t s w i t h i n A g 106 must have i n t e n s i t i e s t h a t are g r e a t e r than or e q u a l t o TTx^, y-jT- But because of e q u a t i o n 5-2.6 t h i s means t h a t R-^ (x,y) must e q u a l one throughout A . Hence by e q u a t i o n 5-2.1, V ^ x ^ , y^) = 0. S i m i l a r l y , i f A i s a t (x^, y 2 ) a n (^ ^ e d i s k i s t o t a l l y w i t h i n a w h i t e a r e a , V - ^ X g , ^2^ = 0- The v a l u e of V a t (x-^, y^) and (x2, i s a l s o z e r o . Only i f the p o i n t A i s i n a w h i t e r e g i o n w h i l e p a r t of c A i s i n a b l a c k . r e g i o n , w i l l V-, (x,y) be non-zero. Under these c o n d i t i o n s we can c o n s i d e r A t o be composed of two r e g i o n s , a w h i t e r e g i o n , A , and a b l a c k r e g i o n , A . W i t h i n A , R, (x,y) = 1, w b w whereas w i t h i n A , R (x,y) = 0 . Hence s b 1 Vn(x,y) = at - y k(x»,y') R,(x,y) dA x Au A ± s c = Tt - f k ( x ' , y ' ) dA - y k(x»,y') • 0 dA A u A A w b V 1 ( x , y ) = % - y k ( x ' , y ' ) dA 5-2.7 A u A s c w But e q u a t i o n 5-2.7 and e q u a t i o n 4-2-3 are i d e n t i c a l i n t h i s s i t u a t i o n . Hence V^(x,y) p r o c e s s e s b l a c k - w h i t e p a t t e r n s i n e x a c t l y the same way as V J op However, V-^(x,y) i s .a much more g e n e r a l o p e r a t o r than V i n t h a t i t can'be used t o p r o c e s s n o n - b i n a r y l ( x , y ) f u n c t i o n s . To i l l u s t r a t e t h i s assume t h a t V^(x,y) has the d i s k geometry and t h a t k ( x ' , y ' ) = 1. L e t l ( x , y ) c o n s i s t of t h r e e r e g i o n s , M-^ , M 2 and M^, as shown i n F i g . 5-2.2. The i n t e n s i t y l e v e l s i n these r e g i o n s are r e l a t e d as 0 £ i ^ < r e s p e c t i v e l y , w i t h the h i g h e r i n t e n s i t y 107 F i g . 5-2.2 A T h r e e - L e v e l l ( x , y ) F u n c t i o n l e v e l a p p l y i n g a t any p o i n t o f c o n t a c t between r e g i o n s . I f t h i s l ( x , y ) f u n c t i o n i s p r o c e s s e d by the a r e a o p e r a t o r , then V-^(x,y) w i l l be i d e n t i c a l l y z e r o w i t h i n M^. I t w i l l be non z e r o w i t h i n p r o v i d e d the d i s t a n c e from the c e n t e r of the opera t o r t o the l i n e s e p a r a t i n g and i s l e s s than one. I t w i l l be non-zero i n w i t h i n u n i t d i s t a n c e of e i t h e r of the l i n e s s e p a r a t i n g f r o m the o t h e r r e g i o n s . V-^(x,y) w i l l be e q u a l t o i t / 2 a l o n g any of the t h r e e d i v i d i n g l i n e s p r o v i d e d the d i s t a n c e to the j u n c t i o n of the t h r e e l i n e s .is g r e a t e r than one. A l o n g the t h r e e l o c i marked AA', V^(x,y) has the form shown i n F i g . 5.2.3(a). A l o n g the l o c i marked BB", CC' and DD' i t w i l l have the form shown i n F i g s . 5.2.3(b), (c) and (d) r e s  p e c t i v e l y . Note t h a t i n a l l cases V 1 ( x , y ) i s a maximum a t the bo u n d a r i e s between r e g i o n s . The v a l u e of t h i s maximum i s dependent 108 TC -_ v, (*./) TC 2 A -\ I -2 -1 0 1 2 F i g . 5.2.3 ( a ) O p e r a t o r R e s p o n s e a l o n g ' A A ' TC -. V, (x,y) TC 2 / 1 B f - P i g . - 1 0 1 2 5.2.3 ( b ) O p e r a t o r R e s p o n s e a l o n g B B 1 TC -_ V, (x,y) 7t ' 2 \ c 1 1 c 1 -2 1 1 1 2 5.2.3 ( c ) O p e r a t o r R e s p o n s e a l o n g CC TC -_ V, (x,y) TC "2 D 1 / D I 1 -2 1 - / < I 0 1 l 2 F i g . 5.2.3 ( d ) O p e r a t o r R e s p o n s e a l o n g D D ' 109 on the l o c a l p r o p e r t i e s of the boundary. In the case of c o n t i n u o u s , non-uniform l ( x , y ) f u n c t i o n s the v a l u e of V^(x,y) i s dependent on the l o c a l p r o p e r t i e s of the l ( x , y ) c o n t o u r s . Por example, i f the I ( x , y ) c o n t o u r s are s t r a i g h t l i n e s i n the x y - p l a n e , V-^(x,y) w i l l e q u a l rc/2. I f the I ( x , y ) f u n c t i o n i s cone-shaped w i t h the open end down, V-^(x,y) w i l l be g r e a t e r t h a n JC/2 everywhere on the cone, and w i l l have the v a l u e it a t the t i p of the cone. I n f a c t , V^(x,y) has the v a l u e Tt a t any l o c a l l ( x , y ) maximum. C o n v e r s e l y , i t i s z e r o a t any l o c a l minimum of l ( x , y ) . g e n e r a l V-^(x,y) o p e r a t o r . I t w i l l not d e t e c t an i l l u m i n a t i o n boundary t h a t c u t s A G u n l e s s A q happens t o l i e a t l e a s t p a r t i a l l y w i t h i n the r e g i o n of h i g h e r i n t e n s i t y . I f one goes back to' the (6 7) work of Hubel and W i e s e l ' , i t i s seen t h a t b i o l o g i c a l l y t h i s • problem i s s o l v e d by h a v i n g a d u a l system of p r o c e s s o r s . They found t h a t i n the r e t i n a l g a n g l i o n and l a t e r a l g e n i c u l a t e of the c a t , neurons had one of two types of r e c e p t i v e f i e l d . There were neurons w i t h an "ON" c e n t e r , "OFF" surround, r e c e p t i v e f i e l d , analogous t o our V-^(x,y) o p e r a t o r . But t h e r e were a l s o neurons w i t h an "OFF" c e n t e r , "ON" surround r e c e p t i v e f i e l d . These are analogous to. the f o l l o w i n g o p e r a t o r : I n t u i t i v e l y , t h e r e i s one d i s t u r b i n g f e a t u r e about the A uA s c where R?(x,y) = max(0. TTx.y) - I(x,y)). 5.2.9 Kx,y) - I(x,y) 110 and where k ( x ' , y ' ) , l ( x , y ) , I ( x , y j , A and A are the same as s c f o r V-jJx.y) . ~^2^x'^^ s u f f e r s from a f a u l t s i m i l a r to that described above f o r V-^(x,y). Namely, v" 2(x,y) w i l l not detect the presence of a boundary between regions of d i f f e r e n t i n t e n s i t y unless A c happens to l i e at l e a s t p a r t i a l l y w i t h i n the region of lower i n t e n s i t y . However, i f we define the combined operator, V(x,y), to be V(x,y) = max(V 1(x,y), V 2 ( x , y ) ) f 5-2.10 •then V(x,y) w i l l be non-zero whenever A i s cut by an i l l u m i n a t i o n boundary. 5•3 Curves i n a Plane In t h i s s e c t i o n we digress from our d i s c u s s i o n of area operators i n order to present some aspects of the theory of curves ^27 28) i n a plane v ' . This m a t e r i a l w i l l be u s e f u l i n the next s e c t i o n where we develop some of the theory associated with area operators. Let C be a smooth arc of a curve along which a c e r t a i n d i r e c t i o n has been chosen as the p o s i t i v e d i r e c t i o n . Such a curve i n a plane i s defined by saying i t i s an ordered c o n f i g u r a t i o n of points (x,y) given by two continuous f u n c t i o n s of a parameter: x = x ( t ) , y = y ( t ) . 5-3-1 Given any such parametric r e p r e s e n t a t i o n f o r C one can always obtain a r e p r e s e n t a t i o n i n terms of the arc l e n g t h , s, along C, x = x ( s ) , y = y(s) 5-3,2 given that s as a f u n c t i o n of t , s ( t ) , i s known. At any point of C define the tangent ve c t o r . This i s a I l l v e c t o r of u n i t l e n g t h a l o n g the tangent l i n e i n the p o s i t i v e d i r e c t i o n as shown i n F i g . 5-3-1. I t i s denoted by T. I f the a n g l e s which. T makes w i t h the p o s i t i v e c o o r d i n a t e axes are a, p, one can w r i t e T = cosod + cos(3j 5 . 3 . 3 I f s i s the a r c l e n g t h measured a l o n g C i n the p o s i t i v e d i r e c t i o n , then f d x T d y . T > ds ds d I t can be shown t h a t f • dT = 0 5 . 3 . 5 "ds which means t h a t dT/ds i s e i t h e r the z e r o v e c t o r or i s p e r p e n d i c u l a r t o T. A u n i t v e c t o r i n the d i r e c t i o n of dT/ds i s c a l l e d the p r i n  c i p a l normal t o C a t the p o i n t i n q u e s t i o n . The p r i n c i p a l normal i s denoted by N and the l e n g t h of dT/ds i s denoted by K. T h e r e f o r e ff = K N. 5 . 3 . 6 The s c a l a r Z i s c a l l e d the c u r v a t u r e of C; i t i s g i v e n by 1 2 \ • 5 . 3 . 7 . d s N In the p r e c e d i n g d i s c u s s i o n the curve C was always r e p  r e s e n t e d i n the form, x ( t ) , y ( t ) , where the a r b i t r a r y a l l o w a b l e parameter t was sometimes s p e c i f i e d by the a r c l e n g t h s of C. C l e a r l y , the a n a l y t i c form of such a r e p r e s e n t a t i o n depends on the c h o i c e of the c o o r d i n a t e system i n the p l a n e . T h e r e f o r e , the q u e s t i o n a r i s e s whether t h e r e i s a p o s s i b i l i t y of c h a r a c t e r i z i n g a curve i n a manner independent of c o o r d i n a t e s , except f o r the p o s i t i o n of the c u r v e i n the p l a n e , t h a t i s , t o w i t h i n d i r e c t ,,2 \2 K = " • 1 Y 112 congruent t r a n s f o r m a t i o n s . When t r y i n g t o f i n d such a r e p r e s e n t a t i o n we have t o l o o k f o r q u a n t i t i e s which are independent of the c h o i c e of c o o r d i n a t e s and parameter, but depend o n l y on the n a t u r e of the c u r v e , t h a t i s , on i t s g e o m e t r i c shape. The a r c l e n g t h s, and the c u r v a t u r e K a r e q u a n t i t i e s of t h i s k i n d , and as such are c a l l e d the i n t r i n s i c  c o o r d i n a t e s of c u r v e s i n a p l a n e . The f u n c t i o n a l r e l a t i o n : K = K ( s ) 5.3.14 i s c a l l e d the n a t u r a l or i n t r i n s i c e q u a t i o n of the c o r r e s p o n d i n g c u r v e . I t can he shown t h a t i f K ( s ) i s c o n t i n u o u s i n an i n t e r v a l , i t d e termines an a r c of a curve u n i q u e l y , except f o r i t s p o s i t i o n and o r i e n t a t i o n i n the p l a n e . As a consequence of t h i s , any i n v a r  i a n t w i t h r e s p e c t t o d i r e c t congruent t r a n s f o r m a t i o n s which can be a s s o c i a t e d w i t h a curve i s c o m p l e t e l y determined where the c o r  r e s p o n d i n g K ( s ) f u n c t i o n i s g i v e n . Note t h a t i n s p e c i f y i n g a curve by g i v i n g i t s i n t r i n s i c e q u a t i o n , the c u r v a t u r e , K = K ( s ) , can take on b o t h p o s i t i v e and n e g a t i v e v a l u e s . I f the c o u n t e r - c l o c k w i s e d i r e c t i o n i s . the d i r e c t i o n of p o s i t i v e change of t a n g e n t , therj f o r t r a v e l i s the p o s i t i v e d i r e c t i o n a l o n g a c u r v e , c u r v a t u r e t o the l e f t i s p o s i t i v e , and c u r v a t u r e t o the r i g h t i s n e g a t i v e . 5.4 Shape D e s c r i p t i o n U s i n g Area Op e r a t o r s I n s e c t i o n 5.2 i t was p o i n t e d out t h a t an a r e a o p e r a t o r responds t o the l o c a l p r o p e r t i e s , i . e . the g e o m e t r i c shape, of the l ( x , y ) c o n t o u r s . I n the l a s t s e c t i o n we saw t h a t i n o r d e r to s p e c i f y a curve i n a p l a n e , c u r v a t u r e as a f u n c t i o n of a r c l e n g t h i s -113 P i g . 5.4.1 A Mesa I n t e n s i t y F u n c t i o n , l ( x , y ) s u f f i c i e n t . A c o n t o u r on an l ( x , y ) f u n c t i o n i s a p l a n a r c u r v e . C l e a r l y , the dependence of the a r e a o p e r a t o r response on the l o c a l g e o m e t r i c shape of the c o n t o u r must be r e l a t e d i n some way t o the l o c a l c u r v a t u r e of t h a t c o n t o u r . I n t h i s s e c t i o n we s t u d y t h i s r e l a t i o n s h i p i n a p r e l i m i n a r y f a s h i o n . We demonstrate t h a t the a r e a o p e r a t o r can be used t o d e t e c t p o i n t s of maximum c u r v a t u r e on a c o n t o u r . I n o t h e r words, we demonstrate t h a t the a r e a o p e r a t o r i s capable of the same type of d i s c r i m i n a t i o n t h a t A t t n e a v e v ; found i n human b e i n g s . (See s e c t i o n 1.3.) The I ( x , y ) f u n c t i o n s d e a l t w i t h i n t h i s s e c t i o n w i l l be two-valued f u n c t i o n s of the type shown i n F i g . 5.4.1. I n the r e g i o n M-j^  and a l o n g the c l o s e d curve C, l ( x , y ) = i - j j . i n the r e g i o n M 2, I ( x , y ) = i 2 , where i 1 > i 2 . T h i s type of. f u n c t i o n w i l l be r e f e r r e d t o as a mesa f u n c t i o n . The curve C has the p a r a m e t r i c r e p r e s e n t a t i o n , x ( s ) , y ( s ) , 114 j B» X P i g . 5 . 4 . 2 . The R e l a t i o n between N and N a t Two P o i n t s XT on a Curve from which one can o b t a i n i t s i n t r i n s i c e q u a t i o n , K(s). D e f i n i t i o n 5 . 4 - 1 1) I f one i s f a c i n g a l o n g the p o s i t i v e d i r e c t i o n f o r C, M-j^  w i l l l i e t o the l e f t . 2) The p o s i t i v e normal t o C, X , extends from C p i n t o M1- In F i g . 5 - 4 - 2 we show the r e l a t i o n between the p o s i t i v e n o r m al, N , and the p r i n c i p a l normal, N, f o r the cases where C i s c u r v i n g away from M^, and towards M^. I n the f i r s t case K(s) i s n e g a t i v e , and i n the second, p o s i t i v e . On the b a s i s of D e f i n i t i o n 5 - 4 . 1 we can prove the f o l l o w i n g g e n e r a l theorem about a r e a o p e r a t o r s . Theorem 5 . 4 . 1 The v a l u e of an a r e a o p e r a t o r a t any p o i n t on a mesa f u n c t i o n i s dependent o n l y on the ge o m e t r i c r e l a t i o n s h i p of the 115 o p e r a t o r and the curve C. P r o o f The v a l u e of I(x,y7 i n equation. 5 . 2 . 4 must l i e i n the range i - ^ >. ijx,y) ^ i ^ . The l e f t - h a n d e q u a l i t y h o l d s o n l y i f A Q i s t o t a l l y w i t h i n M^; the r i g h t - h a n d e q u a l i t y , i f A C i s t o t a l l y w i t h i n M^. I f the curve C i n t e r s e c t s A , the n i^ > TTx ,*y7 > 1^' C l e a r l y , the v a l u e of I(x,y7 i n e q u a t i o n 5 . 2 . 4 i s dependent on t h e g e o m e t r i c r e l a t i o n s h i p o f C and. A , and on the magnitude o f i - ^ and But the v a l u e o f R-^(x,y) i n e q u a t i o n 5 . 2 . 3 or R 2(x,y) i n e q u a t i o n 5 . 2 . 9 i s dependent o n l y on the r e l a t i v e magnitude o f I ( x , y ) and lTx"7y7 and hence i s independent o f t h e a c t u a l v a l u e s o f i - ^ and i ^ . However, s i n c e i t . i s dependent on the r e l a t i v e magnitude of I (x,y"5, i t must he dependent on the geometric r e l a t i o n  s h i p "between A and C, and the p o i n t (x,y) and C. Si n c e R-, (x,y) • and R2(x,y) a r e the v a r i a b l e p a r t s i n the i n t e g r a l s o f e q u a t i o n s 5 . 2 . 2 and 5 . 2.8, i t f o l l o w s t h a t V ^ ( x , y ) , Y^(x,j) and c o n s e q u e n t l y V(x,y) a r e dependent only.on. the geometric r e l a t i o n s h i p between C and the a r e a o p e r a t o r . Q.E.D. We now r e s t r i c t our a t t e n t i o n t o a p a r t i c u l a r a r e a o p e r a t o r . I t i s the V- L(x,y) o p e r a t o r d e f i n e d i n eq u a t i o n s 5 - 2 . 2 t o 5 . 2 . 6 . The geometry of the o p e r a t o r i s a u n i t r a d i u s d i s k where A i s the c e n t e r of the d i s k and A i s the remainder. The c s k ( x ' , y ! ) w e i g h t i n g f u n c t i o n i s c o n s t a n t and equals one. We w i l l demonstrate t h a t t h i s o p e r a t o r can be used t o d e t e c t p o i n t s on th e . c u r v e C a t which the a b s o l u t e v a l u e of the K ( s ) f u n c t i o n i s a l o c a l maximum. That i s , g i v e n an I ( x , y ) mesa f u n c t i o n t h i s a r e a o p e r a t o r can d e t e c t p o i n t s of maximum l o c a l c u r v a t u r e ( p o s i t i v e or n e g a t i v e ) on the curve C. 116 D e f i n i t i o n 5.4.2 The d i s t a n c e between two p o i n t s , P. and P., i n the J x y - p l a n e i s denoted by D(P^,P_.) where 1 D f p ^ p . ) = { ( - - x / + ( y ^ y / } 2 5.4.1 - The two f o l l o w i n g lemmas f o r m a l i z e some of the d i s c u s s i o n i n s e c t i o n 5-2. They are t r u e f o r any a r e a o p e r a t o r p o s s e s s i n g the p o i n t c e n t e r , d i s k s u r r o u n d geometry, independent of the w e i g h t i n g f u n c t i o n . Lemma 5.4.1 I f the p o i n t (x,y) i s anywhere i n IVL^ ,. • V 1 ( x , y ) = 0. Lemma 5-4.2 I f the p o i n t P Q i s i n and the p o i n t P C i s on C, then V 1 ( x , y ) = 0 a t P q i f and o n l y i f f o r a l l P , D ( P 0 , P c ) > . l D e f i n i t i o n 5-4.3 Le t us a s s o c i a t e w i t h any p o i n t ( x ( s ) , y ( s ) ) on C the f i r s t p o i n t ( x m > y m ) a l o n g the p o s i t i v e normal t o C a t which V ^ ( x , y ) i s a maximum. L e t us denote the curve d e f i n e d by the s e t of such p o i n t s as C and l e t i t have the p a r a m e t r i c r e p r e s e n t a t i o n x = x ( s ) , r m * • m m y m = y ( s ) . Then V (x,y) a l o n g C i s g i v e n by V (x ( s ) , y m ( s ) ) which we w i l l denote by V™(s), and which we s h a l l c a l l the ope: r i d g e f u n c t i o n , or the r i d g e f u n c t i o n f o r s h o r t . Lemma 5.4-3 Por any p o i n t ( x ( s ) , y ( s ) ) on C t h e r e e x i s t s a p o i n t (x ( s ) , y ( s ) ) on C . m ' ' Jm m P i g . 5 . 4 . 3 An I ( x , y ) Mesa F u n c t i o n w i t h a Convex M 2 Region T h i s lemma i s t r u e by v i r t u e of the f a c t t h a t V^(x,y) i s bounded, 0 $ V^(x,y) $ i t , and by Lemmas.5-4.1 and 5 . 4 . 2 . D e f i n i t i o n 5 . 4 . 4 A r e g i o n i s s a i d t o be convex i f g i v e n any two p o i n t s i n the r e g i o n , a l l p o i n t s on the s t r a i g h t l i n e c o n n e c t i n g the two p o i n t s a l s o l i e i n the r e g i o n . C o n s i d e r the case where l ( x , y ) i s a mesa f u n c t i o n i n w hich M 2 i s a convex r e g i o n as shown i n F i g . 5 . 4 - 3 - Theorem 5 - 4 - 2 I f M 2 i s convex then V^(x,y) < TC/2 f o r a l l x and y. P r o o f The p o i n t c e n t e r , d i s k s u r r o u n d of the V^(x,y) o p e r a t o r i s a convex r e g i o n . The i n t e r s e c t i o n of convex r e g i o n s i s a con vex r e g i o n . Hence i f the o p e r a t o r d i s k i n t e r s e c t s K^, the i n t e r  s e c t i o n must be a convex r e g i o n . I f the c e n t e r of the d i s k i s i n M 2, V^(x,y) = 0 by Lemma 5 . 4 . 1 . I f the c e n t e r of the d i s k i s on C, then the. a r e a of i n t e r s e c t i o n between the d i s k and M 0 must 118 be l e s s than, o r equ a l t o o t / 2 . I f the c e n t e r of the d i s k i s i n M^, the area of i n t e r s e c t i o n of the d i s k and must be l e s s t h a n tc/2. But V^(x,y) i s e q u a l t o t h i s a r e a of i n t e r s e c t i o n . Hence, V 1 ( x , y ) ^  J t / 2 . Q.E.D. Theorem 5.4-3 I f M 0 i s convex, C and C are c o i n c i d e n t . 2 ' m P r o o f There a re two p o s s i b l e c a s e s . I n the f i r s t case shown i n F i g . 5.4.4(a) the c e n t e r of the o p e r a t o r d i s k i s on C a t a p o i n t (x ,y ) and i s c o m p l e t e l y c o n t a i n e d by the d i s k . I f (x ,y ) i s any p o i n t on the p o s i t i v e normal t o C a t (x , y ^ ) , n xi 0 0 t h e n a l o n g N , V-, (x ,y ) = V-, (x ,y ) , as l o n g as M 0 remains to p' 1 n' Jn 1 o , J o to 2 c o m p l e t e l y c o n t a i n e d by the d i s k . As soon as a p o i n t on N i s reached 'at which some p a r t of i s not c o n t a i n e d i n the d i s k (dashed c i r c l e i n F i g . 5 . 4 . 4(a)), the a r e a of i n t e r s e c t i o n between and the d i s k w i l l have dec r e a s e d . Hence, V n (x -,y ) > V n (x ,y ). S i n c e V n (x ,y ) > V n (x ,v ), C and C l v o' Jo 1 n' Jn 1 o' Jo 1 n'^-n7' m are c o i n c i d e n t i n t h i s case. In t he second and more u s u a l case shown i n F i g . 5 .4 .4(b), the o p e r a t o r i s a t some p o i n t ( x 0 > y o ) on C, and i s not t o t a l l y c o n t a i n e d w i t h i n the o p e r a t o r d i s k . By Theorem 5-4.2 the a r e a of i n t e r s e c t i o n of and the o p e r a t o r d i s k i s l e s s t h a n or e q u a l t o a t / 2 . T h i s i m p l i e s t h a t when the c e n t e r of the d i s k i s on C a t (x ,y ), the ar e a of i n t e r s e c t i o n must l i e i n one o f 0 0 the h a l f - d i s k s formed by the tangent d i a m e t e r a t ( x Q , y o ) . (See F i g . 5.4.4(b) ..) Let-us c a l l i t the h a l f - d i s k , and the o t h e r the M-^  h a l f - d i s k . O b v i o u s l y , as the o p e r a t o r moves away from ( x Q , y o ) a l o n g the p o s i t i v e normal, the a r e a of i n t e r s e c t i o n between and the h a l f - d i s k must d e c r e a s e . The o n l y way i t c o u l d i n c r e a s e i s i f a p a r t o f M moved a c r o s s the tangent d i a m e t e r i n t o the h a l f - d i s k . But t h i s would c o n t r a d i c t the h a l f - d i s k d e f i n i t i o n . Hence, i f ( x n > y n ) i - s a n y p o i n t on the p o s i t i v e normal t o C a t (x ,y ), V, (x ,y ) > V-, (x , y ). v o'Jo ' 1 o' Jo 1 n ' ° n Thus, i n bo t h . c a s e s C and C are c o i n c i d e n t i f 1YL i s ' m 2 convex. Q.E.D. I n the case where i s t h e convex r e g i o n two theorems can he proven t h a t are the e q u i v a l e n t s of Theorems 5.4.2 and 5.4. Theorem 5.4.4 I f M n i s a convex r e g i o n , C and C a r e c o i n c i d e n t 1 to ' m and Cffi i s c o n t i n u o u s everywhere except a t most a t one p o i n t . P r o o f Assume t h a t the c e n t e r of the d i s k o p e r a t o r i s on C and t h a t the r e g i o n c o n t a i n e d hy C i s convex. I n o r d e r t o prove t h i s theorem i t i s s u f f i c i e n t t o show t h a t as the d i s k c e n t e r . moves i n f i n i t e s i m a l l y away from C a l o n g the p o s i t i v e normal, the a r e a of i n t e r s e c t i o n between and the d i s k s t a y s c o n s t a n t or i n c r e a s e s except i n one v e r y s p e c i a l s i t u a t i o n . I f the d i s k c e n t e r i s a t some p o i n t on C, and the r e g i o n and the curve C are c o m p l e t e l y c o n t a i n e d by the d i s k , i t i s obvious t h a t the a r e a of i n t e r s e c t i o n w i l l not change as the d i s k c e n t e r moves an i n f i n i t e s i m a l d i s t a n c e a l o n g the p o s i t i v e normal. The o t h e r p o s s i b l e s i t u a t i o n i s i l l u s t r a t e d i n F i g . 5.4 The c e n t e r o f the d i s k i s on C a t the p o i n t ( x o , y Q ) . The curve C i n t e r s e c t s the edge of the d i s k a t a number of p l a c e s , f o r m i n g the a r c s L^, L ^ e t c . which l i e c o m p l e t e l y w i t h i n M^. The p r o j e c t i o n s o f thes e a r c s on the tangent t o C a t (x ,y ) are denoted 121 OPERA TOR DISK TANGENT TO C AT (x0/y0) F i g . 5.4e5 The O p e r a t o r D i s k on C w i t h a Convex R e g i o n T T "by I ^ f L , e t c . I f the c e n t e r of the d i s k moves an i n f i n i t e s i m a l d i s t a n c e An a l o n g N , the change i n the a r e a o f i n t e r s e c t i o n P between the d i s k and i s g i v e n by m AA = T L d An. I f the a r c L_. l i e s I n the p o s i t i v e d i r e c t i o n w i t h r e s p e c t T t o the tangent l i n e , the q u a n t i t y L. An must be p o s i t i v e . S i n c e T — i s convex, a l l the L.. must l i e i n the p o s i t i v e d i r e c t i o n . Hence the q u a n t i t y AA must be p o s i t i v e . I n computing AA, second o r d e r e f f e c t s due t o changes T i n the 1. can l e g i t i m a t e l y be i g n o r e d p r o v i d e d a t l e a s t one o f «J T the L. i s non-zero. F o r a v e r y s p e c i a l c l a s s o f f i g u r e s i t i s j p o s s i b l e t o have a s i t u a t i o n i n w h i c h t h e r e are no L. a r c s when the d i s k c e n t e r i s on C, and y e t such an a r c i s formed i n moving an i n f i n i t e s i m a l d i s t a n c e An a l o n g IT . The s i t u a t i o n i s i l l u s t r a t e d 122 OPERATOR DISK J\ c F i g . 5.4.6(a) The Operator D i s k (b) The D i s k Moved a D i s t a n c e i n F i g . 5.4.6. I n F i g . 5.4.6(a) the c e n t e r of t h e d i s k i s a t ( x 0»y o) o n The tangent t o C a t t h i s p o i n t i s c o i n c i d e n t w i t h C out t o the edge of the d i s k . The edge o f the d i s k i s t h e n c o i n c i d e n t w i t h C f o r a s h o r t d i s t a n c e , t h e remainder o f C l y i n g w i t h i n the d i s k . S i n c e the curve C i s not p a r t of the M, r e g i o n , t h e r e a r e no L. a r c s . Consequently, the f o r m u l a f o r AA becomes But i t i s e v i d e n t i n F i g , 5.4.6(b) t h a t the AL^ segment l i e s i n the n e g a t i v e N d i r e c t i o n w i t h r e s p e c t t o the tangent l i n e . C o n s e q u e n t l y , AA i s n e g a t i v e . ment on C i n e i t h e r d i r e c t i o n away from (x ,y ) e i t h e r causes 17 o o the f o r m a t i o n of an a r c or causes and C t o be c o m p l e t e l y c o n t a i n e d w i t h i n the d i s k . a t ( x o , y o ) An a l o n g N. P T h i s s i t u a t i o n o c c u r s o n l y a t one p o i n t on C. Move-123 The a r e a of i n t e r s e c t i o n between the d i s k and a t ( x 0 > y o ) i s a maximum. I t de c r e a s e s s t e a d i l y as the d i s k moves i n the p o s i t i v e d i r e c t i o n away from ( x 0»y o)« I t was p o i n t e d out i n the p r o o f o f Theorem 5.4.2 t h a t V-^(x,y) i s equal, t o the a r e a o f i n t e r s e c t i o n between the d i s k and the r e g i o n p r o v i d e d t he d i s k c e n t e r i s on C or i n 1YL^ . Prom the above d i s c u s s i o n , the a r e a of i n t e r s e c t i o n between the convex r e g i o n and the d i s k s t a y s c o n s t a n t or i n c r e a s e s , except a t most a t one p o i n t , as the d i s k c e n t e r moves away from C a l o n g N . Consequently, V^(x,y) d e c r e a s e s or s t a y s c o n s t a n t . Prom lemma 5.4.1, V (x,y) =0 i n . C l e a r l y a t any p o i n t on C, V (x,y) a l o n g N i s a l o c a l maximum on C, except -L P a t most a t one p o i n t . By d e f i n i t i o n 5.4.3, &m i s the sequence of such l o c a l maxima. Hence, C and C are c o i n c i d e n t . m At t he s i n g u l a r p o i n t f o r t h e s p e c i a l c l a s s of f i g u r e s , V,(x,y) a l o n g N i s a maximum a t the i n t e r s e c t i o n of H and the o p p o s i t e s i d e of C. Thus, C and C m a r e s t i l l c o i n c i d e n t , but C m i s d i s c o n t i n u o u s a t the s i n g u l a r p o i n t . Q.E.D. The second theorem f o r a convex r e g i o n i s e a s i e r t o prove. Theorem 5.4.5 I f M 1 i s convex, V ^ ( s ) ? - f t / 2 . P r o o f By d e f i n i t i o n 5.4.3, V™(s) i s the v a l u e of the o p e r a t o r a l o n g C . By theorem 5.4.4, i t i s the v a l u e a l o n g C. I f the c e n t e r of the o p e r a t o r d i s k i s on C, the a r e a o f i n t e r s e c t i o n § of the d i s k and the convex r e g i o n , , must be l e s s t h a n or e q u a l t o %/2. Hence, V ^ ( s ) M x/2. Q.E.D. As mentioned e a r l i e r , we want t o show the r e l a t i o n  s h i p between the v a l u e of the V-^(x,y) o p e r a t o r and the c u r v a t u r e f u n c t i o n , K ( s ) , of the curve C. Up t o t h i s p o i n t we have demonstrated t h a t i f e i t h e r o r i s convex, V ^ ( x , y ) a l o n g the normal t o C has a l o c a l maximum a t C. Thus, t h e r e i s a r i d g e i n the V ^ ( x , y ) f u n c t i o n a l o n g the curve C. T h i s r i d g e f u n c t i o n has been d e s i g n a t e d the V™(s) f u n c t i o n . I f V ^ ( s ) - it/2 f o r a l l s, we know the r e g i o n i s convex. C o n s e q u e n t l y , K ( s ) cannot take on p o s i t i v e v a l u e s . S i m i l a r l y , i f V ^ (S)^JE / 2 f o r a l l s, M-^  i s the convex r e g i o n and K ( s ) i s a non-negative f u n c t i o n . I f v ^ s o ^ = % ^ 2 , ^ e a r c o f " t i l e c u r v e ^ w i t h i n t h e o p e r a t o r d i s k c e n t e r e d a t s Q must be a s t r a i g h t l i n e and c o n s e q u e n t l y K ( s ) = 0. We can now demonstrate t h a t t h e r e i s a one-to-one c o r r e l a t i o n between the l o c a l maxima and minima of the K ( s ) f u n c t i o n f o r a curve C, and those of t h e r e s u l t i n g V ^ ( s ) f u n c  t i o n , p r o v i d e d t h e s e f e a t u r e s a r e " w i d e l y " spaced on C. B e f o r e d o i n g t h i s , we r e q u i r e some d e f i n i t i o n s . D e f i n i t i o n 5.4.5 1) A f e a t u r e on the curve C i s any p o i n t on C a t w h i c h i t s c u r v a t u r e f u n c t i o n K ( s ) has a l o c a l maximum or minimum t h a t i s non-zero. 2) A f e a t u r e a t a p o i n t P^ on C i s s e p a r a b l e (by the o p e r a t o r ) i f : ' . (a) the d i s t a n c e between. P^ and any o t h e r f e a t u r e , or p o i n t a t which K ( s ) changes s i g n , i s g r e a t e r t h a n the d i a m e t e r of the o p e r a t o r d i s k , i . e . g r e a t e r than two; 125 (b) when the c e n t e r of the o p e r a t o r d i s k i s on C at P, , such t h a t D(P. P ) £ 1 , t h e r e i s o n l y one a r c o f C K I K i n t e r s e c t e d by the d i s k . Theorem 5.4.6 I f a f e a t u r e a t P^ on C i s s e p a r a b l e t h e n C and a r e c o i n c i d e n t a t a l l p o i n t s P-^  on C such t h a t D(P^,P^) $ 1. P r o o f Draw a c i r c l e of r a d i u s 1+e , e >0, c e n t e r e d on i n such a f a s h i o n t h a t o n l y a s i n g l e a r c o f C i s i n t e r s e c t e d . One of the two r e g i o n s formed by the a r c of C c u t t i n g t h i s d i s k , must be convex. I f i t i s a convex r e g i o n , a p p l y Theorem 5.4.4 t o complete the p r o o f o f t h i s theorem. I f i t i s a convex r e g i o n , a p p l y Theroem. 5.4.3- Q.E.D. D e f i n i t i o n 5.4.6 An i n d i c a t o r i s any p o i n t on the V™(s) f u n c t i o n a t which i t has a l o c a l maximum or minimum t h a t i s not e q u a l to Jt/2. U s i n g the above d e f i n i t i o n s we can s t a t e the f o l l o w i n g theorem: Theorem 5.4.7 P o r every s e p a r a b l e f e a t u r e on C t h e r e i s a c o r r e s  ponding i n d i c a t o r on V ^ ( s ) . I n o r d e r t o prove the above theorem, a n o t h e r theorem i n plane geometry i s r e q u i r e d . Theorem 5.4.8 Assume t h a t t h e a b s o l u t e v a l u e of the c u r v a t u r e f u n c t i o n f o r a curve G- i s m o n o t o n i c a l l y d e c r e a s i n g . A convex r e g i o n i s d e f i n e d i f a d i s k of a p p r o p r i a t e s i z e i s c e n t e r e d a t some p o i n t on G. I f the c e n t e r of the d i s k moves a l o n g G i n 126 (+) (+) F i g c 5.4.7(a) A D i s k a t Two P o s i t i o n s (b) The D i s k s Superimposed on a Curve w i t h D e c r e a s i n g C u r v a t u r e t h e d i r e c t i o n o f d e c r e a s i n g a b s o l u t e v a l u e of c u r v a t u r e , the a r e a of t h i s convex r e g i o n i n c r e a s e s . P r o o f The s i t u a t i o n p o s t u l a t e d i n the statement of the theorem i s d e p i c t e d i n P i g . 5.4.7(a). A geometric p r o o f f o r t h e theorem i s i n d i c a t e d i n P i g . 5.4.7(b). The d i s k and i n t e r s e c t e d a r c o f G- a t p o s i t i o n #1 i n F i g . 5.4.7(a) are r i g i d l y - t r a n s l a t e d t o p o s i t i o n #2 so t h a t the tangent v e c t o r s T-^  and T^ a r e c o i n c i d e n t . S i n c e the a b s o l u t e v a l u e o f t h e c u r v a t u r e d e c r e a s e s between p o s i t i o n s #1 and #2, i t i s e v i d e n t t h a t A^ must c o n t a i n e d by A^. Q.E.D. Proof, o f Theorem 5.4.7 The p o i n t P Q i n F i g . 5.4.8 i s a s e p a r a b l e f e a t u r e . Thus, the a b s o l u t e v a l u e of the c u r v a t u r e f u n c t i o n i n the ranges P t o P„ and o 2 P Q t o P^ i s m o n o t o n i c a l l y d e c r e a s i n g . C o n s i d e r the d i s k of r a d i u s one c e n t e r e d a t P-. . I f the c e n t e r of t h e d i s k i s moved 127 P i g . 5.4.8 The D i s k Operator i n the Neighborhood of a Se p a r a b l e F e a t u r e a l o n g C toward P^, t h e convex a r e a A-^  i n c r e a s e s by Theorem 5.4.8. S i m i l a r l y as the center, o f the d i s k a t P^ moves toward P^, the convex a r e a A^ i n c r e a s e s . C l e a r l y , t h e convex a r e a must be a minimum a t some p o i n t on C between and P^. I t f o l l o w s t h e n t h a t V ^ ( s ) has e i t h e r a l o c a l maximum o r minimum i n t h e same r e g i o n . I f 1VL and IYL are and r e g i o n s r e s p e c t i v e l y , t h e p o s i t i v e d i r e c t i o n a l o n g C i s from P^ t o P^, and the p o i n t P Q i s a l o c a l c u r v a t u r e maximum. I t f o l l o w s t h a t V ^ ( s ) ^ ft/2 between P^ and P-^  and has a l o c a l maximum i n t h i s range. C o n v e r s e l y i f and 1VL a r e and M^, r e g i o n s r e s p e c t i v e l y , the p o i n t P Q i s a l o c a l c u r v a t u r e minimum, and V ^ ( s ) - ^ f t / 2 and has a l o c a l minimum i n the range t o P^. Q.E.D. 128 I n the p r o o f of Theorem 5.4.7 i t i s seen t h a t a s e p a r a b l e l o c a l maximum or minimum of the c u r v a t u r e f u n c t i o n g i v e s r i s e t o a c o r r e s p o n d i n g l o c a l maximum or minimum of t h e V m ( s ) f u n c t i o n . I n a d d i t i o n , i f t h e s e p a r a b l e f e a t u r e i s a t a p o i n t P 0 on C, t h e i n d i c a t o r on V™(s) must occur a t a p o i n t P ± on C such t h a t D ( P o , P ± ) < 1. T h i s completes our- developement of t h e t h e o r y a s s o c i a t e d w i t h the V ^(x,y) o p e r a t o r and I ( x , y ) mesa f u n c t i o n s . A s i m i l a r s e t o f theorems c o u l d be developed f o r the Y^(x,y) o p e r a t o r . The t h e o r y f o r the o p e r a t o r V(x,y) - max(V-^, V^) would c o n s i s t o f some c o m b i n a t i o n o f the t h e o r y f o r V ^(x,y) and V^Cxjy). L e t us c o n s i d e r f o r a moment the case where I ( x , y ) i s a c o n t i n u o u s f u n c t i o n . Assume an a r e a o p e r a t o r i s t r a c k e d a l o n g some c o n t o u r o f a c o n t i n u o u s I ( x , y ) f u n c t i o n , say I ( x , y ) = I c . .The r e s u l t i n g response f u n c t i o n i s i d e n t i c a l w i t h t h a t o b t a i n e d a l o n g the curve C o f the mesa f u n c t i o n I ' ( x , y ) , where f l i f Kx,y)>. I - I ' ( x , y ) = j c [ o i f I ( x , y ) ? I c . T h i s f o l l o w s from the d e f i n i t i o n of R-^(x,y) i n e q u a t i o n 5.2.3, and the f a c t t h a t the contour l i n e I ( x , y ) = I and the curve C c f o r I ' ( x , y ) a r e i d e n t i c a l . Thus, the theorems g i v e n above r e l a t i n g f e a t u r e s on C t o t h e V™(s) f u n c t i o n can be a p p l i e d t o the c o n t o u r l i n e s on a c o n t i n u o u s I ( x , y ) f u n c t i o n . I n p a r t i c u l a r the V^(x,y) f u n c t i o n a l o n g an i n t e n s i t y c o ntour l i n e can d e t e c t p o i n t s of l o c a l maximum c u r v a t u r e . I t a l s o r e f l e c t s the l o c a l c o n c a v i t y or c o n v e x i t y o f the r e g i o n e n c l o s e d by the contour l i n e . 6. SECONDARY PROCESSING OP RECEPTOR ACTIVITY FUNCTIONS 6.1 I n t r o d u c t i o n In the p r e v i o u s c h a p t e r s we have seen t h a t the r e c e p t o r a c t i v i t y i n a l a t e r a l i n h i b i t o r y network p r o v i d e s a graded enhance ment o f i l l u m i n a t i o n d i s c o n t i n u i t i e s . I t has been shown t h a t over a c e r t a i n range of o p e r a t i o n t h i s graded enhancement p r o  p e r t y can be modelled by a weighted a r e a o p e r a t o r . I n a t h e o r e t i c a l s t u d y -of a r e a o p e r a t o r s i t was shown t h a t a f u n c t i o n , V ^ ( s ) , c o u l d be d e r i v e d which p r o v i d e d a d e s c r i p t i o n o f an i l l u m i n a t i o n boundary. T h i s d e s c r i p t i o n i s r e l a t e d t o the c u r v a t u r e f u n c t i o n of the boundary. As such, f o r a g i v e n o p e r a t o r , i t i s an i n t r i n s i c d e s c r i p t i o n dependent o n l y on the shape o f the boundary. I n t h i s c h a p t e r we d i s c u s s two s t u d i e s i n v o l v i n g the p r o c e s s i n g o f r e c e p t o r a c t i v i t y f u n c t i o n s e l i c i t e d by v a r i o u s p a t t e r n s of i l l u m i n a t i o n . Two examples of such a c t i v i t y f u n c t i o n s are g i v e n i n P i g . 6.1.1(a) and ( b ) . The i l l u m i n a t i o n p a t t e r n i n F i g . 6.1.1(a) was a w h i t e , f i v e - s i d e d p o l y g o n on a b l a c k background. Note the peaks i n r e c e p t o r a c t i v i t y near the c o r n e r s ; the l o w e r a c t i v i t y a l o n g the edges; and the complete l a c k of a c t i v i t y i n the c e n t e r o f t h e polygon. The i l l u m i n a t i o n p a t t e r n . i n F i g . 6.1.1(b) was a w h i t e d i s k on a b l a c k background. I n t h i s case the r e c e p t o r a c t i v i t y i s a maximum around the p e r i p h e r y o f the whole f i g u r e , d e c r e a s i n g u n i f o r m l y t o z e r o i n the c e n t e r . The f i r s t s t udy d e a l t w i t h i n t h i s c h a p t e r demonstrates t h a t i s i s p o s s i b l e t o p r e d i c t a p a t h around the boundary of a p o l y g o n a l p a t t e r n on the b a s i s of the i n f o r m a t i o n c o n t a i n e d i n the r e c e p t o r a c t i v i t y f u n c t i o n . The second s t u d y d e a l s w i t h an F i g . 6.1.1(a) I s o m e t r i c o f the ("b) I s o m e t r i c o f the R e c e p t o r Receptor A c t i v i t y F u n c t i o n A c t i v i t y F u n c t i o n f o r a f o r a White Polygon White D i s k a l g o r i t h m t h a t has heen developed f o r o b t a i n i n g the r e c e p t o r a c t i v i t y e q u i v a l e n t ' of the V ^ ( s ) d e s c r i p t i o n of a boundary. I n b o t h o f the f o l l o w i n g s t u d i e s , the r e c e p t o r a c t i v i t y f u n c t i o n s were o b t a i n e d u s i n g the minimum a m b i g u i t y s i m u l a t o r con f i g u r a t i o n determined i n c h a p t e r J>. T h i s i s ' the u n i f o r m k ^ ( d ) c o n f i g u r a t i o n i n which the r e c e p t o r f i e l d of view i s 1.5u, and the u n i f o r m i l l u m i n a t i o n a c t i v i t y f o r a rounded 9x9 a r r a y i s zero p u l s e s / s e c . The r e c e p t o r a c t i v i t y f u n c t i o n s f o r the v a r i o u s i l l u m i n a t i o n p a t t e r n s were punched onto paper t a p e . T h i s tape served as the i n p u t t o a PDP-9 computer w h i c h was programmed i n assembler language t o c a r r y out t h e subsequent p r o c e s s i n g o f the f u n c t i o n s . 6.2 Boundary P r e d i c t i o n U s i n g Peak Receptor A c t i v i t y I n F i g . 6.2.1(a) the e^ = e^(x,y) f u n c t i o n f o r a-white polygon on a b l a c k background i s shown. T h i s f u n c t i o n s e r v e s 131 F i g . 6.2.1(a) The P o l y g o n a l e^(x,y) (b) The Contcured x ^ ( x , y ) F u n c t i o n F u n c t i o n F i g . 6.2.1(c) The L o c a l Maxima of (d) The Expanded L o c a l Maxima the x ^ ( x , y ) F u n c t i o n as the i n p u t t o the s i m u l a t o r which computes the r e c e p t o r response f u n c t i o n x^ = x ^ ( x , y ) a t i n t e g e r v a l u e s of x and y. F i g . 6.1.1(a) and F i g . 6.2.1(b) show i s o m e t r i c and contour p l o t s of the x ^ ( x , y ) f u n c t i o n f o r t h e w h i t e p o l ygon. T h i s f u n c t i o n i s read i n t o a PDP-9 d i g i t a l computer f o r f u r t h e r p r o c e s s i n g . I n t h i s s e c t i o n a method i s d e s c r i b e d which uses the l o c a l maxima of x ^ ( x , y ) f u n c t i o n s t o t r a c e the boundary of p o l y g o n s . The f i r s t s t age i n v o l v e s the d e t e c t i o n of t h e s e l o c a l maxima. T h i s i s a c c o m p l i s h e d by comparing the a c t i v i t y , x^, of each r e c e p t o r w i t h t h a t o f n e i g h b o r i n g r e c e p t o r s . I n F i g . 6.2.1(c) the l o c a l 132 maxima f o r the x ^ ( x , y ) f u n c t i o n of F i g . 6.2.1(b) are shown. Only t h e s e l o c a l maxima are r e t a i n e d . I n the next stage o f the p r o c e s s , use i s made of a p r i o r i n f o r m a t i o n t h a t r e l a t e s peak r e c e p t o r a c t i v i t y and wedge a n g l e a. ( T h i s i n f o r m a t i o n was o b t a i n e d t h r o u g h the experiments d e s c r i b e d i n c h a p t e r 3.) Hence, a wedge a n g l e , a ^ ( x , y ) , can be a s s o c i a t e d w i t h each o f the l o c a l maxima of t h e x ^ ( x , y ) f u n c t i o n . I n the computer the a p p r o p r i a t e v a l u e i s i n s e r t e d i n t o each of t h e non-zero x^ l o c a t i o n s . The address of each ct^ v a l u e i s t h e n spread through a s m a l l b l o c k of memory s u r r o u n d i n g t h a t address. T h i s compensates f o r the u n c e r t a i n t y t h a t a r i s e s i n a s s o c i a t i n g peak r e c e p t o r a c t i v i t y and- wedge a n g l e . T h i s u n c e r t a i n t y i s caused by the o r i e n t a t i o n dependence problem d e a l t w i t h i n c h a p t e r s 2 and 3, and by the d i s c r e t e n a t u r e of t h e x ^ ( x , y ) f u n c t i o n . The p r o c e s s of p r e d i c t i n g the p a t h o f the polygon boundary can now b e g i n . The p o i n t a s s o c i a t e d w i t h the a b s o l u t e maximum of the x ^ ( x , y ) f u n c t i o n i s a r b i t r a r i l y chosen as the s t a r t i n g p o i n t . C a l l i t the f i r s t peak p o i n t , a n ^ "Le"^ v a l u e a s s o c i a t e d w i t h i t be denoted by a^. I n the neighborhood of PP-^ the po l y g o n boundary c o n s i s t s of two s t r a i g h t edges which form a wedge of ang l e a^. The o r i e n t a t i o n of t h i s wedge i n the x y - p l a n e i s unknown. In o r d e r t o o b t a i n the o r i e n t a t i o n o f one s i d e of t h e wedge, a s p i r a l s e a r c h through the d a t a b l o c k i n memory i s c a r r i e d o u t u n t i l a non-zero l o c a t i o n i s encountered. T h i s o p e r a t i o n i s i l l u s t r a t e d i n F i g . 6.2.2(a).. The second s i d e of the wedge, demean have one of two o r i e n t a t i o n s , 2a-, degrees a p a r t , w i t h 133 DETECTS THIS SIDE DETECTS THIS SIDE 6.2.2(a) Search P a t t e r n f o r (h) T e s t i n g f o r the Second l o c a t i n g One Side of the Wedge Side 6.2.2(c). P a r t of the P o l y g o n (d) A Change i n the D i r e c t i o n Boundary of the Boundary F i g . 6.2.2(e) The Complete Poly g o n Boundary 134 r e s p e c t t o the. f i r s t s i d e . L i n e s are extended from PP-^ i n b o t h d i r e c t i o n s through, the d a t a b l o c k u n t i l a non-zero l o c a t i o n i s encountered. T h i s o p e r a t i o n i s i l l u s t r a t e d i n P i g . 6.2.2(b). I t e s t a b l i s h e s the r e l a t i v e o r i e n t a t i o n of the two s i d e s of the wedge. ' The wedge a n g l e and o r i e n t a t i o n a re now known. Con s e q u e n t l y , the c l o c k w i s e d i r e c t i o n around the boundary can be determined. A l i n e i s drawn i n the c l o c k w i s e d i r e c t i o n from PP-j t o t h e f i r s t l o c a l maximum. T y p i c a l l y , i t w i l l be a s a d d l e p o i n t (denote i t by SP^) of the x ^ ( x , y ) f u n c t i o n caused by a l o n g s t r a i g h t p o r t i o n of the boundary. I f so, i t w i l l have an oc^ o f 180°. The l i n e j o i n i n g PP-^ t o SP^ i s extended u n t i l a n o t h e r l o c a l maximum i s encountered. Assume i t a l s o has an of 180°, and hence denote i t by SP^. The boundary l i n e i s redrawn t o connect PP-^ t o . S P ^ and i s a g a i n extended. Assume t h a t t h i s t i me a l o c a l maximum i s encountered h a v i n g an oc^ / 180°. Let i t be and denote the p o i n t by PP,-,. The boundary l i n e i s redrawn t o connect PP^ t o PP 2- T h i s determines the f i r s t p o r t i o n of the polygon boundary as i l l u s t r a t e d i n P i g . 6 . 2 . 2 ( c ) . A t t h i s p o i n t the f o l l o w i n g i n f o r m a t i o n i s a v a i l a b l e . The a n g l e a s s o c i a t e d w i t h P P 2 ; the o r i e n t a t i o n of one s i d e o f the P?2 wedge; and the d i r e c t i o n of p r o g r e s s i o n around the boundary. Hence, the o r i e n t a t i o n of the second s i d e of the wedge can be p r e d i c t e d . I f i t i s c o n s i d e r e d as a d i r e c t e d l i n e s t a r t i n g a t PP 2> i t must make an a n g l e of (rc-a^) w i t h t h e d i r e c t e d l i n e PP-^—»-PP2. A l i n e i s extended i n t h i s new d i r e c t i o n u n t i l i t encounters e i t h e r a s a d d l e p o i n t or a peak p o i n t . T h i s i s i l l u s t r a t e d i n F i g . 6.2.2(d). I n t h i s f a s h i o n the polygon boundary 135 i s t r a c e d . The completed t r a c e i s shown i n P i g . 6.2.2(e). I n the above r o u t i n e , a p r i o r i i n f o r m a t i o n p e r m i t s the a s s o c i a t i o n o f the l o c a l maxima of the r e c e p t o r a c t i v i t y f u n c t i o n w i t h wedge a n g l e s . Once t h i s a s s o c i a t i o n i s made, each l o c a l maximum p r o v i d e s a d e s c r i p t i o n of the nearby boundary. Co n s e q u e n t l y , the sequence of such l o c a l maxima a l o n g the boundary, and the d i s t a n c e between them.,, p r o v i d e a good d e s c r i p t i o n f o r t h e bo undary. 6.3 Concave-Convex F i g u r e s I n the r o u t i n e d e s c r i b e d i n the p r e c e d i n g s e c t i o n , the i n i t i a l o p e r a t i o n a f t e r o b t a i n i n g the x ^ ( x , y ) f u n c t i o n was t o d e t e c t the l o c a l maxima of the f u n c t i o n . However, the peak response a s s o c i a t e d w i t h a wedge a n g l e g r e a t e r t h a n 180° i s not a l o c a l maximum, because the wedge i s the . i n t e r s e c t i o n o f two s t r a i g h t edges, and the s t r a i g h t edge response i s g r e a t e r than the response a s s o c i a t e d w i t h such a wedge. polygon s k e t c h e d i n F i g . 6.3.1(a) i s shown. The responses a s s o c  i a t e d w i t h the f o u r c o r n e r s o f the p o l ygon t h a t are concave w i t h r e s p e c t t o the w h i t e r e g i o n a r e s a d d l e p o i n t s on the x ^ ( x , y ) f u n c t i o n . They a r e not l o c a l maxima. Thus, the procedure d e s c r i b e d i n s e c t i o n 6.2 c o u l d not be used t o t r a c e the boundary of a concave-convex f i g u r e . T h i s problem can be a v e r t e d by s e t t i n g up a second r e c e p t o r network which responds a c c o r d i n g t o the s e t o f e q u a t i o n s n , . • • n 6.3.1 136 F i g . 6.3.1(a) A B l a c k P o l y g o n with, (b) R e c e p t o r A c t i v i t y F u n c t i o n a Convex "Vertex f o r the B l a c k P o l y g o n where e! = e(e.) - e.. ( I n the case of the s i m u l a t o r 1 1 max 1 (e.) = 242 p.p.s.) Note t h a t even w i t h b l a c k and w h i t e p a t t e r n s , 1 max e^, and hence e^, can have any v a l u e i n the range 0< e^£ 242' p.p.s. due t o the f i n i t e s i z e o f t h e r e c e p t o r f i e l d of v i e w . The x^ s e t of e q u a t i o n s are the e q u i v a l e n t of t h e V^(x,y) o p e r a t o r d e s c r i b e d i n c h a p t e r 5. They a r e a l s o analogous to the i n h i b i t o r y - c e n t e r , e x c i t a t o r y - s u r r o u n d neurons found i n the r e t i n a l g a n g l i o n and l a t e r a l g e n i c u l a t e o f t h e cat and 1 (6,11) monkey . I n p r a c t i c e i t was a s i m p l e m a t t e r t o modify the ; . s i m u l a t o r so t h a t b o t h the x ^ x . y ) and the x ^ ( x , y ) a c t i v i t y f u n c  t i o n s were o b t a i n e d i n the course of a s i n g l e s c a n over a pa.ttern. I f the c e n t r a l r e c e p t o r of the rounded 9x9 a r r a y of r e c e p t o r s was i n an i l l u m i n a t e d a r e a , the x^ e q u a t i o n s were used. I f i t was i n a dark a r e a , the x1^ e q u a t i o n s were used, and a marker*Was put put on the punched ' paper tape o u t p u t . The x | ( x , y ) f u n c t i o n f o r t h e b l a c k p o l y g o n of F i g . 6.3.1(a) i s shown i n F i g . 6.3.2(a). At each, p o i n t where the x i ( x , y ) 137 F i g . 6.3.2(a) I s o m e t r i c of the s ! ( x , y ) (b) I s o m e t r i c View o f the F u n c t i o n f o r the Black:. P o l y g o n Combined A c t i v i t y F u n c t i o n f u n c t i o n ( F i g . 6..3.1(b)) had a s a d d l e p o i n t , the x | ( x , y ) f u n c t i o n has a l o c a l maximum. The r e v e r s e i s t r u e as w e l l . The combined a c t i v i t y f u n c t i o n , x ^ ( x , y ) , i s shown i n P i g . 6.3.2(b). T h i s f u n c t i o n i s e q u i v a l e n t t o the V(x,y) o p e r a t o r i n s e c t i o n 5.2. I t i s the c o m b i n a t i o n of the x ^ ( x , y ) and x | ( x , y ) f u n c t i o n s . C o n s equently, t h e r e i s a l o c a l a c t i v i t y maximum a t a l l f i v e c o r n e r s of the polygon. Hence, the a l g o r i t h m o f s e c t i o n 6.2 would t r a c e the boundary of t h i s concave-convex polygon. 6.4 Ridge P o i n t O perators At t h e end of s e c t i o n 6.2 i t was p o i n t e d out t h a t the sequence of l o c a l maxima o f the r e c e p t o r a c t i v i t y f u n c t i o n a l o n g the boundary of a p a t t e r n p r o v i d e s a good d e s c r i p t i o n o f t h a t boundary. There i s another way of d e s c r i b i n g a p a t t e r n boundary i n terms o f r e c e p t o r a c t i v i t y w h i c h i s more d e t a i l e d , and which c o n t a i n s the l o c a l maxima d e s c r i p t i o n . An i s o m e t r i c of the r e c e p t o r a c t i v i t y f u n c t i o n f o r a polygon i s shown i n P i g . 6.4.1. A l i n e has been ske t c h e d on t h i s f i g u r e w hich f o l l o w s the r i d g e i n the r e c e p t o r a c i t i v t y f u n c t i o n F i g . 6.4.1. I s o m e t r i c o f an A c t i v i t y F u n c t i o n w i t h Superimposed Ridge L i n e a d j a c e n t t o the "boundary. The sequence of p o i n t s making up t h i s r i d g e i s c a l l e d the r e c e p t o r a c t i v i t y r i d g e f u n c t i o n , or the r i d g e . f u n c t i o n f o r s h o r t . T h i s r i d g e f u n c t i o n i s the r e c e p  t o r a c t i v i t y e q u i v a l e n t of the V ^ ( s ) f u n c t i o n f o r a boundary d e a l t w i t h i n s e c t i o n 5.4. The sequence of l o c a l maxima d e s c r i b e d e a r l i e r i n t h i s c h a p t e r i s the sequence of t u r n i n g p o i n t s of the r i d g e f u n c t i o n . I n o r d e r t o study the p r o p e r t i e s of the r i d g e f u n c t i o n , and t o see i f i t p r o v i d e s a u s e f u l s t a r t i n g p o i n t f o r a p a t t e r n r e c o g n i t i o n scheme, an a l g o r i t h m f o r e x t r a c t i n g i t was d e v e l o p e d . The f i r s t stage of the a l g o r i t h m d e t e c t s any r i d g e p o i n t s i n the r e c e p t o r a c t i v i t y f u n c t i o n . The second stage t h e n o p e r a t e s on the s e t o f r i d g e p o i n t s t o perform the e x t r a c t i o n . Three b a s i c a l l y s i m i l a r o p e r a t o r s are used f o r the e x t r a c t i o n of r i d g e p o i n t s . They a l l r e l y on the e x a m i n a t i o n of an odd number of c o l l i n e a r p o i n t s t o determine whether the c e n t r a l p o i n t of the l i n e i s a maximum. They d i f f e r i n the number 139 • X ® y c X e P i g . 6.4.2(a) The H o r i z o n t a l Ridge (b) The D i a g o n a l Ridge P o i n t O p erator P o i n t Operator x • • x • • A A © A A • • X • • X (c) The Combined Ridge P o i n t Operator of l i n e s examined and/or the o r i e n t a t i o n of the l i n e s w i t h r e s p e c t t o the r e c e p t o r g r i d . The h o r i z o n t a l , d i a g o n a l and combined o p e r a t o r s a r e shown s c h e m a t i c a l l y i n P i g s . 6.4.2(a), (b) and (c) r e s  p e c t i v e l y . I n the case of t h e h o r i z o n t a l and d i a g o n a l o p e r a t o r s the c e n t r a l p o i n t i s d e s i g n a t e d t o be a r i d g e .point i f i t s a c t i v i t y exceeds a p r e s e t t h r e s h o l d and i s a maximum on a t l e a s t one of the two l i n e s . P o r the combined o p e r a t o r t h e c e n t r a l p o i n t a c t i v i t y must exceed th e t h r e s h o l d and be a maximum on a t l e a s t two of t h e f o u r l i n e s . (A t h r e s h o l d i s i n c l u d e d i n the o p e r a t o r so t h a t u n i f o r m l y i l l u m i n a t e d a r e a s do not g i v e r i s e t o s p u r i o u s r i d g e p o i n t s . ) A contoured d i s p l a y of the p o l ygon a c t i v i t y f u n c t i o n i s shown i n P i g . 6.4.3(a). The s e t o f r i d g e p o i n t s d e t e c t e d on t h i s f u n c t i o n by the h o r i z o n t a l , d i a g o n a l , and combined o p e r a t o r s i s shown i n P i g . 6.4.3(b), ( c ) , and (d) r e s p e c t i v e l y . P o r the 140 • • • • • • • • P i g . 6.4.3(a) Contoured A c t i v i t y F u n c t i o n f o r Polygon (b) Ridge P o i n t s D e t e c t e d by H o r i z o n t a l Operator • • • • • • • • • • • F i g . 6.4.3(c) Ridge P o i n t s D e t e c t e d (d) Ridge P o i n t s D e t e c t e d by by D i a g o n a l Operator • ' Combined Operator h o r i z o n t a l and d i a g o n a l o p e r a t o r s whenever one of the o p e r a t o r l i n e s i s p a r a l l e l t o a p o r t i o n o f the boundary one g e t s a number of s p u r i o u s r i d g e p o i n t s . For example, i n the case o f the h o r i z o n t a l o p e r a t o r t h e r e are c l u s t e r s of r i d g e p o i n t s a l o n g the h o r i z o n t a l and v e r t i c a l s i d e s of the polygon. S i m i l a r l y , the d i a g o n a l o p e r a t o r g i v e s c l u s t e r s of s p u r i o u s p o i n t s a l o n g the d i a g o n a l s i d e s of the p o l y g o n . T h i s c l u s t e r i n g of r i d g e p o i n t s occurs because, by d e f i n i t i o n , t h e a c t i v i t y of the c e n t r a l p o i n t of an o p e r a t o r l i n e must be g r e a t e r than or e q u a l t o the a c t i v i t y of any o f the r e m a i n i n g p o i n t s . I f an o p e r a t o r l i n e i s 141 F i g . 6.4.4(a) Contoured Receptor A c t i v i t y (b) Ridge P o i n t s D e t e c t e d F u n c t i o n f o r a B l a c k "S" . by H o r i z o n t a l O p e r a t o r F i g . 6 .4 .4(c) Ridge P o i n t s D e t e c t e d (d) Ridge P o i n t s D e t e c t e d by D i a g o n a l O p e r a t o r by Combined Operator p a r a l l e l t o an a c t i v i t y c o n t o u r , the c o n d i t i o n o f e q u a l i t y i s f r e q u e n t l y met by p o i n t s l y i n g o f f t h e a c t u a l r i d g e l i n e of the a c t i v i t y f u n c t i o n . Such p o i n t s a r e d e s i g n a t e d t o be r i d g e p o i n t s by the h o r i z o n t a l or d i a g o n a l o p e r a t o r s . However, the combined o p e r a t o r r e q u i r e s t h a t the c e n t r a l . p o i n t be a maximum a l o n g a t l e a s t two of the o p e r a t o r l i n e s . Hence, i t does not d e s i g n a t e such p o i n t s , t o be r i d g e p o i n t s . T h i s i s e v i d e n t i n 6 .4 .3(b). A contoured d i s p l a y of the r e c e p t o r a c t i v i t y f u n c t i o n i n re sponse to a b l a c k "S" i s shown i n F i g . 6 .4 .4(a). I n F i gs..6 . 4 . 4(b) 142 and (c) the c l u s t e r s of s p u r i o u s r i d g e p o i n t s d e t e c t e d by the h o r i z o n t a l and d i a g o n a l o p e r a t o r s a r e a g a i n p r e s e n t . I n F i g . 6.4.4(d) the combined o p e r a t o r has g i v e n a much c l e a n e r s e t of r i d g e p o i n t s . Note the gap i n d i c a t e d by the arrow. A l t h o u g h t h i s gap a l s o o c c u r s i n F i g . 6 . 4 . 4(c), i t does i l l u s t r a t e a f a u l t of the • combined o p e r a t o r . S i n c e t h e p a t t e r n i s a b l a c k "S", the r e c e p  t o r a c t i v i t y f u n c t i o n a t the p o i n t i n d i c a t e d has a s a d d l e p o i n t of t h e type mentioned i n s e c t i o n 6.3. The combined o p e r a t o r does not d e s i g n a t e such p o i n t s t o be r i d g e p o i n t s s i n c e they are n o r m a l l y a maximum a l o n g o n l y one of t h e o p e r a t o r l i n e s , i n t h i s case the v e r t i c a l l i n e . Hence, i n c h o o s i n g a r i d g e p o i n t o p e r a t o r , t h e r e i s an u n a v o i d a b l e t r a d e - o f f between s p u r i o u s r i d g e p o i n t s and the d e t e c t i o n o f s a d d l e p o i n t s . The r i d g e p o i n t o p e r a t o r s can a l s o be used on t h e combined r e c e p t o r a c t i v i t y f u n c t i o n d i s c u s s e d i n s e c t i o n 6.3. T h i s type of f u n c t i o n f o r a l e t t e r "E" i s shown i n contour i n F i g . 6 .4 .5(a). As one would e x p e c t , t h e r e are a g r e a t number o f s p u r i o u s r i d g e p o i n t s d e t e c t e d by t h e h o r i z o n t a l o p e r a t o r , F i g . 6 .4 .5(b). The p o i n t s d e t e c t e d by the d i a g o n a l and the com bin e d o p e r a t o r s a r e shown i n F i g s . 6 .4 .5(c) and (d) r e s p e c t i v e l y . As u s u a l , the combined o p e r a t o r g i v e s the c l e a n e s t s e t o f r i d g e p o i n t s . There a r e , however, gaps a t two c o r n e r s on the l e t t e r . I n the next s e c t i o n , an a l g o r i t h m i s d e s c r i b e d which o p e r a t e s on a g i v e n s e t of r i d g e p o i n t s t o e x t r a c t the r i d g e f u n c t i o n . 143 P i g . 6.4.5(a) Combined Receptor A c t i v i t y (b) Ridge P o i n t s D e t e c t e d F u n c t i o n f o r an "E" i n Contour by H o r i z o n t a l O p e r a t o r F i g . 6.4.5(c) Ridge P o i n t s D e t e c t e d (d) Ridge P o i n t s D e t e c t e d by by D i a g o n a l Operator Combined Operator 6.5 The "Ridge-Runner" A l g o r i t h m The " r i d g e - r u n n e r " a l g o r i t h m d e s c r i b e d i n t h i s s e c t i o n t r a c e s around a p a t t e r n boundary by f o l l o w i n g the l i n e of maximum a c t i v i t y t h r o u g h a s e t of r i d g e p o i n t s . The a c t i v i t y a t each p o i n t on t h i s l i n e i s noted, and the r e s u l t i n g a c t i v i t y sequence forms the r i d g e f u n c t i o n f o r the p a r t i c u l a r p a t t e r n . The t r a c i n g a l g o r i t h m o p e r a t e s on a " s e a r c h and d e s t r o y " p r i n c i p l e . Assume t h a t the t r a c e i s p a r t i a l l y complete, h a v i n g t h a r r i v e d a t the j p o i n t , P., on the r i d g e l i n e . A l l the p o i n t s i n a sqiiare of s i d e n, n odd, c e n t e r e d on P.. , are examined. A l i s t i s made of the l o c a t i o n s o f a l l the r i d g e p o i n t s encountered The P. l i s t i s compared w i t h the l i s t c ompiled f o r the p r e v i o u s 3 p o i n t , > "to see i f i t c o n t a i n s any p o i n t s not i n the P^ . l i s t . I f i t does, the p o i n t i n t h i s s u b l i s t h a v i n g the g r e a t e s t a c t i v i t y i s chosen, and i t becomes ^ n o t , the p o i n t i n the complete P. l i s t h a v i n g the g r e a t e s t a c t i v i t y - i s chosen 3 and becomes P . , . I n b o t h cases the l o c a t i o n and a c t i v i t y o f J + l J. P . i i s noted i n the r i d g e f u n c t i o n t a b l e and t h e n a z e r o i s i n s e r t e d i n the P-,n l o c a t i o n . The p o i n t P.,n i s d e a l t w i t h i n J + l 0+1 the same manner as P.. 3 S e t t i n g the a c t i v i t y of r i d g e - l i n e p o i n t s t o z e r o a f t e r t hey have been d e t e c t e d p r e v e n t s the a l g o r i t h m from d o u b l i n back a l o n g the r i d g e - l i n e . A z e r o i s a l s o i n s e r t e d i n t o a l l l o c a t i o n s i n the l i s t . T h i s tends t o wipe out s p u r i o u s 3 r i d g e p o i n t s which can cause problems i f a gap i n the r i d g e - l i n e i s e ncountered. G i v e n the above b a s i c p r o c e d u r e , the " r i d g e - r u n n e r " a l g o r i t h m s t a r t s a t the p o i n t of maximum a c t i v i t y on the s e t o f r i d g e p o i n t s . I t proceeds a l o n g the r i d g e - l i n e ' f o r a p r e s e t number of points, I t t h e n i n s e r t s the a p p r o p r i a t e a c t i v i t y v a l u e s back i n t o the l o c a t i o n s of the f i r s t few p o i n t s on the r i d g e - l i n e . I t goes back t o the p o i n t where i t stopped and proceeds as d e s c r i b e d above except t h a t i t now checks each new p o i n t on the r i d g e - l i n e t o see i f i t i s the s t a r t i n g p o i n t . I f i t i s , t h e a l g o r i t h m i s f i n i s h e d and the r i d g e f u n c t i o n f o r the p a t t e r n boundary sh o u l d be c o n t a i n e d i n the r i d g e f u n c t i o n t a b l e . 145 The above paragraphs p r o v i d e a d e s c r i p t i o n of the b a s i c o p e r a t i n g p r i n c i p l e s of the " r i d g e - r u n n e r " a l g o r i t h m . I n implementing t h i s a l g o r i t h m on the computer we found t h a t some s p e c i a l r o u t i n e s had t o be i n c l u d e d . One of t h e s e enables the a l g o r i t h m t o get i n t o and out of v e r y sharp c o r n e r s i n a p a t t e r n boundary, e.g. t h e i n t e r n a l a n g l e s of a "W". I t a l s o p e r m i t s i t t o get back on the r i d g e - l i n e i f i t has s t r a y e d . Another r o u t i n e a l l o w s the a l g o r i t h m t o span gaps i n the r i d g e - l i n e . S t i l l a t h i r d i s used i f the p a t t e r n has more th a n one boundary, e.g. the b l o c k l e t t e r s " 0 " and "B". The a l g o r i t h m has been t e s t e d on the r i d g e p o i n t s e t s of numerous p a t t e r n s . I t works e q u a l l y w e l l on s e t s o b t a i n e d from the combined a c t i v i t y f u n c t i o n or t h e i n d i v i d u a l f u n c t i o n s . I t works b e s t when the number o f s p u r i o u s r i d g e p o i n t s i s kept t o a minimum, but w i l l work even i n such extreme cases as t h a t of F i g . 6.4.5(h). The p ath t r a c e d by the a l g o r i t h m through t h i s s e t o f r i d g e p o i n t s i s shown i n F i g . 6.5.1(a). Note the l i n e r u n n i n g o f f from the bottom bar on the "E". The a l g o r i t h m s t r a y e d from the r i d g e - l i n e h e r e , and t h e n r e t u r n e d . The p o i n t s p i c k e d up a l o n g t h i s s p u r i o u s l i n e would not be i n c l u d e d i n the r i d g e f u n c t i o n . The p a t h t r a c e d t h r o u g h the s e t o f r i d g e p o i n t s i n F i g . 6.4.5(d) i s shown i n F i g . 6.5.1(b). Note t h a t t h i s p a t h i s b a s i c a l l y s i m i l a r t o t h a t of F i g . 6.5.1(a). Two more examples o f paths t r a c e d by the a l g o r i t h m are g i v e n i n F i g s . 6.5.1(c) and ( d ) . I n t h i s and the p r e c e e d i n g s e c t i o n we have demonstrated an a l g o r i t h m f o r o b t a i n i n g the r i d g e f u n c t i o n o f a r e c e p t o r a c t i v i t y 146 » P i g . 6.5.1(a) R i d g e - l i n e Traced Through, (b) R i d g e - l i n e Traced through. Ridge P o i n t s of P i g . 6.4.5(b) Ridge P o i n t s of Pig.6..4.5(d F i g . 6.5.1(c) R i d g e - L i n e f o r P o i n t s (d) R i d g e - L i n e f o r P o i n t s i n F i g . .6.4.3(d) i n F i g . 6.4.4(c) f u n c t i o n . In the next c h a p t e r we d i s c u s s the p r o p e r t i e s of t h i s r i d g e f u n c t i o n and suggest some p o s s i b l e methods f o r u s i n g i t i n a p a t t e r n r e c o g n i t i o n scheme. 147 7. THE RIDGE FUNCTION AND APPLICATIONS 7.1 I n t r o d u c t i o n I n t h i s c h a p t e r vie p r e s e n t a number of examples of r i d g e f u n c t i o n s o b t a i n e d as a r e s u l t of the p r o c e s s i n g d e s c r i b e d i n the l a s t c h a p t e r . We p o i n t out how v a r i o u s f e a t u r e s o f these r i d g e f u n c t i o n s r e l a t e t o f e a t u r e s on the pa.tterns from which they were o b t a i n e d . I n p a r t i c u l a r we show how the r i d g e f u n c t i o n r e l a t e s t o the c u r v a t u r e of the a p t t e r n . T h i s g i v e s us a p r a c t i c a l d e m o n s t r a t i o n o f some o f the theorems i n c h a p t e r 5. F i n a l l y , we d i s c u s s some of the i m p l i c a t i o n s these r i d g e f u n c t i o n s may have i n the f i e l d o f p a t t e r n r e c o g n i t i o n . 7•2 The Re c e p t o r A c t i v i t y Ridge F u n c t i o n I n c h a p t e r 5 some o f t h e t h e o r y a s s o c i a t e d w i t h a r e a o p e r a t o r s was dev e l o p e d . I t was shown t h a t near p o i n t s on the boundary a t which the c u r v a t u r e f u n c t i o n had a non-zero l o c a l maximum or minimum, the V ^ ( s ) f u n c t i o n a l s o had a l o c a l maximum or minimum. I n c h a p t e r 4 we saw t h a t -the r e c e p t o r a c t i v i t y i n a l a t e r a l i n h i b i t o r y network can be modelled t o a f i r s t a p p r o x i m a t i o n by an a r e a o p e r a t o r . Thus, some o f the t h e o r y o f a r e a o p e r a t o r s s h o u l d be a p p l i c a b l e . I n p a r t i c u l a r , t he V ^ ( s ) f u n c t i o n s h o u l d have a r e c e p t o r a c t i v i t y e q u i v a l e n t . As i n d i c a t e d i n c h a p t e r 6, the r e c e p t o r a c t i v i t y r i d g e f u n c t i o n i s t h i s e q u i v a l e n t . . In o r d e r t o demonstrate t h a t the t h e o r y f o r the V ^ ( s ) f u n c t i o n can be a p p l i e d t o the r i d g e f u n c t i o n , c o n s i d e r an example. The r i d g e l i n e p a t h f o r a l e t t e r "G" t r a c e d by the a l g o r i t h m o f 148 F i g . 7.2.1 The R i d g e - l i n e P a t h and Ridge F u n c t i o n f o r a B l a c k "G" s e c t i o n 6.5, and the r e s u l t i n g r i d g e f u n c t i o n , a re g i v e n i n F i g . 7.2.1. (The s m a l l c i r c l e i n d i c a t e s the s t a r t and f i n i s h o f the c l o c k w i s e t r a c e . The numbers i n d i c a t e e q u i v a l e n t p o i n t s on the p a t h and f u n c t i o n . ) The l e t t e r was b l a c k on a w h i t e back ground. Hence the r e g i o n w i t h i n the path i s an r e g i o n ; o u t  s i d e i s an M-^  r e g i o n . The s m a l l t a b s a t the b e g i n n i n g and end of the r i d g e f u n c t i o n i n d i c a t e the nominal s t r a i g h t edge v a l u e , ft/2. The r i d g e l i n e p a t h b e g i n s a t a l o c a l maximum of c u r v a t u r e . By Theorem 5.4.5 the r i d g e f u n c t i o n i s g r e a t e r t h a n ft/2; by Theorem 5.4.7 i t has a l o c a l maximum. As the path proceeds i n a c l o c k w i s e d i r e c t i o n away from the s t a r t i n g p o i n t , the boundary 149 becomes l o c a l l y concave. By Theorem 5.4.2 the r i d g e f u n c t i o n s h o u l d , and does, decrease u n t i l i t i s l e s s than. JC/2. (The " n o i s i n e s s " of the f u n c t i o n i s caused by the d i s c r e t e n a t u r e of the r e c e p t o r a r r a y . ) I t remains below at/2 a l l the way around the concave e x t e r i o r of the l e t t e r . At the end of t h i s e x t e r i o r p o r t i o n o f the "G-", a t (3) , the boundary has two c o n s e c u t i v e l o c a l minima of c u r v a t u r e . The r i d g e f u n c t i o n by Theorem 5.4.7 s h o u l d have two l o c a l minima. I t does. (Note t h a t the magnitudes of t h e s e minima r e f l e c t the magnitude of the change i n the tangent t o t h e boundary a t t h e s e two p o i n t s . See below.) The p a t h a l o n g the boundary becomes l o c a l l y convex. The v a l u e o f the r i d g e f u n c  t i o n i n c r e a s e s u n t i l i t i s , on the average, g r e a t e r t h a n Jt/2 (Theorem 5.4.5). The path proceeds around the i n s i d e of the "G" u n t i l a n o t h e r l o c a l maximum of c u r v a t u r e i s encountered a t (5). The r i d g e f u n c t i o n has a l o c a l maximum at (5) (Theorem 5-4.7). (Note the d i f f e r e n c e i n magnitude between t h i s l o c a l maximum and the one a t the b e g i n n i n g o f the f u n c t i o n . ) I n a s i m i l a r f a s h i o n the f e a t u r e s on the remainder of the boundary of the "G-" are m i r r o r e d by the c o r r e s p o n d i n g i n d i c a t o r s ( l o c a l maxima and minima) on the r i d g e f u n c t i o n . As noted above i n p a r e n t h e s e s , the r i d g e f u n c t i o n , and indeed the V ^ ( s ) f u n c t i o n c o n t a i n s more i n f o r m a t i o n about an i l l u m i n a t i o n boundary t h a n i s i n d i c a t e d by the t h e o r y i n c h a p t e r 5. The experiments i n c h a p t e r 3 demonstrated t h a t the peak r e c e p t o r a c t i v i t y near a wedge v e r t e x gave a r e a s o n a b l y a c c u r a t e measure of the a n g l e . Thus the magnitudes of the i n d i c a t o r s on the r i d g e f u n c t i o n s h o u l d v a r y depending on the wedge a n g l e w i t h which they, are a s s o c i a t e d . C o n s i d e r the l e t t e r M and i t s r i d g e f u n c t i o n '150 F i g . 7.2.2 The R i d g e - L i n e and Ridge F u n c t i o n f o r an "M" shown i n F i g . 7.2.2. (The s m a l l c i r c l e on the boundary a g a i n i n d i c a t e s the b e g i n n i n g of the c l o c k w i s e t r a c e . ) The wedges marked ( l ) and (3 ) have e q u a l a n g l e s t h a t are l e s s t h a n the a n g l e of the wedge a t ( 2 ) . S i m i l a r l y , the l o c a l maxima of the r i d g e f u n c t i o n a t (l) and (3) a r e g r e a t e r t h a n t h a t a t ( 2 ) . Theorem 5.4.1 i s a theorem f o r a r e a o p e r a t o r s t h a t i s made t a n g i b l e by an e x a m i n a t i o n of r i d g e f u n c t i o n s . P a r a p h r a s e d , the theorem s t a t e s t h a t the r e c e p t o r a c t i v i t y f u n c t i o n a s s o c i a t e d w i t h a b l a c k - w h i t e p a t t e r n i s determined by the g e o m e t r i c a l shape of the curve s e p a r a t i n g the b l a c k and w h i t e r e g i o n s . . The theorem has a number o f i m p l i c a t i o n s about the p r o p e r t i e s of r i d g e , f u n c  t i o n s . F i r s t , t he r i d g e f u n c t i o n o f a p a t t e r n must be independent of the p a t t e r n o r i e n t a t i o n . I n F i g . 7.2.3(a) the r i d g e l i n e s and r i d g e f u n c t i o n s f o r a b l a c k "N" and "Z" a r e shown. I t i s e v i d e n t t h a t t h e r i d g e f u n c t i o n s are i d e n t i c a l i n the type and sequence of l o c a l maxima and minima. But the l e t t e r s a r e a l s o i d e n t i c a l , one b e i n g r o t a t e d 90° w i t h r e s p e c t t o the o t h e r . 151 V F i g . 7.2.3(a) The R i d g e - L i n e s and Ridge F u n c t i o n s f o r an "N" and a "Z" F i g . 7.2.3(h) The R i d g e - L i n e s and Ridge F u n c t i o n s f o r an "M" and a "W" S i m i l a r l y , i n F i g . 7.2.3(b) the r i d g e f u n c t i o n s f o r "M" and "W" have an i d e n t i c a l sequence of major f e a t u r e s . A second and more i n t e r e s t i n g i m p l i c a t i o n of Theorem 5.4.1 i s t h a t i f a p a t t e r n has an a x i s of symmetry, t h e r i d g e f u n c t i o n must have two p o i n t s . a b o u t which i t i s symmetric. ( I n m a t h e m a t i c a l t e r m i n o l o g y the f u n c t i o n i s s a i d t o be even w i t h r e s p e c t t o t h e s e two p o i n t s . ) Two examples of symmetric p a t t e r n s and t h e i r r i d g e f u n c t i o n s are g i v e n i n F i g . 7.2.4. The s t a r t i n g p o i n t , ( l ) , f o r t h e r i d g e f u n c t i o n of the l e t t e r "C" i n F i g . 7.2.4(a) i s on the a x i s o f symmetry i n d i c a t e d by the l i n e . The r i d g e f u n c t i o n i s symmetric about the b e g i n n i n g , and t h e m i d d l e . The 152 F i g . 7.2.4(a) The Ridge F u n c t i o n f o r •(h) The Ridge F u n c t i o n f o r a a L e t t e r w i t h One A x i s of L e t t e r w i t h Two Axes o f Symmetry Symmetry l e t t e r "X" i n F i g . 7.2.4(h) has two axes of symmetry as i n d i c a t e d . The r i d g e f u n c t i o n c o n s e q u e n t l y has f o u r p o i n t s o f symmetry, namely the f o u r l a r g e l o c a l maxima. I t i s e v i d e n t t h a t the r i d g e f u n c t i o n i n d i c a t e s b o t h the presence and the number of axes of symmetry i n a p a t t e r n . Another consequence of Theorem 5.4.1 i s t h a t i f two or more p a t t e r n s have a" common subsequence o f boundary f e a t u r e s , t h e y must have a c e r t a i n s e c t i o n o f t h e i r r i d g e f u n c t i o n s i n common. The r i d g e l i n e paths and r i d g e f u n c t i o n s f o r the l e t t e r s "E" and "F" are g i v e n i n F i g . 7.2.5. I f one proceeds from ( l ) i n a c l o c k w i s e d i r e c t i o n a l o n g the boundary t o ( 6 ) , the same sequence of boundary f e a t u r e s i s encountered i n b o t h c a s e s . Sim.ila.rly, the r i d g e f u n c t i o n s f o r the two l e t t e r s have a common sequence o f i n d i c a t o r s between ( l ) and ( 6 ) . There a re one or two o t h e r p o i n t s about r i d g e f u n c t i o n s t h a t s h o u l d be s e l f - e v i d e n t . F i r s t , t he r i d g e f u n c t i o n f o r a c l o s e d c ontour i s p e r i o d i c . Second, s i n c e a c l o s e d f i g u r e may have more th a n one c l o s e d c o n t o u r , more than one r i d g e f u n c t i o n 153 P i g , 7.2.5 Two l e t t e r s w i t h a Common Sequence.of F e a t u r e s may be n e c e s s a r y t o d e s c r i b e a f i g u r e a d e q u a t e l y . In. F i g . 7.2.6 t h e r i d g e f u n c t i o n s f o r the o u t s i d e and i n s i d e c o n t o u r s of the l e t t e r "0" are shown. The two f u n c t i o n s are s e p a r a t e d by a s h o r t l i n e i n d i c a t i n g the nominal s t r a i g h t edge v a l u e , TC/2. Note t h a t the average v a l u e o f the f u n c t i o n f o r the o u t s i d e , concave c o n t o u r i s l e s s t h a n i t / 2 , whereas f o r the i n s i d e , convex contour i t i s g r e a t e r t h a n %/2. F i g . 7.2.6 The Two Ridge F u n c t i o n s f o r an "0" 154 7.3 Three Types of Ridge F u n c t i o n s I n s e c t i o n 5.2 t h r e e d i f f e r e n t g e n e r a l i z e d a r e a o p e r a t o r s were proposed. Two of t h e s e , V-^(x,y) and " ^ ( ^ y ^ ^ a d d i r e c t n e u r o p h y s i o l o g i c a l c o u n t e r p a r t s . The t h i r d , V ( x , y ) , was s i m p l y a c o m h i n a t i o n of the f i r s t two such t h a t V (x,y) = m a x ( ( Y 1 ( x , y ) , V 2 ( x , y ) ) . 7.3.1 I f one c o n s i d e r s o n l y b l a c k and w h i t e p a t t e r n s , . t h e V^(x,y) o p e r a t o r i s a model f o r a H a r t l i n e l a t e r a l i n h i b i t o r y r e c e p t o r network w i t h a c t i v i t y , x^, d e f i n e d by the s e t of e q u a t i o n s 1,4,1. S i m i l a r l y , t h e v" 2(x,y) o p e r a t o r i s a model f o r the l a t e r a l i n h i b i t o r y r e c e p t o r network d e s c r i b e d i n s e c t i o n 6.3, the a c t i v i t y , x|, b e i n g d e f i n e d by the s e t of e q u a t i o n s 6.3.1. F i n a l l y , the V ( x , y ) o p e r a t o r s e r v e s as a model f o r the c o m b i n a t i o n of the above two l a t e r a l i n h i b i t o r y r e c e p t o r networks. The a c t i v i t y , X^, of t h i s combined network i s s p e c i f i e d by •X±(x,y) = m a x ( x ± ( x , y ) , x | ( x , y ) ) . 7.3.2 Thus, f o r any g i v e n b l a c k and w h i t e p a t t e r n i t i s p o s s i b l e t o o b t a i n t h r e e r e c e p t o r a c t i v i t y f u n c t i o n s . A w h i t e "H" on a w h i t e background g i v e s r i s e t o the t h r e e a c t i v i t y f u n c t i o n s shown i n i s o m e t r i c v i e w i n F i g s . 7.3.1 ( a ) , (b) and ( c ) . I f any of these f u n c t i o n s are o p e r a t e d on by t h e a l g o r i t h m s d e s c r i b e d i n s e c t i o n s 6.4 and 6.5, a r i d g e f u n c t i o n i s o b t a i n e d . The r i d g e f u n c t i o n s o b t a i n e d from the a c t i v i t y f u n c t i o n s i n F i g . 7 .3 .1(a), (b) and (c) a r e shown i n F i g s . 7 .3 .2(a), (b) and ( c ) r e s p e c t i v e l y . ' I f the f u n c t i o n i n F i g . 7.3.2(a) i s r e f l e c t e d t h r o u g h a l i n e a t the s t r a i g h t edge v a l u e , one o b t a i n s a f u n c t i o n s i m i l a r t o t h a t of F i g . 7 .3 .2(b). I n a sense, they are m i r r o r 155 F i g . 7.3.1(a) I s o m e t r i c of x ^ ( x , y ) (b) I s o m e t r i c of x | f o r a f o r a White "H" White "H" F i g . 7.3.1(c) I s o m e t r i c o f X_^(x,y) f o r a White "H" images. The r i d g e f u n c t i o n i n F i g . 7.3.2(c) i s a c o m b i n a t i o n of the l o c a l maxima s p i k e s of the o t h e r two f u n c t i o n s . T h i s l a s t f a c t i n d i c a t e s a l o s s of i n f o r m a t i o n i n t h a t a l o c a l maximum on the X^(x,y) r i d g e f u n c t i o n can correspond t o a l o c a l minimum of c u r v a t u r e o f the boundary. However, assume t h a t i n a d d i t i o n t o t a b u l a t i n g the magnitude of the p o i n t s making up the r i d g e f u n c t i o n , the " r i d g e - r u n n e r " a l g o r i t h m a l s o r e c o r d s whether they come from the x^ or x^ a c t i v i t y f u n c t i o n s . W i t h . t h i s e x t r a b i t of i n f o r m a t i o n i t i s p o s s i b l e t o make the X^(x,y) r i d g e f u n c t i o n i n F i g . 7.3.2(c) t a k e on the form o f e i t h e r of 156 U F i g . 7.3.2(a) The x ± ( x , y ) Ridge (b) The xj_(x,y) Ridge F u n c t i o n F u n c t i o n f o r a White "H" F i g . 7.5.2(c) ' The X (x,y) Ridge (d) Transformed X ± ( x , y ) F u n c t i o n Ridge F u n c t i o n t h e o t h e r two. For example i f each of the x^ p o i n t s i s s u b t r a c t e d from t w i c e the no m i n a l edge v a l u e , one o b t a i n s the f u n c t i o n shown i n F i g . 7.3.2(d). T h i s f u n c t i o n i s p r a c t i c a l l y i d e n t i c a l w i t h t h e one shown i n F i g . 7.3.2(b). The o n l y advantage i n w o r k i n g w i t h the X (x,y). r i d g e f u n c t i o n i s t h a t the r i d g e - r u n n e r a l g o r i t h m can be made more e f f i c i e n t . 157 7.4 The Ridge F u n c t i o n and P a t t e r n R e c o g n i t i o n I n the p r e v i o u s s e c t i o n s i t has been shown t h a t t h e r e i s a one-to-one c o r r e l a t i o n between the major f e a t u r e s on a p a t t e r n and the major f e a t u r e s , or i n d i c a t o r s , on i t s r i d g e f u n c t i o n . I t i s p o s s i b l e t h e n t h a t the r i d g e f u n c t i o n d e s c r i p t i o n of p a t t e r n s might form a b a s i s f o r a v e r y g e n e r a l p a t t e r n r e c o g  n i t i o n a l g o r i t h m . I n t h i s s e c t i o n f o u r d i f f e r e n t methods f o r i n c o r p o r a t i n g r i d g e f u n c t i o n s i n t o such an a l g o r i t h m a r e d i s c u s s e d . A l t h o u g h the p a t t e r n samples d e a l t w i t h are b l o c k c a p i t a l l e t t e r s , c h a r a c t e r r e c o g n i t i o n i s not the g o a l . The methods d i s c u s s e d a r e i n no'way e x p l i c i t l y d e s i g n e d f o r t h i s p a t t e r n s e t . They can be a p p l i e d i n g e n e r a l t o any p a t t e r n t h a t w i l l g i v e r i s e t o a'rid'ge f u n c t i o n or f u n c t i o n s . B e f o r e d e a l i n g w i t h the f o u r methods a few samples of r i d g e f u n c t i o n s from s i m i l a r p a t t e r n s are p r e s e n t e d . The p a t t e r n s were w h i t e a l p h a b e t i c c h a r a c t e r s on a b l a c k background. They were chosen from a v a r i e t y o f L e t t r a s e t a l p h a b e t s . In. F i g s . 7.4.1 t o 7.4.3 f i v e samples each of the l e t t e r s " P , "X", and "H" are g i v e n a l o n g w i t h the c o r r e s p o n d i n g r i d g e f u n c t i o n s . I n F i g s . 7.4.4 t o 7.4.6 f i v e samples of the l e t t e r s "C", "U", and " J " are p r e  sented. I n a l l the f i g u r e s the s t a r t and end of the r i d g e f u n c t i o n have a s m a l l tab i n d i c a t i n g t h e nominal s t r a i g h t edge v a l u e . The b e g i n n i n g of the c l o c k w i s e p ath around the p a t t e r n s i s i n d i c a t e d t y . a (1). One p o s s i b l e method f o r i n c o r p o r a t i n g r i d g e f u n c t i o n s i n t o a p a t t e r n r e c o g n i t i o n a l g o r i t h m a r i s e s from t h e i r p e r i o d i c n a t u r e . I f the r i d g e f u n c t i o n i s denoted by R ( s ) w i t h p e r i o d T R, i t can be expanded i n a F o u r i e r s e r i e s 158 © (a) l e t t r a s e t A l p h a b e t #109 (b) l e t t r a s e t A l p h a b e t #756 I M .A H ft (D * !/ © J u ® I. (D 1® (c) L e t t rase_t A l p h a b e t #171 (d) L e t t r a s e t A l p h a b e t #587 f 7 © fi A A A ft M H ® - y© u ® (e) L e t t r a s e t Alphabet #183 P i g . 7.4.1 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples o f the L e t t e r "K", White on B l a c k 159 (a) L e t t r a s e t A l p h a b e t #109 (L) L e t t r a s e t A l p h a b e t #756 ( c ) L e t t r a s e t A l p h a b e t #171 (d) L e t t r a s e t A l p h a b e t #587 ry® I 'I (e) L e t t r a s e t Alphabet #183 F i g . 7.4.2 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples o f the L e t t e r "X", White on B l a c k 160 7)u (a) L e t t r a s e t A l p h a b e t #109 (b) L e t t r a s e t A l p h a b e t #756 @ ® ® — AH |.V© (c) L e t t r a s e t A l p h a b e t #171 n @ LnJ (d) L e t t r a s e t A l p h a b e t #587 ®- I ® y ® (e) L e t t r a s e t Alphabet #183 F i g . 7.4.3 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r "H", White on B l a c k 161 (a) L e t t r a s e t A l p h a b e t #109 ©- © © (c) L e t t r a s e t A l p h a b e t #171 ©— (D ® (b) L e t t r a s e t A l p h a b e t #756 ©— © If © (d) L e t t r a s e t A l p h a b e t #587 (e) L e t t r a s e t Alphabet #183 P i g . 7.4.4 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r "C", White on B l a c k 162 CD— @ w ' © ®- © " © (a) L e t t r a s e t Alphabet #109 (b) L e t t r a s e t A l p h a b e t #756 ®- © ^ ® ® — © , ¥ © (c) L e t t r a s e t A l p h a b e t #171 (d) L e t t r a s e t A l p h a b e t #587 © - © v © (e) L e t t r a s e t Alphabet #183 . 7.4.5 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r "U", White on B l a c k 163 c 7 - 0 J © — © (a) L e t t r a s e t A l p h a b e t #109 (b) l e t t r a s e t A l p h a b e t #756 © Oh* ©V 0 A ® — ©1 .© (c) L e t t r a s e t A l p h a b e t #171 (d) L e t t r a s e t A l p h a b e t #587 ©-> ©— a>y © (e) l e t t r a s e t Alphabet #183 7.4.6 The R i d g e - L i n e s and Ridge F u n c t i o n s f o r F i v e Samples of the L e t t e r " J " , White on B l a c k 7.4.2 164 CO R(s) = / (a c o s n u D s + b sinnWps) 7.4.1 E=~0 n n K where T p/2 T R/2 1 / 2 /" a Q = ~ / R(s)ds , .a n = ^ / R ( s ) c o s n ^ R s ds . R -T R/2 R -T R/2 T R/2 b = ^— / R ( s ) s i n n w R s ds R =4/2 and ^>R = |^ 7.4.3 E q u a t i o n 7.4.1 can be r e w r i t t e n as R ( s ) - YL cos(no;.Rs - 0 ) 7.4.4 n=6 where c n = ( a ^ + b 2 ) l / 2 7.4.5 and t a n 0 = ^n 7.4.6 a n V a r i o u s people ' have suggested t h a t a v e c t o r space r e p r e s e n t a t i o n f o r p a t t e r n s p r o v i d e s a convenient m a t h e m a t i c a l •framework f o r r e c o g n i t i o n a l g o r i t h m s . The F o u r i e r s e r i e s r e p r e  s e n t a t i o n f o r R ( s ) t i e s i n v e r y n e a t l y i n t h a t the components o f the s e r i e s form an o r t h o g o n a l s e t of b a s i s v e c t o r s . The s e t of c o e f f i c i e n t s £c ^ f o r a f u n c t i o n R(s) d e f i n e a p o i n t i n t h i s v e c t o r space. T h i s p o i n t c orresponds t o the p a t t e r n from which the R ( s ) r i d g e f u n c t i o n was o b t a i n e d . H o p e f u l l y , s i m i l a r p a t t e r n s , e.g. the f i v e d i f f e r e n t v e r s i o n s o f the l e t t e r "K" i n F i g . 7.4.1, occupy n e i g h b o r i n g p o i n t s i n t h i s v e c t o r space.. I f so, t h i s " c l u s t e r " should be s e p a r a b l e from the " c l u s t e r " c o r r e s p o n d i n g t o an "X", or an "H", e t c . The F o u r i e r s e r i e s e x p a n sion a l l o w s one t o determine a 165 g r e a t d e a l about the symmetry of the p a t t e r n from which the R ( s ) f u n c t i o n was o b t a i n e d . I n s e c t i o n 7.2 i t was p o i n t e d out t h a t g i v e n a p a t t e r n w i t h m axes of symmetry, the c o r r e s p o n d i n g r i d g e f u n c t i o n would have 2m p o i n t s about which i t i s an even f u n c t i o n . I f such a f u n c t i o n i s expanded i n a F o u r i e r s e r i e s , the component phase s h i f t s a r e r e l a t e d as l j 0± - i 0 j i = hit , h=0,l,2 f 7.4.7 where 0 ^ , 0 . e | 0 ^ . Thus, the presence of an a x i s of symmetry i n a p a t t e r n can be d e t e c t e d by examining the phase s h i f t s o f the components i n the F o u r i e r s e r i e s e x p a n s i o n of i t s r i d g e f u n c  t i o n . R o t a t i o n a l symmetry i n a p a t t e r n can be d e t e c t e d by . an e x a m i n a t i o n of the s e t o f c o e f f i c i e n t s { c n j i n . the F o u r i e r e x p a n s ion o f i t s r i d g e f u n c t i o n . The o r d e r , k, o f r o t a t i o n a l symmetry i n a p a t t e r n i s d e f i n e d t o be k = & 7.4.8 where a i s the minimum a n g l e o f r o t a t i o n necessary, f o r s e l f - c o n g r u e n c e . Thus, a l l p a t t e r n s have at. l e a s t f i r s t o r d e r r o t a t i o n a l symmetry. t h A p a t t e r n w i t h m axes o f symmetry has m o r d e r r o t a t i o n a l symmetry. F o r example, the "X" i n F i g . 7.4.7(a) i s s e l f - c o n g r u e n t i f r o t a t e d through, an a n g l e it. I t thus has second o r d e r r o t a t i o n a l symmetry. I t a l s o has two axes o f symmetry. However, a f i g u r e may have no axes of symmetry and y e t have h i g h e r o r d e r r o t a t i o n a l symmetry. The "N" and the s w a s t i k a i n F i g s . 7.4.7(b) and (c) have no axes of symmetry, but have second and f o u r t h o r d e r r o t a t i o n a l symmetry, r e s p e c t i v e l y . C o n s i d e r f o r a moment the "X" and the "N" i n F i g . 7.4.7. 166 F i g o 7 . 4 . 7(a) The L e t t e r "X" has Second Order R o t a t i o n a l Symmetry and Tv/o Axes of Symmetry (t>) The L e t t e r "N" has Second Order R o t a t i o n a l Symmetry F i g . 7 . 4 . 7(c) The S w a s t i k a has F o u r t h Order R o t a t i o n a l Symmetry Assume one s t a r t s a t the p o i n t ( l ) on each of the f i g u r e s and proceeds i n a c l o c k - w i s e f a s h i o n a l o n g the boundary t o the p o i n t (2). The sequence of f e a t u r e s encountered i s , i n b o t h c a s e s , e x a c t l y t h e same as i f one had s t a r t e d a t (2) and proceeded c l o c k  wise t o ( l ) . Indeed, i f the l e n g t h of t h e boundary i s n o r m a l i z e d t o 2ft u n i t s , the same s i t u a t i o n h o l d s f o r any two p o i n t s a d i s t a n c e jt u n i t s a p a r t a l o n g the boundary. T h i s i m p l i e s t h a t the r i d g e f u n c t i o n , which i s always p e r i o d i c w i t h p e r i o d T^, must f o r t h e s e f i g u r e s have a more fundamental p e r i o d T^. B o t h f i g u r e s have second o r d e r r o t a t i o n a l symmetry, c o n s e q u e n t l y , T-^/T^ - 2. t h I n g e n e r a l i f a f i g u r e has k o r d e r r o t a t i o n a l symmetry, T . = \ 7.4.9 1 k 167 The F o u r i e r s e r i e s e x p a n s i o n f o r i t s r i d g e f u n c t i o n i n t e r r a s o f T R i s g i v e n i n E q u a t i o n s 7.4.1 t o 7.4.3. H o w e v e r , t h e r i d g e f u n c t i o n c a n a l s o h e e x p a n d e d i n a F o u r i e r s e r i e s u s i n g t h e f u n d a m e n t a l p e r i o d , T R ( s ) = / d c o s p w „ s + e s i n p u ) „ s 7.4.10 p^O P f P f Pit w h e r e co „ = m~ 7.4.11 i i f a n d , T f / 2 d Q = ~ - r R ( s ) d s •f - T f / 2 T f /2 d = %r f R ( s ) cospcOpS d s 7.4.12 P f J - T f / 2 T /2 2 r e^ = 7JT~ j R ( s ) s i n p c o ^ s d s - T f / 2 U p o n s u b s t i t u t i o n o f E q u a t i o n 7.4.9 i n t o 7.4.11 we f i n d , w f = k ( | ^ - ) = k cog, 7.4.13 R w h i c h c a n i n t u r n he s u b s t i t u t e d i n t o E q u a t i o n 7.4.10. I f t h e r e s u l t i s e q u a t e d t o E q u a t i o n 7.4.1, one o b t a i n s , ^ a cosntops + s i n n w R s = d c o s k p w R s + e s i n k p w R s 7.4.14 n=0 p=0 p p I f we e q u a t e t h e c o e f f i c i e n t s o f e q u i v a l e n t s i n e a n d c o s i n e t e r m s , we f i n d t h a t t h e a ^ a n d b n v a n i s h f o r a l l n t h a t a r e n o t i n t e g r a l m u l t i p l e s o f k . Hence, i f the r i d g e f u n c t i o n f o r a f i g u r e w i t h k o r d e r r o t a t i o n a l symmetry i s g i v e n hy, . 168 t h t h e n R ( s ) = £ZQ c n c o s ( n w R s - 0 n ) 7 . 4 . 1 5 c n = 0 i f g / i , i = 1,2,... 7.4.16 w C o n v e r s e l y , g i v e n the s e t of F o u r i e r c o e f f i c i e n t s {'c^ s p e c i f y i n g a r i d g e f u n c t i o n , one can determine the o r d e r of r o t a t i o n a l symmetry of the c o r r e s p o n d i n g p a t t e r n . The o r d e r i s the g r e a t e s t common d i v i s o r f o r t h e n a t which t h e c do not v a n i s h . n I f t h i s c o n d i t i o n on the £c n j i s t a k e n i n c o n j u n c t i o n i t h the c o n d i t i o n on the [ 0 n | i n E q u a t i o n 7.4.7, one can d e t e r  mine the number of axes of symmetry of a f i g u r e . I f the ^ c n ^ f o r t h e r i d g e f u n c t i o n o f a p a t t e r n s a t i s f y E q u a t i o n 7.4.16, the p a t t e r n has o r d e r k r o t a t i o n a l symmetry. I f the ^0 n "j s a t i s f y E q u a t i o n 3.4.7, the p a t t e r n a l s o has a t l e a s t one a x i s of symmetry. But i f a p a t t e r n has o r d e r k r o t a t i o n a l symmetry, and a t l e a s t one a x i s of symmetry, i t must have k axes o f symmetry. I f the j~0n^  do not s a t i s f y E q u a t i o n 7.4.7, the p a t t e r n s t i l l has o r d e r k r o t a t i o n a l symmetry, but has no axes of symmetry. F o r example, the [ c n ] f o r b o t h ' t h e "X" and the "N" i n F i g . 7.4.7 sh o u l d s a t i s f y E q u a t i o n 7.4.16 w i t h k = 2, i n d i c a t i n g t h a t they b o t h have second o r d e r r o t a t i o n a l symmetry. However, o n l y the [^ n] f o r the l e t t e r "X" w i l l s a t i s f y E q u a t i o n 7.4.7, s i n c e o n l y i t has a t l e a s t one a x i s o f symmetry. S i n c e b o t h c o n d i t i o n s a r e met by the "X", i t must, and does, have two axes of symmetry. Note t h a t the above d i s c u s s i o n on symmetry can a l s o be a p p l i e d t o the F o u r i e r s e r i e s e x p a n sion o f the c u r v a t u r e f u n c t i o n 169 K ( s ) suggested by Masnikosa . There are a number of o t h e r p o s s i b l e methods b e s i d e s the F o u r i e r s e r i e s f o r i m plementing r i d g e f u n c t i o n s i n t o a p a t t e r n r e c o g n i t i o n a l g o r i t h m . These i n c l u d e the use of c o r r e l a t i o n t e c h n i q u e s , the development of a code word d e s c r i p t i o n o f the f u n c t i o n s , and the use of moments on a t w o - d i m e n s i o n a l r e p r e s e n t a  t i o n f o r the r i d g e f u n c t i o n . The use of c o r r e l a t i o n t e c h n i q u e s would i n v o l v e the s t o r a g e of a " t e m p l a t e " r i d g e f u n c t i o n f o r each of the expected p a t t e r n c l a s s e s . A f t e r s u i t a b l e n o r m a l i z a t i o n , the r i d g e f u n c t i o n from a p a t t e r n would be. c r o s s - c o r r e l a t e d w i t h each of the " t e m p l a t e s " . The unknown p a t t e r n c o u l d t h e n be a s s i g n e d t o the c l a s s w i t h w h i c h i t s r i d g e f u n c t i o n had the g r e a t e s t , weighted c r o s s - c o r r e l a t i o n c o e f f i c i e n t . A code word d e s c r i p t i o n of the f u n c t i o n s c o u l d be (32) ^ 31) developed a l o n g the l i n e s proposed by Clemens , and Masnikosa'-^ . The s i m p l e s t example of such a d e s c r i p t i o n i s one t h a t r e c o r d s the sequence of l o c a l maxima and minima of the f u n c t i o n . - I f a one i n d i c a t e s a l o c a l maximum and a z e r o a l o c a l minimum, the code word f o r the l e t t e r s "K" i n F i g . 7 . 4.1(a), ( c ) , (d) and (e) would be IIIOIIC'1101. The code word f o r the."K" i n (b) would be 111011011001. The code word f o r the l e t t e r s "X" i n F i g . 7.4.2 would be 101101101101; and f o r the l e t t e r s - " H " i n F i g . 7.4.3, 111001111001. Thus, g i v e n a s e t of code words f o r the v a r i o u s p a t t e r n s , r e c o g n i t i o n of an unknown p a t t e r n would i n v o l v e a s i m p l e comparison of code words. The use of moments i s a n o t h e r well-known p a t t e r n • c l a s s i f i c a t i o n t e c h n i q u e . I f the r i d g e f u n c t i o n i s l e f t as a 170 f u n c t i o n o f two v a r i a b l e s , R ( x , y ) , i t c o u l d be t r e a t e d as a l i n e mass of v a r y i n g d e n s i t y . I t would be p o s s i b l e t o o b t a i n a s e t o f n o r m a l i z e d moments f o r t h i s l i n e mass. T h i s s e t c o u l d be t r e a t e d as a v e c t o r — r e c o g n i t i o n of a p a r t i c u l a r p a t t e r n b e i n g dependent on the " c l u s t e r i n g " of p o i n t s i n the moment v e c t o r space. A l l o f the above methods f o r implementing r i d g e f u n c t i o n s i n t o a p a t t e r n r e c o g n i t i o n a l g o r i t h m a r e b e i n g i n v e s t i g a t e d by (•3-5) L. Brown . They by no means exhaust the p o s s i b i l i t i e s . Indeed, the r i d g e f u n c t i o n i t s e l f i s but one o f a number of p o s s i b l e boundary d e s c r i p t i o n s t h a t can be o b t a i n e d from the r e c e p t o r a c t i v i t y f u n c t i o n . 171 8. CONCLUSIONS The a c t i v i t y i n a l a t e r a l i n h i b i t o r y network i s g r e a t e s t where t h e r e i s a s p a t i a l change i n the i l l u m i n a t i o n i m p i n g i n g on i t s r e c e p t o r s . T h i s enhancement of i l l u m i n a t i o n b o u n d a r i e s has been e x t e n s i v e l y s t u d i e d i n the case of b l a c k and w h i t e p a t t e r n s . I n i t i a l s t u d i e s i n d i c a t e d t h a t the magnitude o f the enhancement near a wedge v e r t e x gave a poor measure o f the wedge a n g l e , due t o the o r i e n t a t i o n dependence problem.. T h i s problem, i s g r e a t l y a l l e v i a t e d by employing a . l a r g e r d i r e c t i n h i b i t i o n f i e l d ; by a l l o w i n g a r e c e p t o r t o have a f i n i t e s i z e d f i e l d o f view; and by c o r r e c t c h o i c e of the r e c e p t o r i n t e r a c t i o n c o e f f i c i e n t s . The peak r e c e p t o r a c t i v i t y f o r a rounded 9x9 a r r a y of r e c e p t o r s h a v i n g a f i e l d of view o f D = 1.5u, and a u n i f o r m k..(d) f u n c t i o n , c o u l d be used t o measure a wedge a n g l e t o w i t h i n +5°. I t was shown e x p e r i m e n t a l l y t h a t under c e r t a i n c o n d i t i o n s the peak r e c e p t o r a c t i v i t y n ear a b l a c k - w h i t e boundary can be mode l l e d by a weighted a r e a o p e r a t o r . A g e n e r a l f o r m u l a t i o n f o r weighted a r e a o p e r a t o r s has been p r e s e n t e d . The f o r m u l a t i o n i s such t h a t the response o f the o p e r a t o r i s independent of t h e a b s o l u t e i n t e n s i t y or change i n i n t e n s i t y o f i l l u m i n a t i o n . T h e o r e t i c a l , s t u d i e s demonstrated t h a t f o r two c o n t i g u o u s r e g i o n s of d i f f e r e n t i n t e n s i t y , t h e peak o p e r a t o r response o c c u r r e d i n g e n e r a l a l o n g the boundary. T h i s peak response can be used t o d e t e c t c o n c a v i t y or c o n v e x i t y o f the boundary, and p o i n t s a t wh i c h i t s c u r v a t u r e f u n c t i o n has non-zero l o c a l maxima and minima. S i n c e the a r e a o p e r a t o r was developed as a model f o r the l a t e r a l i n h i b i t o r y network, the same p r o p e r t i e s s h o u l d , 172 and do, h o l d f o r i t . I n p a r t i c u l a r , the peak r e c e p t o r a c t i v i t y a l o n g a b l a c k - w h i t e boundary (the r i d g e f u n c t i o n ) can be used as a d e s c r i p t i o n of the l o c a l shape of the boundary. V a r i o u s methods f o r u s i n g t h i s p r o p e r t y of r i d g e f u n c t i o n s i n a p a t t e r n r e c o g n i t i o n a l g o r i t h m have been proposed. The method i n v o l v i n g a F o u r i e r s e r i e s e x p a n s i o n of the r i d g e f u n c t i o n p e r m i t s the d e t e c t i o n of b o t h b i l a t e r a l and r o t a t i o n a l symmetry i n t h e . o r i g i n a l p a t t e r n . There a r e 'a number of p o s s i b l e e x t e n s i o n s of t h e work d e s c r i b e d i n t h i s t h e s i s : 1. A d e t e r m i n a t i o n o f the u t i l i t y o f the r i d g e f u n c t i o n or some o t h e r d e r i v a t i v e of the r e c e p t o r a c t i v i t y / a r e a o p e r a t o r f u n c t i o n i n v i s u a l p a t t e r n r e c o g n i t i o n ; 2. A s t u d y o f n o n - i s o t r o p i c w e i g h t i n g f u n c t i o n s f o r both the a r e a o p e r a t o r and the H a r t l i n e e q u a t i o n s ; 3. An i n v e s t i g a t i o n o f the r e l a t i o n s h i p , i f any, between the a r e a o p e r a t o r response and the a c t i v i t y i n a l a t e r a l i n h i b i  t o r y network f o r p a t t e r n s w i t h g r e y l e v e l s . I n t h i s t h e s i s we have shown t h a t the a c t i v i t y i n a l a t e r a l i n h i b i t o r y network near a w h i t e - b l a c k boundary i s determined by the l o c a l shape of t h a t boundary. 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