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Biol. Cybern. 61, 103 113 (1989) B i o l o g i c a lC y b e r n e t i c sO Springer-Verlag 1989

G a b o r F il te r s a s T e x t u r e D i s c r i m i n a t o rI. Fogel I and D. Sagi 21 Department of Atmospheric Optics, Soreq Nuclear Research Center, Yavne 70600, Israel2 Department of Applied Mathematics & Computer Sciences, The Weizmann Institute of Science, Rehovot 76100, Israel

Ab stra ct . The present paper presents a model fortexture discrimination based on Gabor functions. Inthis model the Gabor power spectrum of the micro-patterns corresponding to different textures is cal-culated. A function that measures the difference be-tween the spectrum of two micropatterns is introducedand its values are correlated with human performancein preattentive detection tasks. In addition, a two stagealgorithm for texture segregation is presented. In thefirst stage the input image is transformed via Gaborfilters into a representation image that allows dis-crimination between features by means of intensitydifferences. In the second stage the borders betweenareas of different textures are found using a Laplacianof Gaussian operator. This algorithm is sensitive toenergy differences, rotation and spatial frequency andis insensitive to local trans lation. The model was testedby means of several simulations and was found to be ingood corre lation with known psychophysical charac-teristics as tex ton based texture segregation and micro-pattern density sensitivity. However, this simple modelfails to predict human performance in discriminationtasks based on differences in the density of "ter-minators". In this case human performance is betterthan expected.

1 In tro d u ct i o nIn a number of recent papers (Julesz 1984a, 1986;Treisman 1986) it has been suggested that visualperception operates in two modes, one preattentiveand the other attentive. In the preattentive mode(which is parallel), differences in a few local s tructuralfeatures are detec ted over the entire visual field. In theattentive mode (which is sequential), a serial searchover a narrow aperture o f attent ion is performed in faststeps (20-80ms). Only in this mode recognition ispossible. According to the above ment ioned works

texture segregation in the preattentive mode is basedon local feature differences in the picture. Psychophys-ical experiments (Beck 1972, 1983; Julesz 1975, 1981,1986) have shown that texture discrimination in thepreattentive mode depends mainly on some simpleproperties such as brightness, color, size and slope (oflines, blobs) of the basic texture elements tha t composethe picture.Julesz (1980, 1984a, 1986) has suggested that tex-ture discrimina tion could be explained in terms of first-orde r differences between local features called textons,rather than global second-order differences betweenpoints in the image as he had suggested before (Julesz1962). Textons are defined as elongated blobs ofspecific color, orientation, length and width, alongwith their te rminators and line crossings. Differences intexton densities are necessary for discrimination in thepreattentive mode. In a series of experiments Gurnseyand Browse (1987) reported that properties such asterminators and line crossings appear not to play animportant functional role in texture discrimination,hence they speculated that they were not textons. Theyfound tha t performance in texture discrimination tasksdepended mainly on (1) the particular micropatternpair used, (2) which member of the pair forms theforeground and which forms the background, (3 ) thetime available to inspect the image and (4 ) thefamiliarity that the subject has with the patterns andprocedure. They suggested that energy differences inthe minimal circle enclosing the basic features creatingthe picture enable discrimination in the preattentivemode. However, they did not suggest that the humanvisual system computed the minimal enclosing circlefor each micropattern in the display: rather, theysuggested that when two micropatterns composed ofthe same line segments are enclosed by different circles,they will stimulate different sets of simple receptors.Another model was suggested by Krose (1987) whoexamined the (r, 0) chord space for each micropattern

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104( t e x t u r e e l e m e n t ) a n d d e f i n e d a m e a s u r e f u n c t i o n t om e a s u r e t h e d i s s im i l a r i ty b e t w e e n t h e m . ( A c h o r d i s av i r t u a l l i n e b e t w e e n t w o p a t t e r n p o i n t s , t h e c h o r ds p a c e is t h e s i ze a n d o r i e n t a t i o n d i s t r i b u t i o n o f al lt h o s e c h o r d s , f o r b i n a r y p i c t u r e s t h e c h o r d s p a c e i si d e n t i c al t o t h e a u t o c o r r e l a t i o n f u n c t i o n . ) K r o s e c o n -c l u d e d t h a t t h e u s e o f a l o c a l a u t o c o r r e l a t i o n a l g o r i t h mo n b i n a r y i m a g e s i n c o m b i n a t i o n w i t h a m u l t i - l a y e r ,m u l t i - r e s o l u t i o n s a m p l i n g a r r a y p l u s d i s s i m i l a r i t ym e a s u r e f u n c t i o n i s a f l ex i b le m e t h o d f o r t h e e x t r a c t i o no f l o c a l s t r u c t u r e f e a t u r e s . O t h e r m o d e l s s u c h a sF o u r i e r t r a n s f o r m a t i o n h a v e a l s o b e e n i n v e s t i g a t e d(Ba jcsy 1973; Ju le sz and Cae l l i 1979) .

T h e m o d e l p r e s e n t e d h e r e is b a s e d o n G a b o r f i l te r s( G a b o r 1 9 4 6; T u r n e r 1 9 8 6 ; C a e l li a n d M o r a g l i a 1 9 8 5;D a u g m a n 1 9 8 7; B e c k e t al . 19 87 ). T h e c l as s o f G a b o rf u n c t i o n s w a s d e s c r i b e d b y G a b o r ( 1 9 4 6 ) a n d w a se x t e n d e d t o t w o d i m e n s i o n s b y D a u g m a n ( 1 98 0 , 1 98 7) .D a u g m a n s h o w e d t h a t G a b o r f i lt er s a r e o p t im a l i n th es e n se t h a t t h e y m i n i m i z e t h e p r o d u c t o f ef f ec t iv e a r e a so c c u p i e d i n th e 2 - D s p a c e a n d 2 - D f r e q u e n c y d o m a i n s .T h i s c l a s s o f fi l t e rs c a n b e d e s c r i b e d i n t e r m s o f as i n u s o i d a l p l a n e w a v e o f s o m e s p a t i a l f r e q u e n c y a n do r i e n t a t i o n w i t h i n a t w o d i m e n s i o n a l G a u s s i a n e n v e -l o p e . B e c k e t a l. ( 1 9 8 7 ) s u g g e s t e d t h a t G a b o r f i lt e r s p l a ya r o l e i n t r i p a r t i t e s e g r e g a t i o n a n d i n t e x t u r e s e g r e -g a t i o n i n g e n e r a l . W e h a v e c h o s e n G a b o r f i lt e rs b e c a u s et h e y h a v e p r o p e r t i e s s i m i la r t o t h o s e o f e a r ly v i s u alc h a n n e l s , b e i n g l o c a l i z e d i n t h e s p a c e a n d f r e q u e n c yd o m a i n s . I n a d d i t i o n e a c h f i lt e r e n c o d e s b o t h s p a t i a lf r e q u e n c y a n d o r i e n t a t i o n t h u s p r e d i c t i n g p r e a t t e n t i v ed e t e c t i o n o f s p a t i a l f r e q u e n c y a n d o r i e n t a t i o n c o n j u n c -t i o n s ( S a g i 1 9 8 8 ) . I n t h e f i r s t p a r t o f t h i s a r ti c l e w e a p p l yt h e G a b o r f i l t e r m o d e l t o i so l a t e d m i c r o p a t t e r n s a n df i n d t h e p o w e r s p e c t r u m o f e a c h b a s i c e l e m e n t (b ym e a n s o f G a b o r f il te r s) t h e n w e c a l c u l a t e th e d i s -s i m i l a r i t y b e t w e e n t h e t w o s p e c t r a . A c c o r d i n g t o t h er e s u l t o f t h e m e a s u r e f u n c t i o n w e d e c i d e w h e t h e r o r n o tt h e t w o b a s i c e l e m e n t s a r e p r e a t t e n t i v e l y d i s t in g u i s h -a b l e . T h e c a l c u l a t e d m e a s u r e i s c o m p a r e d w i t h d a t af r o m p s y c h o p h y s i c a l e x p e r i m e n t s d o n e b y K r o s e( 19 8 7) . I n t h e s e c o n d p a r t w e d e s c r ib e t h e G a b o r b a s e da l g o r i t h m f o r t e x t u r e s e g r e g a t i o n ( G G L ) f o r t e x t u r e sc o m p o s e d o f m a n y m i c r o p a t t e r n s t h u s c a p t u r i n g s o m ep r o p e r t i es o f m i c r o p a t t e r n g r o u p s . A b a si c a s s u m p t i o nw e m a k e h e r e i s t h a t t h e o n l y i n f o r m a t i o n a v a i l a b le f o rp r e a t t e n t i v e d i s c r m i n a t i o n i s d if f e re n c e s b e t w e e n f ir s to r d e r s t a t i s ti c s o f f i l te r r e s p o n s e s , c o m p u t e d a s s p e c t r a le n e r g y .

2 M easuring Similari ty Usin g Gab or Fi l tersA 2 - D G a b o r f u n c t i o n i s a h a r m o n i c o s c i l l a to r , w h i c hi s a s i n u s o i d a l p l a n e w a v e o f s o m e f r e q u e n c y a n do r i e n t a t i o n w i t h i n a G a u s s i a n e n v e l o p e . I t s f r e q u e n c y ,

o r i e n t a t i o n a n d b a n d w i d t h a r e c o n t r o l l e d b y i t s p a -r a m e t e r s . T h e G a b o r f u n c t i o n i s d e f i n e d a s f o l lo w s :G(x, y lW, O ,~p,X, Y)

- [ ( x - X ) 2 + ( y - y ) 2 ]

=exp 2~2 x s i n ( W ( x c o s O - y s i n O ) + ( p ) ,0)w h e r e a i s t h e G a u s s i a n w i d t h , 0 is t h e fi l te r o r i e n -ta t ion , W i s it s f requ enc y and q~ i s i t s phase sh i f t. X , Yd e f i n e t h e c e n t e r o f t h e f i lt e r . L e t L(x , y ) b e t h e i n p u te l e m e n t m a t r i x a n d G(x, y)] W, O, (p, X , Y) t h e G a b o ro p e r a t o r . W e f i n d t h e G * L s p e c t r a f o r v a r i o u s o r i e n -t a t i o n s a n d s h if ts . T h i s s p e c t r u m i d e n t i f i e s t h e t e x t u r ee l e m e n t . M o r e e x a c t l y :G LI (X , YI W , O) = E G(x, Yl W, O, O, X , Y) x L(x , y)

x y

G Lz( X, Y] W, O) = ~. G(x, Y l W, O, ~/2, X , Y) x L (x , y) (2 )X, yS2(X , Y[ W, O) = GL ](X , Y I W, O) + GL 2(X , Y I W, 0) ,w h e r e x , y a r e i n d i c e s o v e r t h e b a s i c m a t r i x e l e m e n t s( x , y = 0 , 3 2) . q ~ l = 0 , q ~ 2 = n / 2 , W = 2 n w / 1 7 w h e r e w i st h e n u m b e r o f c y c le s in 1 7 p i x e ls ( t h e s iz e o f t h e i n p u tp a t t e r n ) . GL1, GL2 a r e th e c o n v o l u t i o n s o f th e G a b o rf i l t e r a n d t h e t e x t u r e . S i s t h e l o c a l l y s h i f t i n v a r i a n to u t p u t o f th e f i l te r s ( b y m e a n s o f G L 1 a n d G L 2 ) . W en e x t i n t r o d u c e a m e a s u r e f u n c t i o n t h a t e s t i m a t e s t h el e v e l o f d i s s i m i l a ri t i e s b e t w e e n t h e t w o m i c r o p a t t e r n s .

T h e m e a s u r e f u n c t i o n D tb i s d e f i n e d a s f o l l o w s :S2(X, Y IW , O) - ~ . S2(X , Y IW , O)D2b (W) x,Y.O x,Y,O , (3)Y~ S2(X, YIW,,O)+ X S ~ ( X , Y I W , O )

X , Y , O X , Y , O

w h e r e Sb i s t h e c o n v o l u t i o n o f t h e f i rs t in p u t m i c r o -p a t t e r n ( b a c k g r o u n d ) w i t h G a b o r f il te r , S r i s th ec o n v o l u t i o n w i t h t h e s e c o n d m i c r o p a t t e r n ( t a r g e t ) , 0r a n g e s b e t w e e n 0 ~ t o 3 6 0 ~ i n 1 0 ~ s t e ps , X a n d Y w e r ev a r i e d a t s t e p s o f a a c r o s s a r a n g e o f 4 o ( 2 5 s a m p l e s) .T h i s m e a s u r e f u n c t i o n h a s v a l u e s b e t w e e n 0 a n d 1 , a n di n d i c a t e s t h e d i s s i m i l a r i t y b e t w e e n t h e t w o i n p u tm i c r o p a t t e r n s . T h e s e r e s u l ts a r e i n v a r i a n t t o r o t a t i o na n d l o c a l s h if t . I f t h e t w o i n p u t p a t t e r n s a r e i d e n t i c a l u pt o r o t a t i o n a n d s h i f t w i t h i n t h e i n t e g r a t i o n a r e a t h er e s u l t w i l l b e s m a l l ; t h e h i g h e r t h e d i s s i m i l a r i t y b e -t w e e n t h e m t h e c l o s e r t o I t h e r e s u l t g e t s . T h ei n t e g r a t i o n o v e r d i f f e r e n t o r i e n t a t i o n s i s n e c e s s a r y ,s in c e w e a p p l y t h i s m e a s u r e t o c a s e s w h e r e t h e p a t t e r n sa r e r a n d o m l y r o t a t e d . T h e i n t e g r a t i o n o v e r s p a t i a lp o s i t i o n s i s d u e t o t h e u n c e r t a i n t y i n p a t t e r n p o s i t io n .W e i n t e g r a t e f i l t e r r e s p o n s e s ( e n e r g y ) , r e l y i n g o n t h ea s s u m p t i o n t h a t t h e p r e a t t e n t i v e s y s t e m u s e s o n l yd i f fe r e n ce s o f a v e r a g e d r e s p o n s e s a n d n o h i g h e r o r d e rs ta t is t ics .

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3 P s y c h o p h y s i c a l D i s c r i m i n a b i l i t ya s a M e a s u r e o f S i m i la r i t yM o s t o f t h e m o d e l s f o r t e x t u r e d i s c r i m i n a t i o n a r eb a s e d o n f i n d i n g l o c a l d i f f e r e n c e s i n t h e i m a g e . I n t h ep r e s e n t m o d e l t h e l o c a l d if f e re n c e s a r e t a k e n i n t oa c c o u n t b y t h e d i s s i m i l a r i t y f u n c t i o n . W e s h a l l s h o wt h a t t h i s m o d e l i s i n h ig h c o r r e l a t i o n w i t h p s y c h o p h y s -i c a l r e s u l t s . F o r m o r e d e t a i l s a b o u t p s y c h o p h y s i c a le x p e r i m e n t s s e e B e r g e n a n d J u l e s z ( 1 9 8 3 ) . I n b r i e f :

T h e s t i m u l u s c o n t a i n i n g t h e t a r g e t e l e m e n t a n dm a n y d i s t r a c t o r s i s p r e s e n t e d f o r a s h o r t t i m e( 2 0 - 8 0 m s ) a n d i s f o l l o w e d b y a m a s k c o n s t r u c t e d f r o mt h e c o n j u n c t i o n o f t h e t a r g e t a n d d i s t r a c t o r a t av a r i a b l e t i m e ( S O A ) a f t e r t h e o n s e t o f t h e s t i m u l u s .I n c r e a s e i n S O A i s e x p e c t e d t o i n c r e a s e a v a i l a b l es t i m u l u s p r o c e s s i n g t i m e a n d t h u s i m p r o v e p e r f o r -m a n c e . S o m e p o p u l a r t a r g e t / d i s t r a c t o r p a i r s ar e p re -s e n t e d i n F i g . 1 ( f r o m K r o s e 1 98 6) . A c c o r d i n g t oJ u l e s c z ' s t e x t o n t h e o r y p r e a t t e n t i v e d i s c r i m i n a t i o n i se x p e c t e d i n t h e f i rs t p a i r a n d a t t e n t i v e d i s c r i m i n a t i o ni n t h e s e c o n d p a i r . T h e f i ft h p a i r ( w h e r e t h e t w o b a s i ce l e m e n ts h a v e t h e s a m e n u m b e r o f t e r m i n a t o r s a n d l in ec r o s s i n g ) i s p r e a t t e n t i v e l y i n d i s t i n g u i s h a b l e . O n t h eo t h e r h a n d d i s c r i m i n a t i o n i n t h e s i x t h p a i r i s i n t h ep r e a t t e n t i v e m o d e s i nc e t h e b a s i c e l e m e n t s d i f f e r i nf o u r t e r m i n a t o r s a n d i n l in e c ro s s in g . T h e s e p a i r s w e r et e s t e d b y K r o s e ( 1 98 7 ) u s i n g e x p e r i m e n t a l p r o c e d u r e sa s u s e d b y B e r g e n a n d J u l e s z ( 1 9 8 3 ) . K r o s e f o u n d ah i g h d e t e c t a b i l i t y f o r p a i r s 6 , 1 , 4 ( o r d e r e d b y t h e l e v e lo f d e t e c t i o n ) a n d a l o w o n e f o r p a i r s 2 , 3 , 5 . T h e s er e s u lt s a r e o n l y i n p a r t i a l a g r e e m e n t w i t h J u l e sz ' sm o d e l . N e x t w e s h a l l d is c u ss t h e p e r f o r m a n c e o f o u rm o d e l o n t h e s e p a i r s .3 .1 A S ing le Spa t ia l -Frequency F i l t er ModelExper imen t 1 . I n t h i s e x p e r i m e n t w e u s e d s i m p le b a s i ce l e m e n t s ( F ig . 1) e m p l o y e d a l s o b y K r o s e ( 1 98 7 ) a n dJ u l e s z ( 1 9 8 4 b , 1 9 8 6 ) . A l l t h e i n p u t e l e m e n t s h a d t h es a m e n u m b e r o f p i x e ls t o p r e v e n t t e x t u r e d i s c r i m i n a -t i o n b a s e d o n e n e r g y d i f f er e n ce s .

T h e s iz e o f e a c h i n p u t m i c r o p a t t e r n u s e d w a s1 7 x 1 7 a n d i t w a s e m b e d d e d i n a m a t r i x o f o r d e r3 3 x 3 3 t o a l l o w v a r i a t i o n s i n o r i e n t a t i o n s a n d s h i ft s .

P a t t e r n I 2 3 4 5 6 7

Target

G r o u n d

+ T + IZ 2 " O ' : '9 1 [ F / 1 - I - - t - - I - 9

Fig. 1. Com binations of target and distractor microp atterns usedin experim ent 1. (After Krose 1987)

105Table 1. The discrimina bility results of the micropattern pairsused in experiment 1Pat tern Dtb1 0.462 0.243 0.394 0.095 0.276 0.727 0.56

P i x e l s b e l o n g i n g t o t h e b a s i c e l e m e n t w i l l b e a t g r a yl e v e l 2 5 5 ; o t h e r p i x e l s w il l g e t t h e g r a y v a l u e 0 . G a b o rf i l t er p a r a m e t e r s w e r e o - = 8 a n d w = 0 . 8 75 ( o n e c y c l e f o r1 9 pi x el s) . T h e i n p u t f o r t h e m o d e l c o n s i s ts o f t w op a t t e r n s a n d t h e o u t p u t s i s t h e i r d i s si m i l a ri t y . W e s h a l lc o m p a r e t h e s e r e s u l t s t o t h o s e f r o m t h e h u m a n v i s u a ls y s t e m f o r a f i x ed S O A . S i n c e w e d i d n o t i n t r o d u c e t h et e m p o r a l d i m e n s i o n t o o u r m o d e l , w e c h o s e a n a r b i-t r a r y S O A ( 2 40 m s , f r o m K r o s e 1 98 7) .

W e c a l c u l a t e S ( 2 ) f o r t h e t w o b a s i c e l e m e n t s , a s af u n c t i o n o f 0 a n d f i n d th e d i s t a n c e b e t w e e n t h e s p e c t r ao f t h e c o n v o l u t i o n s ( 3) . T h e r e s u lt s o f t h is m o d e l a r eg i v e n i n T a b l e 1 .T h e r e s u l t s d e p i c t e d i n T a b l e 1 a r e i n h i g h c o r r e -l a t i o n w i t h t h e r e s u l t s f o r t h e h u m a n v i s u a l s y s t e m( K r o s e 1 9 8 6 , 1 9 8 7) , t h e d i s c r i m i n a b i l i t y o r d e r s o f t h ep a i r s a r e a l m o s t t h e s a m e . A c o m p a r i s o n b e t w e e n t h e s er e s u lt s is p r e s e n t e d i n F ig . 2. N o t e t h a t o u r m o d e l h a sl o w p e r fo r m a n c e , c o m p a r e d t o t h e h u m a n p e r f o r -m a n c e , i n t h e ca s e w h e r e t h e t w o m i c r o p a t t e r n s d i ff e ri n t h e n u m b e r o f t e r m i n a t o r s ( s e e T a b l e 1 p a i r 4 ).

c,O

1 O 0 - -90 --80 - -70 --60 - -50 - - . 440 --30 --20 - -10 - -

I0.1

, , 6

.7

, ,1

, 5' , 2 "3

I 1 I I I I I 1 I0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

D tbFig. 2. Correlation between ou r model predictions (w =0.875)and K rose's psychophysical data, for the patterns show n in Fig. 1

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106Table 2. The discriminabil i ty results o f the m icropattern pairsused in experiment 2

(%) Dtb9 0.21

18 0.3327 0.3936 0.4250 0.46

Experiment 2. F o l l o w i n g K r o s e ' s e x p e r i m e n t , w e w i llf i n d d i s c r i m i n a t i o n v a l u e s f o r t h e f o l lo w i n g p a i r s : t h eb a c k g r o u n d e l e m e n t s i n t h is e x p e r i m e n t a r e L s h a p e sa n d t h e t a r g e t e l e m e n t v a r y f r o m L to + . S e e t h ev a r i a t i o n a s a f u n c t i o n o f ~ i n F i g . 3 .

A l l t h e t a r g e t e l e m e n t s h a v e t h e s a m e n u m b e r o fl in e c ro s s i n g s a n d t e r m i n a t o r s . H e n c e a c c o r d i n g t o t h et e x t o n t h e o r y w e e x p e c t t h e d i s c r i m i n a t i o n t o b e th es a m e f o r a ll p a i r s , u p t o t h e r e s o l u t i o n o f c r o s s i n gd e t e c t i o n . P s y c h o p h y s i c a l r e s u l t s s h o w d i s c r i m i n a -b i l i t y v a r i a t i o n f r o m 0 , w h e r e t h e t a r g e t i s L , u p t o ac e r t a i n v a l u e i n d i c a t i n g t h e d i s c r i m i n a b i l i ty o f th e( L , + ) p a i r , w h e r e t h e t a r g e t i s + . T h e r e s u l ts o f o u rm o d e l a r e g i v e n i n T a b l e 2. T h e y a r e f o u n d t o b e i nh i g h c o r r e l a t i o n w i t h t h e p s y c h o p h y s i c a l r e s u l t s .

T h e c o r r e l a t i o n b e t w e e n t h e r es u l ts o f t h e G a b o rm o d e l w i t h t h e p s y c h o p h y s i c a l e x p e r im e n t s a r e p r e -s e n t e d i n F i g . 4.

3.2 A Multi Spatial-Frequency Filter ModelI n t h e p r e v i o u s S e c t . (3 .1 ) w e c o m p u t e d t h e m o d e ld i s c r i m i n a t i o n v a l u e s ( i n f r e q u e n c y d o m a i n ) f o r as i n g le G a b o r f il te r w i t h c e n t e r f r e q u e n c y o f w = 0 .8 7 5 .H o w e v e r t h e h u m a n v is u al s y st e m c o nt a in s m a n yf il te r s w i t h d i f fe r e n t p e a k s p a t i a l f r e q u e n c i es o p e r a t i n gi n p a r a l l e l ( W i l s o n 1 9 8 3 ; W a t s o n 1 9 8 3 ) , a n d i t i sr e a s o n a b l e t o a s s u m e t h a t d i s c r i m i n a t i o n is b a s e d o nt h e f i lt e r t h a t y i e ld t h e b e s t d i s c r i m i n a t i o n f o r th e p a i ro f p a t t e r n s i n v o l v e d . W e t e st e d o u r m o d e l o n a s p a t ia lf r e q u e n c y r a n g e o f w = 0 . 5 to w = 4 . 2 5. F o r e a c h p a t t e r nw e s e l e c te d t h e f i l te r w i t h t h e b e s t d i s c r i m i n a t i o n v a l u e(Dtb) a n d t o o k t h i s v a l u e a s th e d i s c r i m i n a t i o n l e v el f o rt h e p a t t e r n . T h e r e s u l t s a r e d e p i c t e d i n F i g . 5 . T h i sm o d i f i e d m o d e l g i v e s s o m e w h a t b e t t e r fi t t o t h ep s y c h o p h y s i c a l d a t a o f K r o s e ( 1 9 8 6, 1 9 8 7) w i t h ac o r r e l a t i o n c o e f f i c i e n t o f r = 0 . 7 1. P a t t e r n 4 is a g a i n o u to f r a n g e , a n d m a y b e c o n s i d e r ed a s a n e x c e p t io n ,w i t h o u t i t th e c o r r e l a t i o n c o e f f i c ie n t i s r = 0 . 9 1 . T h ed e v i a t i o n o f p a t t e r n s 3 a n d 5 a r e d u e t o t h e h i g h s p a t i a lf r e q u e n c y c o m p o n e n t s . L i m i t i n g t h e f r e q u e n c y r a n g et o w < 1 . 2 5 ( h i g h e s t f i lt e r) , w h e r e m o s t o f th e p a t t e r n se n e r g y l i es i m p r o v e s c o r r e l a t i o n u p t o 0 .9 6 . M o s td i s c r ir n i n a b i li t y f u n c t i o n s h a v e a m a j o r p e a k a t w

Patt ern I 2 3 4 5e(% ) 9 18 27 56 5 0

T a r g e t I I'" 1 1 - +G r o u n d l [ I i I....

Fig. 3. Combinations of target and distractor micropatterns usedin experiment 2. (After Krose 1987)8070605 04O3O

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D tb4. Correlation between our model predictions (w=0.875)Krose 's psychophysical data, for the patterns shown in Fig. 3

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0D tbFig. 5. Correlation between ou r multi-filter model predictionsand Kros e's psychophysical data. Textures pairs 1-7 a re the sameas Fig. i, pairs a-e are the same as in Fig. 3. Op en symbols are fordiscriminability values obtain ed u sing only low frequency filters(w < 1.25), solid symbols are for values obtained adding higherfrequencies filters (w

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107around 0.75 and a secondary one at w around 1.5.Patterns 3 and 5 are significantly better discriminatedat w= 1.5 then at w = 0.75.The Gabo r filter model, in its present form, canno taccount for pattern discrimination based on ter-minators. Note that for pattern 6, discrimination alsomay be based on differences in terminators density,and human performance is superior to our modelperformance (in this case human performance is errorfree, thus the performance differences may be largerthan seen). Thus we may have to modify the Gaborfilter model in order to deal with pattern discrimi-nation based on terminators.3.3 SummaryIn the first experiment we tested seven input pairs,where in some of them preattentive discrimination wasgenerated. According to the results of this model thetransition from attentive to preattentive mode isgradual throughout the interval between 0 to 1. If adivision to preattentive/attentive mode existed wewould expect psychophysical data to be close to 100%if the discrimination were preattentive and close to 0%if it were attentive. But the spread of the resultsimplicates that this division is too coarse. Anotherpossibility is that such a division exists and we mustadd a threshold level to the model above whichpreattentive mode operates and below which attentivemode is necessary.The psychophysical performance on the pairs 2 and3 is almost the same. According to the texton theory,supported by many demonstrations, pair 2 (T,L) ispreattentively indistinguishable because texton dif-ferences are very low (one terminator), while in thethird pair (T, + ) preattentive discrimination is possibledue to the high texton density differences (one ter-minator and one line crossing). Introducing a thres-hold between the two pairs ma y resolve this problem.Another resolution to this problem comes from theobservation that (X, T) discrimination requires highfrequency filters while (T, L) and (X, L) discrimina tionare affected only by low frequency filters. This mayresult lower sensitivity to (X, T) differences in experi-ments using short presen tation time, due to the longerintegration time of high frequency filters (Watson1983; Wilson 1983).In the second experiment we held the number ofterminato rs and line crossings constant while varying(the spatial location of the lines that composes themicropattern, see Fig. 3). The psychophysical and theGabor model results have changed from close to zero,which means o segregation, up to the level of thedetectability of the L, + pair. According to the textontheory texture discrimination should not be greatlyaffected by increasing ~, up to the + resolution.

A quantitative comparison based on correlationcoefficients can be made between our model and thelocal cross -co rre lat ion model of Krose (1986, 1987).Taking in to account all the patterns used in Sect. 3.2Krose's model gives a correlation coefficient of 0.65while ours yields 0.71. Our corre lation coefficient canbe improved by no t considering pair 4 (terminators) upto 0.96, his can be improved by not considering pair 5up to 0.84 (Krose 1986). The two models fail ondifferent patterns, an observation t hat may imply thata model combining features from the two models mayhave a better predictive value. For example, as Krose(1987) points out his model may have better resultsusing the chord space of patterns which are low - passfiltered and then thresholded.Finally, the model prediction may be affected bythe choice of the dissimilarity measure. We took thedifferences between energies at a particular spatialfrequency band as a correlate of discriminability. Thiswas done by integrating over all orientations and isjustified since all patterns are rand omly rotated. Mostpattern pairs, although having the same energy whenaveraged across all orientations, differ in the distri-bution of energies across the different orientations(different second order statistics). Krose (1986) used adissimilarity measure (Euclidean distance) tha t is sensi-tive to this differences in some cases. Replacing (3) bythe Euclidean distance measure improves our corre-lation value up to 0.91 (all patterns), mainly due to theimproved discriminability on pair 4. It is interesting tonote that the energy distribution across the differentorientat ion filters is translated in all stimuli consideredhere into spatial distribution. This spatial variationmay limit discrimination, but on the o ther ha nd it canenhance discrimination between textures differing inthe second order statistics of the orientation spectrum.4 T e x t u r e S e g r e g a t i o n B a s e d o n G a b o r F il t er sThe task of the next algorithm (GGL) is to find bordersbetween regions o f different textures. This is a subset ofthe general problem of locating and characterizingdifferent textures. A number of techniques for texturediscrimination have been suggested. Most of themlimit the number of features and fall into one of thefollowing classes: Template matching (Hall 1979),pattern sampling (Fairhurst and Stonham 1976),spatial transforms, geometrical moments and featureextraction (Rosenfeld and Kak 1976). The algorithmhas two stages. In the first stage the input image istransformed into output image by convolution withGabor signals and enables discrimination betweenfeatures by means of intensity differences. The targetregion and the background region will have differentintens ity levels if they cont ain different features, hence

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108they can be seg re ga ted eas i ly on th is bas is . The seco nds tage f inds the bo rders . The a lgo r i thm i s sens i t ive toe n e rg y d i f fe r e nc e s , r o t a t i o n a n d s p a t ia l f r e q u e n c y a n dis in sens i t ive to loca l t rans la t ion .4.1 The GGL Algorithm ( Gabor-Gaussian-Laplacian )A. C o n v o l v e t h e i n p u t i m a g e w i t h o d d a n d e v e nGa b o r f i lt e r a n d c o m p u t e s u m o f s q u a re s fo r e a c hpos i t ion .B . S m o o t h t h e o u t p u t o f A (b y Ga u s s i a n ).C . T h re s h o l d .D . R e m o v e n o i s e (s m a ll c o n n e c t e d c o m p o n e n t s ) .E . F i n d b o rd e r l i n es (b y La p l a c i an ) .F . C h a n g e p a ra m e t e r s i n Ga b o r f i l t e r a n d r e p e a t s te p sA-E.In the fo l lowing s imula t ions we used 32 f i l t e rshav ing para me ters as fo l lows : 0 = 0 ~ 45 ~ 90 ~ 135 ~w= 0.5 , 0 .625, 0 .875, 1 . (p= 0 ~ 90 ~ See the fi l ters inFig. 6.T h e p a ra m e t e r a d e p e n d s m a i n l y o n t h e i n p u timage sca le and was tes ted in the reg ion 6 -18 , the bes tcho ice was 12 . There i s a lmos t no change in thec o n v o l v e d o u t p u t a ro u n d o -= 1 2. P h a s e i n fo rm a t i o n i slos t by tak ing on ly the energy va lues . The inpu tp ic tu res to th i s a lgo r i thm are o f size 512 x 512 p ixe ls

and the g ray leve l a re in the range 0 -255 , a l l the p ic tu resa r e m o n o c h r o m e . W e u s ed c o m p u t e r g e n e r a t e d im a g e sa n d a l s o i m a ge s t a k e n b y a v i d e o c a m e ra . R e s u l t s d i dno t d i f fe r in the two cases even tho ugh the came raimage s are noisy in natu re. A grid o f 32 x 32 is set ont h e i n p u t i m a g e . A c o n v o l u t i o n w i t h S i n e a n d C o s i n eGa b o r f i lt e rs o f s iz e 65 x 6 5 a r e m a d e a ro u n d e a c h g r i dp o i n t . He n c e t h e n u m b e r o f o u t p u t e n e rg y v a l u es a r e( 3 2 - 4 ) * ( 3 2 - 4 ) = 7 8 4 . T h e - - 4 co m e f r om t he f ac ttha t we s ta r t a t l ine and co lu mn 32 and s top a t l ine andco lumn 480 , because the f i l t e r in teg ra t ion s ize i s 65x 65 . Fo r ea ch g r id po in t , the sum of squares o f theSine and Cos ine f i l t e rs ou tpu ts i s ca lcu la ted and thent r a n s fo rm e d i n t o t h e r e g i o n 0 -2 5 5 b y m e a n s o f a l i n e a rt r a n s fo rm a t i o n . T h e o u t p u t p i c t u r e i s t h e n a c o l l e c ti o nof 28 x 28 square s of 16 x 16 pixel s ize, with g rayleve ld i s t r ibu t ion depend ing on the t ex tu re used . The edgesa re c o m p u t e d o n e a c h o f t h e c o n v o l u t i o n o u t p u t s ( fora l l 0, co combina t ions ) . As we m en t i one d ear l i e r theou t pu t o f the f ir s t s tep i s a co l lec t ion o f 28 x 28 squares .Th is i s a low reso lu t ion ou tpu t . In add i t ion , h ighg ra y l ev e l s qu a re s m a y e x i s t i n t h e n e i g h b o rh o o d o f l o wgray leve l ones and v ice versa , th is in t roduc es h ighfreque ncy no ise re la t ive to the t ex tu re s ize . Fo r thesere a s o n s a s m o o t h i n g o p e ra t i o n i s n e c e s s a ry b e fo reca lcu la t ing tex tu re bo rders . A Ga uss ian f i l te r o f size 31x 31 p ixe l s i s convo lved wi th the p ic tu re in o rder tosmooth i t and a th resho ld i s then def ined to be in them i d d l e o f t h e t wo p e a k s o f t h e s m o o t h e d h i s t o g ra m .Since te rmina to rs and l ine c ross ings can no t bet r iv i a ll y c o n v e r t e d i n t o Ga b o r p a ra m e t e r s a n d t h eb a s i c e l e m e n t s a r e r a n d o m l y o rd e re d o n t h e p i c t u r e(which means var iab le loca l dens i ty ) , smal l connec tedc o m p o n e n t s (n o i s e ) m a y b e o c c u r r i n g . T h e s e c o m p o -n e n t s a r e t h e n r e m o v e d u s in g a c o n n e c t e d c o m p o n e n ta lgor i thm (Rosenfe ld and K ak 1976) . At the end o f thi ss tep a b inary p ic tu re i s p resen ted where the g ray leve l sfo r the d i scr iminab le t ex tu res a re d i f fe ren t . Hence bya p p l y i n g a L a p l a c i a n o p e ra t o r t h e b o rd e r s c a n b efo u n d . T h e o p t i m a l s i t u a t i o n i s t o a p p l y Ga b o r f i l t e rjus t once . Usual ly th i s i s no t the case and a fewa p p l i c a t i o n a r e r e q u i r e d . W e r a n t h e a l g o r i t h m a f e wt i m e s v a ry i n g t h e Ga b o r p a ra m e t e r s . I n t h e c a s e o f n od i s c r i m i n a t i o n t h e o u t p u t p i c t u r e s we re s m o o t h (n oin fo rmat ion ) , and were re jec ted . In the remain ingresu l t s we found edges and i f they d id no t d i f fe rs ign i f ican t ly the avera ge ove r a l l o f them w as taken .Otherwise there i s no seg rega t ion .

Fig. 6a and b. The set of Ga bor filters used in the G GLalgorithm. The set contains 4 frequencies (32, 26, 18, and 16pixels/cycle), 4 orientation s (0~ 45 ~ 90~ 135~ and 2 phase pairsfor each orientation and frequency. Variation in frequency alongthe X axis and variation in rot atio n along the Y axis, each figure(a, b) has a different phase

4.2 ResultsT h e n u m b e r o f cy c le s w i t h i n a Ga b o r p a t c h i n c re a s e swi th ~ . Ther efo re fo r a f ixed ~r an op t im al reg ion fo r wcan be def ined , a was tes ted in the reg ion 6 -18 and thebes t cho ise was 12 , 26 p ixe l s per wa ve per iod was fou nd

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Fig. 7. a The input texture pair: +sembedded in Ls distractors, b Theresult of convolving a Gabor filterwith the imag e (w=0.625, 0=90~e T he smoothing ope ration (Gaussian)result, d The threshold operationresult, e The result of removing smallconnected com ponents (noise). f Th eoutput (Laplacian) result of the G GLalgorithm

to be the b est ch oice in mos t cases (e.g . w = 0.625 wh icheffec t ively resembles a D O G funct ion , s ince fo r theGa u s s i a n we u s e d t h e re we re n o e f f ec t iv e s i de b a n d s i nadd i t ion to the f i r s t nega tives ) . In som e cases w = 1 wasfound to be be t te r than 0 .625 . So usua l ly i t wass u f f i c i e n t t o ru n t h e a l g o r i t h m wi t h t h e p a ra m e t e r sa = 1 2 , w= 0 .6 2 5 , 0 = 0 ~ 4 5 ~ 9 0 ~ In t h e fo l l o wi n ge x p e r i m e n t s we a t t e m p t e d t o d i s c r i m i n a t e b e t we e n t h ed i f fe ren t t ex tu res and to f ind the borders be tweent h e m , u s i n g t h e GGL a l g o r i t h m . A p i c t u r e i s ac o m b i n a t i o n o f t wo d i f f e r e nt t e x t u re s a n d t h e b a s i ce lemen ts a re o f s ize 17 17 p ixe ls rand om ly (o r ien ta-t i o n - a n d t r a n s l a t i o n w i se ) e m b e d d e d i n a m a t r i x o fs ize 33 33. The tex tu res in the fo l lowing exper im en tsare a co l lec t ion o f Ls, Os, Ts a n d + s m i c ro p a t t e rn s .T h e s e m i c ro p a t t e rn s p a i r s h a v e b e e n i n v e s t i g a t e d b yBerge n and Ju lesz (1983) and K rose (1987) . Bergen andJ u l e s z (1 9 8 3 ) fo u n d p re a t t e n t i v e d i s c r i m i n a t i o n fo rp i c t u r e s c o m p o s e d o f L e m b e d d e d i n a b a c k g ro u n d o f3 5 + s a n d a t t e n t i v e d i s c r im i n a t i o n w h e n L wa s e m b e d -ded in 35 Ts. Kro s e (1 9 8 7 ) fo u n d h i g h d e t e c t a b i l i t yp e r fo rm a n c e fo r O e m b e d d e d i n + s. I n t h e f i rs t p a ir(F ig . 7a) which i s a co l lec t ion o f + s embed ded in a

b a c k g r o u n d o f Ls we fo u n d h i g h d i s c r i m i n a b i l i t ybe tw een the d i f fe ren t t ex tu re s (F ig . 7b ), in add i t ion theb o rd e r s b e t we e n t h e m we re fo u n d (F i g . 7 a - f) . A l m o s tt h e s a m e r e s u lt s h a v e b e e n o b t a i n e d wh e n + s we ree m b e d d e d i n Os (F ig . 8 a -d ) . L o we r s e g re g a t i o n wa sfo u n d fo r + s e m b e d d e d i n Ts (Fig. 9a--c) while therewas no segrega t ion a t a l l fo r Ts e m b e d d e d i n Ls(Fig. 10a-d).

5 General DiscussionIn g e n e ra l G a b o r fu n c t i o n s c a n e a s il y s e g re g a te i m a g e shav ing reg ions d i f fe r ing in one o f the fo l lowing p roper-t i es: Spa t ia l f requ ency , dens i ty o f e lemen ts , o r ie n ta-t i o n , p h a s e a n d e n e rg y . T h e l a rg e r t h e n u m b e r o fd i f fe rences , the be t te r the separa t ion . These p roper t i esc a n b e c o n v e r t e d t r i v i a l l y t o t h e Ga b o r f i l t e r p a ra m -e ters . In add i t ion , G ab or f i lt e rs d i scr im ina te wel lp ic tu res d i f fe r ing in the i r t ex ton s p rope r t i es exc ep t fo rt e rm i n a t o r s .

T h e f a i l u r e o f t h e Ga b o r f i l t er b a s e d m o d e l t op re d i c t t e rm i n a t o r b a s e d d i s c r i m i n a t i o n i s i n t e r e st i n gin i t se l f . Models based on l inear f i l t e rs a re qu i te

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Fig. 8. a The input texture pair:+ s embedded in Os distractors.b The result o f convolving a G aborfilter with the image, e The resultofGaussian + Threshold+Noiseremoving, d The outp ut(Laplacian) result o f the G GLalgorithm

Fig. 9. a The input texture pair: +s embedded in Ts distractors, b The resultof convolving Gab or filter with the image (w=0.625, 0=90~ e The outpu t(Laplacian) result of the GGL algorithm

success fu l in p red ic t ing v i sua l phenomena , main ly a tc o n t r a s t t h r e s h o l d l e v el (Da u g m a n 1 9 87 ; W a t s o n1983 ; Wi lson 1983) , wh i le above th resho ld somenon l inear i t i es have to be pos tu la ted (Ju lesz 1980 ; Sag iand Ho chs t e in 1983 , 1985) . The g enera l va l id i ty o f thel inear f i lt e r based mod el i s a l so supp or te d by e lec t ro -

phys io log ica l s tud ies re f lec t ing p roper t i es o f s ing lecor t i ca l ce ll s (Mare e l ja 1980 ; Da ug ma n 1980) . Themod el exam ined here perfo r ms su rp r i s ing ly well on a llp a t t e rn s e x a m i n e d e x c e p t o n e , wh i c h, a s c a n b e s e e nfrom Fig . 5 , (pa t te rn no . 4 ), con ta ins t e rmina to rs .T h e re i s a v e ry h i g h c o r r e l a t i o n b e t we e n h u m a n

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Fig. 10. a The input texture pair:Ts embedded in Ls distractors.b The result of convolving Gaborfilter with the image (w = 1,0=90~ e The result for the samefrequency when 0=4 5 ~ d Theresult of w=0.625 and 0=9 0 ~

perform ance and the model perform ance, r = 0.96 whenpat tern 4 is not considered (r= 0.71 with pat tern 4). Inthe l igh t o f the mode l ' s good per fo rmance , the fa i lu reon t e rmina to rs deserves spec ia l a t t en t ion . Th i s fa i lu rei s no t to ta l ly unexpected , s ince t ex tu res composed o fpa t t e rns used in pa i r 4 were shown to have iden t i ca lpower spectrum (Julesz and Cael l i 1979). Our modelco mp u t e s l o ca l p o w er s p ec t ru m an d t h u s may p ro v i d eusefu l loca l in fo rma t ion fo r d i scr iminat ion on pa i r 4 .T h u s w e ma y n o t b e fo r ced t o a cco u n t f o r t e rmi n a t o rde tec t ion by add ing a spec ia l " t e rm inato r de tec to r" tothe Ga bor f i lt e r model . A na tu ra l ex tens ion o f thepresen t model , tha t c an g ive a be t t e r d i scr iminat ion o fpair 4 , i s a model ut i l izing second order s tat is t ics offi l ter responses (across or ienta t ion o r space). Similarly ,d i scr iminat ion on pa i r 4 can be impro ved by in t roduc-ing a s t rong non l inear i ty a t the f i lt e rs ' ou tpu t s . How -ever , more exper imenta l excep t ions have to be foundbefore any o f thi s account s can be implemented .Anoth er p rob lem to the l inear f il t e ring appro ach i sa dem ons t ra t io n by Ju lesz and Kro se (1988) where theyshow tha t t ex tu re d i scr iminat ion be tween t ex tu res madeof Ls and Xs i s possible even when the relevant spat ialfrequencies (according to the f i l tering approach)are miss ing . Tha t Xs an d Ls can be d i scr iminated bythe Gabor f i l t e r model i s c l ear f rom the fac t tha t thedatu m po in t re l a t ing to X/L di scr iminat ion fa ll s a longthe same l ine as the o ther da ta po in t s show n in Fig . 5,an d f ro m t h e f ac t t h a t t h e G G L a l g o r i t h m p re s en t ed

here show a good segregat ion . I t i s a l so known tha tX/L di scr iminat ion depends on the s i ze o f the Xs a n dw h en t h e Xs are ma de l a rger , d i scr iminat ion i s reduced(Bergen and Ade lson 1988; Voor ees and Pog gio 1988).However, as Julesz and Krose (1988) points out , i t i sn o t c l e a r w h e t h e r t h e i r o w n d emo n s t r a t i o n a rg u esagain st a specif ic l inear-f i l ter based m ode l ( laplacian ofgaussian) or agains t the use of l inear-f i l ters in general .The a ssump t ion tha t the rem oval o f cer ta in f requenciesf rom the inpu t image resu l t s the i r removal f rom theoutput of the l inear f i l ters , i s not necessari ly t rue,s imple non l inear it i es as th resho ld ing (o r squar ing) m ayin t roduce them back in to the f i l t e rs ou tpu t s . In ourmod el th i s k ind o f non l inear i ty ( squar ing) ex i st s, t husthe f i l tered image, after the nonl ineari ty , wi l l containfrequencies related to the s ize of the X and L micro-pa t t e rns . Th i s f i l t e red image can be segregated bystages B-F i n the GG L a lgor i thm which essent i a l ly areper fo rming add i t iona l f i l t e r ing (us ing a Lap lac ian o fGauss ian) . The in teres t ing po in t here i s tha t in th i s casethe a lgor i thm's ou tpu t i s a ma p wi th d i f fe ren t g raylevels corresponding to the different textures , whi le inthe s imple case (non-fi l tered original images) theou tpu t i s a map descr ib ing the borders be tween thesegregated areas . Al though th i s exp lanat ion fo r theJulesz and Krose (1988) phenomena is speculat ive, i tcan be t es t ed psychophys ica l ly . Ev idence fo r theessent ial part of i t , the secondary f i l tering s tage,al ready exis ts (Sagi and Hochstein 1985; Sagi 1989).

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112I n th e G G L a l g o r i t h m t h e p a r a m e t e r s a r e s e t t o

c e r t a i n v a l u e s . T h e s iz e o f t h e G a u s s i a n e n v e l o p e i sh e l d c o n s t a n t f o r a l l f r e q u e n c i e s a n d o r i e n t a t i o n s ,w h i c h h a s t h e e f f e c t o f d e c r e a s i n g t h e b a n d w i d t h o f t h ef i l t e r w i t h i n c r e a s i n g f r e q u e n c y . T h i s m a k e s t h e c o m -p u t a t i o n a n d t h e c o m p a r i s o n b e t w e e n d i f f e r e n t f r e -q u e n c y f i lt e rs ea s i e r. F o r m a n y i m a g e s a s m a l l v a r i a -t i o n i n t h e f il t er si z e o r t h e G a b o r e n v e l o p e s i z e d i d n o tp r o d u c e a n y s i g n i fi c a n t d i ff e r e n ce in t h e s e g r e g a t i o nr e s ul ts . I n a d d i t i o n i t w a s f o u n d t h a t t h e d i s c r i m i n a t i o na b i l i ty d i d n o t d e p e n d c r i ti c a l ly o n a s p e c if ic f r e q u e n c yo r o r i e n t at i o n . I n s o m e c a s e s t h e c o m b i n a t i o n o fs e v e r a l f il te r s p r o d u c e s a b e t t e r d i s c r i m i n a t i o n . T h en e c e ss i ty o f s t e p D i n t h e G G L a l g o r i t h m d e p e n d s o nt h e s iz e o f t h e s m o o t h i n g o p e r a t i o n : t h e l a r g e r t h es m o o t h i n g s p r e a d t h e l e ss th e n o i se . O n t h e o t h e r h a n dt h e b o r d e r s b e c o m e l e ss a n d l es s a c c u r a t e , s o f o r h ig hr e s o l u t i o n b o r d e r s s t e p D i s n e c e s s a r y a n d f o r l o w e rr e s o l u t i o n i t c a n b e b y p a s s e d w h i l e in c r e a s i n g t h es m o o t h i n g o p e r a t i o n s c a l e . W e u s e d t h e i s o t r o p i cL a p l a c i a n f i l t e r t o f i n d t h e b o r d e r s i n a l l d i r e c t i o n s .T h e L a p l a c i a n s t e p c o m e s a f t e r t h r e s h o ld i n g t h ei m a g e . T h e s e s te p s c a n b e i n t e r c h a n g e d a n d i n t h a t c a s ea n a l g o r i t h m f o r s k e l e t o n i n g t h e p i c t u r e is n e e d e d a f t e rt h r e s h o l d i n g . W e c h e c k e d a n i t e r a t i v e v e r s io n o f t h eG G L a l g o r i t h m a n d f o u n d t h a t t h e b o r d e r s b e t w e e nd i f f e r e n t t e x t u r e s b e c o m e l e s s a c c u r a t e . T u r n e r ( 1 9 8 6 )u s e d G a b o r f i lt e rs it e r a t i v e l y a n d c o u l d d i s c r i m i n a t ew e l l e v e n t e x u t r e s w i t h i d e n t i c a l t h i r d o r d e r s t a t i s t i c s .N o t h d u r f t ( 1 9 8 5) a n d S a g i a n d J u l e s z ( 19 8 7) h a v es h o w n t h a t t e x t u r e d i s c r i m i n a t i o n i s h i g h ly d e p e n d e n to n t h e d e n s i t y o f t h e i n p u t m i c r o p a t t e r n s w i t h i n t h ei m a g e . D e n s e r t e x t u r e l e a d s t o h i g h e r d i s c r i m i n a t i o n .T h e p e r f o r m a n c e o f o u r m o d e l i m p r o v e s w h e n t h ed e n s i t y o f t h e t e x t u r e e l e m e n t s i s b e i n g i n c r e a s e d .

W e c a n s u m m a r i z e t h e se r e s ul ts a n d c o n c l u d e t h a t :(1 ) V i s u a l d i s c r im i n a t i o n p e r f o r m a n c e i s c o n t i n u o u si n t h e s e n s e t h a t d i s c r i m i n a t i v e a b i l i t y i n c r e a s e s w i t ht h e d i f f e r e n c e s b e t w e e n t h e m i c r o p a t t e r n s . (2 ) T h eh u m a n v i s u al s y s t e m c a n b e d i v id e d i n t o p r e a t t e n -t i v e /a t t e n ti v e m o d e s , b y i n t r o d u c i n g a t h r e s h o ld r a n g et h a t s e p a r a t e s t h e p a r a l l e l a n d s e q u e n t i a l m o d e s .S i m i l ar i ty v a l u es a r o u n d t h e t h r e sh o l d m a y p r o d u c ep r e a t t e n t i v e d e t e c t i o n d e p e n d i n g o n s t i m u l u s c o n -f i g u r a t io n , a n d n o i s e l e v el . P r a c t i c a l ly , t h i s m e a n s t h a ts t i m u l i h a v i n g d i s c r i m i n a b i l i t y v a l u e s w i t h in t h i s r a n g ec a n n o t s e r v e a s g o o d i n d i c a to r s f o r p r e a t t e n ti v e o ra t t e n t i v e p r o c e s s e s . (3 ) T h e f il te r in g a p p r o a c h s e e m st o p r e d ic t q u i te a c c u r a t e l y h u m a n p e r f o r m a n c e inp r e a t t e n t i v e t a s k s , a n d o f f e r s s o m e n a t u r a l w a y s t oa c c o m m o d a t e f o r t he u n a c c o u n t a b l e c a se s.Acknowledgements.We would l ike to thank Nissan Spector andBart Ru benstein for their helpful com men ts and help in edit ing.This work was sup ported b y a grant from the Israel Academy ofSciences and Humanities.

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Received: April 29, 1988Accepted in revised form: Feb ruary 2 , 1989Dr . Dov Sa giDe pa r t me n t o f App l ie d M a t he ma t i cs & Comp u t e r Sc ie nce sThe Weizmann Ins t i tu te of ScienceRe hovo t 76100Israe l