SURGIMIENTO DEL ANÁLISIS DE RESPONSABILIDAD

There are reasons why one might expect differences between the processing of changes in disparity and changes in luminance. The first derivative of disparity, disparity gradient, when calibrated for viewing parameters, is a signal for surface slant. The second derivative, disparity curvature, which needs no such calibration, is a signal for surface cu rv a tu re ^ F Mechanisms for processing contour disparities constitute specialized mechanisms for perceiving slant and curvature, which have no analogue in the luminance domain. In addition, there may well exist cyclopean mechanisms which obtain estimates of the first and second derivatives of disparity by performing operations on a disparity map.

Sensitivity to both slant and curvature is limited by the relationship between disparity and viewing distance. Disparity due to differences in local depth is inversely proportional to the square of the viewing distance (figure 1.4). Thus, the accuracy with which the slant and curvature of real-world objects may be perceived will fall off very rapidly as the distance to an object increases.

Marr's 2V2D Sketch

Marr (1982; Marr and Nishihara, 1978) argued that the function of all depth cues is to generate a representation of the surface orientation of all points in the visual scene: the 2V2D Sketch. Any complete representation of surface orientation requires two parameters for each point. Marr envisaged a representation consisting of a measure of slant and tilt. Slant is defined as the angle between the surface normal and the line of sight. Tilt is defined as the angle formed by the projection of the surface normal onto the fronto-parallel plane. Thus, tilt is the direction of slant. Surface orientation may be uniquely specified relative to an observer by the slant and tilt. Figure 1.16 illustrates this scheme for a variety of surtace slants and tilts.

Slant and curvature provide two very different forms of information about a surface. Slant is one of two free parameters of the orientation of the surface relative to the viewer. Curvature is a property related to the shape of a surface. A point on a surface of a

Under certain circumslanccs the luminance derivatives of an image may also provide information as to the slant and curvature of a surface. Although it is generally stated that the visual system is insenstitive to slow changes in luminance, it remains an empirical question as to under what lighting conditions the derivatives oljluminanceiiight be used as a cue to shape and whether the visual system is sensitive to them.

g i v e n c u r v a t u r e c a n a p p e a r at a n y sl ant a n d tilt to t he o b s e r v e r , w h i l e a v e r y s m a l l p a t c h o f su i f a c e o f a g i v e n sl ant a n d tilt c a n h a v e a l ar ge r a n g e o f c u r v a t u r e s . M a r r di d no t e n v i s a g e a d i r e c t role f o r s t e r e o p s i s in t he p e r c e p t i o n o f c u r v a t u r e . In M a r r ’s s c h e m e , c u r v a t u r e is r e p r e s e n t e d at a f u r t h e r l e v e l o f p r o c e s s i n g a f t e r t h e c o m b i n a t i o n o f d e p t h c u e s . T h i s f u r t h e r r e p r e s e n t a t i o n is c o n s t r u c t e d o u t o f t h e r e p r e s e n t a t i o n o f s l a n t a n d tilt ( M a r r a n d N i s h i h a r a , 1 9 7 8 ) . H e n c e , t h e s c h e m e m a k e s s p e c i f i c p r e d i c t i o n s r e g a r d i n g r e l a t i v e s e n s i t i v i t y to d i s p a r i t y g r a d i e n t s a n d d i s p a r i t y c u r v a t u r e . C u r v a t u r e d i s c r i m i n a t i o n a n d d e t e c t i o n s h o u l d b e p r e d i c t a b l e f r o m t h e s e n s i t i v i t y t o sl ant a n d tilt, w i t h p e r h a p s t he a d d i t i o n o f s o m e e x t r a v a r i a b i l i t y d u e to t he p r o c e s s o f c o m b i n i n g sl a nt a n d tilt m e a s u r e m e n t s f r o m d i f f e r e n t s pat i al l o c a t i o n s .

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F ig u re 1.16. Marr's 2^H i) Sketch. Two dimensions are required to represent each visible surface, in this case slant and tilt. Slant increases with the length o f the arrow. Tilt is given by the orientation o f the arrow.

K o e n d e r i n k ' s s h a p e i n d e x K o e n d e r i n k ( 1 9 9 0 ) e x p l o i t s t he k n o w n f a ct s o f d i f f e r e n t i a l g e o m e t r y t o p r o d u c e a t h e o r e t i c a l p o l a r r e p r e s e n t a t i o n o f s u r f a c e s h a pe . If s e c t i o n s are t a k e n t h r o u g h a p o i n t o n a t h r e e - d i m e n s i o n a l s u r f a c e , the m a x i m u m a n d m i n i m u m c u r v a t u r e s at t ha t p o i n t (Kmax a n d K m i n ) wi l l lie i n p e r p e n d i c u l a r p l a n e s . T h e l o c a l s h a p e o f s u r f a c e s c a n be c h a r a c t e r i z e d by a p o i n t in Kmax, Knim s p a c e . K o e n d e r i n k d e f i n e d t w o q u a n t i t i e s : t he

s h a p e i n d e x ( S) a n d t he c u r v e d n e s s (C) , c o r r e s p o n d i n g to a p o l a r c o o r d i n a t e s y s t e m w i t h i n t h i s s p a ce , as d e p i c t e d in f ig ur e 1.17. A s w i th M a r r ’s 2 V 2 D S k e t c h , K o e n d e r i n k ' s s c h e m e is n o t s p e c i f i c to s t e r e o p s i s , b u t p r o v i d e s a r e p r e s e n t a t i o n o f s h a p e w h i c h c a n be c o n s t r u c t e d f r o m all d e p t h c u e s a n d is p r i m a r i l y b a s e d o n s e n s i t i v i t y to s u r f a c e c u r v a t u r e . It h a s a l s o b e e n p r o p o s e d t h a t s t e r e o p s i s is e s p e c i a l l y s e n s i t i v e t o d i s p a r i t y c u r v a t u r e o n t h e g r o u n d s o f v i e w i n g g e o m e t r y ( R o g e r s a n d C a g e n e l l o , 1 989) a n d c o m b i n a t i o n w i t h m o n o c u l a r p e r s p e c t i v e c u e s ( S t e v e n s a n d B r o o k e s , 1988). min -0.5 m a x 0. 5

Figure 1.17. Koenderink's (1990) representation o f surface shape, based on a polar coordinate system in a space defined by two principle curvatures {K,naxy ^min)- Shape is ^iven by the shape index, S, and curvedness, C.