j physiol 1968 campbell 551 66

8/2/2019 J Physiol 1968 Campbell 551 66

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J . P h y s i o l . ( 1 9 6 8 ) , 1 9 7 , p p . 5 5 1 - 5 6 6 551W i t h 7 t e x t - f i g u r e 8P r i n t e d i n G r e a t B r i t a i n

APPLICATION OF FOURIER ANALYSIS TO THE VISIBILITYOF GRATINGS

BY F . W. CAMPBELL AND J . G . ROBSONFrom t h e P h y s i o l o g i c a l L a b o r a t o r y , U n i v e r s i t y o f C a m b r i d g e

( R e c e i v e d 1 0 N o v e m b e r 1 9 6 7 )SUMMARY

1 . T h e c o n t r a s t t h r e s h o l d s o f a v a r i e t y o f g r a t i n g p a t t e r n s h a v e b e e nm e a s u r e d o v e r a w i d e r a n g e o f s p a t i a l f r e q u e n c i e s .2 . C o n t r a s t t h r e s h o l d s f o r t h e d e t e c t i o n o f g r a t i n g s w h o s e l u m i n a n c ep r o f i l e s a r e s i n e , s q u a r e , r e c t a n g u l a r o r s a w - t o o t h w a v e s c a n b e s i m p l yr e l a t e d u s i n g F o u r i e r t h e o r y .3 . O v e r a w i d e r a n g e o f s p a t i a l f r e q u e n c i e s t h e c o n t r a s t t h r e s h o l d o f ag r a t i n g i s d e t e r m i n e d o n l y b y t h e a m p l i t u d e o f t h e f u n d a m e n t a l F o u r i e rc o m p o n e n t o f i t s wave f o r m .

4 . G r a t i n g s o f c o m p l e x wave f o r m c a n n o t b e d i s t i n g u i s h e d f r o m s i n e -w a v e g r a t i n g s u n t i l t h e i r c o n t r a s t h a s b e e n r a i s e d t o a l e v e l a t w h i c h t h eh i g h e r h a r m o n i c c o m p o n e n t s r e a c h t h e i r i n d e p e n d e n t t h r e s h o l d .5 . T h e s e f i n d i n g s c a n b e e x p l a i n e d by t h e e x i s t e n c e w i t h i n t h e n e r v o u ss y s t e m o f l i n e a r l y o p e r a t i n g i n d e p e n d e n t m e c h a n i s m s s e l e c t i v e l y s e n s i t i v et o l i m i t e d r a n g e s o f s p a t i a l f r e q u e n c i e s .

INTRODUCTIONO u r a b i l i t y t o p e r c e i v e t h e d e t a i l s o f a v i s u a l s c e n e i s d e t e r m i n e d b y t h er e l a t i v e s i z e a n d c o n t r a s t o f t h e d e t a i l p r e s e n t . T h i s i s c l e a r l y d e m o n -s t r a t e d when t h e s c e n e i s a n e x t e n d e d g r a t i n g p a t t e r n w h o s e l u m i n a n c e

p e r p e n d i c u l a r t o t h e b a r s i s m o d u l a t e d s i n u s o i d a l l y a b o u t a f i x e d m e a nl e v e l ( s i n e - w a v e g r a t i n g : F i g . 1 ) . I n t h i s c a s e t h e t h r e s h o l d c o n t r a s t *n e c e s s a r y f o r p e r c e p t i o n o f t h e b a r s i s f o u n d t o b e a f u n c t i o n o f t h e s p a t i a lf r e q u e n c y o f t h e g r a t i n g . T h e r e c i p r o c a l o f t h e t h r e s h o l d c o n t r a s t i s t h e' c o n t r a s t s e n s i t i v i t y ' a n d t h e v a r i a t i o n o f t h e s e n s i t i v i t y o v e r a r a n g e o fs p a t i a l f r e q u e n c i e s i s d e s c r i b e d b y t h e ' c o n t r a s t - s e n s i t i v i t y f u n c t i o n ' .T h e f i r s t m e a s u r e m e n t o f t h e c o n t r a s t - s e n s i t i v i t y f u n c t i o n o f t h e hu m a nv i s u a l s y s t e m w a s r e p o r t e d b y S c h a d e i n 1 9 5 6 . S c h a d e i n t e r p r e t e d h i s

* I n t h i s work w e f o l l o w M i c h e l s o n ( 1 9 2 7 ) , w h o d e f i n e d t h e c o n t r a s t o f a g r a t i n g a s t h em axim u m l u m i n a n c e minus t h e minimum l u m i n a n c e d i v i d e d b y t w i c e t h e m e a n l u m i n a n c e ,a s i l l u s t r a t e d i n F i g . 1 .

) by guest on March 1, 2012jp.physoc.orgDownloaded from J Physiol (
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F . W . CAMPBELL AND J . G . ROBSONf i n d i n g s i n t e r m s o f t h e m o d u l a t i o n t r a n s f e r f u n c t i o n o f a d i f f r a c t i o nl i m i t e d o p t i c a l s v s t e m t o g e t h e r w i t h t h e e f f e c t s o f s p a t i a l i n t e r a c t i o n sw i t h i n t h e n e r v o u s m e c h a n i s m o f t h e r e t i n a . M o r e r e c e n t l y , t h e a c t u a lm o d u l a t i o n t r a n s f e r f u n c t i o n o f t h e d i o p t r i c m e c h a n i s m o f t h e e y e h a sb e e n m e a s u r e d d i r e c t l y a n d t h e c o n t r a s t - s e n s i t i v i t y f u n c t i o n o f t h e v i s u a ln e r v o u s s y s t e m h a s b e e n i n d e p e n d e n t l y d e t e r m i n e d ( A r n u l f & D u p u y ,1 9 6 0 ; C a m p b e l l & G r e e n , 1 9 6 5 ; C a m p b e l l & G u b i s c h , 1 9 6 6 ) .L

0

F i g . 1 . Luminance d i s t r i b u t i o n a c r o s s s i n e - , s q u a r e - , s a w - t o o t h - a n d r e c t a n g u l a r -w a v e g r a t i n g s o f c o n t r a s t 0 - 5 . Th e s p a c e - a v e r a g e l u m i n a n c e i s i n d i c a t e d b y ad a s h e d l i n e . Th e c o n t r a s t , m , i s d e f i n e d b y m = ( L . x - L m t n ) / ( 2 Z ) , w h e r e Li s t h e s p a c e - a v e r a g e l u m i n a n c e . Th e s p a t i a l f r e q u e n c y o f t h e g r a t i n g i s t h e r e c i -p r o c a l o f t h e a n g u l a r s u b t e n s e a t t h e o b s e r v e r ' s e y e o f o n e c o m p l e t e c y c l e . T h er e c t a n g u l a r w a v e - g r a t i n g i s c h a r a c t e r i z e d b y i t s d u t y c y c l e , r , w h i c h i s t h e r a t i o o ft h e a n g u l a r w i d t h o f t h e l i g h t e r b a r t o t h e w i d t h o f o n e c o m p l e t e c y c l e .

U s i n g t h e m e a s u r e d m o d u l a t i o n t r a n s f e r f u n c t i o n o f t h e d i o p t r i c s i t i sp o s s i b l e t o c a l c u l a t e t h e r e t i n a l l i g h t d i s t r i b u t i o n f o r a n y o b j e c t ( e . g .G u b i s c h , 1 9 6 7 ) . H o w e v e r , i t i sn o t k n o w n w h e t h e r t h e c o n t r a s t t h r e s h o l do f a n y t a r g e t o t h e r t h a n a s i n e - w a v e g r a t i n g c a n b e c a l c u l a t e d f r o m ak n o w l e d g e o f t h e c o n t r a s t - s e n s i t i v i t y f u n c t i o n o f t h e v i s u a l s y s t e m . I nc a l c u l a t i n g t h e l i g h t d i s t r i b u t i o n i n t h e r e t i n a l i m a g e , l i n e a r t h e o r y c a nc e r t a i n l y b e u s e d s i n c e t h e p r i n c i p l e o f s u p e r p o s i t i o n n e c e s s a r i l y a p p l i e st o t h e f o r m a t i o n o f o p t i c a l i m a g e s ( e . g . H o p k i n s , 1 9 6 2 ) . H o w e v e r , i n c a l -c u l a t i n g a c o n t r a s t t h r e s h o l d i t i s n o t c e r t a i n t h a t s i m p l e l i n e a r t h e o r y c a nb e u s e d s i n c e t h e p r i n c i p l e o f s u p e r p o s i t i o n d o e s n o t n e c e s s a r i l y a p p l y t ot h e n e r v o u s i n t e r a c t i o n s o f t h e v i s u a l s y s t e m . T h e p u r p o s e o f t h i s i n v e s t i -g a t i o n w a s t o f i n d o u t t o w h a t e x t e n t t h e c o n t r a s t - s e n s i t i v i t y f u n c t i o nc o u l d b e u s e d f o r t h e p r e d i c t i o n o f c o n t r a s t t h r e s h o l d s .A c o n v e n i e n t t e c h n i q u e f o r c a l c u l a t i n g t h e l i g h t d i s t r i b u t i o n i n t h ei m a g e o f a n o n - s i n u s o i d a l g r a t i n g i s t o c o n s i d e r s e p a r a t e l y t h e i m a g i n g o f

e a c h o f t h e F o u r i e r c o m p o n e n t s o f t h e g r a t i n g w a v e f o r m . I n t h i s s t u d y ,

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VISIBILITY OF GRATINGS 5 5 3t h e c o n t r a s t t h r e s h o l d s o f g r a t i n g s w i t h v a r i o u s wave f o r m s ( F i g . 1 ) h a v eb e e n m e a s u r e d a n d t h e r e s u l t s h a v e b e e n c o n s i d e r e d i n t e r m s o f t h e s e n s i -t i v i t y o f t h e v i s u a l s y s t e m t o t h e i n d i v i d u a l h a r m o n i c c o m p o n e n t s o f t h e s ew a v e f o r m s .

METHODSTh e m eth od u s e d t o g e n e r a t e g r a t i n g t a r g e t s w i t h d i f f e r e n t m o d u l a t i o n w a v e f o r m s w a s

b a s e d o n t h a t o f S c h a d e ( 1 9 5 6 ) . A r a s t e r o f 2 0 0 0 l i n e s was p r o d u c e d on t h e s c r e e n o f ac a t h o d e - r a y t u b e a t a f r a m e f r e q u e n c y o f a b o u t 1 0 0 / s e c . The b e a m i n t e n s i t y was m o d u l a t e dby a s i g n a l f r o m a w a v e - f o r m g e n e r a t o r . With s u i t a b l e s y n c h r o n i z a t i o n o f t h e t i m e - b a s e andw a v e - f o r m g e n e r a t o r s , s t a t i o n a r y g r a t i n g s o f a n y d e s i r e d s p a t i a l f r e q u e n c y c o u l d b e d i s -p l a y e d o n t h a s c r e e n o f t h e c a t h o d e - r a y t u b e . Th e c o n t r a s t o f t h e g r a t i n g p a t t e r n c o u l d b ev a r i e d b y a d j u s t i n g t h e m o d u l a t i o n v o l t a g e . I t wa s f o u n d t h a t t h e c o n t r a s t was p r o p o r t i o n a lt o t h e a m p l i t u d e o f t h e m o d u l a t i n g v o l t a g e , a t l e a s t f o r c o n t r a s t s o f l e s s t h a n 0 - 6 . Th e m e a ns c r e e n l u m i n a n c e ( 5 0 0 c d / M 2 ) w a s i n d e p e n d e n t o f t h e c o n t r a s t o r s p a t i a l f r e q u e n c y o f t h eg r a t i n g b e i n g d i s p l a y e d . For making m e a s u r e m e n t s o f c o n t r a s t t h r e s h o l d t h e m o d u l a t i n gv o l t a g e w a s s w i t c h e d on a n d o f f a t 0 * 5 c / s s o t h a t t h e pattem was p e r i o d i c a l l y i n t r o d u c e dand removed w i t h o u t a ny c h a n g e i n t h e m e a n l u m i n a n c e .

I n f r o n t o f t h e w h i t e s c r e e n o f t h e c a t h o d e - r a y t u b e was p l a c e d a s h e e t o f w h i t e c a r d -b o a r d 3 0 c m i n d i a m e t e r h a v i n g a c e n t r a l a p e r t u r e e i t h e r 1 0 x 1 0 c m o r 2 x 2 c m . Th e c a r d -b o a r d s u r r o u n d wa s i l l u m i n a t e d t o match t h e c a t h o d e - r a y t u b e s c r e e n .

U n l e s s o t h e r w i s e s t a t e d , v i e w i n g was m o n o c u l a r , t h e e y e was h o m a t r o p i n i z e d and a na r t i f i c i a l p u p i l o f 2 - 5 mm d i a m e t e r w as u s e d . Th e r e f r a c t i o n was a l w a y s c o r r e c t e d f o r t h ev i e w i n g d i s t a n c e s t o w i t h i n 0 - 2 5 D . C o n t r a s t t h r e s h o l d s w e r e d e t e r m i n e d b y t h e s u b j e c ta d j u s t i n g t h e c o n t r a s t o f a g r a t i n g o f a g i v e n s p a t i a l f r e q u e n c y u n t i l t h e p attem w a s b a r e l yd e t e c t a b l e . E i t h e r f i v e o r t e n o b s e r v a t i o n s w e r e m a d e f o r e a c h t h r e s h o l d d e t e r m i n a t i o n .T h i s g a v e a s t a n d a r d e r r o r o f t h e m e a n o f l e s s t h a n 1 0 %. Th e a u t h o r s a c t e d a s s u b j e c t s . I ne v e r y e x p e r i m e n t b o t h s u b j e c t s g a v e c o n s i s t e n t and v e r y s i m i l a r r e s u l t s .

RESULTSWhe n g r a t i n g s w e r e d i s p l a y e d w h i c h h a d f r e q u e n c i e s o f more t h a n 1 0

c / c m o n t h e s c r e e n i t w a s n o t e d t h a t t h e r e w a s a s i g n i f i c a n t d e c r e a s e i nt h e i r c o n t r a s t d u e t o l i m i t a t i o n s o f t h e c a t h o d e - r a y t u b e . T h i s l i m i t a t i o nmade i t n e c e s s a r y f o r t h e s u b j e c t t o v i e w t h e s c r e e n f r o m t w o d i s t a n c e s i no r d e r t o c o v e r a s u f f i c i e n t l y w i d e r a n g e o f s p a t i a l f r e q u e n c i e s w i t h o u t t h en u m b e r o f v i s i b l e b a r s o f t h e g r a t i n g b e i n g t o o f e w . O b s e r v a t i o n s h o w st h a t c o n t r a s t t h r e s h o l d s a r e s i g n i f i c a n t l y r a i s e d i f l e s s t h a n f o u r c y c l e s o ft h e g r a t i n g a r e v i s i b l e . I n F i g . 2 t h e c o n t r a s t - s e n s i t i v i t y f u n c t i o n s( F . W . C . ) f o r t h e t w o v i e w i n g d i s t a n c e s o f 5 7 a n d 2 8 5 c m a r e s h o w n . T h er e s u l t s s h o w n a s t r i a n g l e s ( A ) w e r e o b t a i n e d a t t h e f u r t h e r d i s t a n c e w h e r et h e 1 0 x 1 0 c m a p e r t u r e s u b t e n d e d 2 x 2 . T h e maximum s e n s i t i v i t yo c c u r r e d a t a s p a t i a l f r e q u e n c y o f 4 c / d e g a n d t h e s e n s i t i v i t y d e c r e a s e da t b o t h h i g h e r a n d l o w e r f r e q u e n c i e s . Some o f t h e d e c r e a s e o f s e n s i t i v i t ya t t h e l o w e r f r e q u e n c i e s m i g h t h a v e b e e n d u e t o t h e s m a l l number o f b a r sb e i n g d i s p l a y e d . T h e r e f o r e t h e o b s e r v a t i o n s w e r e r e p e a t e d a t t h e n e a r e r

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tp://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http://jp.physoc.org/http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4/16

F . W . CAMPBELL AND J . G . ROBSONv i e w i n g d i s t a n c e w h e r e t h e l a r g e a p e r t u r e s u b t e n d e d 1 0 0 x 1 0 0 . T h e s er e s u l t s a r e s h o w n a s s q u a r e s ( E l ) . T h e s e n s i t i v i t i e s a t s p a t i a l f r e q u e n c i e sl e s s t h a n 3 c / d e g w e r e s u b s t a n t i a l l y g r e a t e r t h a n a t t h e f u r t h e r v i e w i n gd i s t a n c e . T h i s c h a n g e w a s a l m o s t c e r t a i n l y d u e t o t h e i n c r e a s e d s i z e o ff i e l d b u t i t m i g h t p o s s i b l y h a v e b e e n c o n n e c t e d w i t h t h e a c t u a l v i e w i n gd i s t a n c e a l t h o u g h t h e e y e w a s a c c u r a t e l y r e f r a c t e d f o r b o t h d i s t a n c e s .

1 0 0 0_ 0

o ho A1 0 0 O OA

V 1 0 =

1I IllI 1 1 10 - 1 1 1 0S p a t i a l f r e q u e n c y ( c / d e g )F i g . 2 . C o n t r a s t s e n s i t i v i t y f o r s i n e - w a v e g r a t i n g s . S u b j e c t F.W. C . , l u m i n a n c e5 0 0 c d / m 2 . V i e w i n g d i s t a n c e 2 8 5 cm and a p e r t u r e 2 0 x 2 0 , A ; v i e w i n g d i s t a n c e5 7 c m , a p e r t u r e 1 0 0 x 1 0 , O; v i e w i n g d i s t a n c e 5 7 c m , a p e r t u r e 2 0 x 2 , 0 .

To e s t a b l i s h t h a t f i e l d s i z e w a s t h e c r u c i a l f a c t o r , f u r t h e r m e a s u r e m e n t sw e r e made a t t h e n e a r d i s t a n c e w i t h t h e 2 x 2 c m a p e r t u r e w h i c hs u b t e n d e d 2 0 x 2 0 . T h e r e s u l t s a r e s h o w n a s c i r c l e s ( o ) i n F i g . 2 a n dc o r r e s p o n d t o t h e r e s u l t s ( A ) o b t a i n e d a t t h e f u r t h e r d i s t a n c e w i t h t h es a m e a p e r t u r e s u b t e n s e . W e t h e r e f o r e f e e l j u s t i f i e d i n c o m b i n i n g r e s u l t so b t a i n e d a t t h e s e two v i e w i n g d i s t a n c e s w h e r e t h i s i s n e c e s s a r y t o o b t a i na w i d e c o v e r a g e o f s p a t i a l f r e q u e n c y . A t s p a t i a l f r e q u e n c i e s o f 0 - 4 c / d e ga n d l e s s , t h e c o n t r a s t s e n s i t i v i t y w i l l p r o b a b l y h a v e b e e n s l i g h t l y r a i s e db e c a u s e o f t h e s m a l l number o f c y c l e s v i s i b l e i n t h e 1 0 0 a p e r t u r e .

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VISIBILITY OF GRATINGSV i s i b i l i t y o f s i n e - a n d s q u a r e - w a v e g r a t i n g s

F o u r i e r t h e o r y s h o w s t h a t a s q u a r e w a v e c a n b e c o n s i d e r e d a s t h e s umo f a n u m b e r o f s i n e - w a v e c o m p o n e n t s w h o s e f r e q u e n c i e s a r e o d d m u l t i p l e so f t h e f u n d a m e n t a l f r e q u e n c y . T h u s a s q u a r e wave w h i c h i s a f u n c t i o n o fx h a v i n g u n i t a m p l i t u d e ( p e a k - t o - p e a k a m p l i t u d e = 2 ) a n d p e r i o d X c a nb e c o n s i d e r e d a s t h e s um o f t h e i n f i n i t e s e r i e s

4 . 2Tx 1. 2 i r x 1 2 r T x- s m X+-- s i n 3 X +- s i n 5 ; . .T h u s t h e a m p l i t u d e o f t h e f u n d a m e n t a l ( f i r s t h a r m o n i c ) c o m p o n e n t o f as q u a r e - w a v e g r a t i n g o f c o n t r a s t m i s 4 m / 7 r , w h i l e t h e a m p l i t u d e s o f t h et h i r d , f i f t h a n d h i g h e r h a r m o n i c s a r e r e s p e c t i v e l y 4 m / 3 m , 4 m / 1 5 T a n d s o o n .T h e e v e n h a r m o n i c s a l l h a v e z e r o a m p l i t u d e .T h e c o n t r a s t - s e n s i t i v i t y f u n c t i o n o f t h e v i s u a l s y s t e m f a l l s o f f r a p i d l ya t s p a t i a l f r e q u e n c i e s a b o v e 3 c / d e g a n d i t m i g h t t h e r e f o r e b e e x p e c t e dt h a t t h e v i s i b i l i t y o f a s q u a r e - w a v e g r a t i n g o f h i g h f r e q u e n c y w o u l d b ed e t e r m i n e d l a r g e l y b y t h e a m p l i t u d e o f i t s f u n d a m e n t a l c o m p o n e n t . T h a ti s , t h e c o n t r a s t s e n s i t i v i t y f o r a s q u a r e - w a v e g r a t i n g w o u l d b e e x p e c t e dt o b e a b o u t 4 / 1 T t i m e s g r e a t e r t h a n t h e c o n t r a s t s e n s i t i v i t y f o r a s i n e - w a v eg r a t i n g o f t h e s a m e s p a t i a l f r e q u e n c y .To t e s t t h i s h y p o t h e s i s w e d e t e r m i n e d t h e r a t i o o f t h e s e n s i t i v i t i e s f o rs i n e - a n d s q u a r e - w a v e g r a t i n g s , b y m e a s u r i n g a l t e r n a t e l y t h e t h r e s h o l d sf o r t h e t w o t y p e s o f g r a t i n g a t v a r i o u s s p a t i a l f r e q u e n c i e s . T h i s p r o c e d u r e

g a v e m o r e p r e c i s e v a l u e s o f t h e r a t i o t h a n w o u l d h a v e b e e n o b t a i n e d b yf i r s t m e a s u r i n g t h e t h r e s h o l d s a t a l l f r e q u e n c i e s f o r o n e wave f o r m a n d l a t e rf o r t h e o t h e r wave f o r m .T h e r e s u l t s s h o w n i n F i g . 3 w e r e o b t a i n e d a t a m e a n l u m i n a n c e o f 5 0 0c d / M 2 w i t h s u b j e c t J . G . R . I t i s c l e a r t h a t a t a l l s p a t i a l f r e q u e n c i e s t h ec o n t r a s t - s e n s i t i v i t y f o r s q u a r e - w a v e g r a t i n g s ( [ ) w a s g r e a t e r t h a n t h a tf o r s i n e - w a v e g r a t i n g s ( o ) . T h e r a t i o o f t h e s e s e n s i t i v i t i e s ( s q u a r e / s i n es e n s i t i v i t y r a t i o ) a t e a c h s p a t i a l f r e q u e n c y i s p l o t t e d b e l o w ( 0 ) T h es t r a i g h t l i n e d r a w n t h r o u g h t h e r a t i o s i s a t t h e p r e d i c t e d v a l u e o f4 1 n = 1 - 2 7 3 . T h e m e a s u r e d r a t i o d i d n o t d e v i a t e s y s t e m a t i c a l l y f r o m t h i sv a l u e u n t i l t h e s p a t i a l f r e q u e n c y w a s r e d u c e d t o l e s s t h a n a b o u t 0 - 8 c / d e gwhen t h e r a t i o b e c a m e r a p i d l y much g r e a t e r .I t i s i n t e r e s t i n g t o n o t e t h a t when t h e s p a t i a l f r e q u e n c y o f a s q u a r e -wave g r a t i n g w h o s e c o n t r a s t i s j u s t a t t h r e s h o l d i s g r e a t e r t h a n 0 - 8 c / d e g ,t h e g r a t i n g i s n o t p e r c e i v e d a s d i f f e r e n t f r o m a s i n e - w a v e g r a t i n g . H o w e v e r ,s q u a r e - w a v e g r a t i n g s w i t h s p a t i a l f r e q u e n c i e s o f l e s s t h a n 0 - 8 c / d e g a r ep e r c e i v e d t o d i f f e r f r o m s i n e - w a v e g r a t i n g s o f t h e s a m e f r e q u e n c y a s s o o na s t h e c o n t r a s t i s g r e a t e n o u g h f o r t h e m t o b e s e e n .T h e e x p e r i m e n t w a s r e p e a t e d o n s u b j e c t F . W . C . a t t w o l u m i n a n c e l e v e l s

5 5 5

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5 5 6 F . W. CAMPBELL AND J . G . ROBSON( F i g . 4 ) . At t h e h i g h e r l u m i n a n c e ( 5 0 0 c d / M 2 ) , t h e s q u a r e / s i n e s e n s i t i v i t yr a t i o was f o u n d t o b e 1 - 2 8 0 + 0 - 0 2 5 S . E . w h e n a l l t h e m e a s u r e m e n t s i n t h ef r e q u e n c y r a n g e 1 - 3 5 c / d e g were l u m p e d t o g e t h e r . T h i s v a l u e d o e s n o td i f f e r s i g n i f i c a n t l y f r o m 4 / 1 f = 1 - 2 7 3 . For f r e q u e n c i e s b e l o w 0 - 8 c / d e g t h es q u a r e / s i n e r a t i o was c l e a r l y g r e a t e r t h a n 4 / i f .

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o rI1111111 I IltIlIll a I0 1 1 1 0S p a t i a l f r e q u e n c y ( c / d e g )F i g . 3 . C o n t r a s t s e n s i t i v i t y f o r s i n e - w a v e g r a t i n g s ( 0 ) i s c o m p a r e d w i t h t h a t f o rs q u a r e - w a v e g r a t i n g s ( E O ) f o r s u b j e c t J . G . R . a t a l u m i n a n c e o f 5 0 0 c d / M 2 . T h er a t i o o f t h e c o n t r a s t s e n s i t i v i t i e s a t e a c h s p a t i a l f r e q u e n c y i s p l o t t e d a t t h e bottomo f t h e f i g u r e ( t h e b a r s show + s . x . o f m e a n ) . A c o n t i n u o u s l i n e i s d r a w n t h r o u g h t h er a t i o s a t 4 / i = 1 - 2 7 3 . Th e d a s h e d l i n e i n d i c a t e s t h e p r e d i c t e d r a t i o a s s u m i n g as i m p l e p e a k d e t e c t o r m e c h a n i s m .

T h e l o w e r p a i r o f c u r v e s i n F i g . 4 w a s o b t a i n e d a t a l u m i n a n c e o f 0 0 5c d / M 2 . A t t h i s l o w e r l u m i n a n c e t h e c o n t r a s t - s e n s i t i v i t y c u r v e s w e r es h i f t e d t o l o w e r f r e q u e n c i e s a n d l o w e r s e n s i t i v i t i e s a n d t h e s q u a r e / s i n es e n s i t i v i t y r a t i o o n l y d e v i a t e d f r o m 4 / i f a t f r e q u e n c i e s b e l o w 0 * 4 c / d e g .

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VISIBILITY OF GRATINGS 5 5 7DePalma & Lowry ( 1 9 6 2 ) m e a s u r e d t h e c o n t r a s t t h r e s h o l d f o r s i n e - a n d s q u a r e - w a v e

g r a t i n g s a t a v a r i e t y o f v i e w i n g d i s t a n c e s . At a d i s t a n c e o f 3 5 i n . ( 8 9 c m ) t h e y f o u n d nos i g n i f i c a n t d i f f e r e n c e i n t h e c o n t r a s t t h r e s h o l d s f o r s i n e - and s q u a r e - w a v e g r a t i n g s . H o w e v e r ,a t v i e w i n g d i s t a n c e s o f 1 0 a n d 1 4 i n . ( 2 5 and 3 6 c m ) t h e y f o u n d t h a t t h e c o n t r a s t t h r e s h o l df o r s q u a r e - w a v e g r a t i n g s was g r e a t e r t h a n f o r s i n e - w a v e g r a t i n g s a t h i g h s p a t i a l f r e q u e n c i e sb u t l e s s a t l o w s p a t i a l f r e q u e n c i e s . Th e i n c o n s i s t e n c i e s i n t h h i r r e s u l t s f o r t h e s e t h r e e d i s t a n c e sm a y h a v e b e e n d u e t o i n c o m p l e t e a c c o m m o d a t i o n a t t h e n e a r e r d i s t a n c e s ( F i n c h a m , 1 9 5 5 )f o r t h e y s t a t e : 'No a r t i f i c i a l p u p i l s w e r e u s e d and o t h e r o p t i c s w e r e p u r p o s e l y o m i t t e d s ot h a t t h e o b s e r v e r l o o k e d d i r e c t l y a t t h e t e s t o b j e c t . ' We f o u n d t h a t o u r r e s u l t s w e r e q u i t ec o n s i s t e n t w h e n a c y c l o p l e g i c and a r t i f i c i a l p u p i l w e r e u s e d a n d t h e o b s e r v e r s were c a r e f u l l yr e f r a c t e d f o r t h e v i e w i n g d i s t a n c e . M e a s u r i n g t h e t h r e s h o l d f o r s i n e - a n d s q u a r e - w a v eg r a t i n g s a l t e r n a te l y a t e a c h s p a t i a l f r e q u e n c y p r o b a b l y a l s o a s s i s te d i n o b t a i n i n g c o n s i s t e n tr e s u l t s .

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1 I II I 1 1 1 1 t 1 1 1111 I I I I0 1 1 1 0S p a t i a l f r e q u e n c y ( c / d e g )

F i g . 4 . C o n t r a s t s e n s i t i v i t y f o r s i n e - w a v e g r a t i n g s ( 0 , 0) i s c o m p a r e d w i t h t h a tf o r s q u a r e - w a v e g r a t i n g s ( r l , *) f o r s u b j e c t F . W . C . U p p e r p a i r o f c u r v e s f o rl u m i n a n c e o f 5 0 0 c d / M 2 n . Lower p a i r o f c u r v e s f o r l u m i n a n c e o f 0 0 5 c d / m 2 .

V i s i b i l i t y o f o t h e r g r a t i n g sR e c t a n g u l a r - w a v e g r a t i n g s . T h e v a r i a t i o n i n l u m i n a n c e a c r o s s a r e c t -a n g u l a r - w a v e g r a t i n g ( F i g . 1 ) i s d e s c r i b e d b y t h e c o n t r a s t , m , a n d t h e d u t yc y c l e , r . T h e a m p l i t u d e s o f t h e f i r s t , s e c o n d , t h i r d a n d s u c c e s s i v e h a r m o n i c s

a r e r e s p e c t i v e l y ( 4 m s i n 7 r r ) / T , ( 4 m s i n 2 7 T r ) / 2 i r , ( 4 m s i n 3 7 T r ) / 3 1 r a n d s o o n .T h e a m p l i t u d e s o f a l l t h e h i g h e r h a r m o n i c c o m p o n e n t s a r e l e s s t h a n t h a t

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F . W . CAMPBELL AND J . G . ROBSONo f t h e f u n d a m e n t a l a l t h o u g h w h e n t h e d u t y c y c l e i s e i t h e r v e r y s m a l l o rv e r y n e a r l y u n i t y t h e a m p l i t u d e s o f t h e f i r s t f e w h a r m o n i c s a r e v e r y n e a r l ya s g r e a t a s t h e a m p l i t u d e o f t h e f u n d a m e n t a l .

T h e s u b j e c t c o u l d b e p r e s e n t e d w i t h e i t h e r a r e c t a n g u l a r - w a v e o r a s i n e -w a v e g r a t i n g o f t h e s a m e s p a t i a l f r e q u e n c y ( l I c / d e g ) a n d t h e same s p a c e -a v e r a g e l u m i n a n c e ( 5 0 0 c d / M 2 ) . T h e s u b j e c t m a d e s e t t i n g s o f t h e t h r e s h o l d

1 . 0 I05

00

I , , , , \0 0 - 5 1

Duty c y c l eF i g . 5 . Th a r a t i o o f t h e t h r e s h o l d c o n t r a s t f o r a s i n e - w a v e g r a t i n g t o t h e t h r e s h o l dc o n t r a s t f o r a r e c t a n g u l a r - w a v e g r a t i n g a s a f u n c t i o n o f t h e d u t y c y c l e ( w i d t h o fb r i g h t b a r / w i d t h o f w h o l e c y c l e ) . Th e b a r s i n d i c a t e + 1 s . E . o f mean. S u b j e c tJ . G . R . , m e a n s c r e e n l u m i n a n c e 5 0 0 c d / M 2 , s p a t i a l f r e q u e n c y 1 1 c / d e g . Th e c u r v et h r o u g h t h e measurements i s t h a t e x p e c t e d t h e o r e t i c a l l y .

c o n t r a s t f o r t h e t w o k i n d s o f g r a t i n g a l t e r n a t e l y . Te n p a i r s o f r e a d i n g sw e r e made a n d u s e d t o c a l c u l a t e t h e r a t i o o f t h e c o n t r a s t s e n s i t i v i t i e s f o rt h e s i n e - w a v e a n d r e c t a n g u l a r - w a v e g r a t i n g . T h i s p r o c e d u r e w a s r e p e a t e df o r r e c t a n g u l a r - w a v e g r a t i n g s w i t h d i f f e r e n t d u t y c y c l e s r a n g i n g f r o m 0 - 0 5( l i g h t b a r s o n a d a r k e r b a c k g r o u n d ) t o 0 - 9 5 ( d a r k e r b a r s o f t h e s a m e w i d t ho n a l i g h t e r b a c k g r o u n d ) . I n t h i s e x p e r i m e n t t h e s u b j e c t v i e w e d t h e d i s -p l a y b i n o c u l a r l y f r o m a d i s t a n c e o f 2 8 5 c m a n d n o c y c l o p l e g i c w a s u s e d .

F i g u r e 5 s h o w s t h e r e s u l t s o b t a i n e d w i t h s u b j e c t J . G . R . T h e m e a s u r e d

5 5 8

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VISIBILITY OF GRATINGSv a l u e s f o r t h e c o n t r a s t - s e n s i t i v i t y r a t i o f o r r e c t a n g u l a r - w a v e a n d s i n e -wave g r a t i n g s ( r e c t a n g u l a r / s i n e s e n s i t i v i t y r a t i o ) h a v e b e e n p l o t t e d a g a i n s tt h e d u t y c y c l e o f t h e r e c t a n g u l a r - w a v e . T h e c o n t r a s t - s e n s i t i v i t y r a t i or e a c h e s a maximum w h e n t h e d u t y c y c l e i s o n e h a l f ( s q u a r e - w a v e g r a t i n g )a n d f a l l s s y m m e t r i c a l l y a s t h e d u t y c y c l e i s r e d u c e d t o w a r d s z e r o o r i n -c r e a s e d t o w a r d s u n i t y .T h e c u r v e s h o w s ( 4 s i n 7 T r ) / r w h i c h i s t h e c o n t r a s t - s e n s i t i v i t y r a t i o t h a tw o u l d b e o b t a i n e d i f o n l y t h e f u n d a m e n t a l c o m p o n e n t o f t h e r e c t a n g u l a r -wave g r a t i n g c o n t r i b u t e d t o i t s v i s i b i l i t y . T h e m e a s u r e d v a l u e s d o n o td e v i a t e s i g n i f i c a n t l y f r o m t h i s c u r v e .We h a v e a l r e a d y s h o w n t h a t t h e r e c t a n g u l a r / s i n e s e n s i t i v i t y r a t i o f o rr e c t a n g u l a r w a v e s w i t h a d u t y c y c l e o f o n e h a l f ( i . e . s q u a r e w a v e s ) i s

c o n s t a n t f o r a l l s p a t i a l f r e q u e n c i e s a b o v e a b o u t 0 - 8 c / d e g , i n d i c a t i n g t h a ta t t h e s e f r e q u e n c i e s o n l y t h e f u n d a m e n t a l o f t h e s q u a r e - w a v e i s i m p o r t a n ti n d e t e r m i n i n g t h e t h r e s h o l d . H o w e v e r , a s q u a r e w a v e h a s n o s e c o n d o ro t h e r e v e n h a r m o n i c s a n d t h e a m p l i t u d e s o f t h e o d d h a r m o n i c s f a l l o f fr a p i d l y w i t h i n c r e a s i n g s p a t i a l f r e q u e n c y . On t h e o t h e r h a n d a r e c t a n g u l a r -wave g r a t i n g w h o s e d u t y c y c l e a p p r o a c h e s z e r o o r u n i t y h a s b o t h e v e n a n do d d h a r m o n i c s , t h e f i r s t f e w o f w h i c h h a v e a m p l i t u d e s n e a r l y a s g r e a t a st h a t o f t h e f u n d a m e n t a l . W e m i g h t t h e r e f o r e e x p e c t t h a t t h e h i g h e rh a r m o n i c s o f s u c h r e c t a n g u l a r w a v e s w o u l d a f f e c t t h e t h r e s h o l d c o n t r a s tt o a g r e a t e r e x t e n t when t h e s p a t i a l f r e q u e n c y w a s l o w e r e d t h a n i s t h ec a s e f o r t h e s q u a r e w a v e .W e h a v e m e a s u r e d t h e r e c t a n g u l a r / s i n e s e n s i t i v i t y r a t i o f o r a r e c t -a n g u l a r wave g r a t i n g w i t h a d u t y c y c l e o f 0 - 1 a t v a r i o u s s p a t i a l f r e q u e n c i e su n d e r t h e s a m e c o n d i t i o n s a s t h e p r e v i o u s e x p e r i m e n t . F i g u r e 6 s h o w s t h er e s u l t f o r J . G . R . T h e c o n t r a s t - s e n s i t i v i t y r a t i o w a s f o u n d t o d e v i a t e f r o mt h e v a l u e o f ( 4 / n ) s i n ( i l / 1 0 ) o n l y a t s p a t i a l f r e q u e n c i e s b e l o w a b o u t 6 c / d e g .

S a w - t o o t h g r a t i n g s . A s a w - t o o t h w a v e i s i l l u s t r a t e d i n F i g . 1 . T h ea m p l i t u d e s o f t h e f i r s t , s e c o n d , t h i r d a n d s u c c e s s i v e h a r m o n i c s o f a s a w -t o o t h o f c o n t r a s t m a r e 2 m / i r , 2 m / 2 7 r , 2 m / 3 r a n d s o o n .T h e t h r e s h o l d c o n t r a s t f o r a s a w - t o o t h g r a t i n g w i t h a s p a t i a l f r e q u e n c yo f 1 1 c / d e g w a s m e a s u r e d a n d c o m p a r e d w i t h t h e t h r e s h o l d c o n t r a s t f o ra s i n e - w a v e g r a t i n g o f t h e s a m e f r e q u e n c y , t h e m e a s u r e m e n t s b e i n g madea l t e r n a t e l y a s b e f o r e . T h e s a w - t o o t h / s i n e s e n s i t i v i t y r a t i o w a s f o u n d t o b e1 * 6 5 + 0 1 1 S . E . T h i s v a l u e d o e s n o t d i f f e r s i g n i f i c a n t l y f r o m t h e r a t i o o ft h e a m p l i t u d e s o f t h e f u n d a m e n t a l s o f t h e t w o w a v e f o r m s ( n j I 2 = 1 . 5 7 ) .T h e p e r c e p t i o n o f s u p r a t h r e s h o l d g r a t i n g sS o f a r w e h a v e c o n s i d e r e d d e t e c t i o n o f t h e p r e s e n c e o f a g r a t i n g when w es w i t c h f r o m a u n i f o r m f i e l d o f a g i v e n l u m i n a n c e t o a f i e l d o f t h e s a m e m e a nl u m i n a n c e h a v i n g a g r a t i n g p a t t e r n o f l o w c o n t r a s t u p o n i t . W e n o w

3 6 P h y . 1 9 7

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