Sampling the Observer’s Psychometric Function

It is com m on in psychophysics to characterise the ability o f observers to perform a task over a range o f stim ulation, building up a m ore revealing picture o f the com petence o f the m echanism s em ployed than w ould be possible with ju st one m easurem ent. Several techniques ex ist for choosing the stim ulus values to

present to the subject (e.g. the m ethod o f adjustm ent, the m ethod o f lim its, the m ethod o f constant stim uli, and adaptive m ethods. See Treutw ein (1995) for a review ). In this thesis two techniques have been used: the m ethod o f constant stim uli, and an adaptive m ethod o f stim ulus selection known as ‘Q uest’ (W atson & Pelli, 1983; see also Ew en K ing-Sm ith et al., 1994). The m ethod o f constant stim uli presents a fixed set o f stim ulus values to the observer that is usually designed to sam ple the com plete range o f their perform ance, from guessing through to the m axim um po ssib le. T his schem e n ecessarily in v olves the experim enter in a large degree o f pilot work to ensure that the stim ulus set does indeed em brace the desired perform ance range. This problem is com pounded by individual differences th at m ay dictate th at som e observers w ith d ifferent sensitivities require a different stim ulus set to others, generating m ore work for the experim enter, and possibly m ore w asted trials for observers. Finally, this technique has been criticised on the grounds o f the relative inefficiency o f its sam pling schem e (Treutwein, 1995). Sam pling evenly throughout the observers’ psychom etric function is held to be w asteful as som e parts o f the function are m ore inform ative o f its shape than others. A m ore e fficien t m ethod w ould sam ple m ore densely at those points, and less so at co m p arativ ely less inform ative regions. The Q uest algorithm (W atson & Pelli, 1983) offers ju st such a sam pling schem e, typically concentrating sam pling o f the stim ulus space in areas leading to high perform ance (e.g. 82% ). This is achieved through a B ayesian p ro b ab ility density function (pdf) th at relates the likelihood o f a p ositive or negative response to stim ulus values. T he experim enter is only required to supply the algorithm with an estim ate o f the subject’s threshold, the algorithm then m aking ‘o n -lin e’ stim ulus value choices based initially on its inbuilt pdf, and subsequently on that distribution m odified by the incorporation o f the observer’s responses during the experim ent.

2.1.2 Stimuli

T he stim uli used th ro u g h o u t this thesis are re fe rre d to as R andom D ot K inem atogram s (RDK ). An R D K com prises a field o f dots, each o f w hich is

capable o f m oving with speed and directional properties that can be assigned independently o f its com panion’s speeds and directions. A typical R D K contains dots with lifetim es lim ited to a short num ber o f anim ation fram es. D ots are ‘reb o rn ’ at a new location when they expire, or when they travel outside the bounds o f the (frequently circular) region o f the display they are required to appear in. These properties o f the R D K m ean that com plex patterns o f m otion can be created w ithout the introduction o f ‘real w o rld ’ geom etric shapes like squares, rectangles and circles. Such shapes have properties that are not relevant to the study o f low -level m otion m echanism s, and m ay introduce undesirable cues that observers could use in place o f the variables under experim ental investigation. For exam ple, chapter 4 exam ines how the perception o f m otion-in- depth varies w ith the spatial frequency o f R D K dots m aking up a radially expanding optic flow. W hen an RD K is m ade up o f lim ited lifetim e dots that can be repositioned in random locations, a continuous flow field can be produced w ithout accom panying ‘object pro p erties’ intruding, or placing constraints on o v erall stim ulus du ratio n . I f m otion in d epth w ere sim u late d usin g an approaching square for an extended period the figure w ould eventually fill the entire display area. A dditionally its spatial frequency content w ould inevitably contain m ore pow er at low frequencies. U sing an R D K with lim ited lifetim e elem ents circum vents these problem s.

P revious investigators have used different designs for the elem ents in their R D K ’s. The sim plest design has a single lum inance value for the body o f the dot, contrasting strongly with that used for the background, w hichever polarity is chosen. T his design is often used for single-pixel dots w here there is no opportunity to m odulate the lum inance profile. G raduated lum inance profiles have also been used, for exam ple H arris & G iachritsis (2000) used a triangular lum inance profile in their RD K , providing a com paratively sm ooth transition betw een background and dot. The D ifference o f G aussian (DoG) elem ent design used in this thesis has been chosen for a com bination o f desirable properties not available with other stimuli. These features are described below.

G a b o r

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Figure 2.1 Construction o f a G abor profile. W h e n a sine w ave (blue curve) is multiplied by a Gauss ian function (red curve) the resulting w indow ed sine wave is referred to as a Gabor. W here the sine w ave e x te n d s infinitely across space in both directions, the G a u ss ia n w in d o w e n s u re s that the resu ltan t w a v e fo rm is spatially restricted. The spectral c o m position o f the G ab o r is limited, being a function o f the sine w av e and Gaussian profile, jointly.

T h is d isc u ssio n starts w ith the ‘G a b o r ’, esse n tia lly a sine w a v e w h o s e a m p litu d e is c u rta ile d th ro u g h m u ltip lic a tio n by a G a u s ia n e n v e l o p e (see f ig u re 2.1). T h e p r o p e rtie s o f the G a b o r s tim u lu s h a v e b e e n t h o r o u g h ly e x a m i n e d and a re well k n o w n in vision sc ie n c e (G a b o r, 1946; M a rc e lja , 1980; K u lik o w s k i, M a r c e lja & B ish o p , 1982), o ffe rin g spatial lo c a lisa tio n an d spatial f r e q u e n c y selectiv ity that are b e lie v e d to m a tc h the p ro p e rtie s o f f r e q u e n c y tu n e d m e c h a n i s m s in p rim a te v i s io n (e.g. A d e l s o n & B e r g e n , 1985; F i e l d & T o l h u r s t , 1 986). A g o o d a p p ro x im a tio n to the G a b o r is a c h ie v e d th ro u g h the s u b tra c tio n o f tw o G a u s s ia n d istrib u tio n s, the ‘D iffe re n c e o f G a u s s i a n ’( D o G ) m o d e l (R o d ie k , 1965; E n ro th - C u g e ll & R o b s o n , 1966; E n r o th - C u g e ll et al. 1983) (see fig u re 2.2). A s lo n g as th e G a u s s i a n c o n s t i t u e n t s d i f f e r in th e ir s t a n d a r d d e v i a t i o n s , th e r e s u l t i n g

w aveform is o f sim ilar form and spatial frequency content to the G abor. The relatio n sh ip betw een the standard deviations o f the com ponent w aveform s determ ines the precise centre frequency and bandw idth o f the stim ulus. Figure 2.2 illu strate s the re su lt o f co m bining G aussian distrib u tio n s o f unequal deviations in one dim ension. C alculation o f the D oG centre frequency can be accom plished using the follow ing form ula (Clem ent, 1993):

f =

ln(& *

^)

a

(■4s *( j * n f * (s^ -

1)

where,

a & b scale the m axim um values o f centre and surround Gaussians, respectively.

s is the ratio o f the standard deviations o f centre and surround Gaussians. a is the standard deviation o f the surround Gaussian.

D iffe re n ce of G a u s s ia n G aussian 1 — Gaus s i an 2 Difference of G au ssian I c

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Fig u re 2.2 C o n s tru c tio n o f a D iffe re n c e o f G a u s s ia n (DoG ). W h e n tw o Gaussian functions (blue and red curves) are subtrac te d the resu ltin g w a v e fo rm (solid green curve) is referred to as a Difference o f Gaussian. The D o G is spatially restricted, like its com ponent parts, and has a limited spectral content d eterm ined jointly by them. See text for details.

F o r the re d u c tio n is t, then, the lim ite d spatial f r e q u e n c y c o n te n t o f the D o G is o n e o f its m o s t s ig n ific a n t a d v a n ta g e s o v e r o th e r d o t d e sig n s. S o lid dots c o n ta in a v e ry b r o a d r a n g e o f s p a tia l f r e q u e n c i e s , a n d a re t h e r e f o r e i n c a p a b l e o f ta rg e tin g sp e c ific n e u ra l s u b - p o p u la tio n s , a n d a ttr ib u tin g e x p e r i m e n t a l resu lts ap p ro p ria te ly is c o n s e q u e n tly m a d e m o r e p r o b le m a tic . O f c o u r s e dots are tw o- d im e n s io n a l entities. A n o t h e r a d v a n ta g e o f u sin g the D if f e r e n c e o f G a u s s ia n is its read y e x te n sio n to tw o d im e n s io n s , w h ic h y ie ld s c irc u la r s y m m e tr y sh a re d by o t h e r d o t d e s ig n s , b u t w h i c h is a b s e n t f r o m 1- d i m e n s i o n a l s tim u li s u c h as G a b o r s and g r a tin g s (a sin e w a v e e x t e n d e d in o n e d i m e n s io n , i.e. w ith o u t the w i n d o w that c u rta ils the G a b o r ) th a t h a v e b e e n u s e d e x t e n s i v e l y in low level vision research. It is also s im ila r to the L a p la c ia n o p e r a to r (M arr, 1982; M a r r &

H ildreth, 1980), in lacking orientation cues that can provide additional cues in studies w here direction o f m otion is the attribute o f interest. A nother advantage o f the dot over the grating is its spatial localisation. A gain, this property is desirable from the point o f view o f m atching stim ulus properties to those o f the m ech an ism s d e te c tin g the stim ulus (M affei et al., 1979; K u likow ski & V idyasagar, 1984).

From inspection o f figure 2.2 it can be seen that a significant portion o f the area under the D ifference o f G aussian curve is below the reference m id-line. In fact the integral o f the waveform approxim ates zero (departure from zero is possible due to quantization errors in digital com puters): the areas above and below the line are equal. This is the third im portant property o f the DoG, it is dc balanced. If a Fourier analysis is perform ed on this stim ulus no pow er exists at 0 cycles per im age. This m eans that the m ean lum inance value o f the dot can be m atched to that o f the background by choosing the m id-level o f the w aveform to be background lum inance. Solid dots, by contrast, have a large dc com ponent that may be detectable by low spatial-frequency m echanism s, especially w here many dots are present, as in a typical RDK. In experim ents that aim to require subjects to integrate over m any individual dot m otions the presence o f a high-am plitude, w idely spread lum inance signal acts as a confounding variable that m ay be exploited unwittingly by participants (Smith, Snow den & M ilne, 1994).