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How do we discriminate relative spatial phase?
Vision Res. Vol. 24. So. I?. pp. 1847-1857. 1984
Printed in
Great
004?-6989;84 53.00 + 0.00
Britain. All rights reserved
Copyright Q 1984Pergamoa Press Ltd
HOW DO WE DISCRIMINATE SPATIAL PHASE?
RELATIVE
DAVID R. BADCOCK* Department of Experimental Psychology, Oxford University, South Parks Road, Oxford OXI 3UD, England (Received 20 May 1983;
in final form 5 June 1984)
Abstract-Discrimination of the relative spatial phase of sinusoids is reformulated as a local contrast discrimination task. This provides a good account of performance with gratings composed of a fundamental and either its second or fourth harmonic. The contrast of the compounds (including a fundamental and its third harmonic) are then varied, keeping the contrast ratio of the components constant, and it is found that performance improves with increasing contrast. The exponent of the power equation relating a base contrast measure to the contrast difference at threshold is derived (assuming the above reformulation to be valid). The average exponent (0.54) is in the range expected from contrast increment detection literature. The effect of contrast on performance is predicted very well. Phase per se does not need to be considered. Spatial phase
Frequency analysis
Local contrast analysis
INTRODU(JTION
Models proposing that the visual system may perform a spatial frequency analysis (Campbell and Robson, 1968; Pollen and Ronner, 1981, 1982) require the coding of the relative spatial phase of components (Oppenheim and Lim, 1981; Brettel et aI., 1982; Piotrowski and Campbell, 1982). Sensitivity to a change of relative phase has been measured with two-component gratings (Burr, 1980; Badcock, 1984) and was found to depend on the relative phase from which the shift was made. Current models of phase coding have difficulty in accounting for this finding. It has been suggested (Badcock and Henning, 1983; Badcock, 1984) that, in fact, the pattern of results is as expected if one were instead measuring sensitivity to changes of contrast in local regions of the stimuli. This suggestion provided the best fit to data from a series of experiments (Badcock, 1984), although some difficulties arose in two of those experiments, because of changes in the luminance profile of the signals, when the contrast of the components was manipulated which meant that slightly different equations had to be used under different conditions. In addition it was assumed previously that contrast difference discrimination obeyed Weber’s Law (cf. Kulikowski and Gorea, 1978; Burton, 1981). Swift and Smith (1983) suggest, however, that this is only true when observers have little familiarity with the signal (for signals like those currently under discussion). Under other circumstances Weber’s Law is not found. This will be. discussed in more detail in the
*Present address: Department of Physiology Durham University, Durham, DHI 3LE. England.
Discrimination
introduction to Experiment 2 but the common finding is that the Weber fraction decreases as the contrast from which the change is made increases. The aim of the current paper is to compare the predictions of the model (proposed in Badcock, 1984) with the behaviour of observers, using waveforms composed of different harmonics (and contrasts) than those from which it was developed. It will then be shown that the deviation from Weber’s Law is as one would expect from a large body of literature and this deviation will be quantified. EXPERIMENT 1 Stimuli
As in the previous paper (Badcock, 1984) the stimuli in this experiment will be two component gratings. They will be composed of a fundamental and either its second (F/2F compound) or its fourth (F/4F compound) harmonic. These two compounds were chosen primarily because a phase shift of 180” in the higher harmonic leaves the luminance profiles sufficiently similar to use the same equations to derive estimates of both local contrast and luminance gradient. The vertical cross-sectional luminance profile of the patterns, L(y), is described by; JY_Y)= I&{ 1 + Acos[Z ~1‘0, + Ay] + Bcos[2 ~JVO~+ Au) + 9, + &lI
(1)
where L, is the mean luminance of the display (7.46cd/m’), A and B are the contrasts of the two components and k is the integer determining the harmonic number of the higher component. The
I841
I s-u
DAVID R. BADC~CK
fundamental frequency cf) was 1.67 c/deg, while the second harmonic was 3.34 g’deg and the fourth was 6.68 cdeg. The reference phase, i.e. the relative phase from which the shift was made, was determined by 4, and the size of the shift was controlled by 4,. The phase of the compound with respect to the edge of the display windows was varied. to avoid cues due to the abrupt truncation of the edges of the stimulus, by randomly choosing a value for A_veach time a pattern was displayed. Figure 1 shows the vertical cross-sectional luminance profiles of the reference stimuli for both Fj2F with 4, = 0’ (Fig. IA) and 4, = 180’ (Fig. IB) and F/4F compounds with &= 0’ (Fig. IC) and 4, = 180” (Fig. ID). As is readily apparent a change in 4, from 0 to 180, merely inverts the luminance profile which changes the sign but not the distribution of the local features. Ordinarily there would also be a change of spatial location but since the overall location of the compound is randomly varied on each presentation this factor will not be salient. Since the luminance profiles are essentially unaltered in shape identical equations can be used for both reference phase conditions. E.rpectations
There is one aspect of the model which limits easy prediction. It is at present, not possible to predict a priori which regions of the compound gratings the observers will use for their judgements. This will be discussed in more detail later but the approach adopted here was to show the very experienced observers the reference pattern and a pattern with a 40’ phase shift of the higher harmonic. The observers then indicated the regions they compared during the task. These regions were used for the calculations to follow. Figure 1 marks the general regions used with small letters. These letters are placed at the y location of local luminance maxima and minima. They indicate the location of zeroes in the first derivative of the appropriate form of equation 1 when no phase shift is employed. When a phase shift is employed the location of the zeroes in the first derivative will move. The values used for the model calculations are the local luminance maxima and minima, within the appropriate region of the compound regardless of their precise location in the reference condition. The region marked a, b and c in Fig. 1 (A, B, C, D) is common to all compounds, although the local sign changes with a 180” phase shift of the higher harmonic and this region was used by the observers for their judgements with both versions of the F/2F compound (Fig. IA, B) and the 0” version of the F/4F compound. With the F/4F compound in the 180” version, however, very experienced observers use the region indicated by the letters e, f and g in Fig. 1D. The equations used here are the same as those used for the “square-wave” version of the F/3F compound in Badcock (1984). The precise formulation used
was chosen because it gave the same pattern of results both before and after practice in the experiments reported in the earlier paper-alternative formulations may also be adequate although a large number were tried and were not. The author only wishes to demonstrate the efficacy of this type of approach. Two local contrast measures are derived and then the difference between them is determined. The local contrast measure for each bar is given by; c, = (c - b)/[(C + b)i2]
(2)
and CI = (a - b)/[(a +
bY21
(3)
The relative contrast difference (RCD) between the local contrast measures is given by; RCD% = I lOO(C, - Q/C,, 1
(4)
These differences are zero in every case when no phase shift, 4,. is used. As the phase shift is increased one bar (denoted a) becomes closer in luminance to b while the other (c) moves further away. With the F/4F compound in the 4, = 180” condition (Fig. 1D) e,fand g are substituted for a, 6 and c respectively. Figure 2 shows the change in relative contrast difference (RCD%) as a function of the phase shift of the higher harmonic, $J,, for both compounds in both
Lo+Lml+Acos[2 I A
k w] + I9 COS[* 2 n fy+ .+J)
=l 0
k:2
k=4
D!stonce
(y )[degrees
of v~suol angle]
Fig. 1. The cross-sectional luminance profile given by equation 1 when the higher harmonic is half the amplitude of the fundamental. The luminance variation is shown for both reference phases for F/ZF (A, 4, = 0; B, 4, = 180) and F/4F (C, I$, = 0; D, 4, = 180) waveforms. The lower case letters indicate local luminance maxima and minima.
How do we discriminate relative spatial phase?
Phase
shtff of htghet
harmon%
Ldeg )
Fig. 2. The functions relating the percentage difference in local contrast (RCD%), from equations 2-4, to the phase shift of the higher harmonic for F/2F (circles) and F/4F (squares) waveforms in both reference phases (4, = 0, solid lines; 4, = 180, dashed lines).
phases. Circles indicate F/2F compounds while squares indicate F/4F. Solid lines indicate the 0” reference phase conditions while dashed lines indicate the 180” conditions. if Weber’s Law applied to this task (i.e. if the contrast change required were a constant proportion of the base contrast) then one would expect the relative contrast difference (RCD%) required for a given performance level to be the same for both reference phases and both FJ2F and F/4F. Since smaller phase shifts, Cp,,are required for a given RCD% value with F/ZF compounds than with F/4F compounds one would then expect greater sensitivity to phase shifts with the former compounds. In addition one would expect a much larger difference between the two reference phase conditions with F/4F than F/2F compounds if one uses Fig. 2 to predict sensitivity to changes of relative phase. referente
Methods Equipment. The stimuli (described above) were generated on a Tektronix 604A oscilloscope (P31 Phosphor) using digital techniques. The frame rate was 250 Hz and there were 850 lines~frame. A Sharp MZ80-I( microcomputer produced a discrete approximation to equation I and loaded it into a local memory unit. It then triggered the unit which generated the voltages for the X, Y and Z axes through 8-bit Digital to Analogue Converters (DACs). The linearity of the relationship between Z-axis voltage and luminance was verified using a digital photometer (Gamma Scientific Model 2400). The experimental session was run by the computer which received response information from the observers via one of two buttons. Display. The screen was viewed at a distance of 114cm and subtended 5 x 5 deg of visual angle. It was in a dark surround. The horizontal gratings were presented in pairs on the screen. Each pattern was 2.25” high and 5” wide and they were separated by a 0.5” high by 5” wide bar of the same space average
1849
luminance as the display (7.46 cd/m’), The patterns were abruptly truncated at the edges but their phase relative to the 2.25 x 5” window was varied from trial to trial using Ay in equation 1. Procedure. After dark adapting for about 5 min the observer pressed a button to begin a trial which consisted of the following sequence: 4.18 set observation interval, 0.5 set pause, second 4.18 set observation interval and then a response interval of unlimited duration. The observer’s response initiated the beginning of the next trial and the onsets of both observation intervals and the response intervals were marked by tones. In each observation interval the observer viewed a pair of patterns consisting of horizontally oriented compound gratings separated by a horizontal bar which had the mean luminance of the display (7.~cd~m2). In one interval the two patterns were identical. These patterns are called the reference stimuli. In the other interval one of the patterns was a reference stimulus but one, the test stimulus, was different in that the relative phase of the two components was varied. The observer’s task was to indicate by pressing one of two buttons which interval contained the pair of stimuli that were different. The test stimulus appeared in one and only one of the four possible locations on each trial and had equal probability of appearing in the top or bottom half of the screen in either the first or second interval. This is a two-alternative temporal forced-choice (2AFC) task with a reference stimulus in both observation intervals. All viewing was binocular with natural pupils and in a darkened room. No feedback was given, In the experiment the value of 4, (the phase shift) was varied using an adaptive two-alternative forcedchoice (2-AFC) procedure with two randomly interleaved staircases. In both staircases the initial value of Cp, was large so that the observer could easily recognize the test stimulus. The same stimuli were used in both staircases and were identical except for the phase shift, Cp,,which for both staircases had the same large initial phase shift of 40”. However, the phase shift was controlled independently and by different rules in each staircase so that the staircases converged on different performance levels. One staircase required two consecutive correct responses for a decrement in Cp,but only one incorrect response for an increase. This staircase converged on the 71% correct performance Ievel. The other staircase required four correct responses for a decrement in 4, and one incorrect response for an increment; it converged on the 84% correct performance level (Levitt, 1971). Both estimates were obtained to provide information about the slope of the psychometric function which relates the percentage of correct discriminations to the phase shift, I$,. The magnitude of the step in 4, was initially set to 10” but was halved the first three times a staircase changed direction (reversed), that is, each time a
1850
DAVID R. BADCOCK
sequence of decrements in $J, was followed by an increment or vice versa. A staircase ran until it had produced six reversals. The values of the phase shifts at the last four reversals were averaged to estimate the phase shift corresponding to the appropriate performance level. The few conditions in which the value obtained with the staircase designed to converge on the 84% level were not higher than the value obtained with the 71% staircase were rejected and repeated. Conditions. The signals were all constructed using discrete approximations to equation 1. The fundamental was 1.67 c/deg and had a Michelson contrast (&n.l, - L,,&/(L,,, + &,) of 0.221. The higher harmonics had a contrast of 0.088 and were thus 0.4 times the contrast of the fundamental. The two components in the reference stimuli were added with a reference phase, 4,. of either 0” or 180” as shown in Fig. 1. The test stimuli had the same 4, value as the reference stimuli but, in addition, had the positive values of 4, determined by the staircases. Obsertrers. Three highly experienced and emmetropic (or corrected to normal) observers participated in this experiment. D.R.B. is the author. AS. and J.G. (females) were naive with respect to the experimental aims and were paid hourly wages. Remits
Figure 3 shows the results for three observers. The phase shift, +,, of the higher harmonic required for 71% correct discrimination is shown as a function of the reference phase, Cp, (4, = 0, open symbols; Qt,= 180, solid symboIs) and the error bars indicate +/one standard deviation. Data for the F/2F condition is shown as solid lines; data for F/4F as dashed lines. Data for D.R.B. and A.S. are based on four staircases per point while those for J.G. are based on Il. For all observers a larger phase shift is required for the same performance level with F/4F than with F/2F ^ z”
(0) Jo-
(b)
ORB F14F
---
AS
compounds. The performance is only slightly worse with F/4F than with F:2F when Cp,=O’ but this difference is much greater for #, = 180” conditions. In both conditions performance with the 4, = 180’ reference phase is worse than with the 4, = 0” condition. Discussion
The results of the present experiment show, as expected from the local contrast difference model, that larger phase shifts are required to reach a criterion performance level with F/4F compounds than with F/2F compounds and also that the difference between the two reference phase conditions is larger with F/4F than F/2F compounds. These results are as expected. For example, the predictions in Fig. 3d show the performance expected if a constant relative contrast difference (RCD%) is required for a constant performance level. The figure was obtained by finding the group mean for the F/4F 4, = 180” condition (the choice is arbitrary but has little effect on the form of the results), calculating its RCD% value (approx. 6.8%) and then finding the phase shift, 4, that would give this value for the other conditions. The agreement between the predictions (Fig. 3d) and the individual data is quite striking (Fig. 3a-c). The Iargest discrepancy between the data and the predictions lies in the finding of poorer performance with F/2F patterns than predicted. The predicted difference between reference phases is also smaller for this waveform than was obtained. One reason for predicting thresholds that are too low for F/2F compounds is found if one considers the C, and C, vatues (the local contrast measures) used to calculate RCD%. These vary considerably among the conditions. It is found that the RCD% value that corresponds to the 71% correct performance level decreases as the size of C, increases; that is, smaller percentage differences are required when the lowest of the local contrast estimates is larger. This would mean that Weber’s law does not hold for these (c)
*
JG F/41: F,2F
-
-
-
PI.(I,CIIO”, F/4!-
”
-
-
F/ZF
Fig. 3. The shift in relative phase, 4,. of the higher harmonic required for 71% correct discrimination for both F/2F (solid lines) and F/4F (dashed lines) waveforms as a function of the reference phase, d. Open symbols denote 4, = 0, solid symbols denote #r = 180. The data for three observers are plotted (a, b and c) along with predictions (d) which are described in the discussion. Error bars indicate t/one standard deviation.
1851
How do we discriminate relative spatial Phase? parameters and thus the predictions of Fig. 3d which are based on this assumption will diverge from the data to the extent that Weber’s law fails. Kulikowski (1976) has shown that under similar conditions the Weber fraction required for detecting a contrast difference does decrease as the reference contrast increases (see his Fig. 4). This behaviour is also consistent with other contrast increment detection studies (e.g. Nachmias and Sansbury, 1974; Legge, 1981) and with demonstrations that contrast matching performance becomes more accurate as the contrast of the reference pattern increases (Georgeson and Sullivan, 1975). Legge (1981) suggests a power law should be used to describe the data from contrast increment detection studies. Following this suggestion exponents were derived for the three observers in this experiment. These exponents, relating the contrast difference to C, at the 71% performance level, were 0.61 (-t/-0.21), 0.61 (-I-/-0.15) and 0.66 (+/-0.09) for J.G., D.R.B. and AS. respectively [the variation indicates +/one standard error of the exponent derived by the method of Mansfield (1973)]. These are all very similar to those found in previous constrast increment detection studies (see Legge, 1981; Swift and Smith, 1983) and suggest that the model may indeed provide a good description of performance in “phase discrimination” studies. F.XPERfiMENT 2
Two related questions still need to be addressed with regard to the efficacy of the relative contrast difference (RCD) model. The first is the ability to predict results obtained with waveforms when the contrasts are varied. This was addressed in Badcock (1984) where the direction of the effect was predicted but the quantitative fit was not good. It was argued there, as in the discussion just presented, that the major discrepancies were caused by assuming that Weber’s Law applied to the current task and that these could be considerably reduced by using a power exponent of less than 1.0. Several recent studies have shown that contrast increment (and decrement) detection follows a complex “dipper shaped” function (see Legge and Foley, 1980) but that at higher base contrasts the “handle” of the dipper is best described by a power function with an exponent of approx. 0.6 (cf. Legge and Kersten, 1983; Swift and Smith, 1983). The exact figure, however, seems to be influenced by spatial frequency (Legge, 1979, 1981), or bar width (Legge and Kersten, 1983; Badcock, 1984) and by temporal parameters of the stimuli (Kulikowski and Gorea, 1978; Pantle, 1983). The current task differs from the contrast increment detection studies just mentioned in that the local variation in contrast is produced by a change in the relative phase of the two components and not by variation in their amplitudes. It is, therefore, of
considerable interest that the exponents derived are so similar to those derived from the contrast increment detection studies. A more detailed evaluation of the magnitude of the power exponent will now be presented. In the previous experiment the exponents were obtained by using vafues obtained from every condition and an estimate of the exponent (0.63) was obtained that was similar to the average values of other experimenters (cf. Legge, 1981; Legge and Kersten, 1983; Swift and Smith, 1983). However, since some dependence on spatial frequency has been demonstrated by some experimenters (e.g. Legge and Foley, 1980; Legge, 1981) it is not clear that averaging performance across patterns with different higher harmonics is as informative as it might be. To examine these issues the three compound gratings (F/2F, F]3F and Ff4F) used previousiy wili be used again in both reference phase conditions. In this study, however, the contrast ratio of the two components remains constant at 2: 1 (the fundamental has twice the amplitude of the higher harmonic) instead of 2.5: 1 as in Experiment 1. The contrast of the compound will be varied over a large range by adjusting the contrast of both components by equal proportions. This procedure for manipulating the contrast avoids the problems noted with F/3F compounds, when only one component’s contrast was varied at a time. It was seen there that some local features disappeared and different equations were required to calculate the focal contrast values. In this study the same equations can be used at all contrast levels for a given pattern and it will therefore be valid to’derive power exponents for each compound and each reference phase individually. This will allow a much more precise estimate for each condition and will also lend support to the ar~rn~t for the previous experiments should the exponents behave as is expected. Method Signals. This experiment uses F/2F, F/3F and F/4F luminance profiles. There is, however, a difficulty with labelling in this experiment. In the previous experiments when F/3F compounds were used (Badcock, 1984) they were generated by;
L~)=L,{l+Asin[2nfCv+A~)] + BsinIx$Cv
+ AY) + #, + tPd)
(5)
whereas when F/2F and F,i4F compounds were used (Experiment 1 of this paper) they were generated by; U_.v) = L,{ 1 + Acos[2 $0,
+ Ay)]
+ Bcos]2 @cV + Au) + #+ + &I)
(6)
where k determines which higher harmonic is used. The only difference between these two equations is that in equation 6 cosines are used instead of sines. Although this difference seems odd it will be maintained in this experiment since it has the advantage
DAVID R 58 ,180’
(0) 80
w
1
0 0'
40 20 -
10
z "
5
0 ;
i (b)
;
80
,"
40
z e
20
: x = .cz v) 2
10 5
JG *180*
(cl 80
0 0’
0 ar
40 20 10
BADC~CK
and they were separated by a 0.74” wide bar of mean luminance. Procedure. The adaptive procedure was identical to that used in the previous experiments. Again, the order of conditions was counterbalanced to control for order and learning effects. Auditory feedback was given for correct responses. Obsercers. Three observers were used. H.P. and J.G. completed three staircases for each performance level in each condition and G.B.H. completed two. Conditions. In each condition the contrast of the higher harmonic (specified by B in equations 5 and 6) was half that of the fundamental (A in equations 5 and 6). The contrast of the fundamental was either 0.442, 0.221, 0.1 I05 or 0.0553. The higher frequency component was either the second (F/2F), third (F/3F) or fourth (F/4F) harmonic of the fundamental (1.67 c/deg) and both 4, = 0” and r#~,= 180” reference phases were used. Results
1y3
5 00553 Contrast
01105
of the
0 221
0442'
Effect of contrast. The results plotted in Figs 4, 5 and 6 show the phase shift of the harmonic corresponding to the 84% correct performance level as a function of the contrast of the compound for the
fundamenfol
Fig. 4. The function relating the phase shift of the second harmonic, I#J,, required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both C#I, = 0” and 4, = 180” reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The second harmonic always had half the contrast of the fundamental. Both axes are logarithmic. Error bars indicate +/- one standard deviation.
the 4, = 180” condition always refers to the reference phase condition for which RCD% would predict poorer performance and therefore it simplifies discussion. Equipmenr. The Sharp MZSO-K also controlled the second display system. The information specifying the signals for each trial was sent to a PDP8/A minicomputer which generated the signals from lookup tables. The signals were then sent via 12-bit DACs through an attenuator to the display screen. The screen was a raster display built according to a design of J. G. Robson. It was masked with black cardboard to subtend 4.99 x 8.51” at the 154 cm viewing distance employed. It had 423 lines per frame, a frame rate of 100 Hz and a mean luminance of 47 cd/m’(P3 1 phosphor). A linearizing table was used to correct the nonlinearity of the Z-axis measured using the digital photometer. Linearity of the attenuator and the DAC was verified using an HP3580A Spectrum Analyzer. With this display the observers were seated with their heads restrained by a chin and forehead rest. Al1 viewing was binocular in a dark room. The patterns for this experiment were vertically oriented but were still presented in pairs on the screen, Each pattern was 3.89” wide and 4.99” high
20
that
.
00553 Contrast
01105
of
the
180'
0221
0442
fundamental
Fig. 5. The function relating the phase shift of the third harmonic, 4,. required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both I#I,= 0” and 4, = 180‘ reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The third harmonic always had half the contrast of the fundamental. Both axes are logarithmic. Error bars indicate +/- one standard deviation.
How do we
discriminate
80
40
20
--__ lx 1
l
J G. 180'
0
0;
(cl
--_
----_‘3-____.
10
5,
0.0553
Conrrast
0 1105
of
the
0 221
0442
fundamental
Fig. 6. The function relating the phase shift of the fourth harmonic, I#,, required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both 4, = 0” and 4, = 180” reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The fourth harmonic always had half the contrast of the fundamental. Both axes are logarithmic, Error bars indicate +/- one standard deviation.
F/2F, F/3F and F/4F compounds respectively. The vertical phase axis is logarithmic (with the error bars indicating + / - one standard deviation) to minimize differences in the slopes of the psychometric functions, for the whole set. Only the 84% performance level is reported in this, and subsequent figures, since. the pattern of results is the same and the higher performance level is inherently less variable, allowing more precise estimation of the exponents to be derived later. The general pattern of results is the same for all observers with each compound. Performance, expressed as degrees of the higher harmonic, improves as the contrast of the components increases, even though the contrast ratio remains constant. Again performance with the (6, = 180” signals is consistently worse than with the 4, = 0” conditions and in general the improvement with increasing contrast of the components is greater with 4, = 0” conditions. This is reflected in the larger separation between the 0 and 180” points at the right hand side of the Figures than on the left hand side. It was also confirmed by a significant interaction between contrast level and
relative
spatial phase?
1853
reference phase (P < 0.01) in an analysis of variance that was conducted. In this analysis of variance the following results were obtained. The simple main effect of waveform (i.e. frequency combination) was significant [F(2,6) = 119.63, P < 0.011 but the main effects of contrast [F(3,3) = 9.08, 0.05 c P < 0.061 and reference phase [F( 1,2) = 14.2, 0.05 < P < 0.071 were not. The interaction between contrast and waveform [F(6,48) = 1.13, P > 0.051 was not significant but the interactions between contrast and reference phase [F(3,48) = 6.54, P c 0.011 and between the waveform and reference phase [F(2,48) = 5.46, P c 0.011 were. The three-way interaction between waveform, contrast level and reference phase was not significant [F(6,48) = 0.628, P > 0.051. This pattern of performance is as predicted by the relative contrast difference model as will be shown. It is necessary, however, to first derive the exponents for the power functions since the predictions are dependent on the exponents’ value. Power law exponents
In the previous experiment, the relationship between C,, the lowest local contrast measure, and AC, the difference between the two local contrast measures, was examined at the 71% correct performance level for each pattern employed. This relationship was found to be described by a power law with an exponent of approx. 0.63. The same relationship has been examined here, except that now there are four contrast levels within each pattern and so the relationship is derived within rather than across patterns and the 84% correct performance level is used. It should, however, be noted that in this experiment the observers reported using a different region of the pattern with F/3F “peaks-add” patterns than in Badcock (1984). It seems that the provision of auditory feedback and the experience with F/4F, 4, = 180” patterns, in the previous experiment, allowed them to determine that the following region was more useful. They fixated the darkest bar in the waveform and compared the two bright bars on either side. This change means that equations with the same number of terms can be used for every pattern in this experiment since the regions used for the other patterns were as in Experiment 1. Figure 7 shows, for observer J.G., the functions relating C,, the lowest local contrast estimate (which is twice the actual contrast) to AC, the contrast difference (also twice the real value), where both C, and AC represent the values at the 84% correct performance level. Both axes are logarithmic which means that the exponent of the power law is equivalent to the slope of the best fitting least-squares regression line. These lines have been calculated and are drawn through the data for each condition. Results for only one observer have been presented but the other two observers gave results that are similar in all respects. The parameters obtained from
DAVID R. BADCOCK
, 0005
001
C,
I
I
0025
I
I,,
I base
I 01
005
confrost
I
025 x 2
05
I
I
I,,
10
1
Fig. 7. The function relating the contrast difference, 1C, - C,,1, to the lowest local contrast value, C,, at the 84% correct discrimination level is plotted for observer. J.G. Both axes are logarithmic. The data points are connected to the best-fitting least-squares regression line for each condition. Solid lines represent F/ZF, dashed lines represent F/3F and dotted lines F/4F. Data for both 4, = 0” (open symbols) and 4, = 180” (filled symbols) reference phase conditions are plotted.
the least squares regression are presented in Table 1. In this Table the estimated values of k, the constant of proportionality, b, the exponent of the power law, and the standard error of the estimate of the exponent (Mansfield, 1973) are listed for each observer in each condition. It is clear, from both Table I and Fig. 7 that sensitivity is relatively lower for F/2F patterns than for either F/3F or F/4F patterns. This is shown by larger values of k in Table 1, and the vertical displacement of the F/2F lines in the Figures. The effects of the variation in k will be discussed later. It is, however, clear from the Table that the data for each condition are reasonably well fitted by straight lines on double logarithmic coordinates. The exponents are plotted in Fig. 8 where the solid horizontal line represents the mean value of the exponent in this study (0.539) and the broken lines indicate +/I (dashed lines), and +I2 (dotted lines), standard deviations (SD = 0.139). It is readily apparent from this Figure that most exponents are within one standard deviation of the mean (13/18). The exceptions to this do not follow any clear trend. The mean obtained here (0.539) is lower than that obtained in the previous experiment Table
I. Parameters
(0.63). The previous estimate is, however, within the one standard deviation range on Fig. 8 as would be the average estimates of Swift and Smith, 0.65, and Legge and Kersten, 0.56. Thus it is concluded that the results obtained in this experiment are compatible with the argument that the observers are detecting differences in local contrast to detect a phase shift and that the relationship between the local contrast measures and the just-detectable contrast difference is best described by a power law with an exponent of approx. 0.539 (+/ - 0.139). Predictions from the local contrast model
The predictions from the local contrast model that has been outlined are tedious to derive. Since both C,, the lowest local contrast measure, and AC, the contrast difference, vary when the relative phase varies, several steps are required to find the phase shift, $J,, that yields the appropriate values in each condition. The predictions were derived in the following way. (a) The mean phase shift, for all three observers, was calculated for the F/4F, 4, = 180” condition with a fundamental contrast of 0.221 in the experiment just presented.
of the power law for three observers with F/ZF, F/3F and F/4F compounds in 0 and 180” reference phases 0'
180’
Condition
Observer
k
b
SE,
k
b
SE.
Fj2F
G.B.H. H.P. J.G.
0.124 0.296 0.21 I
0.43 I 0.61 0.624
0.18 0.02 0.10
0.17 0.651 0.314
0.471 0.816 0.64
0.11 0.1 0. I5
F/3F
G.B.H. H.P. J.G.
0.104 0.I2.5 0.095
0.523 0.587 0.538
0.06 0.07 0.14
0.064 0.069 0.115
0.384 0.193 0.593
0.13 0.03 0.08
Fj4F
G.B.H. H.P. J.G.
0.164 0.084 0.057
0.716 0.517 0.374
0. I 0.07 0.05
0.079 0.079 0.091
0.513 0.593 0.583
0.02 0.09 0.02
How do we discriminate relative spatial phase?
1855
GBH
0
HP
V
JG
0
,
. zso so
-1
0 _-----_~_-_-_-_-_~--_____I
-
0
.
1 SD
-2S0
.
F/2
00
F/2
180’
F/3
Stimulus
0’
F/3
180’
F/4
0’
F/4
180”
condtrton
Fig. 8. The exponents of the power functions are plotted for each stimulus condition using data from the 84% correct performance level, for each observer C.B.H., squares: H.P., inverted triangles; J.G., diamonds. The right hand ordinate indicates the mean value of the exponent (solid line), +/one standard deviation (dashed lines) and +/- two standard deviations (dotted lines).
(b) At this mean shift the value of C, and AC was in the usual manner. (c) The mean value for the exponent, obtained in the experiment (0.54) was used to allow the derivation of the constant of proportionality, k, by using obtained,
AC = kc; which after reorganization
(7)
becomes
k = AC/C:
(8)
(d) Using tables that give the value of C, and AC, for each #, and assuming an exponent of 0.54, the value of k was derived, as in step c above, and a regression line relating (p, to k was created. The value of Q;, that gave the value of k, derived in step b, was then found using this regression line. The expected pattern of results when the contrast of the patterns is increased while the contrast ratio of its components remains constant will now be presented. The expectations are as presented in Fig. 9. This Figure shows the overall means, for the three observers, with the standard deviations calculated from the several estimates for each observer. The lines indicate the performance expected when the exponent is 0.54 and the value of k is 0.09414. The data point used to calculate k is indicated by the star. Two points need to be made about this Figure. First, the pattern of improvement in performance with increasing contrast is predicted extremely well by the local contrast model. It predicts the smaller influence of contrast on the 4p,= 180” patterns for al1 compounds and also the larger difference between retative phases with higher contrasts. This pattern is deter-
mined by the exponent but clearly the average exponent provides a good account of the pooled results. The second point to note about Fig. 9 is that the absolute performance levels, while agreeing with the predictions for the F/3F and F/4F compounds, are at variance with the predictions for the F/2F compounds. While the pattern of improvement with increases in contrast is predicted well by the model, the performance obtained is worse than predicted. The vertical position of the lines is determined by the constant of proportionality, k. With the predictions here this has been determined for the F/4F, 4, = l80”, 0.221 contrast point and the same value has been used everywhere. This effectively predicts F/4F and F/3F performance but in essence moves ail the variability to the F/2F section of the Figure. No single value of k will give a perfect fit for all compounds. Examination of Table 1, and Fig. 7 will show why. The lines for the F/2F compounds in the Figure are shifted vertically compared to those of other compounds. This is reflected in Table 1 by larger values of k, indicating that observers were relatively less sensitive to variations within F/2F compounds. The value of the constant of proportionality, k, is not often discussed in the contrast increment detection literature. More frequently the effect of this constant is eliminated by dividing both AC and C, by the contrast threshold for the particular signal (e.g. Legge, 1981; Legge and Kersten, 1983). If, therefore the vertical position of the lines in Fig. 9 was adjusted, by correcting k to allow for variations in threshold sensitivity, then the predicted performance could be indistinguishable from the data in Fig. 9.
DAVID
R.
BADCOCK
F/4f
i trau~ means
: 0 0’
251,
, 00553
01105
, 0221
,
/
0442 Contrast
,
,
0055301105 of
the
I
I
0221
0442
1
t 0055301105
0’
.
180°
-O*
Predictions *
A
I
---180’ 0221
0442
fundament~i
Fig. 9. In each section of this Figure the symbols represent the mean phase shift, $,, for the three observers from Figs 4, 5 and 6 of the higher harmonic required for 84% correct discrimination as a function of the contrast of the fundamental (the higher harmonic was haIf the contrast of the fundamental). Both axes are logarithmic and the error bars indicate +/ - one standard deviation. The higher harmonic is 2F in the left hand column, 3F in the middle and 4F on the right. Data for both 4, = 0” (open symbols) and 4, = 180” (solid symbols) reference phases are plotted. The lines represent the predictions using the local contrast difference model when the exponent of the power function is assumed to be 0.54 (the group mean) and k, the constant of proportionality, is 0.09414 and is derived from the data point indicated by the star (F/4F. (b, = 180’. fundamental = 0.221). The predictions for &I,= 0” and 4, = 180” reference phase conditions are shown by the solid and dashed lines respectively. This correction has not been made here since it is not clear which threshold is the appropriate one to use for the adjustment. Given so many alternatives it is
unlikely that one could not be found that would give an appropriate correction but this would not justify its choice. It is, however, appropriate to note that the thresholds measured by Lawden (1981) for detecting a harmonic in the presence of a suprathreshold fundamenta1 did give relative sensitivities close to those that would be required here. In any case it is clear that within each compound the Iocal contrast model predicts the pattern of improvement with increasing contrast very well and its performance across compounds would, indeed, be excellent if the vertical position of the predictions were corrected to allow for threshold sensitivity.
Since it has been shown that the model proposing local contrast difference discrimination, provides a good account of the “phase” discrimination performance presented above (and that presented by Burr (1980) whose experimental paradigm was similar; see Badcock, 1984) it is reasonable to ask how generally this formulation may be applied. The main thrust of the model is the reformulation of the phase discrimination task as a contrast difference discrimination task. Arend and Lange (1979) have also argued that the phase dependent behaviour they observed could be regarded as detecting variations in local regions. The local region could as easily be a uniform field as in many contrast threshold paradigms (e.g. Campbell and Robson, 1968; Derrington and Henning, 1980) as the nonuniform region which served as a background here. Swift and Smith (1983) have recently attributed per-
formance in spatial frequency masking studies to a strategy of this type. An example of the detection of variation in a uniform background in the “phase” discrimination literature may have been provided by Burton and Moorhead (1981) who noted that in their experiment “the key difference that emerged as the peak shift increased from zero was the appearance of new spatial features in the modified target. . . ” (p. 1059). An examination of their Fig. 1 shows that new bars appear in the uniform field surrounding the targets. Burton and Moorhead (1981) did not analyze their data in terms of the contrast required for detecting these new features but such an analysis may well have been useful. The current study provides data for a non-uniform field, as did Burr (1980), but perhaps the most extreme example is provided by Caelli and Bevan who used two-dimensional phasequantized textures. It is extremely difficult to analyze their stimuli using the current model since it is very difficult to know which regions to compare but examination of their Fig. 2 suggests that overall contrast variation is a powerful cue with their stimuli. These studies were all designed to measure phase sensitivity but it is not enough to manipulate the phase spectrum of a stimulus and simply assume that the observer must calculate a phase spectrum and then detect differences in that spectrum. Reasonable alternative strategies must be eliminated iirst. The current model helps to explain the wide variation in the estimates of “phase“ sensitivity obtained with different stimuli without reference to the phase spectrum of stimuli. It also draws strong parallels between “phase” discrimination performance and spatial frequency masking studies (Swift and Smith, 1983). In support of this comparison the average exponent is
How do we discriminate relative spatial phase?
lower in the second experiment than the first, showing the same trend with practice as the exponent derived by Swift and Smith. At present the major difficulty with the model lies in determining the regions that will be compared. It was not possible, in the experiments presented here to predict this a priori and to have told the observers which region to use would have introduced circularity. It seems likely that observers who have had experience with stimuli and feedback on their performance, will try to find a region of the stimulus that optimizes their sensitivity. This will be determined by a combination of the base contrast in the region and its rate of change with a phase shift. To optimize his performance the observer should find the region in which the contrast difference threshold is reached most rapidly. The strategies adopted for finding these regions need to be investigated as they may apply very generally (see also Swift and Smith, 1983). Implications for frequency analysis. The arguments presented above do not argue against the existence of multiple size, or frequency, tuned channels (Campbell and Robson. 1968) but they do place limits on the usefulness of the analogy between the visual system and a Fourier analyser. To the extent that only a local region of the stimulus is used to perform the task the rest of the stimulus may be said to be redundant. Thus removing the rest of the stimuius should have little effect on performance. The Fourier description of a stimulus, however, is not the same when the rest of the stimulus is removed and thus would not necessarily predict the same performance. In line with this argument, one would expect that with the current signals in a fixed size screen that “phase” thresholds should be reiatively independent of the fundamental spatial frequency, as long as it is high enough for the critical region to be small enough to still fit on the screen and low enough to allow the smallest detecting units to estimate the luminance at the individual points required. Burr (1980) has shown that these thresholds are, in fact, independent of spatial frequency over a large range with rapid deterioration at both low and high frequencies. The low frequency fall-off is at approximately the correct spatial frequency for the above argument. The appropriate high frequency is hard to determine but the fall-off is as rapid as one would expect. The present author has an unpublished replication of Burr’s (1980) results using longer durations but giving the same pattern of behaviour. Since a Fourier description can neither indicate the critical region of the pattern nor provide a simple account of the observed behaviour it must be concluded that its usefulness in this area, at least, is quite limited. author was supported by a Rhodes Scholarship during this work. He would like to thank the M.R.C. for help in supplying equipment. This paper was greatly improved through the comments of Dr Bruce Henning, Professor G. Westheimer and two anonymous referees. Acknowledgemenfs-The
1857 REFERENCES
Arend L. E. and Lange R. V. (1979) Phase-dependent interactions of widely separated spatial frequencies in pattern discrimination. Vision Ref. 19, 1089-1092. Badcock D. R. (1984) Some aspects of “spatial phase” coding in man. D.Ph. thesis. Oxford. Badcock D. R. (1984) Spatial phase or luminance profile discrimination? Vision Res. 24, 613-623. Badcock D. R. and Henning G. B. (1983) Discrimination of spatial phase relations in compounh gratings. Inuesr. Oahthal. visuai Sci. 24. 47(A). Breitel H., Caelli T., Hilt k: and Rentschler I. (1982) Modelling perceptual distortion: Amplitude and phase transmission in the human visual system. Human Neurobiol. 1, 61-67. Burr D. C. (1980) Sensitivity to spatial phase. Vision Rex 20, 391-396.
Burton G. J. (1981) Contrast di~~mination by the human visual system. Eiol. Cybernet. 40, 27-38. Burton G. J. and Moorhead I. R. (1981) Visual form perception and the spatial phase transfer function. J. opt. Sot. Am. 71, 1056-1063. Caelli T. and Bevan P. (1982) Visual sensitivity to twodimensional spatial phase. J. opt. Sot. Am. 72,13?5-1381. Campbell F. W. and Robson J. G. (1%8) Application of fourier analysis to the visibility of gratings. J. Physiol. Lond. 197, 551-566. Derrington A. M. and Henning G. B. (1981) Pattern discrimination with flickering stimuli. Vision Res. 21, 597-602.
Georgeson M. A. and Sullivan G. D. (197.51 Contrast constancy: Debfurring in human vision by spatial freauencv channels. 1. Phvsioi. 252. 627-656. Kuiikowski J. J. (1976) Bffective contrast constancy and linearity of contrast sensation. Vision Res. 16, 1419-1431. Kulikowski J. J. and Gorea A. (1978) Complete adaptation to patterned stimuli: A necessary and sufficient condition for Weber’s Law for contrast. Vision Res. 18, 1223-1227. Lawden M. C. (1981) Harmonic bandwidth of spatial phase discrimination. Invest. Ophthal. visual Sci. 20, 123(A). Legge G. E. (1979) Spatial frequency masking in human vision: Binocular interaction. J. opt. Sot. Am. 69, 838-847. Legge G. E. (I 98 I) A power law for contrast discrimination. &ion Rei. 21,.457-461. Legge G. E. and Foley J. (1980) Contrast masking in human vision. J. opt. Sot. Am. 70, 1458-1471. Legge G. E. and Kersten D. (1983) Light and dark bars: Contrast discrimination. Vision Res. 23. 475-484. Levitt H. (1971) Transformed up-down methods in psychoacoustics. J. acoust. Sot. Am. 49,467-477. __ Mansfield R. J. W. 11973) Latencv functions in human vision. Vision Res. i3, 2i19-2234: Nachmias J. and Sansbury R. V. (1974) Grating contrast; Discrimination may be better than detection. Vision Res. 14, 1039-1042.
Oppenheim A. V. and Lim J. S. (1981) The importance of phase in signals. Proc. IEEE 69, 529-541. Pantle A. (1983) Temporal determinants of spatial sinewave masking. Vision Res. 23, 749-757. Papoulis A. (I 962) The Fourier Integral and Its Applications. McGraw-Hill, New York. Piotrowski L. N. and Campbell F. W. (1982) A demonstration of the visual importance and flexibility of spatialfrequency amplitude and phase. Perception 11, 337-346. Pollen D. A. and Ronner S. F. (1981) Phase relationships between adjacent simple cells in the visual cortex. Science 212, 1409-1411. Pollen D. A. and Ronner S. F. (1982) Spatial computation performed by simple and complex cells in the visual cortex of the cat. Vision Res. 22, IOf-118. Swift D. J. and Smith R. A. (1983) Spatial frequency masking and Weber’s law. Vision Res. 23, 495-505.
Printed in
Great
004?-6989;84 53.00 + 0.00
Britain. All rights reserved
Copyright Q 1984Pergamoa Press Ltd
HOW DO WE DISCRIMINATE SPATIAL PHASE?
RELATIVE
DAVID R. BADCOCK* Department of Experimental Psychology, Oxford University, South Parks Road, Oxford OXI 3UD, England (Received 20 May 1983;
in final form 5 June 1984)
Abstract-Discrimination of the relative spatial phase of sinusoids is reformulated as a local contrast discrimination task. This provides a good account of performance with gratings composed of a fundamental and either its second or fourth harmonic. The contrast of the compounds (including a fundamental and its third harmonic) are then varied, keeping the contrast ratio of the components constant, and it is found that performance improves with increasing contrast. The exponent of the power equation relating a base contrast measure to the contrast difference at threshold is derived (assuming the above reformulation to be valid). The average exponent (0.54) is in the range expected from contrast increment detection literature. The effect of contrast on performance is predicted very well. Phase per se does not need to be considered. Spatial phase
Frequency analysis
Local contrast analysis
INTRODU(JTION
Models proposing that the visual system may perform a spatial frequency analysis (Campbell and Robson, 1968; Pollen and Ronner, 1981, 1982) require the coding of the relative spatial phase of components (Oppenheim and Lim, 1981; Brettel et aI., 1982; Piotrowski and Campbell, 1982). Sensitivity to a change of relative phase has been measured with two-component gratings (Burr, 1980; Badcock, 1984) and was found to depend on the relative phase from which the shift was made. Current models of phase coding have difficulty in accounting for this finding. It has been suggested (Badcock and Henning, 1983; Badcock, 1984) that, in fact, the pattern of results is as expected if one were instead measuring sensitivity to changes of contrast in local regions of the stimuli. This suggestion provided the best fit to data from a series of experiments (Badcock, 1984), although some difficulties arose in two of those experiments, because of changes in the luminance profile of the signals, when the contrast of the components was manipulated which meant that slightly different equations had to be used under different conditions. In addition it was assumed previously that contrast difference discrimination obeyed Weber’s Law (cf. Kulikowski and Gorea, 1978; Burton, 1981). Swift and Smith (1983) suggest, however, that this is only true when observers have little familiarity with the signal (for signals like those currently under discussion). Under other circumstances Weber’s Law is not found. This will be. discussed in more detail in the
*Present address: Department of Physiology Durham University, Durham, DHI 3LE. England.
Discrimination
introduction to Experiment 2 but the common finding is that the Weber fraction decreases as the contrast from which the change is made increases. The aim of the current paper is to compare the predictions of the model (proposed in Badcock, 1984) with the behaviour of observers, using waveforms composed of different harmonics (and contrasts) than those from which it was developed. It will then be shown that the deviation from Weber’s Law is as one would expect from a large body of literature and this deviation will be quantified. EXPERIMENT 1 Stimuli
As in the previous paper (Badcock, 1984) the stimuli in this experiment will be two component gratings. They will be composed of a fundamental and either its second (F/2F compound) or its fourth (F/4F compound) harmonic. These two compounds were chosen primarily because a phase shift of 180” in the higher harmonic leaves the luminance profiles sufficiently similar to use the same equations to derive estimates of both local contrast and luminance gradient. The vertical cross-sectional luminance profile of the patterns, L(y), is described by; JY_Y)= I&{ 1 + Acos[Z ~1‘0, + Ay] + Bcos[2 ~JVO~+ Au) + 9, + &lI
(1)
where L, is the mean luminance of the display (7.46cd/m’), A and B are the contrasts of the two components and k is the integer determining the harmonic number of the higher component. The
I841
I s-u
DAVID R. BADC~CK
fundamental frequency cf) was 1.67 c/deg, while the second harmonic was 3.34 g’deg and the fourth was 6.68 cdeg. The reference phase, i.e. the relative phase from which the shift was made, was determined by 4, and the size of the shift was controlled by 4,. The phase of the compound with respect to the edge of the display windows was varied. to avoid cues due to the abrupt truncation of the edges of the stimulus, by randomly choosing a value for A_veach time a pattern was displayed. Figure 1 shows the vertical cross-sectional luminance profiles of the reference stimuli for both Fj2F with 4, = 0’ (Fig. IA) and 4, = 180’ (Fig. IB) and F/4F compounds with &= 0’ (Fig. IC) and 4, = 180” (Fig. ID). As is readily apparent a change in 4, from 0 to 180, merely inverts the luminance profile which changes the sign but not the distribution of the local features. Ordinarily there would also be a change of spatial location but since the overall location of the compound is randomly varied on each presentation this factor will not be salient. Since the luminance profiles are essentially unaltered in shape identical equations can be used for both reference phase conditions. E.rpectations
There is one aspect of the model which limits easy prediction. It is at present, not possible to predict a priori which regions of the compound gratings the observers will use for their judgements. This will be discussed in more detail later but the approach adopted here was to show the very experienced observers the reference pattern and a pattern with a 40’ phase shift of the higher harmonic. The observers then indicated the regions they compared during the task. These regions were used for the calculations to follow. Figure 1 marks the general regions used with small letters. These letters are placed at the y location of local luminance maxima and minima. They indicate the location of zeroes in the first derivative of the appropriate form of equation 1 when no phase shift is employed. When a phase shift is employed the location of the zeroes in the first derivative will move. The values used for the model calculations are the local luminance maxima and minima, within the appropriate region of the compound regardless of their precise location in the reference condition. The region marked a, b and c in Fig. 1 (A, B, C, D) is common to all compounds, although the local sign changes with a 180” phase shift of the higher harmonic and this region was used by the observers for their judgements with both versions of the F/2F compound (Fig. IA, B) and the 0” version of the F/4F compound. With the F/4F compound in the 180” version, however, very experienced observers use the region indicated by the letters e, f and g in Fig. 1D. The equations used here are the same as those used for the “square-wave” version of the F/3F compound in Badcock (1984). The precise formulation used
was chosen because it gave the same pattern of results both before and after practice in the experiments reported in the earlier paper-alternative formulations may also be adequate although a large number were tried and were not. The author only wishes to demonstrate the efficacy of this type of approach. Two local contrast measures are derived and then the difference between them is determined. The local contrast measure for each bar is given by; c, = (c - b)/[(C + b)i2]
(2)
and CI = (a - b)/[(a +
bY21
(3)
The relative contrast difference (RCD) between the local contrast measures is given by; RCD% = I lOO(C, - Q/C,, 1
(4)
These differences are zero in every case when no phase shift, 4,. is used. As the phase shift is increased one bar (denoted a) becomes closer in luminance to b while the other (c) moves further away. With the F/4F compound in the 4, = 180” condition (Fig. 1D) e,fand g are substituted for a, 6 and c respectively. Figure 2 shows the change in relative contrast difference (RCD%) as a function of the phase shift of the higher harmonic, $J,, for both compounds in both
Lo+Lml+Acos[2 I A
k w] + I9 COS[* 2 n fy+ .+J)
=l 0
k:2
k=4
D!stonce
(y )[degrees
of v~suol angle]
Fig. 1. The cross-sectional luminance profile given by equation 1 when the higher harmonic is half the amplitude of the fundamental. The luminance variation is shown for both reference phases for F/ZF (A, 4, = 0; B, 4, = 180) and F/4F (C, I$, = 0; D, 4, = 180) waveforms. The lower case letters indicate local luminance maxima and minima.
How do we discriminate relative spatial phase?
Phase
shtff of htghet
harmon%
Ldeg )
Fig. 2. The functions relating the percentage difference in local contrast (RCD%), from equations 2-4, to the phase shift of the higher harmonic for F/2F (circles) and F/4F (squares) waveforms in both reference phases (4, = 0, solid lines; 4, = 180, dashed lines).
phases. Circles indicate F/2F compounds while squares indicate F/4F. Solid lines indicate the 0” reference phase conditions while dashed lines indicate the 180” conditions. if Weber’s Law applied to this task (i.e. if the contrast change required were a constant proportion of the base contrast) then one would expect the relative contrast difference (RCD%) required for a given performance level to be the same for both reference phases and both FJ2F and F/4F. Since smaller phase shifts, Cp,,are required for a given RCD% value with F/ZF compounds than with F/4F compounds one would then expect greater sensitivity to phase shifts with the former compounds. In addition one would expect a much larger difference between the two reference phase conditions with F/4F than F/2F compounds if one uses Fig. 2 to predict sensitivity to changes of relative phase. referente
Methods Equipment. The stimuli (described above) were generated on a Tektronix 604A oscilloscope (P31 Phosphor) using digital techniques. The frame rate was 250 Hz and there were 850 lines~frame. A Sharp MZ80-I( microcomputer produced a discrete approximation to equation I and loaded it into a local memory unit. It then triggered the unit which generated the voltages for the X, Y and Z axes through 8-bit Digital to Analogue Converters (DACs). The linearity of the relationship between Z-axis voltage and luminance was verified using a digital photometer (Gamma Scientific Model 2400). The experimental session was run by the computer which received response information from the observers via one of two buttons. Display. The screen was viewed at a distance of 114cm and subtended 5 x 5 deg of visual angle. It was in a dark surround. The horizontal gratings were presented in pairs on the screen. Each pattern was 2.25” high and 5” wide and they were separated by a 0.5” high by 5” wide bar of the same space average
1849
luminance as the display (7.46 cd/m’), The patterns were abruptly truncated at the edges but their phase relative to the 2.25 x 5” window was varied from trial to trial using Ay in equation 1. Procedure. After dark adapting for about 5 min the observer pressed a button to begin a trial which consisted of the following sequence: 4.18 set observation interval, 0.5 set pause, second 4.18 set observation interval and then a response interval of unlimited duration. The observer’s response initiated the beginning of the next trial and the onsets of both observation intervals and the response intervals were marked by tones. In each observation interval the observer viewed a pair of patterns consisting of horizontally oriented compound gratings separated by a horizontal bar which had the mean luminance of the display (7.~cd~m2). In one interval the two patterns were identical. These patterns are called the reference stimuli. In the other interval one of the patterns was a reference stimulus but one, the test stimulus, was different in that the relative phase of the two components was varied. The observer’s task was to indicate by pressing one of two buttons which interval contained the pair of stimuli that were different. The test stimulus appeared in one and only one of the four possible locations on each trial and had equal probability of appearing in the top or bottom half of the screen in either the first or second interval. This is a two-alternative temporal forced-choice (2AFC) task with a reference stimulus in both observation intervals. All viewing was binocular with natural pupils and in a darkened room. No feedback was given, In the experiment the value of 4, (the phase shift) was varied using an adaptive two-alternative forcedchoice (2-AFC) procedure with two randomly interleaved staircases. In both staircases the initial value of Cp, was large so that the observer could easily recognize the test stimulus. The same stimuli were used in both staircases and were identical except for the phase shift, Cp,,which for both staircases had the same large initial phase shift of 40”. However, the phase shift was controlled independently and by different rules in each staircase so that the staircases converged on different performance levels. One staircase required two consecutive correct responses for a decrement in Cp,but only one incorrect response for an increase. This staircase converged on the 71% correct performance Ievel. The other staircase required four correct responses for a decrement in 4, and one incorrect response for an increment; it converged on the 84% correct performance level (Levitt, 1971). Both estimates were obtained to provide information about the slope of the psychometric function which relates the percentage of correct discriminations to the phase shift, I$,. The magnitude of the step in 4, was initially set to 10” but was halved the first three times a staircase changed direction (reversed), that is, each time a
1850
DAVID R. BADCOCK
sequence of decrements in $J, was followed by an increment or vice versa. A staircase ran until it had produced six reversals. The values of the phase shifts at the last four reversals were averaged to estimate the phase shift corresponding to the appropriate performance level. The few conditions in which the value obtained with the staircase designed to converge on the 84% level were not higher than the value obtained with the 71% staircase were rejected and repeated. Conditions. The signals were all constructed using discrete approximations to equation 1. The fundamental was 1.67 c/deg and had a Michelson contrast (&n.l, - L,,&/(L,,, + &,) of 0.221. The higher harmonics had a contrast of 0.088 and were thus 0.4 times the contrast of the fundamental. The two components in the reference stimuli were added with a reference phase, 4,. of either 0” or 180” as shown in Fig. 1. The test stimuli had the same 4, value as the reference stimuli but, in addition, had the positive values of 4, determined by the staircases. Obsertrers. Three highly experienced and emmetropic (or corrected to normal) observers participated in this experiment. D.R.B. is the author. AS. and J.G. (females) were naive with respect to the experimental aims and were paid hourly wages. Remits
Figure 3 shows the results for three observers. The phase shift, +,, of the higher harmonic required for 71% correct discrimination is shown as a function of the reference phase, Cp, (4, = 0, open symbols; Qt,= 180, solid symboIs) and the error bars indicate +/one standard deviation. Data for the F/2F condition is shown as solid lines; data for F/4F as dashed lines. Data for D.R.B. and A.S. are based on four staircases per point while those for J.G. are based on Il. For all observers a larger phase shift is required for the same performance level with F/4F than with F/2F ^ z”
(0) Jo-
(b)
ORB F14F
---
AS
compounds. The performance is only slightly worse with F/4F than with F:2F when Cp,=O’ but this difference is much greater for #, = 180” conditions. In both conditions performance with the 4, = 180’ reference phase is worse than with the 4, = 0” condition. Discussion
The results of the present experiment show, as expected from the local contrast difference model, that larger phase shifts are required to reach a criterion performance level with F/4F compounds than with F/2F compounds and also that the difference between the two reference phase conditions is larger with F/4F than F/2F compounds. These results are as expected. For example, the predictions in Fig. 3d show the performance expected if a constant relative contrast difference (RCD%) is required for a constant performance level. The figure was obtained by finding the group mean for the F/4F 4, = 180” condition (the choice is arbitrary but has little effect on the form of the results), calculating its RCD% value (approx. 6.8%) and then finding the phase shift, 4, that would give this value for the other conditions. The agreement between the predictions (Fig. 3d) and the individual data is quite striking (Fig. 3a-c). The Iargest discrepancy between the data and the predictions lies in the finding of poorer performance with F/2F patterns than predicted. The predicted difference between reference phases is also smaller for this waveform than was obtained. One reason for predicting thresholds that are too low for F/2F compounds is found if one considers the C, and C, vatues (the local contrast measures) used to calculate RCD%. These vary considerably among the conditions. It is found that the RCD% value that corresponds to the 71% correct performance level decreases as the size of C, increases; that is, smaller percentage differences are required when the lowest of the local contrast estimates is larger. This would mean that Weber’s law does not hold for these (c)
*
JG F/41: F,2F
-
-
-
PI.(I,CIIO”, F/4!-
”
-
-
F/ZF
Fig. 3. The shift in relative phase, 4,. of the higher harmonic required for 71% correct discrimination for both F/2F (solid lines) and F/4F (dashed lines) waveforms as a function of the reference phase, d. Open symbols denote 4, = 0, solid symbols denote #r = 180. The data for three observers are plotted (a, b and c) along with predictions (d) which are described in the discussion. Error bars indicate t/one standard deviation.
1851
How do we discriminate relative spatial Phase? parameters and thus the predictions of Fig. 3d which are based on this assumption will diverge from the data to the extent that Weber’s law fails. Kulikowski (1976) has shown that under similar conditions the Weber fraction required for detecting a contrast difference does decrease as the reference contrast increases (see his Fig. 4). This behaviour is also consistent with other contrast increment detection studies (e.g. Nachmias and Sansbury, 1974; Legge, 1981) and with demonstrations that contrast matching performance becomes more accurate as the contrast of the reference pattern increases (Georgeson and Sullivan, 1975). Legge (1981) suggests a power law should be used to describe the data from contrast increment detection studies. Following this suggestion exponents were derived for the three observers in this experiment. These exponents, relating the contrast difference to C, at the 71% performance level, were 0.61 (-t/-0.21), 0.61 (-I-/-0.15) and 0.66 (+/-0.09) for J.G., D.R.B. and AS. respectively [the variation indicates +/one standard error of the exponent derived by the method of Mansfield (1973)]. These are all very similar to those found in previous constrast increment detection studies (see Legge, 1981; Swift and Smith, 1983) and suggest that the model may indeed provide a good description of performance in “phase discrimination” studies. F.XPERfiMENT 2
Two related questions still need to be addressed with regard to the efficacy of the relative contrast difference (RCD) model. The first is the ability to predict results obtained with waveforms when the contrasts are varied. This was addressed in Badcock (1984) where the direction of the effect was predicted but the quantitative fit was not good. It was argued there, as in the discussion just presented, that the major discrepancies were caused by assuming that Weber’s Law applied to the current task and that these could be considerably reduced by using a power exponent of less than 1.0. Several recent studies have shown that contrast increment (and decrement) detection follows a complex “dipper shaped” function (see Legge and Foley, 1980) but that at higher base contrasts the “handle” of the dipper is best described by a power function with an exponent of approx. 0.6 (cf. Legge and Kersten, 1983; Swift and Smith, 1983). The exact figure, however, seems to be influenced by spatial frequency (Legge, 1979, 1981), or bar width (Legge and Kersten, 1983; Badcock, 1984) and by temporal parameters of the stimuli (Kulikowski and Gorea, 1978; Pantle, 1983). The current task differs from the contrast increment detection studies just mentioned in that the local variation in contrast is produced by a change in the relative phase of the two components and not by variation in their amplitudes. It is, therefore, of
considerable interest that the exponents derived are so similar to those derived from the contrast increment detection studies. A more detailed evaluation of the magnitude of the power exponent will now be presented. In the previous experiment the exponents were obtained by using vafues obtained from every condition and an estimate of the exponent (0.63) was obtained that was similar to the average values of other experimenters (cf. Legge, 1981; Legge and Kersten, 1983; Swift and Smith, 1983). However, since some dependence on spatial frequency has been demonstrated by some experimenters (e.g. Legge and Foley, 1980; Legge, 1981) it is not clear that averaging performance across patterns with different higher harmonics is as informative as it might be. To examine these issues the three compound gratings (F/2F, F]3F and Ff4F) used previousiy wili be used again in both reference phase conditions. In this study, however, the contrast ratio of the two components remains constant at 2: 1 (the fundamental has twice the amplitude of the higher harmonic) instead of 2.5: 1 as in Experiment 1. The contrast of the compound will be varied over a large range by adjusting the contrast of both components by equal proportions. This procedure for manipulating the contrast avoids the problems noted with F/3F compounds, when only one component’s contrast was varied at a time. It was seen there that some local features disappeared and different equations were required to calculate the focal contrast values. In this study the same equations can be used at all contrast levels for a given pattern and it will therefore be valid to’derive power exponents for each compound and each reference phase individually. This will allow a much more precise estimate for each condition and will also lend support to the ar~rn~t for the previous experiments should the exponents behave as is expected. Method Signals. This experiment uses F/2F, F/3F and F/4F luminance profiles. There is, however, a difficulty with labelling in this experiment. In the previous experiments when F/3F compounds were used (Badcock, 1984) they were generated by;
L~)=L,{l+Asin[2nfCv+A~)] + BsinIx$Cv
+ AY) + #, + tPd)
(5)
whereas when F/2F and F,i4F compounds were used (Experiment 1 of this paper) they were generated by; U_.v) = L,{ 1 + Acos[2 $0,
+ Ay)]
+ Bcos]2 @cV + Au) + #+ + &I)
(6)
where k determines which higher harmonic is used. The only difference between these two equations is that in equation 6 cosines are used instead of sines. Although this difference seems odd it will be maintained in this experiment since it has the advantage
DAVID R 58 ,180’
(0) 80
w
1
0 0'
40 20 -
10
z "
5
0 ;
i (b)
;
80
,"
40
z e
20
: x = .cz v) 2
10 5
JG *180*
(cl 80
0 0’
0 ar
40 20 10
BADC~CK
and they were separated by a 0.74” wide bar of mean luminance. Procedure. The adaptive procedure was identical to that used in the previous experiments. Again, the order of conditions was counterbalanced to control for order and learning effects. Auditory feedback was given for correct responses. Obsercers. Three observers were used. H.P. and J.G. completed three staircases for each performance level in each condition and G.B.H. completed two. Conditions. In each condition the contrast of the higher harmonic (specified by B in equations 5 and 6) was half that of the fundamental (A in equations 5 and 6). The contrast of the fundamental was either 0.442, 0.221, 0.1 I05 or 0.0553. The higher frequency component was either the second (F/2F), third (F/3F) or fourth (F/4F) harmonic of the fundamental (1.67 c/deg) and both 4, = 0” and r#~,= 180” reference phases were used. Results
1y3
5 00553 Contrast
01105
of the
0 221
0442'
Effect of contrast. The results plotted in Figs 4, 5 and 6 show the phase shift of the harmonic corresponding to the 84% correct performance level as a function of the contrast of the compound for the
fundamenfol
Fig. 4. The function relating the phase shift of the second harmonic, I#J,, required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both C#I, = 0” and 4, = 180” reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The second harmonic always had half the contrast of the fundamental. Both axes are logarithmic. Error bars indicate +/- one standard deviation.
the 4, = 180” condition always refers to the reference phase condition for which RCD% would predict poorer performance and therefore it simplifies discussion. Equipmenr. The Sharp MZSO-K also controlled the second display system. The information specifying the signals for each trial was sent to a PDP8/A minicomputer which generated the signals from lookup tables. The signals were then sent via 12-bit DACs through an attenuator to the display screen. The screen was a raster display built according to a design of J. G. Robson. It was masked with black cardboard to subtend 4.99 x 8.51” at the 154 cm viewing distance employed. It had 423 lines per frame, a frame rate of 100 Hz and a mean luminance of 47 cd/m’(P3 1 phosphor). A linearizing table was used to correct the nonlinearity of the Z-axis measured using the digital photometer. Linearity of the attenuator and the DAC was verified using an HP3580A Spectrum Analyzer. With this display the observers were seated with their heads restrained by a chin and forehead rest. Al1 viewing was binocular in a dark room. The patterns for this experiment were vertically oriented but were still presented in pairs on the screen, Each pattern was 3.89” wide and 4.99” high
20
that
.
00553 Contrast
01105
of
the
180'
0221
0442
fundamental
Fig. 5. The function relating the phase shift of the third harmonic, 4,. required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both I#I,= 0” and 4, = 180‘ reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The third harmonic always had half the contrast of the fundamental. Both axes are logarithmic. Error bars indicate +/- one standard deviation.
How do we
discriminate
80
40
20
--__ lx 1
l
J G. 180'
0
0;
(cl
--_
----_‘3-____.
10
5,
0.0553
Conrrast
0 1105
of
the
0 221
0442
fundamental
Fig. 6. The function relating the phase shift of the fourth harmonic, I#,, required for 84% correct discriminations as a function of the contrast of the fundamental is plotted for both 4, = 0” and 4, = 180” reference phase conditions for each observer (G.B.H., squares; H.P., inverted triangles; J.G., diamonds). The fourth harmonic always had half the contrast of the fundamental. Both axes are logarithmic, Error bars indicate +/- one standard deviation.
F/2F, F/3F and F/4F compounds respectively. The vertical phase axis is logarithmic (with the error bars indicating + / - one standard deviation) to minimize differences in the slopes of the psychometric functions, for the whole set. Only the 84% performance level is reported in this, and subsequent figures, since. the pattern of results is the same and the higher performance level is inherently less variable, allowing more precise estimation of the exponents to be derived later. The general pattern of results is the same for all observers with each compound. Performance, expressed as degrees of the higher harmonic, improves as the contrast of the components increases, even though the contrast ratio remains constant. Again performance with the (6, = 180” signals is consistently worse than with the 4, = 0” conditions and in general the improvement with increasing contrast of the components is greater with 4, = 0” conditions. This is reflected in the larger separation between the 0 and 180” points at the right hand side of the Figures than on the left hand side. It was also confirmed by a significant interaction between contrast level and
relative
spatial phase?
1853
reference phase (P < 0.01) in an analysis of variance that was conducted. In this analysis of variance the following results were obtained. The simple main effect of waveform (i.e. frequency combination) was significant [F(2,6) = 119.63, P < 0.011 but the main effects of contrast [F(3,3) = 9.08, 0.05 c P < 0.061 and reference phase [F( 1,2) = 14.2, 0.05 < P < 0.071 were not. The interaction between contrast and waveform [F(6,48) = 1.13, P > 0.051 was not significant but the interactions between contrast and reference phase [F(3,48) = 6.54, P c 0.011 and between the waveform and reference phase [F(2,48) = 5.46, P c 0.011 were. The three-way interaction between waveform, contrast level and reference phase was not significant [F(6,48) = 0.628, P > 0.051. This pattern of performance is as predicted by the relative contrast difference model as will be shown. It is necessary, however, to first derive the exponents for the power functions since the predictions are dependent on the exponents’ value. Power law exponents
In the previous experiment, the relationship between C,, the lowest local contrast measure, and AC, the difference between the two local contrast measures, was examined at the 71% correct performance level for each pattern employed. This relationship was found to be described by a power law with an exponent of approx. 0.63. The same relationship has been examined here, except that now there are four contrast levels within each pattern and so the relationship is derived within rather than across patterns and the 84% correct performance level is used. It should, however, be noted that in this experiment the observers reported using a different region of the pattern with F/3F “peaks-add” patterns than in Badcock (1984). It seems that the provision of auditory feedback and the experience with F/4F, 4, = 180” patterns, in the previous experiment, allowed them to determine that the following region was more useful. They fixated the darkest bar in the waveform and compared the two bright bars on either side. This change means that equations with the same number of terms can be used for every pattern in this experiment since the regions used for the other patterns were as in Experiment 1. Figure 7 shows, for observer J.G., the functions relating C,, the lowest local contrast estimate (which is twice the actual contrast) to AC, the contrast difference (also twice the real value), where both C, and AC represent the values at the 84% correct performance level. Both axes are logarithmic which means that the exponent of the power law is equivalent to the slope of the best fitting least-squares regression line. These lines have been calculated and are drawn through the data for each condition. Results for only one observer have been presented but the other two observers gave results that are similar in all respects. The parameters obtained from
DAVID R. BADCOCK
, 0005
001
C,
I
I
0025
I
I,,
I base
I 01
005
confrost
I
025 x 2
05
I
I
I,,
10
1
Fig. 7. The function relating the contrast difference, 1C, - C,,1, to the lowest local contrast value, C,, at the 84% correct discrimination level is plotted for observer. J.G. Both axes are logarithmic. The data points are connected to the best-fitting least-squares regression line for each condition. Solid lines represent F/ZF, dashed lines represent F/3F and dotted lines F/4F. Data for both 4, = 0” (open symbols) and 4, = 180” (filled symbols) reference phase conditions are plotted.
the least squares regression are presented in Table 1. In this Table the estimated values of k, the constant of proportionality, b, the exponent of the power law, and the standard error of the estimate of the exponent (Mansfield, 1973) are listed for each observer in each condition. It is clear, from both Table I and Fig. 7 that sensitivity is relatively lower for F/2F patterns than for either F/3F or F/4F patterns. This is shown by larger values of k in Table 1, and the vertical displacement of the F/2F lines in the Figures. The effects of the variation in k will be discussed later. It is, however, clear from the Table that the data for each condition are reasonably well fitted by straight lines on double logarithmic coordinates. The exponents are plotted in Fig. 8 where the solid horizontal line represents the mean value of the exponent in this study (0.539) and the broken lines indicate +/I (dashed lines), and +I2 (dotted lines), standard deviations (SD = 0.139). It is readily apparent from this Figure that most exponents are within one standard deviation of the mean (13/18). The exceptions to this do not follow any clear trend. The mean obtained here (0.539) is lower than that obtained in the previous experiment Table
I. Parameters
(0.63). The previous estimate is, however, within the one standard deviation range on Fig. 8 as would be the average estimates of Swift and Smith, 0.65, and Legge and Kersten, 0.56. Thus it is concluded that the results obtained in this experiment are compatible with the argument that the observers are detecting differences in local contrast to detect a phase shift and that the relationship between the local contrast measures and the just-detectable contrast difference is best described by a power law with an exponent of approx. 0.539 (+/ - 0.139). Predictions from the local contrast model
The predictions from the local contrast model that has been outlined are tedious to derive. Since both C,, the lowest local contrast measure, and AC, the contrast difference, vary when the relative phase varies, several steps are required to find the phase shift, $J,, that yields the appropriate values in each condition. The predictions were derived in the following way. (a) The mean phase shift, for all three observers, was calculated for the F/4F, 4, = 180” condition with a fundamental contrast of 0.221 in the experiment just presented.
of the power law for three observers with F/ZF, F/3F and F/4F compounds in 0 and 180” reference phases 0'
180’
Condition
Observer
k
b
SE,
k
b
SE.
Fj2F
G.B.H. H.P. J.G.
0.124 0.296 0.21 I
0.43 I 0.61 0.624
0.18 0.02 0.10
0.17 0.651 0.314
0.471 0.816 0.64
0.11 0.1 0. I5
F/3F
G.B.H. H.P. J.G.
0.104 0.I2.5 0.095
0.523 0.587 0.538
0.06 0.07 0.14
0.064 0.069 0.115
0.384 0.193 0.593
0.13 0.03 0.08
Fj4F
G.B.H. H.P. J.G.
0.164 0.084 0.057
0.716 0.517 0.374
0. I 0.07 0.05
0.079 0.079 0.091
0.513 0.593 0.583
0.02 0.09 0.02
How do we discriminate relative spatial phase?
1855
GBH
0
HP
V
JG
0
,
. zso so
-1
0 _-----_~_-_-_-_-_~--_____I
-
0
.
1 SD
-2S0
.
F/2
00
F/2
180’
F/3
Stimulus
0’
F/3
180’
F/4
0’
F/4
180”
condtrton
Fig. 8. The exponents of the power functions are plotted for each stimulus condition using data from the 84% correct performance level, for each observer C.B.H., squares: H.P., inverted triangles; J.G., diamonds. The right hand ordinate indicates the mean value of the exponent (solid line), +/one standard deviation (dashed lines) and +/- two standard deviations (dotted lines).
(b) At this mean shift the value of C, and AC was in the usual manner. (c) The mean value for the exponent, obtained in the experiment (0.54) was used to allow the derivation of the constant of proportionality, k, by using obtained,
AC = kc; which after reorganization
(7)
becomes
k = AC/C:
(8)
(d) Using tables that give the value of C, and AC, for each #, and assuming an exponent of 0.54, the value of k was derived, as in step c above, and a regression line relating (p, to k was created. The value of Q;, that gave the value of k, derived in step b, was then found using this regression line. The expected pattern of results when the contrast of the patterns is increased while the contrast ratio of its components remains constant will now be presented. The expectations are as presented in Fig. 9. This Figure shows the overall means, for the three observers, with the standard deviations calculated from the several estimates for each observer. The lines indicate the performance expected when the exponent is 0.54 and the value of k is 0.09414. The data point used to calculate k is indicated by the star. Two points need to be made about this Figure. First, the pattern of improvement in performance with increasing contrast is predicted extremely well by the local contrast model. It predicts the smaller influence of contrast on the 4p,= 180” patterns for al1 compounds and also the larger difference between retative phases with higher contrasts. This pattern is deter-
mined by the exponent but clearly the average exponent provides a good account of the pooled results. The second point to note about Fig. 9 is that the absolute performance levels, while agreeing with the predictions for the F/3F and F/4F compounds, are at variance with the predictions for the F/2F compounds. While the pattern of improvement with increases in contrast is predicted well by the model, the performance obtained is worse than predicted. The vertical position of the lines is determined by the constant of proportionality, k. With the predictions here this has been determined for the F/4F, 4, = l80”, 0.221 contrast point and the same value has been used everywhere. This effectively predicts F/4F and F/3F performance but in essence moves ail the variability to the F/2F section of the Figure. No single value of k will give a perfect fit for all compounds. Examination of Table 1, and Fig. 7 will show why. The lines for the F/2F compounds in the Figure are shifted vertically compared to those of other compounds. This is reflected in Table 1 by larger values of k, indicating that observers were relatively less sensitive to variations within F/2F compounds. The value of the constant of proportionality, k, is not often discussed in the contrast increment detection literature. More frequently the effect of this constant is eliminated by dividing both AC and C, by the contrast threshold for the particular signal (e.g. Legge, 1981; Legge and Kersten, 1983). If, therefore the vertical position of the lines in Fig. 9 was adjusted, by correcting k to allow for variations in threshold sensitivity, then the predicted performance could be indistinguishable from the data in Fig. 9.
DAVID
R.
BADCOCK
F/4f
i trau~ means
: 0 0’
251,
, 00553
01105
, 0221
,
/
0442 Contrast
,
,
0055301105 of
the
I
I
0221
0442
1
t 0055301105
0’
.
180°
-O*
Predictions *
A
I
---180’ 0221
0442
fundament~i
Fig. 9. In each section of this Figure the symbols represent the mean phase shift, $,, for the three observers from Figs 4, 5 and 6 of the higher harmonic required for 84% correct discrimination as a function of the contrast of the fundamental (the higher harmonic was haIf the contrast of the fundamental). Both axes are logarithmic and the error bars indicate +/ - one standard deviation. The higher harmonic is 2F in the left hand column, 3F in the middle and 4F on the right. Data for both 4, = 0” (open symbols) and 4, = 180” (solid symbols) reference phases are plotted. The lines represent the predictions using the local contrast difference model when the exponent of the power function is assumed to be 0.54 (the group mean) and k, the constant of proportionality, is 0.09414 and is derived from the data point indicated by the star (F/4F. (b, = 180’. fundamental = 0.221). The predictions for &I,= 0” and 4, = 180” reference phase conditions are shown by the solid and dashed lines respectively. This correction has not been made here since it is not clear which threshold is the appropriate one to use for the adjustment. Given so many alternatives it is
unlikely that one could not be found that would give an appropriate correction but this would not justify its choice. It is, however, appropriate to note that the thresholds measured by Lawden (1981) for detecting a harmonic in the presence of a suprathreshold fundamenta1 did give relative sensitivities close to those that would be required here. In any case it is clear that within each compound the Iocal contrast model predicts the pattern of improvement with increasing contrast very well and its performance across compounds would, indeed, be excellent if the vertical position of the predictions were corrected to allow for threshold sensitivity.
Since it has been shown that the model proposing local contrast difference discrimination, provides a good account of the “phase” discrimination performance presented above (and that presented by Burr (1980) whose experimental paradigm was similar; see Badcock, 1984) it is reasonable to ask how generally this formulation may be applied. The main thrust of the model is the reformulation of the phase discrimination task as a contrast difference discrimination task. Arend and Lange (1979) have also argued that the phase dependent behaviour they observed could be regarded as detecting variations in local regions. The local region could as easily be a uniform field as in many contrast threshold paradigms (e.g. Campbell and Robson, 1968; Derrington and Henning, 1980) as the nonuniform region which served as a background here. Swift and Smith (1983) have recently attributed per-
formance in spatial frequency masking studies to a strategy of this type. An example of the detection of variation in a uniform background in the “phase” discrimination literature may have been provided by Burton and Moorhead (1981) who noted that in their experiment “the key difference that emerged as the peak shift increased from zero was the appearance of new spatial features in the modified target. . . ” (p. 1059). An examination of their Fig. 1 shows that new bars appear in the uniform field surrounding the targets. Burton and Moorhead (1981) did not analyze their data in terms of the contrast required for detecting these new features but such an analysis may well have been useful. The current study provides data for a non-uniform field, as did Burr (1980), but perhaps the most extreme example is provided by Caelli and Bevan who used two-dimensional phasequantized textures. It is extremely difficult to analyze their stimuli using the current model since it is very difficult to know which regions to compare but examination of their Fig. 2 suggests that overall contrast variation is a powerful cue with their stimuli. These studies were all designed to measure phase sensitivity but it is not enough to manipulate the phase spectrum of a stimulus and simply assume that the observer must calculate a phase spectrum and then detect differences in that spectrum. Reasonable alternative strategies must be eliminated iirst. The current model helps to explain the wide variation in the estimates of “phase“ sensitivity obtained with different stimuli without reference to the phase spectrum of stimuli. It also draws strong parallels between “phase” discrimination performance and spatial frequency masking studies (Swift and Smith, 1983). In support of this comparison the average exponent is
How do we discriminate relative spatial phase?
lower in the second experiment than the first, showing the same trend with practice as the exponent derived by Swift and Smith. At present the major difficulty with the model lies in determining the regions that will be compared. It was not possible, in the experiments presented here to predict this a priori and to have told the observers which region to use would have introduced circularity. It seems likely that observers who have had experience with stimuli and feedback on their performance, will try to find a region of the stimulus that optimizes their sensitivity. This will be determined by a combination of the base contrast in the region and its rate of change with a phase shift. To optimize his performance the observer should find the region in which the contrast difference threshold is reached most rapidly. The strategies adopted for finding these regions need to be investigated as they may apply very generally (see also Swift and Smith, 1983). Implications for frequency analysis. The arguments presented above do not argue against the existence of multiple size, or frequency, tuned channels (Campbell and Robson. 1968) but they do place limits on the usefulness of the analogy between the visual system and a Fourier analyser. To the extent that only a local region of the stimulus is used to perform the task the rest of the stimulus may be said to be redundant. Thus removing the rest of the stimuius should have little effect on performance. The Fourier description of a stimulus, however, is not the same when the rest of the stimulus is removed and thus would not necessarily predict the same performance. In line with this argument, one would expect that with the current signals in a fixed size screen that “phase” thresholds should be reiatively independent of the fundamental spatial frequency, as long as it is high enough for the critical region to be small enough to still fit on the screen and low enough to allow the smallest detecting units to estimate the luminance at the individual points required. Burr (1980) has shown that these thresholds are, in fact, independent of spatial frequency over a large range with rapid deterioration at both low and high frequencies. The low frequency fall-off is at approximately the correct spatial frequency for the above argument. The appropriate high frequency is hard to determine but the fall-off is as rapid as one would expect. The present author has an unpublished replication of Burr’s (1980) results using longer durations but giving the same pattern of behaviour. Since a Fourier description can neither indicate the critical region of the pattern nor provide a simple account of the observed behaviour it must be concluded that its usefulness in this area, at least, is quite limited. author was supported by a Rhodes Scholarship during this work. He would like to thank the M.R.C. for help in supplying equipment. This paper was greatly improved through the comments of Dr Bruce Henning, Professor G. Westheimer and two anonymous referees. Acknowledgemenfs-The
1857 REFERENCES
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