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Article  |   June 2023
Detection of vertical interocular phase disparities using luster as cue
Author Affiliations
  • Frederick A. A. Kingdom
    McGill Vision Research, Department of Ophthalmology, Montreal General Hospital, Montréal, Quebec, Canada
    fred.kingdom@mcgill.ca
  • Keyvan Mohammad-Ali
    Faculty of Medicine and Health Sciences, Montreal, Quebec, Canada
    keyvan.mohammad-ali@mail.mcgill.ca
  • Camille Breuil
    McGill Vision Research, Department of Ophthalmology, Montreal General Hospital, Montréal, Quebec, Canada
    camille.breuil.pro@gmail.com
  • Deuscies Chang-Ou
    McGill Vision Research, Department of Ophthalmology, Montreal General Hospital, Montréal, Quebec, Canada
    deuscies.chang-ou@mail.mcgill.ca
  • Artur Irgaliyev
    McGill Vision Research, Department of Ophthalmology, Montreal General Hospital, Montréal, Quebec, Canada
    artur.irgaliyev@mail.mcgill.ca
Journal of Vision June 2023, Vol.23, 10. doi:https://doi.org/10.1167/jov.23.6.10
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      Frederick A. A. Kingdom, Keyvan Mohammad-Ali, Camille Breuil, Deuscies Chang-Ou, Artur Irgaliyev; Detection of vertical interocular phase disparities using luster as cue. Journal of Vision 2023;23(6):10. https://doi.org/10.1167/jov.23.6.10.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Interocular disparities in contrast generate an impression of binocular luster, providing a cue for their detection. Disparities in the carrier spatial phase of horizontally oriented Gabor patches also generate an impression of luster, so the question arises as to whether it is the disparities in local contrast that accompany the phase disparities that give rise to the luster. We examined this idea by comparing the detection of interocular spatial phase disparities with that of interocular contrast disparities in Gabor patches, in the latter case that differed in overall contrast rather than phase between the eyes. When bandwidth was held constant and Gabor spatial frequency was varied, the detection of phase and contrast disparities followed a similar pattern. However, when spatial frequency was fixed and Gabor envelope standard deviation (and hence number of modulation cycles) was varied, thresholds for detecting phase disparities followed a U-shaped function of Gabor standard deviation, whereas thresholds for contrast disparities, following an initial decline, were more-or-less constant as a function of Gabor standard deviation. After reviewing a number of possible explanations for the U-shape found with phase disparities, we suggest that the likely cause is binocular sensory fusion, the strength of which increases with the number of modulation cycles. Binocular sensory fusion would operate to reduce phase but not contrast disparities, thus selectively elevating phase disparity thresholds.

Introduction
Two spatially separated eyes with overlapping visual fields form the basis of binocular vision, an arrangement that benefits the user with a wider field of view, stereopsis, binocular summation, and binocular difference detection. The last of these refers to interocular differences in contrast or hue rather than the positional disparities that underpin stereopsis. Sensitivity to interocular differences in contrast or hue is a subject of increasing interest (Julesz & Tyler, 1976; Tyler & Julesz, 1976; Cohn, Leong & Lasley, 1981; Julesz, 1986; Cormack, Stevenson, & Schor, 1991; Stevenson, Cormack, Schor, & Tyler, 1992; Formankiewicz & Mollon, 2009; Yoonessi & Kingdom, 2009; Malkoc & Kingdom, 2012; Georgeson, Wallis, Meese, & Baker, 2016; Jennings & Kingdom, 2016; Kingdom, Jennings, & Georgeson, 2018; Reynaud & Hess, 2018; Kingdom, Seulami, Jennings, & Georgeson, 2019). Interocular differences in contrast or hue have been referred to simply as interocular differences (Kingdom, Jennings, & Georgeson, 2018), interocular (de)-correlations (Cormack et al., 1991; Stevenson et al., 1992; Reynaud & Hess, 2018), binocular luminance disparities (Formankiewicz & Mollon, 2009), and dichoptic differences (e.g. Yoonessi & Kingdom, 2009; Malkoc & Kingdom, 2012). Here, we refer to them simply as disparities. A disparity in either luminance or chromatic contrast can generate an impression of luster, the cue that presumably enables them to be detected (Formankiewicz & Mollon, 2009; Yoonessi & Kingdom, 2009; Malkoc & Kingdom, 2012; Jennings & Kingdom, 2016; Kingdom et al., 2018). Recent studies have suggested models for contrast disparity detection based on luster (Georgeson et al., 2016; Jennings & Kingdom, 2016; Wendt & Faul, 2020), and have demonstrated that contrast disparity detection is an adaptable dimension of vision (Kingdom et al., 2018), consistent with the existence of channels selective for contrast disparity (Cohn et al., 1981; Li & Atick, 1994; Zhaoping, 2014; May & Zhaoping, 2016; Kingdom et al., 2018; Wendt & Faul, 2020; May & Zhaoping, 2022 – note Li and Zhaoping are the same person: Zhaoping Li). 
In this communication, we examine the detection of disparities in the carrier spatial-phase of horizontally oriented Gabor patches, termed here as “phase disparities.” Because the patches are horizontally oriented, their phase disparities are in the vertical direction, and as such do not typically generate an impression of stereoscopic depth, consistent with them being detected instead by luster. It has been suggested that it is not the phase disparity per se that generates luster, but the accompanying disparities in interocular root mean square (RMS) contrast (Kingdom et al., 2018; Kingdom et al., 2019). For this reason, we report our phase disparity results also in terms of RMS contrast (see below for details). We have also measured the detection of disparities in actual spatial contrast, using Gabors with the same carrier spatial phase but with different peak-to-trough contrasts in the two eyes. With these “contrast disparity” Gabors we report our measurements also in terms of RMS contrast to enable a direct comparison with the RMS measures of phase disparity. Note that phase disparities in Gabor patches introduce local but not global disparities in contrast, whereas contrast disparities introduce both local and global disparities in contrast. A comparison of thresholds for detecting phase versus contrast disparities using luster as a cue has helped to reveal the underlying detection mechanisms involved. 
In summary, we have measured thresholds for detecting both phase and contrast disparities in horizontally oriented Gabor patches of various spatial-frequencies and envelope sizes. Results from the two types of thresholds are compared by converting each into RMS contrast disparities. 
Terminology and definitions
The definitions and terminology we have used in this communication are as follows. For convenience, we define the contrast C of the Gabor's sine-wave modulation as the value of C prior to multiplication by the Gabor's Gaussian envelope. If the Gabor contrasts in both eyes are equal, we have for the left (L) and right (R) eye's stimulus:  
\begin{eqnarray*} {S_L}\left( x \right) = C\, \cdot \,cos\left( {fx + {\phi _L}} \right) \end{eqnarray*}
and  
\begin{eqnarray*} {S_R}\left( x \right) = C\, \cdot \,cos\left( {fx + {\phi _R}} \right) \end{eqnarray*}
where f is spatial frequency and ϕL, and ϕR are the left and right eye spatial phases. Thus, phase disparity Pdiff (“diff” is short for the more familiar “difference”) is given by:  
\begin{equation}{P_{diff}} = {\rm{\Delta }}\phi = |{\phi _L} - {\phi _R}|\quad\end{equation}
(1)
Pdiff can be expressed in terms of the RMS disparity in contrast, termed here Cdiff-phase. As Kingdom et al. (2018) showed the relationship between Cdiff-phase and Pdiff is neatly captured by the variance sum law which expresses the sum (or difference) of two correlated variables in terms of their individual variances and their mutual correlation. If the correlation term is expressed as cos(Δϕ), Cdiff-phase is given by:  
\begin{eqnarray*}{C_{diff - phase}} &=& C_{L.rms}^2 + C_{R.rms}^2 - 2{C_{L.rms}}\\ && {C_{R.rms}}\cos(\Delta \phi )\end{eqnarray*}
Where CL.rms and CR.rms are the RMS contrasts of, respectively, the left and right eye signals. If the left and right eye RMS contrasts are equal, this reduces to:  
\begin{equation*}{C_{diff - phase}} = {C_{rms}}\sqrt {[2\left( {1 - \cos \left( {{\rm{\Delta }}\phi } \right)} \right]} \end{equation*}
 
Given that \({C_{rms}} = C/\sqrt 2 \) we can write instead  
\begin{equation}{C_{diff - phase}} = C\sqrt {\left[ {1 - \cos \left( {{\rm{\Delta }}\phi } \right)} \right]}\quad \end{equation}
(2)
 
Now consider the measurement of interocular disparities in peak-to-trough Gabor contrast, ΔC, given by:  
\begin{equation}{\rm{\Delta }}C = |{C_L} - {C_R}|\quad\end{equation}
(3)
where CL and CR are the left and right eye Gabor contrasts. In order to compare thresholds measured for interocular phase disparities and peak-to-trough contrast disparities, we use the measure Cdiff-contrast, which, like Cdiff-phase, is measured in terms of RMS contrast and is calculated thus:  
\begin{equation}{C_{diff - contrast}} = {\rm{\Delta }}C/\sqrt 2 \quad\end{equation}
(4)
 
An issue that potentially arises when using Gabor stimuli to measure interocular disparity thresholds is that the RMS measures Cdiff-phase and Cdiff-contrast might vary with Gabor bandwidth and hence the number of modulation cycles, assuming that the visual system integrates contrast information across the Gabor stimulus as a whole. To determine if they do so vary, we measured Cdiff-phase and Cdiff-contrast for Gabors clipped at six envelope standard deviations (as in the experiments described here), averaged across the full range of absolute Gabor phases (absolute Gabor phase is randomized on each trial in all our experiments) and for various bandwidths. Our simulations have revealed that under these circumstances both Cdiff-phase and Cdiff-contrast are unaffected by bandwidth. 
A potential problem with measuring thresholds for Cdiff-contrast in the experience of the senior author is the considerable difficulty involved in eliminating perceptual contrast cues as to the target stimulus (i.e. the stimulus with a contrast disparity). This is in part due to the “winner-take-all” behavior of dichoptic contrast mixtures, in which the apparent contrast of a dichoptic mixture of contrasts tends toward that of the component with the higher contrast (Meese, Georgeson, & Baker, 2006; Baker, Wallis, Georgeson, & Meese, 2012; Kingdom & Libenson, 2015). To make matters worse, the extent of the winner-take-all behavior is dependent on the average contrasts of the stimuli (Meese et al., 2006; Baker et al., 2012), and, from our observations with Gabors, also dependent on spatial frequency and bandwidth, adding to the difficulty of controlling the artifact when contrast, spatial frequency, and bandwidth are all variables. Our attempts at minimizing contrast cues in the measurement of Cdiff-contrast thresholds will be described in the following section. 
Methods
Observers
Four observers took part in the experiments. All were authors, although at the time of testing, three were naïve as to the purpose of the experiments. All experiments were conducted in accordance with the Declaration of Helsinki and the Research Institute of the McGill University Health Centre (RI-MUHC) Ethics Board. Observer initials on graphs have been anonymized in accordance with requirements of the (RI-MUHC) Ethics Board. 
Stimulus display
All experiments were conducted using a Dell Precision T1650 PC with a VISaGe graphics card (Cambridge Research Systems [CRS], UK). The visual stimuli were displayed on a gamma-corrected Sony Trinitron Multiscan F500 flat-screen CRT Monitor. Stimulus generation and experimental control used custom software written in C. Participants viewed the dichoptic pairs through a custom-built eight-mirror stereoscope with an aperture of 10 degrees × 10 degrees and a viewing distance along the light path of 100 cm. During the experiments observers were seated in a darkened room and their responses were recorded via a keypad. 
Stimuli
Example Gabor pairs are illustrated in Figure 1. They were horizontally oriented and of various spatial frequencies and bandwidths. The horizontal separation of the two members of each pair on the monitor was adjusted so that they appeared fused in the center of the aperture. A surrounding dark grey circle one pixel wide with a radius of 3.5 standard deviations of the Gabor standard deviation was presented at all times and served to maintain fixation and help to minimize any vergence misalignment. 
Figure 1.
 
Example dichoptic Gabor stimulus pairs. (a) Identical stimuli to left (LE) and right (RE) eyes. (b) Gabor pair with an interocular disparity in spatial-phase Pdiff. (c) Gabor pair with an interocular disparity in contrast Cdiff. Gabors are all 1.5 octaves in bandwidth.
Figure 1.
 
Example dichoptic Gabor stimulus pairs. (a) Identical stimuli to left (LE) and right (RE) eyes. (b) Gabor pair with an interocular disparity in spatial-phase Pdiff. (c) Gabor pair with an interocular disparity in contrast Cdiff. Gabors are all 1.5 octaves in bandwidth.
Procedure
We used a conventional 2IFC procedure in conjunction with a three-up-one-down staircase that adjusted the relevant stimulus dimension according to the previous responses. Gabors were presented within a raised cosine temporal envelope with full stimulus exposure duration of 400 ms. A button press in response to the previous trial initiated each trial sequence enabling observer control over timing. The first stimulus appeared 250 ms following the button press and the second stimulus 500 ms after the offset of the first. After a further 500 ms interval, feedback was provided in the form of a 150 ms five pixel diameter red circle for an incorrect response. After 50 trials, the session was terminated. The task for the observer was to select the interval containing the “lustrous” stimulus. The Δϕ or ΔC either increased or decreased proportionately on each trial by a factor of 2.5 for the first five trials and 1.3 thereafter. There were between four and eight sessions for each of the spatial frequency/contrast/size conditions in each experiment resulting in a total of between 200 and 400 trials per condition. The condition order in each experiment was randomized. 
Phase-disparity conditions
The absolute spatial phase of the test was randomized on each trial but the phase disparity Δϕ was determined by the staircase. The component phase for the left eye was (ϕL + kΔϕ/2) and for the right eye (ϕLkΔϕ/2), where k was set randomly to 1 or −1 on each trial. To minimize cues as to the interval with the phase disparity, contrast jitter was selected from a range ±15% of the designated contrast for that condition. An upper limit of 180 degrees phase disparity was imposed on the staircase procedure. 
Contrast-disparity conditions
In the attempt to minimize extraneous contrast cues to the detection of interocular contrast as opposed to phase disparities, we introduced a contrast randomization that was dependent on whether the interval was target or comparison. Let the baseline contrast in the target interval be C and the target interocular contrast ΔC. For the target interval, we randomly selected from the following four conditions: one eye's contrast C the other eye's contrast C − ΔC, or vice-versa; one eye's contrast C + ΔC/5 and the other eye's contrast C + ΔC/5 − ΔC, or vice versa. For the non-target interval, the contrasts in both eyes were set equal but randomly selected from four values: C − ΔC/4, C − ΔC/5, C − ΔC/6, and C − ΔC/7. These choices of target and non-target interval contrasts are somewhat arbitrary and were arrived at after a considerable amount of pilot work in an attempt to find a suitable arrangement of contrasts that perceptually reduced perceived overall contrast as a cue to the interval with ΔC. As such, they should not be regarded as definitive for minimizing perceptual contrast cues to ΔC detection. 
Analysis
The raw Δϕ and ΔC values were first log transformed then pooled into between seven and 11 bins that spanned between the minimum and maximum stimulus values. Psychometric functions of proportion correct against stimulus value were fitted with Gumbel (log Weibull) functions using a maximum likelihood criterion, using routines customized from the Palamedes toolbox (Prins & Kingdom, 2018). Thresholds on all graphs are the antilogs of the fitted thresholds and error bars are upper and lower standard errors derived using bootstrap analysis. Pdiff thresholds were equal to Δϕ thresholds, whereas Cdiff-phase thresholds were calculated from Δϕ thresholds using Equation 2 and Cdiff-contrast thresholds from ΔC thresholds using Equation 4
Results
Experiment 1: Contrast dependence of Pdiff
In the first experiment, we measured threshold Pdiff as a function of contrast for 1.5 octave bandwidth Gabors at two spatial frequencies, 0.25 and 0.5 cpd. Results for two observers are shown in Figure 2. The left panels present thresholds in terms of Pdiff, whereas the right panels show the same thresholds converted to Cdiff-phase. Pdiff thresholds decline rapidly with contrast to a contrast of about 10% after which there is, in most cases, a small increase in thresholds. However, when converted to Cdiff-phase the thresholds increase monotonically with contrast. The difference in the pattern of thresholds for the two measures lies in the fact that whereas Pdiff is contrast independent (Equation 1) Cdiff-phase is not (Equation 2). In the following experiments, all contrasts are set to either the minimum of the Pdiff function or to a greater value. Finally, note that thresholds for both spatial frequencies are similar, a point to which we will return. 
Figure 2.
 
Results for Experiment 1 for two observers. Interocular phase difference are plotted as a function of contrast for 1.5 octave bandwidth Gabors at two spatial frequencies (0.25 and 0.5 cpd), in terms of Pdiff (left panels) and Cdiff-phase (right panels). Error bars are standard errors obtained by bootstrap analysis and are not shown if smaller than the data symbols. The symbol # indicates conditions in which error bars were unobtainable.
Figure 2.
 
Results for Experiment 1 for two observers. Interocular phase difference are plotted as a function of contrast for 1.5 octave bandwidth Gabors at two spatial frequencies (0.25 and 0.5 cpd), in terms of Pdiff (left panels) and Cdiff-phase (right panels). Error bars are standard errors obtained by bootstrap analysis and are not shown if smaller than the data symbols. The symbol # indicates conditions in which error bars were unobtainable.
Experiment 2: Spatial frequency dependency of Pdiff /Cdiff-phase and Cdiff-contrast
In this experiment, phase disparity thresholds were measured as a function of Gabor spatial frequency, for 1.5 octave bandwidth Gabors set to 10% contrast. As in the previous experiment, Pdiff thresholds were also converted to Cdiff-phase thresholds. Unlike in the previous experiment, however, we also measured threshold contrast disparities Cdiff-contrast. Results for three observers are shown in Figure 3, with blue symbols for Pdiff and Cdiff-phase and orange symbols for Cdiff-contrast. There is a moderate increase in thresholds as a function of spatial frequencies greater than 0.5 cpd (consistent with the results of the first experiment which found similar Pdiffs for 0.25 and 0.5 cpd), irrespective of the measure used (Pdiff or Cdiff-phase) or the type of disparity (Pdiff or Cdiff-contrast). 
Figure 3.
 
Effect of spatial frequency (SF) on interocular disparities. Blue circles in the left panel are threshold Pdiff and in the right panel Cdiff-phase. Orange circles in the right panel are threshold Cdiff-contrast. Errors bars and markings as in the previous figure (Figure 2).
Figure 3.
 
Effect of spatial frequency (SF) on interocular disparities. Blue circles in the left panel are threshold Pdiff and in the right panel Cdiff-phase. Orange circles in the right panel are threshold Cdiff-contrast. Errors bars and markings as in the previous figure (Figure 2).
Unlike in the previous experiment, in which the independent variable was contrast, the contrast was fixed in this experiment at 10% and, hence, as a result, the pattern of results is very similar for Pdiff and Cdiff-phase. The implications of these results will be discussed later. 
Experiment 3: Size dependency of Pdiff/Cdiff-phase and Cdiff-contrast
In this experiment, we measured disparity thresholds as a function of the size of Gabors of given spatial frequency, where size is given by the standard deviation of the Gabor's Gaussian envelope. As the standard deviation increases for a given spatial frequency, so too does the number of modulation cycles. The Gabor's spatial frequency bandwidth on the other hand declines with standard deviation. After discovering an unexpected effect of standard deviation, we expanded the experiment to include a range of Gabor spatial frequencies (0.5, 1, 2, and 4 cpd) and two contrasts (0.1 and 0.8). 
Figure 4 shows the results for Pdiff. In all combinations of spatial frequency and contrast, there is a consistent U-shaped function of Pdiff against standard deviation. 
Figure 4.
 
Results of Experiment 3 for four observers. Threshold Pdiff is plotted as a function of the standard deviation of the Gabor's Gaussian envelope, for spatial frequencies SF and contrasts C as indicated on each graph. Error bars and associated symbols as in previous figures.
Figure 4.
 
Results of Experiment 3 for four observers. Threshold Pdiff is plotted as a function of the standard deviation of the Gabor's Gaussian envelope, for spatial frequencies SF and contrasts C as indicated on each graph. Error bars and associated symbols as in previous figures.
Finally, for four of the conditions shown in Figure 4, we also measured Cdiff-contrast, and for the 2.0 cpd and 0.8 contrast condition we measured both Pdiff and Cdiff-contrast at exposure durations of 200 ms and 100 ms (all other conditions were at exposure durations of 400 ms). Figure 5 shows the results, with Pdiff converted to Cdiff-phase in blue symbols and Cdiff-contrast shown in orange symbols. In the previous experiment, when spatial frequency was the independent variable Cdiff-phase and Cdiff-contrast produced similar results (see Figure 3), but, in this experiment, the two measures take on a different pattern. The U-shaped functions in Figure 4 with Pdiff are, as one would expect, also evident in their Cdiff-phase counterparts shown in blue in Figure 5 (remember for each function in Figure 5 contrast is held constant). However, the Cdiff-contrast plots show little or no U-shape. Although the two measures are similar at low standard deviations, the weak to non-existent upturn with Cdiff-contrast after the middle of the standard deviation range results in much lower thresholds for Cdiff-contrast compared to Cdiff-phase at the high end of the standard deviation range. 
Figure 5.
 
Results of Experiment 3 comparing Cdiff-phase with Cdiff-contrast for four observers. SF = spatial frequency, C = contrast. As in all previous experiments, data were collected at a stimulus exposure time of 400 ms, expect for the bottom pair of panels where exposure durations were 200 ms and 100 ms as indicated.
Figure 5.
 
Results of Experiment 3 comparing Cdiff-phase with Cdiff-contrast for four observers. SF = spatial frequency, C = contrast. As in all previous experiments, data were collected at a stimulus exposure time of 400 ms, expect for the bottom pair of panels where exposure durations were 200 ms and 100 ms as indicated.
Discussion
Relevance of findings to the role of binocular luster and its physiological basis
It is evident from observers’ reports of all the experiments described, that vertical phase disparities in the carriers of horizontally oriented Gabor stimuli are detectable by binocular luster. However, the amount of luster and, hence, ease of detectability in this study has been shown to depend on a number of factors besides interocular phase disparity, specifically contrast, spatial frequency, and bandwidth/number of cycles. As we noted in the Introduction, luster is a perceptual signal that appears to originate from local or global interocular disparities in contrast, so it is reasonable to conclude that it is the local contrast disparities that accompany interocular phase disparities that underpin the latter's detection. The question arises therefore as to why ostensibly equivalent amounts of phase disparity and contrast disparity do not always generate equivalent amounts of luster, as the results from the experiments manipulating cycle number in Figure 5 show. We will shortly return to consider why this is the case. 
How might binocular luster in general be detected? Wendt and Faul (2020) have suggested that interocular pairings of on-center and off-center neurons provide the physiological basis for the impression of luster. They showed how the impression of luster in various spatio-luminance arrangements of a test patch, surround ring and background correlated with the modeled activity of on- and off-center neuron pairs in the two eyes. Single-unit recordings of monkey cortical neurons have inferred the presence of on- and off-center neuron pairings in what are termed “tuned inhibitory” neurons (Poggio & Fischer, 1977; Prince, Cumming, & Parker, 2002; Read & Cumming, 2004). Tuned-inhibitory neurons appear to sum on- and off-center responses from the two eyes and, as a result, are minimally responsive or unresponsive to binocularly matched stimuli while maximally responsive to stimuli of opposite luminance polarity, such as a luminance increment in one eye and a luminance decrement in the other. Tuned-inhibitory neurons therefore could provide the physiological basis for the perception of binocular luster. 
A question that arises is whether there is something “special” about interocular opposite polarities in terms of the magnitude of luster they elicit. A number of investigators using suprathreshold stimuli have provided evidence in the affirmative (Anstis, 2000; Georgeson et al., 2016; Wendt & Faul, 2019; Wendt & Faul, 2022). On the other hand, Kingdom, Sato, Chang, and Georgeson (2020) suggest the contrary, at least for stimuli whose contrasts are at or close to detection threshold. They compared the detection and identification of interocular equal, unequal-same-polarity and unequal-opposite-polarity contrast patches, in which luster was the cue for identifying interocular disparities. When compared to detection they found that interocular same-polarity pairs were as identifiable as interocular opposite-polarity pairs. In the present study, disparities in Gabor phase, as measured either by Pdiff or Cdiff-phase, will presumably stimulate both interocular opposite polarity as well as interocular same polarity center-surround mechanisms, but preferentially for opposite polarity mechanisms at large phase disparities. On the other hand, disparities in Gabor contrast, as measured by Cdiff-contrast, will only stimulate interocular same-polarity mechanisms. Given that in none of our experiments were Cdiff-phase thresholds lower than Cdiff-contrast thresholds, the results of the present study do not support the idea that interocular opposite-polarity stimuli are special. 
Dependency of phase disparity detection on cycle number
We now consider the main and notably unexpected result of the present study: the U-shape pattern of Pdiff as a function of Gabor standard deviation/number of modulation cycles (see Figures 45). We consider four possible reasons for the U-shape: bandwidth, surround suppression, vergence eye movements, and sensory fusion. 
Bandwidth
As the number of modulation cycles increases, the spatial-frequency bandwidth decreases, with the result that the low spatial frequencies in the stimulus, which in general better support Pdiff detection (see Experiment 2 and Figure 3) increasingly drop out. However, the bandwidth explanation would predict the same U-shape function for Cdiff-contrast, whereas in Experiment 3 (see Figure 5) there is little or no U shape. Hence, bandwidth would seem to be an unlikely explanation. 
Surround suppression
A second possibility for the U-shape is surround suppression. Surround suppression is evidenced in a wide variety of visual dimensions. For example, duration thresholds for motion discrimination in large noise patterns can be a factor of two or more higher than in small noise patterns (Tadin, Lappin, Gilroy, & Blake, 2003; Tadin, 2015). Both neurophysiology and psychophysics has revealed that maximum surround suppression often occurs when surround and test have similar orientations (reviewed by Gheorghiu, Kingdom, & Petkov, 2014), as with our stimuli. The suppression of Pdiff detection might therefore be an example of such “iso-orientation” surround suppression operating in channels sensitive to interocular disparities. However, as with the bandwidth explanation above, we would expect surround suppression to occur equally for Cdiff-contrast and Cdiff-phase thresholds, unlike what we found. Thus, again, it seems unlikely that the U shape is a result of surround suppression. 
Vergence eye movements
In principle, vergence eye movements are a strong candidate for the U-shape. A difference in the vertical vergence angle of the two eyes of appropriate sign could serve to reduce any phase disparity and thereby increase Pdiff thresholds. Moreover, the greater the number of carrier cycles the more there is available to drive eye movements, with the result that Pdiff would be expected to increase with cycle number, at least after any initial decline. With contrast disparities on the other hand, vergence eye movements would not be expected to affect thresholds and hence we would not expect Cdiff-contrast to increase with cycle number. Both the above considerations are in keeping with our results. The main evidence against vergence eye movements being responsible for the U-shape function of Pdiff is the effect of reducing the exposure duration of the stimuli from 400 ms to 200 ms and then further to 100 ms, as in the bottom two panels of Figure 5. Howard, Allison, and Zacher (1997; discussed in Howard, 2002), measured the amplitude of vertical vergence eye movements for textured stimuli that oscillated in vertical disparity at various temporal frequencies. Assuming that our cosine temporal-enveloped stimuli are a close approximation to half a temporal cycle of disparity modulation, we would expect, on the basis of Howard et al.’s results, that the effectiveness of vertical vergence eye movements for reducing phase disparities would be negligible at 100 ms exposure duration. Yet, we observed no narrowing of the difference in Cdiff-phase and Cdiff-contrast results at the two shorter exposure durations. Therefore, we are inclined to conclude that vergence eye movements, however attractive as an explanation, are unlikely to be the cause of the U-shaped function of Pdiff
Sensory fusion
The final possibility we consider is binocular sensory fusion, the process whereby an internal process serves to reduce binocular disparities. Binocular sensory fusion has been revealed in a number of recent psychophysical studies demonstrating that horizontally oriented sinusoidal grating patches with vertical interocular phase disparities are perceived as gratings intermediate in phase between the two eye phases (Ding & Sperling, 2006; Huang, Zhou, Zhou, & Lu, 2010). Zhaoping Li (personal communication) has suggested that for the horizontally oriented Gabors with interocular vertical carrier disparities used in the present study, sensory fusion will tend to reduce the amount of interocular difference, importantly more so for Gabors with many compared to few numbers of cycles. Critical, however, to any sensory fusion explanation of our results, is that for sensory fusion to reduce luster and hence increase Pdiff /Cdiff-phase thresholds, luster must follow not precede binocular sensory fusion. 
Why might the effect of sensory fusion be greatest for Gabors with relatively large numbers of carrier cycles? Consider two Gabor pairs, one pair with a single cycle the other pair with several cycles, with each pair having the same phase disparity. If the members of each Gabor pair were vertically shifted via sensory fusion to bring them into carrier phase alignment, they would still be imperfectly matched because of the disparity of their contrast envelopes. However, the magnitude of the resulting envelope disparities declines as the number of carrier cycles increases, presumably making them harder to detect. In the extreme case of a pair of Gabors with opposite carrier phase, but with the Gabor in one eye positionally shifted to make its carrier bars line up with those of the other eye's Gabor, the interocular correlation between the pair increases from about 0.33 to 0.99 as bandwidth ranges from 0.49 to 0.17 (i.e. as the standard deviation and number of carrier cycles increases), the range from about the minimum to the maximum threshold Pdiff at the high standard deviation end in the experiments with the 2 cpd Gabors (see Figures 45). A corollary to this observation is the relative difficulty one experiences when attempting to discriminate side-by-side opposite-phase Gabor pairs with large numbers of cycles, as can be seen in Figure 6. We suggest, therefore, that there is likely to be less resistance to binocular sensory phase alignment in Gabors with relatively large numbers of cycles, resulting in the increase in thresholds with standard deviation in Experiment 3. However, notwithstanding the above argument based on the physical properties of positionally shifted Gabors with different carrier phases, it is perfectly possible that binocular sensory fusion might simply be more efficient the greater the number of modulation cycles. 
Figure 6.
 
Gabor pairs with different bandwidths/number of cycles. The two members of each pair are in opposite spatial phase, hard to perceive in the top pair but easily perceived in the bottom pair.
Figure 6.
 
Gabor pairs with different bandwidths/number of cycles. The two members of each pair are in opposite spatial phase, hard to perceive in the top pair but easily perceived in the bottom pair.
Efficient coding
Experiment 3 showed a modest increase in both Pdiff /Cdiff-phase as well as Cdiff-contrast for 1.5 octave bandwidth, 10% contrast Gabors as a function of spatial frequency (SF) in the range 0.25 to 4.0 cpd. Using very different methods and stimuli Reynaud and Hess (2018) found a similar result with contrast disparities. Together with the fact that in our experiments the increase in thresholds with SF was observed for both phase and contrast disparities it seems unlikely that the increase was due to an increase in the effectiveness of either vergence eye movements or sensory fusion. 
An alternative possibility is that based on considerations of efficient coding. Li and Atick's (1994) efficient coding theory of binocular vision posits that binocular summing and binocular differencing channels would be expected to have different spatial-frequency-dependent sensitivities, with the differencing channel declining in sensitivity with SF at relatively high SFs, in accord with the results here. 
Conclusion
We have demonstrated an unexpected effect of the number of a Gabor's carrier cycles on the ability of observers to detect interocular disparities in the Gabor's spatial phase. We suggest that the increase in phase disparity thresholds as the number of carrier cycles increases beyond an initial minimum value is likely due to binocular sensory fusion acting to reduce the Gabor's effective phase disparity. 
Acknowledgments
Funded by a Canadian Institute of Health Research grant (MOP 123349) given to F.K. and by a McGill summer medical bursary given to K.M.-A. Special thanks also to Zhaoping Li for informal discussions and Scott Stevenson's review of an earlier version of the study, both suggesting that binocular sensory fusion might underpin the main finding of this study. 
Commercial relationships: none. 
Corresponding author: Frederick A. A. Kingdom. 
Email: fred.kingdom@mcgill.ca. 
Address: McGill Vision Research, Department of Ophthalmology, Montreal General Hospital, 1650 Cedar Ave., Rm. L11.520, Montréal, Quebec, H3G 1A4, Canada. 
References
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Figure 1.
 
Example dichoptic Gabor stimulus pairs. (a) Identical stimuli to left (LE) and right (RE) eyes. (b) Gabor pair with an interocular disparity in spatial-phase Pdiff. (c) Gabor pair with an interocular disparity in contrast Cdiff. Gabors are all 1.5 octaves in bandwidth.
Figure 1.
 
Example dichoptic Gabor stimulus pairs. (a) Identical stimuli to left (LE) and right (RE) eyes. (b) Gabor pair with an interocular disparity in spatial-phase Pdiff. (c) Gabor pair with an interocular disparity in contrast Cdiff. Gabors are all 1.5 octaves in bandwidth.
Figure 2.
 
Results for Experiment 1 for two observers. Interocular phase difference are plotted as a function of contrast for 1.5 octave bandwidth Gabors at two spatial frequencies (0.25 and 0.5 cpd), in terms of Pdiff (left panels) and Cdiff-phase (right panels). Error bars are standard errors obtained by bootstrap analysis and are not shown if smaller than the data symbols. The symbol # indicates conditions in which error bars were unobtainable.
Figure 2.
 
Results for Experiment 1 for two observers. Interocular phase difference are plotted as a function of contrast for 1.5 octave bandwidth Gabors at two spatial frequencies (0.25 and 0.5 cpd), in terms of Pdiff (left panels) and Cdiff-phase (right panels). Error bars are standard errors obtained by bootstrap analysis and are not shown if smaller than the data symbols. The symbol # indicates conditions in which error bars were unobtainable.
Figure 3.
 
Effect of spatial frequency (SF) on interocular disparities. Blue circles in the left panel are threshold Pdiff and in the right panel Cdiff-phase. Orange circles in the right panel are threshold Cdiff-contrast. Errors bars and markings as in the previous figure (Figure 2).
Figure 3.
 
Effect of spatial frequency (SF) on interocular disparities. Blue circles in the left panel are threshold Pdiff and in the right panel Cdiff-phase. Orange circles in the right panel are threshold Cdiff-contrast. Errors bars and markings as in the previous figure (Figure 2).
Figure 4.
 
Results of Experiment 3 for four observers. Threshold Pdiff is plotted as a function of the standard deviation of the Gabor's Gaussian envelope, for spatial frequencies SF and contrasts C as indicated on each graph. Error bars and associated symbols as in previous figures.
Figure 4.
 
Results of Experiment 3 for four observers. Threshold Pdiff is plotted as a function of the standard deviation of the Gabor's Gaussian envelope, for spatial frequencies SF and contrasts C as indicated on each graph. Error bars and associated symbols as in previous figures.
Figure 5.
 
Results of Experiment 3 comparing Cdiff-phase with Cdiff-contrast for four observers. SF = spatial frequency, C = contrast. As in all previous experiments, data were collected at a stimulus exposure time of 400 ms, expect for the bottom pair of panels where exposure durations were 200 ms and 100 ms as indicated.
Figure 5.
 
Results of Experiment 3 comparing Cdiff-phase with Cdiff-contrast for four observers. SF = spatial frequency, C = contrast. As in all previous experiments, data were collected at a stimulus exposure time of 400 ms, expect for the bottom pair of panels where exposure durations were 200 ms and 100 ms as indicated.
Figure 6.
 
Gabor pairs with different bandwidths/number of cycles. The two members of each pair are in opposite spatial phase, hard to perceive in the top pair but easily perceived in the bottom pair.
Figure 6.
 
Gabor pairs with different bandwidths/number of cycles. The two members of each pair are in opposite spatial phase, hard to perceive in the top pair but easily perceived in the bottom pair.
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