Stereopsis and mean luminance

Abstract

Abstract

Abstract:

Abstract  Stereopsis is dependent on the average level of illumination, especially if it differs between the two eyes. We manipulate the mean luminance seen by both eyes or the interocular difference in mean luminance by using neutral density (ND) filters placed in front of both eyes or just one eye respectively. Stereo acuity was measured using a one temporal interval forced choice task for detecting the sign of a Gaussian depth perturbation in a noise field with a comparable spectrum to that found in natural images. We show that the effect of changing mean luminance is spatial scale independent within the range of 0.5 to 4 cpd, certainly not larger at higher spatial scales. To investigate its origin we manipulate two factors, the temporal synchrony between the two eyes and the interocular contrast. Both factors are implicated in the loss of stereo performance when the mean luminance is different between the eyes, suggesting an underlying explanation in terms of temporal low-pass filtering resulting in the combination of a luminance-dependent temporal delay and a luminance-dependent change in contrast gain.

Introduction

Our ability to perceive small differences in depth of objects depends on the processing of horizontal retinal disparity derived from having two horizontally separated eyes (i.e., stereopsis). For optimal stereopsis, the retinal images in the two eyes should be comparable in terms of spatial detail (Hess, Liu, & Wang, 2003), contrast resolution (Halpern & Blake, 1989; Legge & Gu, 1989), and temporal presentation (Hess & Wilcox, 2006). While it is known that stereopsis is diminished (Livingstone & Hubel, 1994) but still present over a wide range of illumination from photopic to scotopic (Mueller & Lloyd, 1948; Nagel, 1902), what is not known is the extent to which the two eyes' images have to be matched in term of mean luminance. This is surprising since often luminance is not equal in the two eyes in everyday vision due to uneven illumination. 

Figure 1 provides a demonstration of perceived depth derived from stereograms where the mean luminance of both or one stereo pairs differs by a factor of two. The stereo pairs have been computed to have the same contrast values but of reduced mean luminance in both eyes (Figure 1a) or in one eye (Figure 1b). In both conditions, the perceived depth is still apparent and vivid. 

Figure 1

 

Stereopsis is robust over small changes in mean luminance whether they be binocular or monocular. (a) A stereo pair that has the same contrast features (RMS = 0.178) but mean luminance reduced to the same extent (0.26). This simulates what viewing through neutral density (ND) filters of 0.3 would look like if placed over the two eyes. (b) A stereo pair that also has the same contrast features (RMS = 0.178) but where one image differs in mean luminance by a factor of 2 (left 0.26, right 0.5). This simulates what viewing through a ND filter of 0.3 would look like if placed over the left eye.

Previous work that assessed the impact that unequal illumination in the two eyes has on binocular function suggests that large interocular changes in mean luminance (larger than that illustrated in Figure 1) do affect binocular performance in general and stereopsis in particular (Heravian-Shandiz, Douthwaite, & Jenkins, 1991; Katsumi, Tanino, & Hirose, 1986; Li et al., 2012; Zhang, Bobier, Thompson & Hess, 2011). 

General binocular effects have been shown in two studies: Heravian-Shandiz et al. (1991) showed that binocular summation is reduced when the mean illumination differs substantially in the two eyes, and Zhang et al. (2011) showed that large changes in the mean luminance between the two eyes disrupt sensory dominance. Li et al. (2012) showed that reducing the mean luminance in one eye increases binocular suppression and reduces stereoacuity compared with comparable levels of (acuity matched) lens blur. Zhou et al. (2013) also found that decreasing mean luminance in one eye reduces its contribution in binocular combination. This suggests that mean luminance can have a strong influence on stereoscopic processing if it is large enough and that this may not be simply explained as a secondary consequence of the resolution changes that occur. This leaves open the possibility that the effects are due to one or more other factors. 

A number of possibilities can be entertained for any reduction in stereopsis resulting from large interocular differences in mean luminances. Firstly, although changes in mean luminance of the sort we are considering here using neutral density (ND) filters are not accompanied by physical contrast changes, nevertheless spatial contrast sensitivity is itself dependent on mean luminance (Van Nes & Bouman, 1967). If the stimuli are of high spatial frequency or if the mean luminance changes are extreme, then spatial contrast sensitivity is reduced (Hess, 1990; Van Nes & Bouman, 1967), as is perceived contrast (Hess, 1990). If this is the explanation for the reduced stereopsis, stimuli composed of higher spatial frequencies should be more vulnerable to the effects of reduced mean luminance since their thresholds are differentially affected (Van Nes & Bouman, 1967) and the explanation can be couched in terms of secondary interocular contrast differences (Halpern & Blake, 1989, Legge & Gu, 1989). As an alternate explanation, the interocular mean luminance difference could result in delayed processing of the lower intensity image, as has been postulated in the Pulfrich effect (Pulfrich, 1922). An interocular asynchrony would be expected to disrupt binocular processing by virtue of the fact that it would reduce the temporal correlation between the left and right eye stimulation. If this is the case, not only would there be an increase in the optimal temporal asynchrony with increasing ND filter but also any reduction in stereo performance introduced by a reduction in interocular mean luminance should be able to be fully compensated for by a suitable change in the physical interocular synchrony of the presented stereo pair. Finally, the effects could be a result of the mean luminance per se, possibly as a result of an active inhibition emanating from the more dark-adapted eye (Denny, Frumkes, Barris & Eysteinsson, 1991). The stereo effects in this case would be expected to be greater at the lowest spatial scale reflecting a direct DC influence and not able to be compensated for by changing the interocular temporal asynchrony. 

We measured stereoacuity using a stimulus composed of broadband two-dimensional (2-D) fractal noise as well as bandpass versions from low to high scale. Mean luminance was varied by the use of ND filters placed before one or both eyes. These filters, within a dark room, reduce the mean luminance without altering the physical contrast. The results show that any explanation for the loss of stereopsis resulting from large interocular differences in mean luminance needs to consider both temporal (asynchrony) and amplitude (contrast) factors. 

Methods

Observers

Observers included two of the authors and two naive students in the laboratory, all with normal or corrected-to-normal vision. All subjects participated in all experiments and completed all conditions. The study was approved by the Institutional Review Board of McGill University. 

Apparatus

Stimuli and experimental procedures were programmed with Matlab R2010a (MathWorks, Natick, MA) using the Psychophysics toolbox (Brainard, 1997; Kleiner, Brainard & Pelli, 2007; Pelli, 1997). Experiments were run on a PC computer and stimuli were displayed on an Iiyama VisionMaster Pro 513 CRT monitor (Ø51 cm, 1600 × 1200 px, 85 Hz; IIyama, Hoofddorp, The Netherlands) gamma corrected. Mean luminance of the screen was 54 cd/m², experiments were run in a dark room. 

The subject viewed these images with an eight mirror modified Wheatstone stereoscope so that the left image was only seen by the left eye and the right image by the right eye; the resulting viewing distance was 54 cm. Kodak Wratten ND filters (Kodak, Rochester, NY) could be placed in front of left (ND_l) and right eye (ND_r) or both eyes. When the ND filter was placed only over one eye, owing to the consensual reflex, the pupil size only varied about 30% over the range we tested (Bartleson, 1968). A physical delay D_r in the presentation of the right stimulus could be added (see Figure 2a). 

Figure 2

 

Methods. (a) Experimental design. The right and left stimuli were shown independently to each eye through an eight-mirror modified Wheatstone stereoscope. ND filters could be put in front of each eye. (b) Stimuli. Stimuli consisted of 2-D fractal noise stereograms in different frequency bands. A Gaussian target area in the center contained a disparity component. Up-row: broadband stimulus, target is 2°. Middle-row: high-frequency stimulus, target is 0.5°. Down-row: low-frequency stimulus, target is 2°. Stimuli are shown with 100% contrast. (c) Psychometric functions for threshold disparity detection measured with the method of constant stimuli for broadband stimuli for subject JZ. The three sets are measured without or with an ND filter in front of the left eye only: light gray: no filter; mid-gray: 1.7 ND; black: 2.1 ND. (d) Schematic representation of the input from the two eyes. The correlated energy between the two eyes depends on two parameters, the minimal signal amplitude (E) and the timing difference between the two signals (U).

Stimuli

Stimuli were stereo image pairs composed of spatially filtered or unfiltered 2-D fractal noise. Two-dimensional fractal noise was generated by weighting the amplitude spectrum of the uniformly distributed noise by one over spatial frequency (1/f). Stereo acuity at three spatial frequency conditions—broadband, low frequency (bandpass 0.5–1 cpd), and high frequency (bandpass 2–4 cpd)—was measured in the current study. Stimuli contrasts (except where specified in the “Effect of contrast” section; Figure 7) were 100% Michelson contrast. 

Horizontal disparity was introduced by shifting the fractal noise contained within a Gaussian patch at the center of each stereogram. Thus, the central noise test patch was embedded within a surround field containing identical fractal noise that served as a reference for zero disparity. Subpixel resolution was achieved through linear interpolation with graphical OpenGL functions. 

The size of the whole stimulus was 7.9°; the sigma of the Gaussian patch was 0.5° for the high-frequency condition and 2° for the other conditions. The use of a Gaussian patch ensured that the edges of the target were not visible. Examples of unfiltered and filtered stereograms are shown in Figure 2b. 

Procedure

A single stimuli discrimination paradigm (Morgan, Dillenburger, Raphael, & Solomon, 2012) was employed to estimate stereoacuity. In a trial, a pair of stereo images was presented on the screen for 0.4 s. The Gaussian patch at the center of the cyclopean image was perceived either in front of the zero-disparity reference plane or behind it. The subjects' task was to identify the direction of the disparity displacement: inward or outward. Each run consisted of twenty trials (10 crossed and 10 uncrossed) for each of seven disparities (0.16, 0.32, 0.64, 1.27, 2.55, 5.10, and 10.20 arcmin). Three to six runs were performed for each condition, depending on the variability. Audio feedback about the correctness of the response was provided after each trial. 

Data analysis

Data were analyzed offline using Matlab R2012a (MathWorks) using the Psychophysics (Brainard, 1997; Kleiner et al., 2007; Pelli, 1997) and Palamedes (Prins & Kingdom, 2009) toolboxes. 

Psychometric functions of correct responses versus disparity displacement were generated; 82% detection thresholds were determined by fitting a Weibull curve (60 to 120 repetitions per level for each subject) to the psychometric datasets (Figure 2c), and standard deviations (SD) of the thresholds were estimated by a bootstrap procedure. 

Modeling

The data were fitted by a simple model built to describe threshold changes that occurred for the different conditions of stimulation. The response measure was taken as the integrated correlated energy between the two eyes. Consider that the inputs from the two eyes are described by top-hat functions over time, these two responses having different amplitudes depending on signal energy and different onset time depending on luminance asynchrony and actual physical delay. In a simple way, the internal response of the system could be described as the area overlapping the two monocular response functions (Figure 2d). The larger the area, the larger the internal response and hence the lower the detection threshold. 

As supposed by Katsumi et al. (1986), the integrated energy depends on the minimal signal energy coming from the two eyes. We therefore modeled the amplitude of the minimum signal E by a divisive effect of the ND filter (see illustration in Figure 2d):  

 

in which the minimum signal E depends on both luminance and contrast: It decreases when stronger ND filters are used and increases with contrast. The exponential form comes from the fact that ND filters have a log effect on luminance. This energy definition is consistent with the contrast energy defined in Ding and Sperling (2006). A nonlinear gain k is added to the contrast value. 

In our measurements, given the ND filter in front of the left eye is always stronger than the one in front of the right eye, we always have ND_l ≥ ND_r. In most cases the contrast is equal to 1 in the two eyes, the only condition in which contrast is different in the two eyes is when C_l = 1 and C_r = 0.5. In this condition, ND_r = 0. So we always have (ND_l / Display Formula ) ≥ (ND_r / Display Formula ) and we then simplify Equation 1 as:  

 

 

This exponential effect of the ND filters accounts well for the influence of the luminance, as can be verified on the data fits (see Figures 3 and 4; Appendix). 

Figure 3

 

Effect of interocular luminance difference on stereopsis for broadband stimuli. Disparity detection thresholds as a function of monocular (left eye, blue triangles) and binocular (red squares) mean luminance reduction using ND filters. Red dashed and blue solid curves represent model fits for respectively binocular and monocular conditions. (a) Subject JZ. (b) Subject AR. (c) Subject YJK. (d) Subject DW. Error bars represent SD. Complete model parameters are available in the Appendix.

Figure 4

 

Spatial frequency dependence. Disparity detection thresholds as a function of monocular ND filters placed in front of the left eye for three spatial frequency conditions: broadband stimuli (bb, medium blue vertical triangles), high-frequency stimuli (hf, light blue rightward triangles), and low-frequency stimuli (lf, dark blue leftward triangles). (a) Subject JZ. (b) Subject AR. (c) Subject YJK. (d) Subject DW. Error bars represent SD. Solid lines indicate model fits for this condition. Complete model parameters are available in the Appendix.

Figure 5

 

Detection thresholds for monocular (1e, blue) and binocular (2e, red) ND filters averaged over the four subjects at the maximum respective visible condition (see Table 1) for each spatial frequency condition. Error bars represent SD. Asterisks indicate significant differences between the monocular and binocular condition, one-tailed pairwise t test, α = 0.05.

The timing difference parameter U is modeled to account for both the luminance asynchrony U_d and actual physical delay D_r as:  

 

 

It includes two components: a luminance asynchrony U_d = (ND_l – ND_r), describing the luminance-driven asynchrony difference between the two eyes, and a timing one D_r, which corresponds to the compensatory delay that could be added in front of the right eye. This delay would compensate for the luminance difference between the two eyes, where w is the weight of the time difference relative to the luminance difference. Therefore, Equation 3 can be rewritten as:  

 

 

So the two key parameters of this model—the timing difference parameter U and the amplitude of the minimum signal E—and the three stimulus input parameters—the neutral density filters used in front of each eye, ND_l and ND_r, and the physical temporal delay added to the stimulus of the right eye, D_r—jointly describe the effect of the filters on the monocular response functions of the two eyes. 

The model we developed describes the effect of the filters on the threshold response T of the subject. The threshold can be described with a baseline and a luminance-dependent component:  

 

 

The parameter b of the model accounts for the baseline when no ND filters were added on both of the eyes. The luminance dependent component includes a linear parameter a and the timing difference between the two eyes U. The balance parameter U is weighted by a coefficient m and a nonlinear gain g. 

So the final model can account for all the conditions with a single equation and six free parameters by substituting Equations 2 and 4 to 5:  

 

where, to summarize, the inputs are: D_r, the physical delay of the right stimulus, ND_l and ND_r, the neutral density filter intensities in front of the left and right eyes, respectively; and where the six free parameters are b the baseline, a the linear luminance component, m the balance weight, w the delay weight, g the asynchrony nonlinear gain, and k the contrast gain. All the parameters are forced to be positive. 

It is important to note that the model accounts for all the conditions at the same time for a given spatial frequency, with only one set of parameters (reported in Table A1) that fits all the data presented in Figures 3, 6, and 7. Fitting details are provided in the Appendix). 

Figure 6

 

Time delay compensation. Disparity detection thresholds as a function of the delay applied to the stimulus in front of the right eye in the three viewing conditions: no ND filter (0e, green circle symbols), monocular ND filter, ND filter in front of the left eye (1e, blue triangle symbols), and binocular ND filters, ND filter in front of both eyes (2e, red square symbols). Results are shown for the threshold elevation corresponding to the maximum ND filter for each subject. (a) 2.3 ND for subject JZ. (b) 2.6 ND for subject AR. (c) 1.8 ND for subject YJK. (d) 2.1 ND for subject DW. Red dashed, blue solid, and green dotted curves represent model fits for binocular ND filter, monocular ND filter, and no ND filter, respectively. Error bars represent SD. Model parameters are available in the Appendix.

Figure 7

 

The effect of contrast energy. Disparity detection thresholds as a function of monocular ND filters placed in front of the left eye for three different contrasts and contrast balance conditions. Medium blue: Stimulus was 100% contrast in both eyes. Light blue: Stimulus was 50% contrast in both eyes. Purple: Stimulus was 100% in the left eye (the filter eye) and 50% in the right eye. (a) Subject JZ. (b) Subject AR. (c) Subject YJK. (d) Subject DW. Error bars represent SD. Solid lines indicate model fits; model parameters are available in the Appendix.

Results

Interocular difference in luminance

We first investigated the effect of interocular luminance differences on stereopsis for the spatially broadband stimulus. The data in Figure 2c show the full psychometric functions of disparity detection obtained for three different interocular luminance differences for one subject. No filter, 1.7 ND filter, and 2.1 ND filter were put in front of the left eye. We can clearly see a rightward shift of the psychometric function, i.e., an increase in detection threshold with increasing neutral density and hence interocular luminance difference. 

The thresholds obtained as a function of different ND filters placed in front of one (interocular luminance difference) or both eyes (binocular luminance difference) are plotted in Figure 3. We can see from the four subjects' data that the binocular luminance difference generates only a small increment in the threshold. However, when the filter is placed in front of only one eye, the relation of the threshold to the interocular luminance difference stays constant until ∼1.5 ND, and thereafter grows exponentially. This finding indicates that disparity processing is little affected by a binocular change in luminance, but is greatly affected by a luminance mismatch between the two eyes, particularly when it is large. 

Effect of spatial frequency

To assess the influence of stimulus spatial frequency, disparity detection thresholds were measured for bandpass stimuli containing either high or low spatial frequencies. Similar to the measurement for the broadband stimuli, the thresholds measured for each subject for interocular luminance differences were obtained by placing ND filters in front of the left eye, and these data are shown in Figure 4. It is immediately obvious that there is a comparable loss in performance for the high-frequency and broadband stimuli. Furthermore, at smaller neutral density values, these two conditions represent much lower thresholds than that for the low-frequency condition (see parameter b in the Table A1, Appendix), suggesting that the high-frequency components dominate the low-frequency ones for minimal disparity detection in broadband images. In addition, we observe that the threshold loss with increasing ND filter is, to a first approximation, similar at the different spatial scales, certainly not greater for the high spatial frequency stimulus (see parameters m and g of the model in Table A1, Appendix). 

The monocular ND filter that produces the largest visible elevation of disparity threshold for the four subjects is lower (see values reported in Table 1) for the low-frequency condition compared with the other two conditions (broadband and high-frequency conditions), but it is also true that stereopsis is worse for the low spatial frequency stimulus at 0 ND. Thus, while the detectability of high-frequency spatial components is more affected by a given luminance reduction compared with that of low-frequency components (Van Nes & Bouman, 1967), disparity processing of high-pass images is not differentially affected by reductions in luminance. The thresholds measured for monocular and binocular ND filters at these maximum visible values are reported in Figure 5. It shows that thresholds are significantly (α = 0.05, one tailed pairwise t test) lower for binocular compared with interocular luminance changes for each spatial frequency condition. 

Table 1

 

The monocular neutral density that produced the largest (or almost largest) visible elevation of disparity threshold for each subject and each spatial frequency. Note: These neutral density values were used to establish the comparison between monocular and binocular conditions in Figures 5 and 6.

	JZ	AR	YJK	DW
bb	2.3	2.6	1.8	2.1
hf	2.3	2.6	1.6	2.0
lf	1.6	2.6	0.8	0.8

Effect of time delay

To investigate the potential delaying effect produced by ND filters, we tried to compensate for this delay by adding a physical delay in the presentation of the right stimulus. Figure 6 represents detection thresholds as a function of the physical delay introduced for the stimulus presented to the right eye for three viewing conditions: without ND filter, with an ND filter in front of the left eye, or with ND filters in front of both eyes. ND filters were chosen to correspond to maximum threshold elevation within the range of our measurements (2.3 ND for JZ, 2.6 ND for AR, 1.8 ND for YJK, and 2.1 ND for DW; see Table 1). 

A first observation is that, in the studied conditions, adding a temporal delay does not affect the detection threshold when ND = 0 (green symbols). Indeed, our stimuli are 400 ms long, so in this condition, with maximum delay of 210 ms, there is still 190 ms of overlap between the two eyes; this duration has been shown to be sufficient for the processing of stereo (Hess & Wilcox, 2006) 

However, when both eyes are seeing the stimulus through a same ND filter, the threshold is increased by a certain amount and then increases monotonically with the delay (red symbols). Most interestingly, when the ND filter is placed in front of the left eye only (blue symbols), a biphasic behavior is observed. Adding a small delay decreases the detection threshold to a minimum D₀ = ND_l / w for all subjects (respectively 70, 106, 107, and 104 ms; Figure 6a through d) and then increases it back, with the same shape, to the values corresponding to the binocular ND filter condition (all the conditions are fitted by the model at the same time). 

The model doesn't capture perfectly the monocularly ND filter data (interocular luminance difference) before the trough, although given the number of conditions and data compared to the number of free parameters, the fit is encouraging. Any departure of data from model is because the model imposes a symmetrical behavior before and after the trough, whereas the data suggest that the slope might be stronger before the trough. 

It is noteworthy that this minimal level reached corresponds to that found for the binocular ND filter condition, but is not as low as that found without filters (ND = 0). Thus, this behavior indicates that the ND filter induces a delay in the system that can be partly compensated for by adding a physical delay in the stimulus presented to the unfiltered eye but that this is not a complete explanation, as even with the optimal compensating delay, sensitivity has not returned to the baseline condition (ND = 0). There must be an additional luminance-dependent factor. 

Effect of contrast

One possibility is that luminance not only induces a delay but also reduces the amplitude of neural responses. To test this, we measured the disparity threshold with combinations of different contrasts and ND filters (Figure 7). Using a 50% contrast, an ND filter applied monocularly for all subjects produces a larger elevation of the thresholds (downward triangles, light blue) compared to a stimulus at 100% contrast (upward triangles, dark blue). This elevation can be compensated for by having a contrast of 100% in both eyes or more interestingly just in front of the eye with the ND filter (hexagrams, purple). Under this condition, and as suggested by our model, the thresholds are equivalent to the 100% condition in the two eyes, and the model fits are then identical. These finding suggest that ND filters may result in reductions of the neural signal amplitude (i.e., contrast gain) and in particular the relative left/right eye amplitudes. 

It is noteworthy that, as shown in Figure 6 for temporal differences, this contrast recovery is also not a complete explanation as the threshold doesn't completely return to its baseline. 

Discussion

Stereoacuity is best under photopic conditions and is reduced under mesopic and scotopic conditions (Livingstone & Hubel, 1994; Mueller & Lloyd, 1948; Nagel, 1902). This dependence is unlikely to be solely the result of either changes in spatial resolution or changes in spatial contrast sensitivity as the reduction is comparable at high and low scales, and yet a given reduction in luminance affects the detectability of fine spatial scales much more than coarse scales (Van Nes & Bouman, 1967). 

Larger changes in stereoacuity are found when the mean luminance is reduced in one eye only. This change is also no greater at the fine scales compared with coarse scales and is therefore unlikely to be a secondary consequence of changes in resolution and/or spatial contrast sensitivity (Georgeson & Sullivan, 1975; Van Nes & Bouman, 1967). The strong influence that interocular differences in mean luminance have on binocular function is well known. Such interocular differences alter ocular dominance (Zhang et al., 2011; Zhou et al., 2013), binocular summation (Heravian-Shandiz et al., 1991; Katsumi et al., 1986), and stereopsis (Li et al., 2012). The effects on stereopsis have been shown to be much greater than one would expect from the associated changes in resolution (Li et al., 2012), supporting the present claim that the loss of stereopsis under such conditions is not simply predicted by the associated resolution changes. 

One proposal that could potentially account for the additional influence an interocular difference in mean luminance has on stereopsis is that of a luminance-dependent temporal delay. Such a delay to one eye would result in an interocular temporal asychroncy that would be likely to affect stereoscopic processing detrimentally. The Pulfrich effect whereby a pendulum swinging in the frontoparallel plane is perceived to rotate in depth when viewed through an ND filter over one eye is currently understood in terms of an induced temporal delay. Visual evoked potential (VEP) studies on humans (Heravian-Shandiz et al., 1991; Katsumi et al., 1986) and single cell recording in monkeys have demonstrated that such delays are induced by reductions in mean luminance (Geisler, Albrecht, & Crane, 2007) and that they would be expected to affect binocular summation and disparity tuning (Cumming & Read, 2005). We also show that by manipulating the interocular asynchrony, the stereo deficits that result from interocular differences in mean luminance can be greatly reduced, suggesting a major component to the stereo loss is consistent with a pure delay. However, this is not the whole story because the stereo deficit resulting from an interocular difference in mean luminance could not be completely compensated for by manipulating the temporal asynchrony. There remains a factor of two loss in stereopsis that is due to an additional mean luminance-dependent effect separate from a delay. A similar loss is seen for bilateral differences in mean luminance also suggesting that there is an additional factor that is primarily dependent on light level. 

It is possible because this is observed at a mesopic luminance (neutral density from 1.8 to 2.6), that the temporal delay that we observe could be due to a transition from cone to rod dynamics. However, on the basis of previous studies (Mueller & Lloyd, 1948; Livingstone & Hubel, 1994), this seems unlikely. The rod-cone transition in stereoscopic sensitivity occurs around 0.02 cd/m² (Mueller & Lloyd, 1948) to 0.03 cd/m² (Livingstone & Hubel, 1994), which is an order of magnitude lower than the luminances (0.43–0.18 cd/m²) at which we observe temporal delays in our experiments, suggesting that the temporal delays observed here are a property of the cone system. 

Returning to the issue of the additional luminance dependent loss of a factor of two that could not be compensated for with a stimulus temporal delay, data from single cell physiology have shown that ND filters also reduce contrast gain (Geisler et al., 2007; Hess, 1990) and response duration (Geisler et al., 2007). Such changes would be expected to reduce interocular correlation (see Figure 2d) and to result in additional losses for stereoscopic functions for monocular or binocular ND filters. We show that by manipulating the contrast in the eye viewing through the ND filter that some, but not all, of the loss in stereopsis observed under the low luminance conditions (interocular luminance difference) can be recovered. This suggests that signal amplitude may also be reduced as a consequence of ND filters, though this is not simply a consequence of elevated spatial contrast detectability, which is strongly scale-dependent, a feature not reflected in the stereoscopic effects reported here. 

Our model provides some insight into the possible effects of ND filters at a theoretical rather than physiological level. The model assumes that the degree of interocular correlation determines stereo thresholds, and the model fits are encouraging based on the idea that reduced mean luminance alters the temporal shape of the neural response producing a reduction in amplitude and delay. However, the exact nature of the luminance-dependent temporal changes are unclear because similar fits can be obtained by a range of different temporal changes ranging from low-pass filtering, temporal shape changes to changes in temporal duration. These modeling predictions are illustrated in Figure 8 and should be compared to the data presented in Figure 6. While we can rule out that a pure temporal delay is solely responsible (Figure 8a), given that the temporal delay could not fully recover baseline sensitivity, the other examples of changes in the temporal aspect of the neural response all produce comparable changes in interocular correlation, whether it be a delay and amplitude reduction (Figure 8b) or a delay and response shortening (Figure 8c). The most parsimonious explanation is that reduced luminance produces a low-pass temporal filtering of the neural response that in turn results in secondary amplitude reduction and temporal delay that are particularly disruptive for stereopsis if they occur monocularly (Figure 8d). Reduced luminance is known to result in a low-passing both spatially and temporally (van Nes, Koenderink, Nas, & Bouman, 1967) and we suggest that it is the temporal rather than the spatial changes that impact stereopsis. 

Figure 8

 

Filtering simulations illustrating putative effects of ND filters on signal and thresholds. For each panel on the left is represented the temporal signal shape of the stimulus (black) and the putative shape of the signal after ND filtering (blue). On the right are presented the simulated threshold curves as a function of the delay applied to the right stimulus (same format as Figure 5) with no ND filter (green dotted line), interocular luminance difference, ND filter in front of the left eye (blue continuous line), and binocular luminance difference, ND filter in front of both eyes (red dashed line). The delay is supposed to have an effect only when luminance is decreased; therefore, the no ND filter baseline condition is flat (green dotted line). (a) pure delay effect. (b) delay and amplitude reduction. (c) delay and shortening of the response. (d) low-pass temporal filtering.

Conclusions

Changes in stereoacuity are observed when there are large changes in mean luminance, especially if these affect only one eye. These changes are no greater at fine compared to coarse spatial scales. Therefore, they are unlikely to be a secondary consequence of changes in resolution and/or spatial contrast sensitivity although our results suggest that signal amplitude may be reduced as a consequence of ND filters. On the other hand, we also show that by manipulating the interocular asynchrony, a major component to the stereo loss is consistent with a pure delay. We conclude that reduced luminance mainly results in a temporal low-passing which strongly impacts stereopsis for naturalistic images of the type used here. 

Acknowledgments

This work was supported by an NSERC grant (#46528) to RFH. We are grateful to Bruce Hansen for image processing (Figure 1) and our naive subjects Yeon Jin Kim (YJK) and Danny Wang (DW). 

Commercial relationships: none. 

Corresponding author: Jiawei Zhou. 

Email: [e-mail]. 

Address: McGill Vision Research, Department of Ophthalmology, McGill University, Quebec, Canada. 

Appendix: Model fitting

The model is able to capture all the luminance, contrast, and time delay conditions used here. It fits all these conditions (data presented in Figures 3, 6, and 7) at the same time for each subject at each spatial frequency, with only one set of parameters (reported in Table A1). 

To better illustrate this point, the Figure A1 shows the individual data and model fits for one example subject in a three-dimensional (3-D) space. The thresholds are represented in function of interocular delay on the x-axis (data from Figure 3) and interocular neutral density on the y-axis (data from Figure 6). The model surface fit represents a continuum in this parameter space. The “contrast” and “two eyes” conditions were also fit at the same time but could not be represented here. 

The six free parameters estimated values are presented for each condition and subject in Table A1. The baseline b, linear luminance effect a, balance weight m, and nonlinear gain g parameters (see Methods, Equation 6) are presented for each condition. The temporal w and contrast k coefficients are only used in timing and contrast experiment for broadband spatial frequency conditions. 

Table A1

 

Model fitting. Notes: Parameter values, see Equation 6, coefficient of determination R², and degrees of freedom df. Data are shown for each spatial frequency condition: broadband (bb), high frequency (hf), and low frequency (lf), for each subject (JZ, AR, YJK, and DW).

	JZ			AR			YJK			DW
	bb	hf	lf	bb	hf	lf	bb	hf	lf	bb	hf	lf
b	2.234	3.234	4.234	1.617	0.728	2.341	1.992	2.245	4.953	4.084	1.228	4.845
m	0.007	0.001	0.062	0.001	4.1356E-06	0.009	0.013	0.096	0.210	0.007	0.052	0.650
a	0.618	1.318	2.04E-10	2.119	2172.000	0.325	0.733	0.163	0.457	1.374	1.088	0.151
g	0.391	2.213	9.92E-11	0.668	3.284	9.23E-12	0.536	3.1001E-10	7.945E-11	1.043	1.6E-10	1.051
w	0.033	NA	NA	0.025	NA	NA	0.017	NA	NA	0.020	NA	NA
k	0.482	NA	NA	0.428	NA	NA	0.180	NA	NA	0.215	NA	NA
R2	0.919	0.954	0.804	0.851	0.951	0.887	0.704	0.954	0.764	0.754	0.990	1.000
df	41.000	9.000	6.000	49.000	10.000	11.000	48.000	5.000	8.000	44.000	1.000	0.000

Figure A1

 

Measured thresholds and model fit for different delay and monocular neutral density conditions for subject JZ, broadband spatial frequency. The thresholds are represented in a 3-D space in function of the delay of the right stimulus on the x-axis, and the neutral density filter in front of the left stimulus on the y-axis. Error bars represent SD. The surface represents the model fit as a continuum in this parameter space.

Fitting procedure was nonlinear least-squares regression, with an average coefficient of determination R² of 0.88 and additive weight on the trough of the time parameter (see Figure 6). Therefore, this value is not very high because we intended to provide a very simple model with very few parameters to account for all the conditions, so with a high number of degrees of freedom (df). Individual values are reported in Table A1. R² values are better for high- and low-frequency conditions compared to broadband because there were much fewer datapoints (and therefore df) in these conditions given there was no delay and contrast parameters.