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Research Article  |   December 2004
Stereomotion speed perception: Contributions from both changing disparity and interocular velocity difference over a range of relative disparities
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Journal of Vision December 2004, Vol.4, 6. doi:10.1167/4.12.6
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      Kevin R. Brooks, Leland S. Stone; Stereomotion speed perception: Contributions from both changing disparity and interocular velocity difference over a range of relative disparities. Journal of Vision 2004;4(12):6. doi: 10.1167/4.12.6.

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

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Abstract

The role of two binocular cues to motion in depth—changing disparity (CD) and interocular velocity difference (IOVD)—was investigated by measuring stereomotion speed discrimination and static disparity discrimination performance (stereoacuity). Speed discrimination thresholds were assessed both for random dot stereograms (RDS), and for their temporally uncorrelated equivalents, dynamic random dot stereograms (DRDS), at relative disparity pedestals of −19, 0, and +19 arcmin. While RDS stimuli contain both CD and IOVD cues, DRDS stimuli carry only CD information. On average, thresholds were a factor of 1.7 higher for DRDS than for RDS stimuli with no clear effect of relative disparity pedestal. Results were similar for approaching and receding targets. Variations in stimulus duration had no significant effect on thresholds, and there was no observed correlation between stimulus displacement and perceived speed, confirming that subjects responded to stimulus speed in each condition. Stereoacuity was equally good for our RDS and DRDS stimuli, showing that the difference in stereomotion speed discrimination performance for these stimuli was not due to any difference in the precision of the disparity cue. In addition, when we altered stereomotion stimulus trajectory by independently manipulating the speeds and directions of its monocular half-images, perceived stereomotion speed remained accurate. This finding is inconsistent with response strategies based on properties of either monocular half-image motion, or any ad hoc combination of the monocular speeds. We conclude that although subjects are able to discriminate stereomotion speed reliably on the basis of CD information alone, IOVD provides a precise additional cue to stereomotion speed perception.

Introduction
There are at least two binocular (or stereomotion) cues to the motion of an object whose depth changes over time. First, the retinal disparity of the moving object, relative to stationary features, changes over time. To use this changing disparity (CD) signal to determine stereomotion speed, positional information must first be combined from both monocular images to establish a disparity signal, which can then be differentiated over time to yield the speed of the object in depth (Figure 1a). Second, each monocular image moves at a different velocity, providing a second potential source of information, known as the interocular velocity difference (IOVD) cue. To use this signal, two separate and independent monocular velocity signals must first be derived, and subsequently combined vectorially across the two eyes (Figure 1b). 
Figure 1
 
Two mechanisms for stereomotion speed perception. a. The differential of stimulus disparity with respect to time yields the changing disparity (CD) cue. b. The vectorial combination of two monocular velocity signals yields the interocular velocity difference (IOVD) cue.
Figure 1
 
Two mechanisms for stereomotion speed perception. a. The differential of stimulus disparity with respect to time yields the changing disparity (CD) cue. b. The vectorial combination of two monocular velocity signals yields the interocular velocity difference (IOVD) cue.
Three experiments were performed to assess how IOVD and CD influence the precision of stereomotion speed judgments, while controlling for stimulus visibility, the precision of static disparity inputs to the CD cue, the displacement of individual stimuli, and the motion of individual monocular half-images. In the first experiment, we compare stereomotion speed discrimination thresholds for stimuli either containing or lacking the IOVD cue at three relative disparity pedestals, finding that when the IOVD cue is included, precision is enhanced. This difference is preserved across relative disparity pedestals, with subjects showing no evidence of responding on the basis of stimulus displacement for any condition. The finding of equivalent stereoacuity for the two stimulus types in Experiment 2 assures that this improvement in precision cannot be accounted for by any difference in the precision of the underlying disparity signal. Finally, by independently varying the relative directions and speeds of monocular half-image motions, Experiment 3 shows that our observers did not respond on the basis of the speed of monocular half-images or on any ad hoc combination of the monocular half-image speeds. Our results are consistent with the notion that the RDS improvement in the precision of stereomotion speed perception is related to the difference of velocities between the two eyes, or IOVD. 
Experiment 1: Stereomotion speed discrimination for RDS and DRDS stimuli
Though Rashbass and Westheimer first pointed out the existence of IOVD and CD as distinct cues in 1961, as they always correspond perfectly in natural stimuli, their relative contributions to the perception of stereomotion have, until recently, remained obscure. However, more modern stimulus generation technology has allowed them to be isolated in the laboratory. The random dot stereogram (RDS), first established as a tool for investigations of static depth perception, has also served well in the field of stereomotion. Static RDSs are said to be cyclopean in the sense that no object features (except the dots themselves) are visible in either monocular view. However, when binocularly fused, the depth of a central object (comprising many dots) can be perceived through binocular disparity. As with any natural stimulus, when this central object undergoes a change of disparity, simulating motion directly toward or away from an observer, its monocular half-images translate at different velocities, and hence the stimulus provides both the CD and IOVD cues to stereomotion. This RDS motion in depth stimulus is however non-cyclopean, because the motion of the central object breaks the camouflage of the surrounding stationary dots. The central object can thus be detected even in one monocular image. However, if we maintain the same rate of CD, but replace both of the entire monocular dot arrays on each frame with an independent (but binocularly correlated) pair, the stimulus is devoid of any coherent monocular motion, either in the central object or the surround. Though the CD cue is identical to that in the RDS, this stimulus, referred to as a dynamic RDS (or DRDS) lacks any IOVD information. The DRDS can also be genuinely described as cyclopean, because the central stimulus object is once again entirely monocularly camouflaged against a background of dots that are also dynamic. It is also possible to create a stimulus containing the IOVD cue but lacking CD, by providing entirely spatially independent long-lifetime images to the left and right eyes, and ensuring that they move at the appropriate velocities. We will refer to this third stimulus as an uncorrelated RDS (or URDS).1 
Several studies have employed such stimuli to investigate the mechanisms of stereomotion speed perception, with disparate results. Harris and Watamaniuk (1995) showed that speed discrimination thresholds are higher for DRDS stimuli moving in depth compared to RDS stimuli, suggesting the importance of the IOVD cue. However, Portfors-Yeomans and Regan (1996; also see Portfors and Regan, 1997) reported no difference in thresholds when comparing “cyclopean” (DRDS) with equivalent “non-cyclopean” stimuli.2 The discrepancy between these studies may be due to any of a number of differences in experimental details. 
The first issue that has made interpretation difficult is the transient invisibility of some of the stimuli. Because all of Harris and Watamaniuk’s (1995) stimuli passed through the plane of the surrounding pattern, the DRDS (whose central stereomotion object was defined purely by its disparity relative to a dynamic surround) became momentarily invisible. As the object neared the region of zero relative disparity, its disparity difference with the surround dropped below disparity detection threshold, at which point the boundaries of the central object disappeared, only to reappear when it exceeded detection threshold on the other side of the surround pattern’s disparity plane. The RDS pattern, however, whose central feature can be detected monocularly due to motion cues, did not suffer this problem. Portfors-Yeomans and Regan (1996) claim that this difference in detectability explains subjects’ inferior performance for DRDS stimuli in the Harris and Watamaniuk (1995) study. By conducting a similar experiment at a disparity pedestal, Portfors-Yeomans and Regan (1996) provided stimuli whose visibility was never compromised, and found equivalent performance for their “cyclopean” (DRDS) and “non-cyclopean” stimuli. However, it is not clear whether the crucial difference between the studies is the visibility difference between stimuli per se, or the antecedent difference in disparity pedestals. 
This second potential reason for the discrepancies between previous studies—that there might be an effect of the relative disparity pedestal itself—must also be addressed. Brooks (2002a) investigated the issue of the relative cue contributions to stereomotion speed perception at two disparity pedestals with a motion adaptation paradigm using stimuli whose stereomotion object was visible throughout the presentation. After adapting to either an RDS (CD and IOVD adaptation) or URDS (IOVD adaptation only) stimulus featuring horizontal monocular motion that differed in direction for each eye, a subsequently seen RDS at the same mean disparity as the RDS adaptation stimuli appeared to move at a reduced rate, implying use of the IOVD cue. Though the sizes of the aftereffects for both adaptation stimuli were equal for stimuli near to the fixation plane, they were larger for RDS compared to URDS adaptation when stimuli appeared at a disparity pedestal. It was hypothesized that although the IOVD cue is employed for stimuli at all disparities, the CD cue has a significant influence only when stimuli are away from zero relative disparity. This hypothesis can reconcile the earlier discrepancy, as Harris and Watamaniuk’s (1995) stimuli passed through zero relative disparity, whereas Portfors-Yeomans and Regan’s (1996) moved at a disparity pedestal. For stimuli moving near zero relative disparity, we would expect lower speed discrimination thresholds for RDS than for DRDS stimuli, whereas for stimuli at a disparity pedestal the difference should be far smaller, perhaps even non-existent. 
Another critical issue is the possibility that subjects might perform the task using displacement and not speed information, and might do so to different extents in different conditions. Harris and Watamaniuk (1995) observed that for the DRDS stimuli in their experiment, whose durations were randomized, “observers tended to choose the longer duration stimuli as ‘faster’, and the shorter duration stimuli as ‘slower’. This suggests that observers were using a position or static disparity cue as well as, or instead of, a speed cue,” because the longer duration stimuli for a given speed would entail a greater displacement, and a greater initial and final disparity. However, Portfors-Yeomans and Regan showed that subjects were able to ignore systematic variations in displacement for stimuli not traversing the zero relative disparity plane, and base their speed discriminations on the task-relevant variable alone. This suggests that for such stimuli there exists a specialized speed-sensitive CD mechanism. However, due to the visibility issues in the Harris and Watamaniuk (1995) stimuli, it remains to be seen whether such a mechanism operates for stimuli near the zero relative disparity plane whose visibility is not compromised. 
In Experiment 1, we investigate the possibility that the relative contributions of the IOVD and the CD cue are a function of relative disparity by measuring stereomotion speed discrimination thresholds or just noticeable differences (JNDs) for RDS or DRDS stimuli at disparity pedestals of −19, 0, and +9 arcmin. We use a novel variant of the DRDS stimulus featuring a stationary surround to the stereomotion target, ensuring that no stimulus suffers any loss of visibility at any point. In addition, we explicitly assess the extent to which subjects are able to ignore systematic variations in stimulus duration, and hence displacement, and base their results instead on a percept of stereomotion speed per se. 
Methods
Apparatus and stimuli
Stereoscopic stimuli were created by alternately displaying the monocular half-images on an Image Systems 240 Hz monitor (120 Hz per eye) using P46 fast phosphor, driven by a Matrox G400 video card. These were viewed through high-speed (switching time 50 μs), high-transmittance (30%) ferro-electric shutter glasses also running at 240 Hz synchronized to the vertical refresh of the monitor. Pilot tests confirmed that at the contrast levels used, there was no perceptible flicker or bleed-through of the unwanted monocular image. 
The visible area of the screen subtended 7.3 (H) × 6.2 (V) deg at the viewing distance of 2.5 m. The mean luminance of the screen was 12.5 cd/m2, and all tests took place in a room devoid of extraneous light. Responses were recorded from a two-button mouse. Subjects wore their best optical corrections for all experimental sessions. 
In each stereo half-image, identical and stationary surround patterns comprised 50% density bright/dark dots at a Michelson contrast of 99.7%, each subtending 2.5 × 3.7 arcmin (4 pixels square). Each surround half-image comprised two 7.3 × 0.66 deg strips centered 1 deg above and below a small, bright, central fixation ring, which was extinguished during stimulus presentation. These features were in identical positions in each stereo half-image, and hence were located binocularly in the fixation plane (see Figure 2). Target stimuli were also random dot patterns (same size, density, and contrast as the surround), which, when visible, entirely filled the vertical gap between the two background strips, and extended horizontally to the edges of the image (7.3 × 1.3 deg) to minimize any effects of monocular half-occlusion, or stereo from motion-defined boundaries (e.g., Lee, 1970). For RDS stimuli, the monocular half-images moved in opposite directions, at a variety of speeds to simulate the appropriate IOVD and CD. The speed of the two monocular half-images in any single stimulus presentation was always equal to simulate directly receding or approaching motion in depth. Dots were always moved either zero or one pixel between frames (we therefore did not attempt to adjust dot location to reflect the 4-ms asynchrony between the frames). No anti-aliasing of stimuli was necessary for subjects to perceive smooth linear motion in all stimulus examples. Although moving RDS patterns remained unchanged throughout their duration, DRDS stimuli featured an entirely novel random array of binocularly correlated elements at the appropriate disparity in each stereo frame (i.e., at 120 Hz) to generate identical CD information without any IOVD. 
Figure 2
 
General stimulus arrangement in dimetric projection. Figure not to scale. See text for detailed parameters.
Figure 2
 
General stimulus arrangement in dimetric projection. Figure not to scale. See text for detailed parameters.
Observers
A total of six subjects contributed data, each of whom had normal or corrected-to-normal vision, and passed preliminary screening tests for stereo deficiencies. Except for one subject (the author LS), all participants were naïve as to the purpose of the experiment. Only one subject (PE) was inexperienced at performing psychophysical tasks before beginning practice sessions. 
Design and procedure
We used an adaptive staircase method to determine the threshold for stereomotion speed discrimination in a three-factor repeated-measures design. The two principal factors were stimulus type (RDS or DRDS), and relative disparity pedestal (−19, 0, or +19 arcmin). The value of the pedestal represents the relative disparity of the stimulus at the mid-point of its motion with respect to the background plane. A third factor, test stimulus duration, was also included to control for responses made on the basis of displacement, as opposed to speed per se. 
After practice, subjects completed either 6 (subjects AK, CN, SS, and PE) or 12 (subjects LS and BB) sessions of testing. These sessions consisted of 6 blocks (2 stimulus types and 3 disparity pedestals), each of which lasted approximately 8 arcmin. The order of presentation of blocks was counterbalanced across sessions. Each block of trials comprised 4 randomly interleaved up-down staircases, which were terminated after 12 reversals. We used a two-interval forced-choice (2IFC) procedure, sequentially presenting the subject with a receding standard and test stimulus in random order, and requiring them to indicate in which interval (the first or the second) the stimulus appeared to move with greater speed. The rate of change of disparity (equal to the difference in monocular velocities for RDSs) of the standard stimulus was always at 0.62 deg/s, while the staircase determined the speed of the test stimulus from a set of 9 possible values: 0.25, 0.31, 0.41, 0.50, 0.62, 0.75, 0.83, 0.93, and 1 deg/s. For fixed duration conditions, the test and standard stimuli lasted 600 ms with an inter-stimulus of 1500 ms. While test or standard stimuli were visible, fixation was unconstrained.3 In variable duration trials, the duration of the test stimulus was 500, 550, 650, or 700 ms in the respective 4 interleaved staircases. Pedestals, speeds, and durations were chosen to prevent substantial diplopia at the beginnings and ends of the stimulus presentations, and to ensure that the stimuli at non-zero pedestals did not traverse the zero relative disparity plane. Data were collected in separate variable duration and fixed duration sessions. On any given day, subjects were unaware of whether they received fixed or variable duration stimuli. Last, two subjects (AK and LS) were also tested with stimuli that approached as opposed to receded. Details of this task were identical to those described above in all other respects. 
Data analysis
Data were analyzed for each subject individually. Responses from trials in each of the 4 interleaves within each block were combined and a cumulative Gaussian curve was fitted to the data by probit analysis (Finney, 1971). This yielded a SD of the underlying distribution for each condition. JNDs, the semi-interquartile distances, were then derived and the Weber fraction computed from this parameter. Unless stated otherwise, the point of subjective equality (PSE) was assumed to be unbiased, and was not a free parameter in the curve fit. This was repeated for each session of testing to allow mean thresholds and associated SEs across sessions to be calculated. Statistical significance was assessed using repeated measures ANOVAs, and two planned contrasts. These were employed to specifically address the hypothesis that there might be a larger difference between RDS and DRDS JNDs at the zero relative disparity pedestal than at both the crossed and the uncrossed relative disparity pedestals. 
The effect of duration was analyzed in two ways. In the first analysis, fixed duration raw data were combined within each stimulus type and disparity pedestal condition, and fit with Gaussian curves, as described above. The same procedure was followed for variable duration data. If subjects’ decisions were influenced by the unhelpful displacement information, one would expect higher thresholds for variable duration conditions. However, if subjects made judgments purely on the basis of perceived stimulus speed, thresholds in the two conditions should be the same. In the second analysis, curve fitting was performed for each duration condition separately with both curve SD and PSE as free parameters. Raw data collected for the variable duration condition were combined across sessions within each of the test durations, to yield four psychometric functions for each stimulus type and relative disparity pedestal. The PSEs for each of these curves were plotted for each condition as a function of test duration, along with fixed duration data (600-ms duration). A linear regression between PSEs and stimulus duration was performed for each plot, and the slopes assessed. If subjects respond on the basis of stimulus displacement, there should be a clear correlation between PSE and stimulus duration. 
Results and discussion
Results showing mean speed discrimination Weber fractions (+SEM) as a function of relative disparity pedestal can be seen in Figure 3 for all six observers. Thresholds for RDS stimuli are lower than those for DRDS stimuli in all but two of 18 instances (negative pedestal for subject SS, positive pedestal for subject BB). Despite small individual differences, there is a clear effect of stimulus type (RDS vs. DRDS), but no obvious general effect of relative disparity pedestal. In line with these informal observations, the results of 2 × 3 repeated measures ANOVAs revealed a statistically significant effect of stimulus type for all subjects [LS: F(1,11) = 62.52, p < .0001; AK: F(1,5) = 21.22, p = .0058; SS: F(1,5) = 8.41, p = .0338; PE: F(1,5) = 35.07, p = .002; CN: F(1,5) = 59.18, p = .0006] except BB, for whom the effect was borderline [F(1,11) = 4.252, p = .064]. Though the planned contrast analyses showed a significant effect for one comparison in subjects BB and LS [BB: negative pedestal, F(1,11) = 20.476, p = .001; and LS: positive pedestal, F(1,11) = 7.114, p = .022], for LS this effect was in the opposite direction to that hypothesized. No subjects showed statistically significant differences in the RDS/DRDS advantage between zero and both pedestal conditions. In general, the results shown in Figure 3 demonstrate that stereomotion speed discrimination performance for RDS stimuli is superior compared to that for DRDS stimuli, and that relative disparity pedestal has no systematic effect. 
Figure 3
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions for receding RDS and DRDS motion plotted as a function of relative disparity pedestal for all six observers. Error bars represent +1 SEM.
Figure 3
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions for receding RDS and DRDS motion plotted as a function of relative disparity pedestal for all six observers. Error bars represent +1 SEM.
Control for displacement
Figure 4 plots the mean threshold (±SEM) for the fixed duration condition against the mean variable duration thresholds for all subjects. From simple inspection, it is clear that thresholds for the two conditions are similar and generally lie along a line of slope 1 and intercept zero, indicated by the dashed line. Constraining the intercept to be zero, a line of best fit has a slope of 0.967 (r2 = 0.36, p < .001). This analysis shows that thresholds are not systematically higher for variable duration stimuli. 
Figure 4
 
Speed discrimination thresholds for fixed duration trials versus variable duration trials for all conditions and subjects. The dotted line represents identical performance on each. Error bars represent +1 SEM.
Figure 4
 
Speed discrimination thresholds for fixed duration trials versus variable duration trials for all conditions and subjects. The dotted line represents identical performance on each. Error bars represent +1 SEM.
Figure 5 plots the PSE data as a function of test duration. Regression analyses were performed on each of these data sets. The negative slope of the bold grey curve represents predicted performance for observers responding only on the basis of stimulus displacement. It should be noted that performance predicted by the displacement response is not quite linear. It is, in fact, a small section of a hyperbolic function, with a correlation coefficient of −0.995 and a slope of −1.06. Horizontal lines intercepting the y-axis at 0.62 deg/s would indicate veridical performance. Of the 36 data sets fit by linear regression, only 4 showed a significant correlation at the uncorrected α = 0.05 level. Of these, 3 lines showed a positive slope, while the other (subject AK, DRDS −19, p = .0499) had a slope of −0.35. This analysis, together with the preceding threshold analysis, demonstrates that none of our subjects’ responses were systematically influenced either by stimulus duration, or by the consequent variations in displacement, or the initial/final static depth. 
Figure 5
 
Stereomotion speed discrimination PSEs plotted as a function of stimulus duration for all conditions and subjects. The bold grey line represents predicted performance based entirely on stimulus displacement.
Figure 5
 
Stereomotion speed discrimination PSEs plotted as a function of stimulus duration for all conditions and subjects. The bold grey line represents predicted performance based entirely on stimulus displacement.
Approaching stimuli
For approaching stimuli, subjects LS and AK show an unambiguous pattern of results, as can be seen in Figure 6. Both subjects show systematically higher thresholds for DRDS stimuli, a fact which is reflected in a statistically significant main effect [LS: F(1,10) = 11.44, p = .0196; AK: F(1,10) = 27.40, p = .0034], while all other effects failed to reach significance. Like Figure 3, Figure 6 shows that subjects exhibit superior performance for RDS stimuli, with no apparent influence of relative disparity pedestal. Interestingly, neither subject showed a statistically significant difference between thresholds when comparing approaching and receding motion for either stimulus type, across relative disparity pedestals. 
Figure 6
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions, for approaching stimuli plotted as a function of relative disparity pedestal for all conditions and two subjects. Error bars represent +1 SEM.
Figure 6
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions, for approaching stimuli plotted as a function of relative disparity pedestal for all conditions and two subjects. Error bars represent +1 SEM.
Experiment 2: Static disparity discrimination for RDS and DRDS stimuli
The relative precision of the disparity signal in RDS versus DRDS stimuli is an important issue that must not be overlooked. To hypothesize logically that the difference in performance between these two stimulus types shown in Experiment 1 reflects the contribution of an additional IOVD cue in RDSs, one must be sure that the disparity information provided in each is equivalent. Although the CD information in each stimulus is mathematically equivalent, the temporal details of their presentation are very different. It is possible that the short lifetime and rapid replacement of each individual dot at a high temporal frequency in a DRDS causes its disparity signal to be degraded relative to the low temporal frequency disparity signal in an RDS. This would, in turn, cause any CD measure derived from the initial disparity signal to be less precise. A difference in the effectiveness of the CD cue between the two stimulus types could therefore entirely account for any observed difference in speed discrimination performance, without any need to appeal to an IOVD mechanism. Here we address this issue directly by testing subjects’ stereoacuity for stationary DRDS and RDS stimuli at each of the relative disparity pedestals used in Experiment 1
Methods
Methods for stereoacuity tasks were identical to those for Experiment 1 except in the following respects. Here we used four of the original subjects. Each trial sequentially presented a pair of stationary RDS or DRDS stimuli. Observers were required to indicate in which of the two intervals the stimulus appeared closer to them. Stimulus duration was constant at 600 ms for each of the 4 interleaved staircases. The disparity of the standard stimulus was varied slightly from trial to trial within each pedestal condition to reduce the extent to which subjects could use the background dots in the zero pedestal condition as an omnipresent standard. In the 4 separate interleaves, the disparity of the standard stimulus was displaced either forward or backward from the nominal pedestal disparity by either one or two pixels (0.6 or 1.2 arcmin). The staircase determined the relative disparity of the test stimulus compared to the standard from a set of 11 possible values: −10, −5, −2.5, −1.24, −0.62, 0, 0.62, 1.24, 2.5, 5, and 10 arcmin. After practice, subjects completed six sessions of data collection. 
Results and discussion
Figure 7 shows mean static disparity discrimination (or stereoacuity) thresholds (+SEM) in stationary RDS and DRDS images as a function of relative disparity pedestal for all subjects. There is no evidence for a difference in stereoacuity for the two stimulus types. Indeed, for all subjects, there was no statistically significant effect of stimulus type on stereoacuity. It is also clear from the data that subjects performed better at the zero relative disparity pedestal, compared to crossed or uncrossed disparities, for both RDS and DRDS stimuli. This main effect of disparity pedestal was statistically significant in all subjects in a 2 × 3 repeated measures ANOVA [LS: F(2,10) = 30.05, p < .0001; AK: F(2,10) = 26.12, p = .0001; CN: F(2,10) = 11.27, p = .0027; BB: F(2,10) = 8.06, p = .0082]. 
Figure 7
 
Stereoacuity thresholds for stationary RDS and DRDS stimuli as a function of relative disparity pedestal for all subjects. Note the difference in vertical axis scale for subject BB (bottom right).
Figure 7
 
Stereoacuity thresholds for stationary RDS and DRDS stimuli as a function of relative disparity pedestal for all subjects. Note the difference in vertical axis scale for subject BB (bottom right).
The finding that stereoacuity varies with disparity pedestal for RDS stimuli confirms previous reports using a variety of different stimuli and techniques (e.g., Ogle, 1953; Blakemore, 1970; Badcock & Schor, 1985). Though the finding that this general pattern of results is preserved for DRDS stimuli has not previously been described, it is perhaps not surprising, given reports of degraded stereopsis at a disparity pedestal with similar stimuli (Stevenson, Cormack, Schor, & Tyler, 1992). Furthermore, that stereoacuity for DRDS targets is no worse than for RDS stimuli is a critical finding. Though Ziegler and Roy (1998) showed that performance for detecting depth in stimuli with either crossed or uncrossed disparity is as good for dynamic dot patterns as it is for static, their patterns were low in dot density, and featured several degrees of disparity. To our knowledge, such data have never been presented for the discrimination of small disparities, or for dense random dot stimuli. It is an impressive property of the human visual system that a sequence of 72 stereo-pairs of distinct luminance patterns, each visible for less than 1/100th of a second, but specifying the same disparity, can be combined into a unified percept whose depth can be perceived as accurately as an unchanging stimulus. These results also allow us to reject with confidence the possibility that the inferior speed discrimination performance for DRDS stimuli relative to that for RDS stimuli simply reflects a difference in the precision of the disparity inputs to the CD system. 
The results of BB are noteworthy for their departure from the pattern established by the group, because stereoacuity for this subject is vastly inferior to that shown for our other subjects (note the scale difference in Figure 7). This is especially apparent at the +19 arcmin pedestal, which may explain BB’s somewhat anomalous stereomotion results in Figure 3. Indeed, if this subject’s data for Experiment 1 are reanalyzed excluding the data from the +19 arcmin pedestal conditions, a highly significant main effect of stimulus type (p < .001) becomes apparent. 
Experiment 3: Testing non-IOVD strategies
In many investigations of stereomotion speed discrimination, including Experiment 1 above, the trajectory of stereomotion is direct (i.e., approaching toward or receding from the binoculus, along the mid-line). With such a trajectory, both monocular images translate at identical speeds, albeit with opposite directions of horizontal motion. Because of this, the speed of one retinal image alone correlates perfectly with the speed of stereomotion, as shown in Equation 1. This expresses vz, the speed of the stimulus in depth (in metres per second), in terms of viewing distance, D, interpupillary distance, I, and monocular velocities vR and vl (in radians per second):  
(1)
This raises the possibility that our subjects could have performed the speed discrimination without any percept of depth or motion in depth even being derived. Although subjects employing such a strategy4 need not have relied on any impression of motion in depth, the comparison of monocular lateral speeds between one half-image in the first interval and one in the second would, in principle, have provided the necessary information for a correct response on each trial. In this and many other previous stereomotion studies, subjects were instructed to view the stimuli binocularly at all times, and each reported a clear percept of motion in depth throughout testing. However, it remains possible that the motion in one monocular image alone dominated their response on each trial. 
Realising this possibility, Harris and Watamaniuk (1995) attempted to address the issue by recording speed discrimination thresholds while varying the trajectory of a stereomotion stimulus (a lone dot) to break the correlation between monocular image speed and 3D stimulus speed. Equation 2 demonstrates the relationship between stimulus trajectory, β, and monocular image velocities, vr and vl, again in terms of the interpupillary distance, I, and viewing distance, D (for a derivation, see Regan, 1993):  
(2)
 
Harris and Watamaniuk’s 2IFC experiment was arranged such that all stimuli had oblique trajectories. Either both stimuli in each pair were equally inclined toward the same eye (the “same” condition), or both stimuli were equally inclined toward opposite eyes (the “different” condition). It was hypothesized that if subjects were responding on the basis of the left monocular half-image alone, or the right image alone, or even if the monocular image was determined randomly on each trial, then thresholds would be higher when standard and test stimuli had different trajectories. They found this not to be the case, and, while their data do indeed rule out consistent monocular strategies using one eye alone, they fail to rule out at least three other non-IOVD strategies. 
First, because in all of their conditions the two comparison stimuli had equal (or equal and opposite) trajectories, this leaves open the possibility that subjects could have based their responses not on the speed of stereomotion, but instead on the percept of cyclopean lateral translation (the rate of change of binocular visual direction, equal to the vector average of left and right eye velocities). In every trial on each condition, the binocular stimulus that moved more rapidly to either side was also the stimulus that moved faster in depth. Though this strategy also requires binocular combination of images, subjects could theoretically have produced equal thresholds while remaining oblivious to any percept of motion in depth. 
Second, it is possible that subjects could have reliably identified the faster (or indeed, the slower) of the two monocular half-images for each stimulus, and then compared the two intervals on the basis of these monocular signals. Though the subject could not have known in advance which eye would receive the more rapid motion, it is possible that the faster (or the slower) motion signal might in some way inhibit or suppress the signal from the other eye, hence allowing subjects to respond in this way. 
Third, the data discussed above do not rule out the possibility that subjects could have made appropriate speed discrimination responses by using a strategy involving the sum of the unsigned speeds. We will refer to this coincidental correlate of stereomotion speed (specifically vz) as the speed sum, which is expressed mathematically in z: Equation 3:  
(3)
 
Note that though this measure need not yield a veridical percept of vz, it could still have produced the data observed in many previous stereomotion studies where the trajectories of stimuli featured monocular images with opposite directions of motion. For such trajectories, referred to as “hitting the head,” the sum of the unsigned speeds is fortuitously equal to the difference of the velocity vectors (i.e., IOVD correlates perfectly with speed sum). Because it is simply half of the speed sum, the average of the two unsigned monocular speeds also correlates perfectly with the veridical IOVD solution in such cases. 
In Experiment 3, we extend the approach of Harris and Watamaniuk (1995) to investigate the issue of monocularly based judgments while controlling for the possibility that subjects could respond on the basis of the speed of either the cyclopean lateral translation, or the monocular lateral motion of one half-image, or the sum (or average) of the unsigned monocular speeds. More specifically, we asked subjects to make 2IFC speed comparisons for stimuli whose trajectories differed, with at least one interval in every test-standard pair having a direct trajectory (and hence no lateral translation). In addition, our matched oblique trajectory conditions had the same monocular speeds (i.e., the values of |vR| and |vl| were identical) yet different stereomotion velocities (i.e., different v z). This ensured that neither the speed of cyclopean lateral translation, nor the speed of individual monocular images, nor the speed sum (or any ad hoc combination of monocular speed information) had any consistent relationship with stereomotion speed. Subjects using the cyclopean lateral motion cue would consider at least one stimulus in each pair to have a speed of zero, and hence show immeasurably high PSEs. Subjects using the speed sum/average cue would show identical PSEs for the matched oblique conditions despite their differing velocities of motion in depth. Furthermore, we can make explicit quantitative predictions as to which pattern of PSEs would emerge if the subject were to use the left image alone, the right image alone, the faster (or slower) of the two, a monocular image chosen at random, or the sum/average of the monocular speeds. Our data allowed us to rule out all of these non-IOVD strategies. 
Methods
In our 2IFC stereomotion speed discrimination task, the speed of a test stimulus was manipulated during testing to match the speed of the standard, which remained constant. Test stimuli in all conditions featured identical speeds of monocular motion in each eye, moving in opposite directions, and hence moved directly away from the binoculus along the mid-line (see stimulus D, Figure 8). This stimulus had no lateral motion, because the average of left and right eye velocities was zero. 
Figure 8
 
Stereomotion trajectory and its relationship to monocular speeds. Oblique stimuli (L and R) feature a ratio of monocular speeds of 2:1 (or 1:2). The stimulus recedes from the eye with the slower motion. While stimuli with opposite monocular directions have trajectories intercepting the interocular axis between the eyes (Hit stimuli), those with the same direction do not (Miss stimuli). Miss stimuli are objectively slower in 3D space, despite containing identical monocular speeds. Stimulus D moves directly away from the binoculus along the mid-line, and features identical monocular speeds.
Figure 8
 
Stereomotion trajectory and its relationship to monocular speeds. Oblique stimuli (L and R) feature a ratio of monocular speeds of 2:1 (or 1:2). The stimulus recedes from the eye with the slower motion. While stimuli with opposite monocular directions have trajectories intercepting the interocular axis between the eyes (Hit stimuli), those with the same direction do not (Miss stimuli). Miss stimuli are objectively slower in 3D space, despite containing identical monocular speeds. Stimulus D moves directly away from the binoculus along the mid-line, and features identical monocular speeds.
Standard stimuli featured several possible trajectories of receding stereomotion in separate conditions, all of which lay on the horizontal meridian (i.e., contained no vertical motion). These trajectories were created in two orthogonal manipulations by varying (a) the relative speeds and (b) the relative directions of the left and right eye monocular image motion (see Figure 8). First, we created stimuli with either equal monocular speeds or with a ratio of 2:1 (or 1:2). When speeds were equal in both eyes, standard stimuli, like their test counterparts, receded directly away from the binoculus, along the mid-line (referred to as condition D). When monocular image speed was faster in the left compared to the right eye, the stimulus trajectory was angled obliquely away from the right eye (referred to as condition R). Conversely, with a faster speed in the right eye, the stimulus trajectory was angled away from the left eye (referred to as condition L). 
In a second manipulation, we created two types of obliquely moving stimuli. One featured monocular motions in opposite directions with an extrapolated trajectory that intercepted the interocular axis in between the two eyes. The other featured monocular motions in the same direction with an extrapolated trajectory that did not. These two classes of trajectory will be referred to as “hitting the head” (or Hit) and “missing the head” (or Miss), respectively. 
The details of Experiment 3 differed from those of Experiment 1 only in the following respects. All stimuli moved with a relative disparity pedestal of zero (i.e., passed through the plane of the background dots at the midpoint of its motion). They simulated a real-world trajectory inclined at + 0.25° for Hit stimuli, or + 2.23° for Miss stimuli, with respect to the midline. 
For the small trajectory angles used here, the difference between vz and the total 3D speed, or v, is negligible (<0.0001% for Hit stimuli; <0.1% for Miss stimuli). We kept the sum of the (unsigned) monocular speeds for all standard stimuli identical in all conditions. This meant that the amplitude of the (signed) velocity difference between the monocular motions in the two eyes (vRvl), and hence the speeds of motion in depth (vz) were different for Hit and Miss conditions. For Hit stimuli, the standard speed was 0.622 deg/s (equivalent to a vz value of 1.04 m/s), and for Miss conditions, 0.207 deg/s (equivalent to a vz value of 0.347 m/s). 
The staircase routine determined the speed of the test stimulus from a set of 9 possible values. For Hit stimuli, possible speeds of motion experienced by each eye simultaneously were 0.178, 0.207, 0.249, 0.276, 0.311, 0.355, 0.373, 0.414, and 0.466 deg/s toward the nasal side, which correspond to interocular velocity differences of 0.355, 0.414, 0.497, 0.553, 0.622, 0.710, 0.746, 0.829, and 0.932 deg/s (or real-world vz values between 0.63 and 1.56 m/s). For Miss stimuli, monocular speeds were 0.083, 0.104, 0.138, 0.207, 0.311, 0.373, 0.414, 0.466, and 0.533 corresponding to velocity differences of 0.166, 0.207, 0.276, 0.414, 0.622, 0.710, 0.746, 0.829, and 0.932 deg/s (or real-world vz values between 0.28 and 1.56 m/s). The Michelson contrast of stimulus patterns was reduced to 20%, to ensure that both monocular images featured perceptually smooth motion at the slower stimulus speeds needed here. In all trials, stimuli were visible for 600 ms, with an inter-stimulus interval of 600 ms. 
The same four subjects who participated in Experiment 2 contributed data. Except for one subject (the author LS), all participants were naïve as to the purpose of the experiment. 
Subjects completed three sessions of testing, with each session consisting of 8 blocks (4 using Hit stimuli and 4 using Miss in random order), and each block lasting approximately 5 min. Each block contained 3 randomly interleaved staircases (12 reversals each), one for each of the 3 trajectory conditions (D, L and R). We sequentially presented the standard and test stimuli in a randomized order. Subjects were asked to indicate whether the first or second stimulus receded in depth more rapidly, regardless of the degree of lateral translation. 
Data were analyzed for each subject individually. Responses from trials in a given condition were combined across the 4 blocks within a session and a cumulative Gaussian curve was fitted to the data by probit analysis (Finney, 1971). Both the mean (PSE) and the SD of the underlying Gaussian distribution were free parameters in the curve fitting routine. PSEs and thresholds were then averaged across sessions and their associated SEs calculated. Statistical significance was assessed using ANOVAs, with specific comparisons performed using Bonferroni corrected t tests. 
Predictions
If subjects experience a veridical stereomotion percept, recorded PSEs should not be significantly different from the speed of their respective standards (i.e., 0.622 deg/s for the Hit conditions and Direct Miss condition and 0.207 deg/s for the oblique Miss conditions). However, if subjects respond instead on the basis of the speed of a monocular image in each interval, or on the basis of the sum or average of monocular velocities, PSEs should show a predictable pattern of biases across trajectory conditions. 
For all monocular strategies, the predicted PSEs for Hit and Miss conditions would be identical, because the same monocular speeds are used in either case. However, when we consider the manipulation of relative monocular image speed, we can predict a variation in perceived speed between the three conditions, the pattern of which is determined by the consistent strategy employed (see Figure 9). If subjects employ an inconsistent monocular strategy (i.e., randomly choosing one of the eyes or strategies on each trial), PSEs should average out to show little or no bias, but JNDs would be greatly increased for conditions L and R compared to those for condition D. 
Figure 9
 
Qualitative PSE predictions for consistent monocular strategies (i.e., judgments based exclusively on the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image). If subjects use the speed of the left eye image, PSEs should be decreased in condition L (because in this condition the left eye’s image is faster in the test stimulus than in the standard stimulus with the equivalent stereomotion speed) and increased in condition R (because here the left eye’s image is slower) compared to condition D. This pattern of results would be reversed if the subject were to respond on the basis of right monocular half-image speeds. If subjects respond on the basis of the faster of the two monocular half-images in each interval, PSEs should always be higher for conditions L and R, because for equivalent stereomotion speeds, oblique trajectories contain one monocular image with a speed lower than either monocular image in the direct stimuli. If subjects were always to use the slower of the two half-images, the opposite pattern of results should be apparent.
Figure 9
 
Qualitative PSE predictions for consistent monocular strategies (i.e., judgments based exclusively on the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image). If subjects use the speed of the left eye image, PSEs should be decreased in condition L (because in this condition the left eye’s image is faster in the test stimulus than in the standard stimulus with the equivalent stereomotion speed) and increased in condition R (because here the left eye’s image is slower) compared to condition D. This pattern of results would be reversed if the subject were to respond on the basis of right monocular half-image speeds. If subjects respond on the basis of the faster of the two monocular half-images in each interval, PSEs should always be higher for conditions L and R, because for equivalent stereomotion speeds, oblique trajectories contain one monocular image with a speed lower than either monocular image in the direct stimuli. If subjects were always to use the slower of the two half-images, the opposite pattern of results should be apparent.
A strategy involving the computation of the sum of monocular image speeds would produce unbiased results across conditions L, D, and R, at least for Hit stimuli. However, the same cannot be said for Miss stimuli. Here, a speed sum strategy no longer makes the same prediction as the veridical IOVD cue. In fact, the speed sum model predicts that the PSE for vz should be the same whether monocular image directions are the same (Hit) or different (Miss). However, for our oblique Miss stimuli, where the speed of one eye’s image is double the speed of the other, a veridical IOVD model predicts a PSE for vz only one third of that predicted for our Hit stimuli (see Figure 8). 
In Experiment 3, we exploit these contrasting predictions. By performing speed discriminations between direct standard stimuli and tests with either Miss trajectories (same monocular directions), or Hit trajectories (opposite monocular directions), we can explicitly test the speed sum strategy, in addition to the monocular strategies involving the exclusive use of the left or the right eye, and the faster or slower monocular image. The quantitative pattern of predicted PSE results based on each of these strategies is shown in Figure 10. Note that in terms of ANOVA effects, while the monocular strategies all predict a statistically significant effect of the manipulation of relative monocular speed (L, D and R) only, the speed sum model predicts no significant effects at all. Only the IOVD model predicts statistically significant main effect of relative monocular speed (L vs. D vs. R), of relative monocular direction (Hit vs. Miss), and a statistically significant interaction. 
Figure 10
 
Quantitative PSE predictions for monocular strategies (the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image), speed sum strategy, and the veridical IOVD cue.
Figure 10
 
Quantitative PSE predictions for monocular strategies (the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image), speed sum strategy, and the veridical IOVD cue.
Results and discussion
PSE data
Stereomotion speed discrimination PSEs are plotted as histograms in Figure 11. PSE data from condition D for both Hit and Miss conditions were unremarkable as expected, all showing near-veridical matches, as predicted by all models and strategies. For the oblique standard conditions, data do not appear similar to any of the non-IOVD predictions shown in Figure 10. However, the pattern of PSE results does resemble the predictions of a veridical IOVD system. Data for all four subjects show near veridical PSEs for all conditions. Analysis of variance found statistically significant main effects of relative monocular speed [LS: F(2,4) = 18.10, p = .0099; AK: F(2,4) = 7.505, p =.0443; CN: F(2,4) = 7.021, p = .0492; BB: F(2,4) = 71.98, p = .0007], relative direction [LS: F(1,2) = 208.2, p = .0048; AK: F(1,2) = 73.59, p = .0133; CN: F(1,2) = 488.9, p = .0020; BB: F(1,2) = 111.0, p = .0089], and of their interaction [LS: F(2,4) = 105.2, p = .0003; AK: F(2,4) = 52.38, p = .0014; CN: F(2,4) = 13.66, p = .0163; BB: F(2,4) = 50.18, p = .0015]. The presence of a main effect of relative direction and an interaction effect confirm the fact that no subject based their responses either on individual monocular image speed or on the (unsigned) sum of the two monocular speeds in each interval. More specifically, the Miss data for all four subjects clearly differ from the Hit data, being far lower than the speed sum prediction of 0.622 deg/s, and hovering just above the veridical IOVD cue predicted value of 0.207 deg/s. 
Figure 11
 
Stereomotion speed discrimination PSEs for Experiment 3. Dark blue bars represent PSEs for Hit trajectories, while light blue bars represent Miss trajectories. Error bars represent +1 SEM.
Figure 11
 
Stereomotion speed discrimination PSEs for Experiment 3. Dark blue bars represent PSEs for Hit trajectories, while light blue bars represent Miss trajectories. Error bars represent +1 SEM.
Though this experiment was explicitly designed to control for monocular and speed-sum strategies, it also allows us to exclude several other strategies that make the same predictions. For example, a strategy where the average of unsigned speeds were computed would make identical predictions to those of the speed sum model, and can hence also be ruled out. In fact, because any strategy based on the unsigned speeds of images will reach the same predictions for Hit and Miss conditions, this whole class of strategies can be eliminated regardless of the combination rule that they later employ. Clearly, any model wishing to account for our data must retain directional information, and hence operate on the monocular (signed) velocity vectors. 
JND data
Speed discrimination thresholds are shown in Figure 12. No statistical significance was shown for main effects or interactions in any subject in our repeated measures ANOVA (p > .22). Speed discrimination thresholds are not significantly smaller for direct stimuli compared to their oblique counterparts. This shows that subjects did not respond on the basis of a monocular image selected at random on every stimulus presentation. 
Figure 12
 
Stereomotion speed discrimination JNDs for Experiment 3. Light green bars represent JNDs for Hit trajectories, while dark green bars represent Miss trajectories. Error bars represent +1 SEM.
Figure 12
 
Stereomotion speed discrimination JNDs for Experiment 3. Light green bars represent JNDs for Hit trajectories, while dark green bars represent Miss trajectories. Error bars represent +1 SEM.
General discussion
The utility of the interocular velocity difference cue
The finding that thresholds for the discrimination of stereomotion speed are lower for RDS stimuli (which contain both IOVD and CD cues to stereomotion speed) than for DRDSs (containing only the CD cue), when potential artifacts have been systematically ruled out, is strong evidence for the use of the IOVD cue in the perception of stereomotion speed. The omission of the IOVD cue in DRDS stimuli causes systematically poorer performance. Though subjects are able to discriminate stereomotion speed when provided with the CD cue alone, thresholds are, on average, 1.4 to 2.4 times worse than when both cues are available. 
One hypothesis, developed to reconcile discrepant previous findings, suggested that although equivalent performance for the two stimulus types (or slight superiority for RDS performance) might be shown at non-zero disparity pedestals, the difference should be larger for stimuli that pass through zero disparity. In fact, our data showed little sign of this pattern of results. Instead, they show a general increase in precision when stimuli include the IOVD cue at all relative disparity pedestals. 
The replication of this finding with approaching stimuli allows us to extend the generality of our conclusions. Advantages have previously been reported for approaching compared to receding motion, both in the perception of motion in depth through looming cues (Perrone, 1986) and in terms of the percentage of primate cortical neurons selective for expansion as opposed to contraction (Tanaka & Saito, 1989; Graziano, Andersen, & Snowden, 1994). However, we find no evidence of such an advantage for approaching compared to receding stimuli for stereomotion speed discrimination. 
Elimination of artifacts
One might think that the precise performance observed in our task could have been achieved by sensing the stimulus’ initial disparity, final disparity, and/or calculating a displacement. It is possible that subjects could have performed well using such a strategy, masking the lack of any ability to accurately perceive stereomotion per se. It is for this reason that the variable duration condition was run. Our finding that thresholds were not significantly different for conditions where duration was fixed compared to those where duration was variable indicates that subjects did not simply respond on the basis of stimulus displacement. This was true for both stimulus types, and for all disparity pedestals. The data showing the lack of the predicted variation of PSE with stimulus duration provide even stronger confirmation. These findings show that both CD and IOVD related information is processed by specialized motion mechanisms, across all of the relative disparity pedestals investigated. In addition, the static stereo-acuity control of Experiment 2 demonstrates that the superior performance with RDS stimuli cannot be due to any difference in the raw disparity information per se between our RDS and DRDS stimuli. Last, the results of Experiment 3 discount any possibility that our results could be due to subjects making responses on the basis of monocular half-images alone, or on the basis of any combination of the unsigned speeds of these images, regardless of their manner of combination. Velocity signals, where the speed and direction of stimulus monocular motion are taken into account, must be used in stereomotion speed estimation. 
Crude difference or crude average?
It has also been posited that the visual system might combine a pair of opposing monocular motion signals in a different way to the IOVD cue. Westheimer (1990) first suggested that equal and opposite monocular motion signals, such as those that are provided from a directly approaching stereomotion stimulus, might cancel each other in some way. This suggestion, renamed the crude average hypothesis, has been extended by Harris and colleagues to explain their results in visual search experiments, where a single MID dot stimulus must be detected amidst a distributed collection of disparity noise dots (Harris, McKee, & Watamaniuk, 1998; Sumnall & Harris, 2000; Harris & Rushton, 2003). These studies found evidence that for a high number of noise dots, the detection of monocular motion was far superior to that of stereomotion (for the same monocular stimulus speed): a result reminiscent of the stereomovement suppression effect (Tyler, 1971). Under the crude average scheme, the two monocular velocities (importantly, the vector, as opposed to the directionless speed) are averaged. This is precisely the calculation required to yield the rate of change of binocular visual direction: the cyclopean lateral velocity of the stimulus. It has been argued that this averaging may be performed in an imperfect fashion, such that even when speeds are physically identical, the result of the neural computation is not zero, but has some small residual value from which a lateral motion percept can result. When the average monocular velocity is zero or near zero (e.g., for direct trajectories), the visual system may have to resort to the CD cue for stereomotion detection, or to this small residual from the crude averaging process to detect motion. 
It is clear that a simple vector average (or sum) of velocity signals cannot produce the results seen here, because for every comparison in our speed discrimination experiments, at least one stimulus had a direct trajectory, and hence the average of velocities (equal to their rate of cyclopean translation) was zero. Use of this cue in discriminations would therefore have lead to immeasurable PSEs, because the oblique stimulus in every pair would have always been regarded as faster, regardless of the actual stimulus speed in depth. This was not observed. Furthermore, if stereomotion speed were perceived exclusively via the CD cue when the physical average is zero, we would expect no difference between JNDs for RDS and DRDS stimuli in Experiment 1. This was clearly not the case. Last, without specifying in what sense or to what extent the averaging process is crude, the possibility of a useable residual signal is difficult to assess. Nonetheless, if this residual signal is used in stereomotion speed discrimination tasks, and the size of this residual is consistently related to the absolute magnitude of monocular speed signals, such a process may possibly be able to accommodate the data of Experiment 1, but it cannot explain those of Experiment 3. Such a scheme predicts that PSEs in Experiment 3 should be far higher for conditions L and R than for condition D, because any crude average of two numbers that are equal and opposite in sign must be smaller than the crude average of two numbers whose magnitude differs by a factor of two. This was not observed. We can therefore reject the crude averaging hypothesis or any variant that relies on lateral cyclopean motion to explain performance in our stereomotion speed discrimination task. 
Though the crude averaging hypothesis cannot account for the data presented here, the idea of the differencing computation (i.e., the IOVD cue) being a somewhat crude process, introducing additional noise at the stage where monocular velocities are subtracted from one another, can explain the results of Harris and colleagues in addition to ours. When stereo targets with direct trajectories are presented, detection is more difficult than in the monocular case. However, when targets with a component of cyclopean lateral motion (a change of binocular visual direction) are presented, they can be more easily detected using this percept. It has recently been proposed that stereomotion trajectory perception is mediated on the basis of binocular visual direction alone (Harris & Dean, 2003; though see also Brooks, 2002b). 
Relationship to previous studies
In terms of pure performance levels, the thresholds in this study are comparable with those reported previously by Harris and Watamaniuk (1995). Though performance superior to that reported in both of these studies has been noted by Portfors-Yeomans and Regan (1996), several factors may explain this. First, the latter study featured longer stimulus durations and shorter ISIs—both factors known to influence performance in the 2IFC paradigm—than those used here. Finally, and perhaps most crucially, unlike our study, Portfors-Yeomans and Regan (1996) used feedback to train their subjects. 
The work presented here reinforces and extends the findings of Harris and Watamaniuk (1995), who showed a superiority of RDS stimuli over their dynamic counterparts in supporting accurate speed percepts near the zero relative disparity point. Though their stimulus was criticized for its lack of visibility during portions of the presentation (Portfors-Yeomans & Regan, 1996; Portfors & Regan, 1997), our study confirms their finding using stimuli whose visibility was not compromised at any point. Furthermore, we have extended this finding to relative disparity pedestals away from zero. Portfors-Yeomans and Regan (1996) and Portfors and Regan (1997) dismissed slightly lower thresholds away from the point of zero relative disparity for the perception of stereomotion speed in “non-cyclopean” targets over their “cyclopean” (DRDS) counterparts due to a lack of statistical significance. The lower thresholds observed in our study, however, did achieve significance. A major difference between their study and ours lies in the “non-cyclopean” stimuli used. For our RDS stimuli, the dot patterns in each eye were unchanging as they drifted laterally. However, in the aforementioned studies, the “non-cyclopean” targets were actually modified DRDSs where the central target feature was presented either on a blank background (Portfors & Regan, 1997), or a background featuring stationary dots (Portfors-Yeomans & Regan, 1997). Consequently, though a dynamic patch of dots could be seen to drift laterally in each monocular image, individual dots did not carry any IOVD information, as they do in a conventional RDS undergoing stereomotion. Though these stimuli can indeed be seen monocularly, they may have been incapable of supplying adequate motion information to an IOVD system. Each monocular motion input would need to be derived either from the reduced contrast signal from some low-pass filtering of the entire patch when presented on the blank background, or from a second-order motion signal (defined by the boundary between dynamic and non-dynamic areas) when presented on the background of stationary dots. The extent to which the IOVD system can process second-order motion is presently unknown. It is also true that the reduction of image contrast has been shown to compromise the perception of speed for both monocular lateral motion (Stone & Thompson, 1992; Thompson, 1982; Thompson, Stone & Brooks, 1995) and for stereomotion (Brooks, 2001). The similar performance levels shown for Portfors-Yeomans and Regan’s cyclopean and non-cyclopean stimuli may therefore have been due to an impoverished IOVD cue. 
Our data agree with the findings of Portfors-Yeomans and Regan (1996) and Portfors and Regan (1997), that subjects do not base their responses on disparity displacement, but instead respond specifically to the speed of each stimulus. However, these results conflict with a finding of Harris and Watamaniuk (1995), who state that though there were no observed trends linking trial duration and perceived speed for RDS stimuli, the same could not be said for DRDS stimuli. Though we have shown that the visibility issue highlighted by Portfors-Yeomans and Regan does not affect the general pattern of threshold results, it may have forced Harris and Watamaniuk’s subjects to respond on the basis of stimulus displacement. However, in our study, perhaps because the constant stationary background renders the stereomotion stimulus visible throughout its duration, even when it traverses the zero relative disparity plane, our subjects did not need to resort to using displacement. 
To reconcile previous discrepant results, we originally postulated that there might be a large performance difference between RDS and DRDS performance near zero relative disparity, and a far smaller difference at disparity pedestals. However, our results show no such systematic pattern. Instead, performance for RDS stimuli is generally elevated above that for DRDS at all relative disparity pedestals. 
This finding may be difficult to resolve with the interpretation of Brooks (2002a), who used motion adaptation to reduce the perceived speed of stereomotion. He found that adaptation was equally effective near zero relative disparity, whether induced by a 100% correlated RDS or an uncorrelated RDS, (or URDS). However, equivalent adaptation for these two stimuli was not the case at a disparity pedestal, where correlated images were more effective. This was interpreted as reflecting the impotence of the CD cue near zero relative disparity, compared to its importance beyond this region. However, an alternative explanation, consistent with our current findings, is that the adaptation effect for URDS stimuli was compromised, to some extent, by its disparity, or its perceived depth. It is known that disparity channels exist for the processing of depth (Richards, 1970; Stevenson et al., 1992 and for motion in depth (Hong & Regan, 1989). It is entirely possible that the extent of the adaptation shown in the Brooks (2002a) study was contingent upon the mean disparity, or the consequent perceived depth of the adapting stimulus. In his third experiment, Brooks presented motion at a disparity pedestal, and adapted with either an RDS stimulus set in the same disparity plane, or a URDS stimulus, that had a mean disparity (defined by random correspondences of dots) of zero. The visual system, when matching pairs of dots in a URDS, would be expected to preferentially select dot matches that are as close to the fixation plane as possible (i.e., the closest match). Though subjects did not report any clear percept of depth for the URDS stimulus, such matching would produce a distribution of disparities centered on zero disparity. Hence, it is possible that URDS adaptation was less effective than RDS adaptation for comparison stimuli presented at non-zero disparity pedestals because of a difference between the disparities (or perceived depths) of the comparison and adaptation stimuli, rather than a difference in the effectiveness of the CD cue as a function of disparity pedestal. 
However, there are methodological differences between Brooks (2002a) and this study that must not be overlooked. In the experiments reported here, no fixation details were visible during actual trials. Though a fixation mark was present in between trials, to which the subjects were asked to direct their eyes when no stimulation was given, it is likely that the subjects tracked the targets with vergence eye movements, at least to some degree, during the course of the stereomotion stimuli. Though the relative disparity of the stimulus is unaffected by such eye movements, the absolute disparity of the target is likely to be near zero for all conditions. Brooks (2002a), however, provided a fixation mark and nonius lines that were visible at all times, and subjects were instructed to fixate. In addition, moving stimuli were presented slightly above and below this fixation mark. As such, any vergence responses were likely to be small. It is possible that the relative influence of the two stereomotion cues varies not as a function of relative disparity, but instead as a function of absolute disparity, explaining the lack of a pedestal effect in this study. However, we consider this unlikely due to the general lack of influence of absolute disparity on stereomotion perception. Despite evidence that small stimuli undergoing a change of absolute disparity can occasionally be seen to move in depth (Regan, Erkelens, & Collewijn, 1986)—albeit with very high thresholds—this cue is ineffective for motion in depth perception for larger patterns of multiple random dots such as ours (Collewijn, Erkelens, & Regan, 1985; Erkelens & Collewijn, 1985; Regan, Erkelens, & Collewijn, 1986). 
Previous attempts to stabilize gaze in the presence of stereomotion stimuli have met with difficulties. Harris and Watamaniuk instructed subjects to fixate a stationary reference mark, but noted that subjects found this impossible. In common with their study, our subjects found it impossible to fixate while viewing our stimuli. It was for this reason that no fixation details were provided. Therefore, one might also be tempted to postulate that vergence eye movements per se might have played a role in the perception of motion in depth in our study and other previous studies in which fixation was not monitored. However, Portfors and Regan (1997) point out “that neither δ, the target’s disparity relative to the fixation plane, nor dδ/dt (the effective stimulus for the perception of motion in depth) is, by definition, affected by binocular convergence of the left and right eyes. As well, it has been shown empirically that a rate of change of binocular convergence angle has no effect on the perception of motion in depth produced by a rate of change of disparity.” Though recent reports have shown examples of motion in depth perception on the basis of vergence monitoring for small stimuli travelling large distances (e.g., Heuer 1993; Backus & Matza-Brown, 2003, Regan et al. (1986), using a wide variety of stereomotion stimuli and a binocular eye-coil tracker, found that neither vergence nor absolute disparity elicited a percept of motion in depth for extended random-dot stereomotion stimuli similar to ours. When their subjects tracked large-field stimuli with no background reference, though large vergence responses were made, no motion in depth was perceived. Furthermore, relative disparity thresholds (measured under open loop conditions) for the detection of motion in depth were the same whether simultaneous large changes in vergence angle were in phase or in antiphase with the disparity changes. These findings show that, for extended random-dot stereomotion stimuli, vergence per se neither generates a percept of motion in depth nor does it appear to alter the percept generated by changes in relative disparity. Even if vergence monitoring does provide information about stereomotion speed (as opposed to detection), one would need to further postulate a difference in the precision of vergence responses to RDS and DRDS stimuli. The possibility of differential precision in the vergence response, as well as the question of whether the absolute disparity of the target stimulus influences the relative contribution of stereomotion cues, can only be fully resolved by the high-precision monitoring of binocular eye movements during a motion-in-depth speed discrimination task, because vergence responses would need to be resolved to within a few minutes of arc. 
Conclusions
In this study, we have shown that RDS speed discrimination performance is superior to DRDS performance, for all relative disparity pedestals investigated, and for both approaching and receding motion. Furthermore, we have demonstrated that this difference cannot be accounted for by the differences in the visibility of the stimuli, or in the precision of the underlying disparity signal in either stimulus, and is not contaminated by responding on the basis of stimulus displacement. In addition, the lack correlation of perceived stereomotion speed with the monocular stimulus speeds, or their sum or average, shows that our results cannot be explained by any strategy based on the individual monocular motions or ad hoc combination thereof. Our data show that neural signals related to both CD and IOVD are used to support speed perception of stereomotion stimuli and join a growing body of evidence for a role for these two cues in other aspects of the perception of motion in depth through binocular cues. There is evidence supporting the use of the CD cue (Cumming 1995; Cumming & Parker, 1994) and the IOVD cue (Allison, Howard, & Howard, 1998; Fernandez & Farrell, 2003; Howard, Allison, & Howard, 1998; Shioiri, Saisho, & Yaguchi, 2000; Shioiri, Kakehi, Tashiro, & Yaguchi, 2003) in stereomotion detection. There is also evidence supporting the use of the IOVD cue (Brooks, 2002b) and the CD cue (Portfors-Yeomans & Regan, 1997) in the context of stereomotion trajectory discrimination. It is perhaps not surprising that the human visual system uses all of the information at its disposal to perform at the remarkable level exhibited in everyday activities. 
Acknowledgments
The authors would like to thank Rami Ersheid and Chad Netzer for technical assistance, and Brent Beutter, Barbara Chapman, and anonymous reviewers for helpful comments. This research was supported by NASA Airspace Systems (711-80-03) and Biomedical Research and Countermeasures (111-10-10) programs. 
Commercial relationships: none. 
Corresponding author: Kevin Brooks. 
Email: k.brooks@unsw.edu.au. 
Address: School of Psychology, The University of New South Wales, Sydney, 2052, Australia. 
Footnotes
Footnotes
1 It is theoretically possible that random dot correspondences in a moving URDS stimulus could form a CD cue for individual dots in a cloud distributed around zero disparity. However, such random matches have been shown to influence stereomotion perception only minimally (for a detailed discussion, see Brooks, 2002a).
Footnotes
2 Though this non-cyclopean stimulus was not strictly a conventional RDS, it did include visible lateral motion signals in each monocular image, as an IOVD cue to stereomotion.
Footnotes
3 Though a perfect vergence response would bring the target stimulus on to the horopter, and abolish any absolute disparity, the relative disparity pedestal of the stimulus with respect to the background is unaffected by eye movements of any kind.
Footnotes
4 Here, the word “strategy” is not intended to imply any deliberate effort on the part of the observer.
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Figure 1
 
Two mechanisms for stereomotion speed perception. a. The differential of stimulus disparity with respect to time yields the changing disparity (CD) cue. b. The vectorial combination of two monocular velocity signals yields the interocular velocity difference (IOVD) cue.
Figure 1
 
Two mechanisms for stereomotion speed perception. a. The differential of stimulus disparity with respect to time yields the changing disparity (CD) cue. b. The vectorial combination of two monocular velocity signals yields the interocular velocity difference (IOVD) cue.
Figure 2
 
General stimulus arrangement in dimetric projection. Figure not to scale. See text for detailed parameters.
Figure 2
 
General stimulus arrangement in dimetric projection. Figure not to scale. See text for detailed parameters.
Figure 3
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions for receding RDS and DRDS motion plotted as a function of relative disparity pedestal for all six observers. Error bars represent +1 SEM.
Figure 3
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions for receding RDS and DRDS motion plotted as a function of relative disparity pedestal for all six observers. Error bars represent +1 SEM.
Figure 4
 
Speed discrimination thresholds for fixed duration trials versus variable duration trials for all conditions and subjects. The dotted line represents identical performance on each. Error bars represent +1 SEM.
Figure 4
 
Speed discrimination thresholds for fixed duration trials versus variable duration trials for all conditions and subjects. The dotted line represents identical performance on each. Error bars represent +1 SEM.
Figure 5
 
Stereomotion speed discrimination PSEs plotted as a function of stimulus duration for all conditions and subjects. The bold grey line represents predicted performance based entirely on stimulus displacement.
Figure 5
 
Stereomotion speed discrimination PSEs plotted as a function of stimulus duration for all conditions and subjects. The bold grey line represents predicted performance based entirely on stimulus displacement.
Figure 6
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions, for approaching stimuli plotted as a function of relative disparity pedestal for all conditions and two subjects. Error bars represent +1 SEM.
Figure 6
 
Stereomotion speed discrimination thresholds, expressed as Weber fractions, for approaching stimuli plotted as a function of relative disparity pedestal for all conditions and two subjects. Error bars represent +1 SEM.
Figure 7
 
Stereoacuity thresholds for stationary RDS and DRDS stimuli as a function of relative disparity pedestal for all subjects. Note the difference in vertical axis scale for subject BB (bottom right).
Figure 7
 
Stereoacuity thresholds for stationary RDS and DRDS stimuli as a function of relative disparity pedestal for all subjects. Note the difference in vertical axis scale for subject BB (bottom right).
Figure 8
 
Stereomotion trajectory and its relationship to monocular speeds. Oblique stimuli (L and R) feature a ratio of monocular speeds of 2:1 (or 1:2). The stimulus recedes from the eye with the slower motion. While stimuli with opposite monocular directions have trajectories intercepting the interocular axis between the eyes (Hit stimuli), those with the same direction do not (Miss stimuli). Miss stimuli are objectively slower in 3D space, despite containing identical monocular speeds. Stimulus D moves directly away from the binoculus along the mid-line, and features identical monocular speeds.
Figure 8
 
Stereomotion trajectory and its relationship to monocular speeds. Oblique stimuli (L and R) feature a ratio of monocular speeds of 2:1 (or 1:2). The stimulus recedes from the eye with the slower motion. While stimuli with opposite monocular directions have trajectories intercepting the interocular axis between the eyes (Hit stimuli), those with the same direction do not (Miss stimuli). Miss stimuli are objectively slower in 3D space, despite containing identical monocular speeds. Stimulus D moves directly away from the binoculus along the mid-line, and features identical monocular speeds.
Figure 9
 
Qualitative PSE predictions for consistent monocular strategies (i.e., judgments based exclusively on the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image). If subjects use the speed of the left eye image, PSEs should be decreased in condition L (because in this condition the left eye’s image is faster in the test stimulus than in the standard stimulus with the equivalent stereomotion speed) and increased in condition R (because here the left eye’s image is slower) compared to condition D. This pattern of results would be reversed if the subject were to respond on the basis of right monocular half-image speeds. If subjects respond on the basis of the faster of the two monocular half-images in each interval, PSEs should always be higher for conditions L and R, because for equivalent stereomotion speeds, oblique trajectories contain one monocular image with a speed lower than either monocular image in the direct stimuli. If subjects were always to use the slower of the two half-images, the opposite pattern of results should be apparent.
Figure 9
 
Qualitative PSE predictions for consistent monocular strategies (i.e., judgments based exclusively on the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image). If subjects use the speed of the left eye image, PSEs should be decreased in condition L (because in this condition the left eye’s image is faster in the test stimulus than in the standard stimulus with the equivalent stereomotion speed) and increased in condition R (because here the left eye’s image is slower) compared to condition D. This pattern of results would be reversed if the subject were to respond on the basis of right monocular half-image speeds. If subjects respond on the basis of the faster of the two monocular half-images in each interval, PSEs should always be higher for conditions L and R, because for equivalent stereomotion speeds, oblique trajectories contain one monocular image with a speed lower than either monocular image in the direct stimuli. If subjects were always to use the slower of the two half-images, the opposite pattern of results should be apparent.
Figure 10
 
Quantitative PSE predictions for monocular strategies (the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image), speed sum strategy, and the veridical IOVD cue.
Figure 10
 
Quantitative PSE predictions for monocular strategies (the left eye, the right eye, the eye receiving the faster image, and the eye receiving the slower image), speed sum strategy, and the veridical IOVD cue.
Figure 11
 
Stereomotion speed discrimination PSEs for Experiment 3. Dark blue bars represent PSEs for Hit trajectories, while light blue bars represent Miss trajectories. Error bars represent +1 SEM.
Figure 11
 
Stereomotion speed discrimination PSEs for Experiment 3. Dark blue bars represent PSEs for Hit trajectories, while light blue bars represent Miss trajectories. Error bars represent +1 SEM.
Figure 12
 
Stereomotion speed discrimination JNDs for Experiment 3. Light green bars represent JNDs for Hit trajectories, while dark green bars represent Miss trajectories. Error bars represent +1 SEM.
Figure 12
 
Stereomotion speed discrimination JNDs for Experiment 3. Light green bars represent JNDs for Hit trajectories, while dark green bars represent Miss trajectories. Error bars represent +1 SEM.
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