Motion parallax thresholds for unambiguous depth perception

Abstract

The perception of unambiguous depth from motion parallax arises from the neural integration of retinal image motion and extra-retinal eye movement signals. It is only recently that these parameters have been articulated in the form of the motion/pursuit ratio. In the current study, we explored the lower limits of the parameter space in which observers could accurately perform near/far relative depth-sign discriminations for a translating random-dot stimulus. Stationary observers pursued a translating random dot stimulus containing relative image motion. Their task was to indicate the location of the peak in an approximate square-wave stimulus. We measured thresholds for depth from motion parallax, quantified as motion/pursuit ratios, as well as lower motion thresholds and pursuit accuracy. Depth thresholds were relatively stable at pursuit velocities 5-20 deg/sec, and increased at lower and higher velocities. The pattern of results indicates that minimum motion/pursuit ratios are limited by motion and pursuit signals, both independently and in combination with each other. At low and high pursuit velocities, depth thresholds were limited by inaccurate pursuit signals. At moderate pursuit velocities, depth thresholds were limited by motion signals.

1. Introduction

Motion parallax (MP), produced during observer or scene translation, serves as an important cue for perceiving relative depth. To illustrate depth perception from MP, consider the case of observer translation: as an observer translates through space, objects appear to move relative to one another, creating relative image motion on the retina. At the same time, the observer maintains fixation on a particular object in the scene, generating smooth pursuit eye movements (Miles & Busettini, 1992). Both the retinal image motion and pursuit eye movements generated during translation are necessary cues for the visual system to generate an unambiguous percept of depth from MP (Nadler et al., 2009; Naji & Freeman, 2004; Nawrot, 2003a; Nawrot & Joyce, 2006). Retinal image motion, by itself, is ambiguous in regards to depth-sign; the extra-retinal pursuit signal is necessary to disambiguate depth (Nawrot & Joyce, 2006; Nawrot & Stroyan, 2009).

Recently, the geometric relationship between proximal and distal parameters for motion parallax was quantitatively described by the Motion/Pursuit Ratio (M/PR) (Nawrot & Stroyan, 2009; Stroyan & Nawrot, 2011). For a particular non-fixated object in space, the relationship among the object's retinal image motion velocity (dθ/dt, here called dθ), the observer's pursuit eye movement velocity (dα/dt, here called dα), viewing distance to the point of fixation (f), and object distance from fixation (d_MP), has been described as

$$d_{MP}∕f\approx \mathrm{d}\theta ∕\mathrm{d}\alpha$$ (1)

The significance of the model is that the relationship between the proximal visual cues, dθ and dα, inform us of the relationship between the two distal cues, d_MP and f. The M/PR is an approximation of the Motion/Pursuit Law (M/PL; Stroyan & Nawrot, 2011), and can be used to approximate relative depths of objects within a scene with given parameters: for example, if one knows dα and f, one can estimate all of the different distances (d_MP) to different objects relative to fixation from each object's particular retinal image motion, or its “parallax” (dθ). Of course, it is important to note that, like binocular disparity, MP is a cue to relative, not absolute, depth (Rogers & Graham, 1979). That is, MP provides no information about an object's absolute distance from the observer, without information about the distance from the observer to the point of fixation.

Because the precise roles of pursuit velocity and retinal image motion for the neural mechanism generating unambiguous depth from MP have only recently been articulated, it is important to re-examine past research and extend these findings within the framework of the M/PR. One particular area of study, MP thresholds, is relevant to the M/PR but has generally received little attention, having been systematically examined in only a handful of studies (Graham et al., 1948; Ono & Ujike, 2005; Rogers & Graham, 1982; Steinbach, Ono, & Wolf, 1991; Ujike & Ono, 2001; Zegers, 1948). In one of the earliest studies of MP thresholds, Graham et al. (1948) measured the minimum z-axis separation between two translating points that observers perceived as being at different depths due to the MP cue. In this study, two needles were placed at slightly different distances on a motorized mechanism, and then the entire mechanism was translated laterally at systematically varying velocities. It is important to note that their observers were stationary while the stimulus needles translated, a paradigm similar to that used in the current study. Observers maintained fixation on one of the translating needles and adjusted the other needle until both needles appeared to be aligned with each other (i.e., there was no relative depth difference). The variance in this setting was used to derive a depth threshold.

Because Graham et al. (1948) and the current study both describe the same underlying geometry, the terminology and measurements Graham et al. used to quantify their measurements are similar to those used with the M/PR. Graham and colleagues sought to articulate and quantify the relationships among viewing distance (what they termed R, comparable to f in the M/PR), angular displacement of fixation (θ/t, comparable to dα), depth threshold magnitude (δ_t, comparable to d_MP), and the angular velocity of the needles relative to one another (ω, comparable to dθ, assuming fixation on one of the needles). Graham and colleagues’ goal was to use the ratio of δ_t to R, multiplied by θ/t and a constant, to determine the threshold of ω(ω_t), the minimum amount of angular velocity, needed to perceive the needles at different depths. Graham and colleagues determined threshold ω_t for different rates of needle translation (i.e., pursuit velocity, if the observer was fixating one of the needles) and found that increasing the pursuit velocity resulted in increased ω_t. However, the increase was approximately constant for both rate and ω_t, implying a constant relationship of the two parameters. Because of the direct comparability of ω_t and needle translation to our terminology, dθ and dα , it is possible to approximate the M/PR by estimating the ratio of ω_t to the rate of needle translation. Based on Figure 5 in Graham et al., we found the rate of needle rate translation (dα) in deg/sec by multiplying the values in Figure 5 by 57.3 (as described in Graham et al., p. 215). We found dθ by taking the anti-log of the log ω_t values in Figure 5 and converting these from arcsec/sec to deg/sec. The ratios of these ω_t values over rate of movement give M/PRs. These derived threshold M/PR values in Graham et al.'s study were stable at about 0.002 for pursuit velocities ranging from 5.73 to 20.0 deg/sec.

Zegers (1948) used methods identical to Graham et al. (1948) to explore the role of field size on MP thresholds (ω_t). A conversion of his data into M/PRs reveals the same pattern of results—thresholds were stable for pursuit velocities up to approximately 20 deg/sec (the fastest velocity Graham et al. measured). However, Zegers extended Graham and colleagues’ findings by measuring ω_t at much higher pursuit velocities, and found that with pursuit greater than 20 deg/sec, the ratio between ω_t and θ/t increased. For instance, in one condition a pursuit velocity of 29 deg/sec produced a near 10-fold increase in M/PR, and a pursuit velocity of 36 deg/sec produced a 40-fold increase in M/PR. Remember these M/PRs were typically flat between rates of 5 and 20 deg/sec. These results suggest that MP thresholds increase sharply above a pursuit speed of about 20 deg/sec.

Graham and colleagues (1948) and Zegers (1948) wondered about the limiting factors in the threshold values they found. They questioned whether their thresholds, and the changes in thresholds, were “...due to imperfect following of the needles by the eye, or whether they are due to retinal activities to be correlated with the rapid movement of the needles” (Graham et al., 1948, p. 218). These are precisely the questions we sought to address in the current study.

In the current study, we address Graham et al.'s (1948) question by systematically manipulating the parameters of MP (i.e., dθ and dα) in order to measure the minimum M/PR that reliably generates unambiguous depth perception. In addition to depth thresholds, motion thresholds and pursuit accuracies (quantified as gains) were measured to determine when each factor might limit the lower thresholds of depth perception from MP.

2. Material and Methods

2.1. Observers

Eight student volunteers (age range: 19-28 years) from North Dakota State University participated in the experiment. All had normal or corrected-to-normal visual acuity, and gave informed consent to participate. The procedures were overseen by the local Institutional Review Board and in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki).

2.2. Apparatus

Stimuli were generated on a Macintosh computer and presented on a 21” flat screen NEC CRT monitor subtending 15.11 deg of visual angle, with a resolution of 2048 pixels × 1053 pixels × 60 hz for the depth and motion threshold tasks, and 1600 × 1200 × 85 hz for the pursuit tasks. An ASL Eye-trac 6000 (Applied Science Laboratory, Bedford, MA) with remote optics and a sampling rate of 120 Hz was used to measure eye position. The eye tracking system has an accuracy of less than 0.5 deg, and error of less than 1.0 deg. The system communicated eye position information to the stimulus computer through a 16-bit analog connection with a National Instruments multifunction I/O board.

Experiments were conducted in a dimly-lit (~ 1 lux) room, with a viewing distance of 196 cm (except during four pursuit task conditions, which were conducted with a shorter viewing distance of 57 cm; see below). At 196 cm each pixel subtended 20 arcsec; at 57 cm each pixel subtended 90 arcsec. During all three tasks, observer movement was restricted by a chinrest. During the depth and motion threshold tasks, an eye patch occluded the observers’ left eyes for monocular viewing.

2.3. Stimuli

Stimuli were composed of 6,400 40 × 40 arcsec black dots positioned randomly on a white background, contained within a 5.82 × 5.82 deg stimulus window. A small black square (6 arc min) in the center of the stimulus window served as a fixation point. In separate conditions, the stimulus window translated leftward or rightward at one of eight velocities (1.3 – 25 deg/sec) to generate pursuit eye movements (dα) as the observer maintained fixation on the center point. Stimulus window translations began with a positional offset opposite the direction of translation so that the fixation point at the center of the stimulus window would traverse the vertical midline midway through the window's translation. To generate smooth stimulus window translation, an updated window position was calculated and redrawn every 16.7 ms.

Stimulus parallax was achieved through relative motion of the dots, translating in opposite directions, within two different regions of the stimulus window during translation of the entire stimulus window (see Figure 1). Dots within the bounds of one region of the stimulus moved in the same direction as the stimulus window, thereby generating dθ in the same direction as dα. This region should appear nearer than the fixation point. Dots within the bounds of the other region moved in the direction opposite the direction of stimulus window movement, and should appear farther than the fixation point. The two regions corresponded to the upper and lower halves of the stimulus window, and the direction of dot movement within the two regions varied randomly between trials. (To help delineate the different motions in the two regions, a border of dots within 0.25 degrees of the outside edge moved in the opposite direction. However, due to the small size of the border, the small number of dots, the brief stimulus duration, the 3 degree eccentricity, and the requirement to maintain fixation at the center of the stimulus, none of the observers reported seeing this border.) The speed of dot movement (0.025-0.92 deg/sec) was the same for both directions of dot motion, and was determined for each trial using an adaptive procedure (see below); retinal image motion was created by translating stimulus dots laterally every 100 ms. To clarify, the dots were translated seven times throughout the 800 ms stimulus presentation; the dots within the depth stimulus appeared to move smoothly. The overall perception of the motion parallax stimulus was approximately that of a square-wave grating.

Figure 1.

Schematic of the stimulus used in the depth threshold task. In this example, background motion is leftward. A rectangle, positioned flush with the horizontal meridian, contains dot motion in the rightward direction. This rectangle appears to be nearer in depth to the observer. Note that in the stimulus, there is no shadow visible behind the rectangle in depth; the shadow in the schematic is meant to represent the fact that this rectangle is nearer in depth compared to the background. The leftward, rightward, and top edges of the rectangle are 50 pixels from the sides of the stimulus.

To determine how the limits of motion perception affect the limits of motion parallax, the motion stimuli in both the depth task and the motion task (below) were made as similar as possible given the 200 ms motion threshold stimulus duration. Our motion threshold stimulus and method were modeled after Snowden's (1992) experiment measuring minimum displacements. The stimulus was identical to the depth threshold stimulus, except the stimulus window did not translate and remained stationary at the center of the display. As in the depth threshold stimulus, the dots in the region above fixation moved to the left or right, and the dots in the region below fixation moved in the direction opposite those in upper half of the stimulus. When the top half of the stimulus moved rightward and the bottom half moved leftward, “clockwise” (CW) two-dimensional (2D) motion was perceived; when the top half moved leftward and the bottom moved rightward, “counterclockwise” (CCW) motion was perceived. The dots moved once during the stimulus presentation, at 100 ms, as in the depth task; the magnitude of displacement that could be presented on a given trial ranged from 20 arcsec to 380 arcsec. It is important to note that though the motion threshold and depth stimuli were identical, except for the presence of pursuit, in no case did observers report a depth percept in viewing the motion stimulus, nor did either of the authors. The motion stimulus contained only shearing motion, with the dots in the lower and upper regions of the stimulus moving at the same velocity; this lack of a relative velocity gradient coupled with the absence of a pursuit signal precluded any cues to depth. Note, also, that the lack of a depth percept was not due to the short stimulus duration—observers perceive depth from MP for viewing times as short as 70 ms (Nawrot & Stroyan, 2012).

In the pursuit condition, the pursuit target was a single white dot presented on a black background. The target subtended 0.07 × 0.07 deg at 196 cm, and .23 × .23 deg at 57 cm viewing distance. The target translated leftward or rightward at one of 11 velocities (see below).

2.4. Procedure

2.4.1. Depth thresholds

To begin a trial a fixation spot was presented at the center of the display. Following the observer's button press to initiate a trial, the fixation spot jumped either leftward or rightward to indicate the starting position of the stimulus translation. The particular starting position was calculated from the pending stimulus velocity such that the stimulus movement would span the vertical midline 400 ms after stimulus onset. Following a variable interval of 1.5 - 3.5 sec, the stimulus appeared and began to move. Observers maintained fixation on the spot at the center of the translating stimulus throughout the 800 ms duration trial. The stimulus window translated to the left or to the right at one of eight velocities (dα: 1.3, 2.3, 5.0, 6.6, 10.0, 15.0, 18.3, and 25.0 deg/sec), with each velocity run in separate blocks. The order of velocity condition presentation was randomly determined for each observer using a Latin Square.

Observers performed a depth-phase discrimination task in which they indicated by button press the location of the half-cycle of the stimulus (above or below fixation) that appeared nearer in depth relative to the other half-cycle. The threshold M/PR (dθ/dα) for unambiguous depth was determined for each direction of stimulus window translation (leftward or rightward) with an interleaved staircase procedure using a three-down, one-up decision rule (Wetherill & Levitt, 1965). The initial stimulus dot motion (dθ) was set to 0.24 deg/sec and the staircase used steps of 0.049 deg/sec. Each staircase ended with either 6 reversals, which should track to an estimate of the 79% threshold, or with the ceiling (0.92 deg/sec) or floor (0.025 deg/sec) value of dθ. Observers completed 10 blocks for each of the eight different dα (pursuit) velocities.

2.4.2. Motion thresholds

As mentioned above, our motion threshold stimulus and procedure were modeled after Snowden (1992). Snowden measured motion thresholds as small dot displacements in random dot stimuli. Our stimulus parameters and the observers’ task were very similar to his. To begin each trial a fixation spot was presented at the center of the screen. Following a button press by the observer, the stimulus was presented, centered on the fixation spot for a duration of 200 ms. Observers were to maintain fixation on the spot during the stimulus presentation. The observer's task for each trial was to indicate by button press the direction of 2D dot motion (CW or CCW) within the stimulus. For each trial the presentation of CW or CCW motion was determined randomly with an equal probability of either direction of motion.

Similar to the motion parallax threshold procedure, a three-down, one-up staircase procedure (Wetherill & Levitt, 1965) was used to determine an estimate of the 79% motion threshold. Staircases started with a dot displacement of 100 arcsec, moved in 20 arcsec steps, and ended with 6 reversals. Observers completed 10 blocks of trials in the motion perception condition.

2.4.3. Pursuit gains

Data collection began with a 9-point calibration of the ASL eye tracking system, followed by a 2-point calibration of the experimental computer's recording of the eye position signal, and a final 5-point calibration along the horizontal axis of the pursuit target's movement. To initiate a pursuit trial, observers fixated a central fixation target and pressed a button, causing the pursuit stimulus to “step” either to the left or to the right (Rashbass, 1961), and then begin translating across the screen in the opposite direction. The amplitude of the step varied across pursuit velocity conditions, such that for each trial the translating pursuit target passed through the original fixation spot 100 ms after onset of translation. The target was erased from the display at 870 ms, regardless of the distance of translation, and eye position was recorded for an additional 306 ms. The observers’ task was to maintain fixation on the target as it translated. The pursuit target translated to the left or right at one of 11 different velocities, resulting in 11 different conditions. Target velocities of 10.6, 14.9, 19.1, and 25.0 deg/sec were collected at 57 cm viewing distance, thereby keeping the pursuit target away from the edges of the screen for the 870 ms stimulus duration. Target velocities of 0.6, 1.2, 1.8, 3.0, 4.3, 5.5, and 7.3 deg/sec were collected at 196 cm viewing distance, enabling us to present very low velocities (i.e., less than 2.1 deg/sec) using the same temporal update rate. Gains at these lower velocities were of interest because most studies investigate pursuit using target velocities 5-30 deg/sec (Leigh & Zee, 1983; but see Khurana & Kowler, 1987; Kowler & McKee, 1987; Kowler et al., 1984; and Santos, Gnang, & Kowler, 2012). Four recordings were taken at each different velocity: two for leftward translation, and two for rightward translation, resulting in 44 total trials. Four of the observers completed 44 more trials at each of the 11 velocities, resulting in 484 additional trials per observer. A comparison of gains from the initial trials and those from the additional trials will be detailed in section 3. The order of pursuit velocity presentation was randomly determined for each observer.

2.5. Data Analysis

Analyses were conducted in Microsoft Excel and SPSS 21 (SPSS IBM, New York, NY). Lower depth thresholds were quantified using the M/PR (Nawrot & Stroyan, 2009). For each of the 10 blocks, in each condition, the threshold dθ was determined from the mean of the last four reversals (measured in deg/sec), or from the floor (0.025 deg/sec) if the observer's performance did not generate the requisite reversals. One observer consistently reached ceiling in the depth threshold task, and therefore this observer's data were not included in the subsequent analyses. For each of the seven remaining observers, threshold dθ was found for each velocity condition by computing the mean of the 10 threshold dθs from each block. Threshold dθs were then converted to M/PRs by dividing by the particular pursuit velocity (dα) presented. In section 3, we present threshold dθ for each pursuit velocity, as well as observer M/PRs. In the motion perception task, observer thresholds were averaged from the 10 blocks of trials.

For eye tracking analysis, eye velocity was derived from eye position using a 2-point central difference algorithm. Data were then low-pass filtered at 40 Hz using a 3-term moving average filter. Saccades were identified using a velocity threshold of 40 deg/sec (Burke & Barnes, 2006), and removed from further analysis. The first 223 ms and the last 306 ms were discarded, in order to exclude open-loop pursuit and pursuit after target disappearance. Gains were analyzed by averaging velocity over the remaining 54 data points (647 ms), and computing the average velocity/target velocity.

3. Results

The results of the depth task are presented here in two, slightly different, ways. First, the results may be plotted as threshold dθ, the minimum amount of retinal image motion necessary to perceive depth, for different pursuit (dα) velocities. Such a presentation is similar to the results of Graham et al. (1948) and Zegers (1948). However, the link to perceived depth and to the motion/pursuit ratio is more easily appreciated with the motion/pursuit ratio (dθ/dα) plotted on the vertical axis.

Figure 2 shows the mean results of the depth perception task. The mean threshold dθ is plotted for each of the eight different pursuit velocities. Similar to Graham et al. (1948), the threshold dθ values remain relatively constant across pursuit velocities of 5-18.3 deg/sec. Thresholds increase with lower pursuit velocities, and with higher pursuit velocities. At a pursuit velocity of 25.0 deg/sec, the mean threshold dθ value is comparable to those at 1.3 and 2.3 deg/sec.

Figure 2.

Threshold dθs obtained in the depth task. Observer dθ (in deg/sec) values are plotted against pursuit velocity. Bars denote standard errors.

Further analysis demonstrates an asymmetry in threshold dθ values for opposing directions of pursuit. Figure 3 shows the mean threshold dθ values from Figure 2, separated into conditions with naso-temporal (NT) eye movements and temporo-nasal (TN) eye movements. With monocular right-eyed viewing, these correspond to rightward-moving and leftward-moving stimuli respectively. For all pursuit velocities, it appears that TN pursuit is accompanied by higher mean threshold dθ values the for pursuit eye movements in the opposite, NT, direction.

Figure 3.

Threshold dθs split by direction. Thresholds are plotted for NT (white squares) and TN (dark squares) pursuit directions, for all pursuit velocity conditions. Bars denote standard errors.

A 2 (pursuit direction: NT and TN) × 8 (pursuit velocity: 1.3-25.0 deg/sec) ANOVA revealed a significant main effect of direction (F(1,94) = 11.07, p < 0.01) and of velocity (F(7,94) = 6.31, p = 0.00). The interaction was not significant (F < 1.00). dθs were significantly higher for TN stimuli (M = 0.27, SE = 0.04) than for NT stimuli (M = 0.17, SE = 0.03). Pairwise comparisons (Bonferroni corrected) showed that dθ at 1.3 deg/sec was significantly higher than dθs at velocities between 5.0 and 15.0 deg/sec (all p < 0.05), and was marginally higher than dθ at 18.3 deg/sec (p = 0.057) but was not significantly different from dθs at 2.3 and 25.0 deg/sec (p > 0.05). dθ at 2.3 deg/sec was significantly higher than dθ 10.0 deg/sec (p < 0.05), but not significantly different from dθs at any of the other velocities (p > 0.05). dθ at 25.0 deg/sec was significantly higher than dθs at velocities between 5.0 and 15.0 deg/sec (p < 0.05).

In addition to these threshold dθ values determined with the depth task, the motion perception task generated very similar dθ threshold estimates, but in the absence of pursuit eye movements. In the motion threshold task, observers could determine the direction of relative stimulus movement with an average step of 54.3 arcsec (SE = 1.68) midway through a 200 ms presentation. Extrapolating from the 1 step/200 ms presentation of the motion threshold task to the 7 step/800 ms presentation of the depth threshold task gives a mean threshold dθ value of 0.13 deg/sec. This is very close to the best dθ thresholds shown in Figure 2. More importantly, this value can help us predict the optimal depth perception threshold if we assume it represents one limit on the perception of depth from motion.

As mentioned above, another way to describe these motion parallax thresholds is in terms of the M/PR. The M/PR, unlike dθ alone, provides a more direct link to perceived depth. That is, the same M/PR (dθ/dα) means the same relative depth (d/f), regardless of the particular values of dθ or dα . This makes M/PR a more meaningful depth threshold measure.

Figure 4 shows the mean M/PR threshold value for each of the eight velocity conditions. The square symbols show the mean M/PR threshold values determined from the motion parallax depth task. The triangular symbols show the “ideal” M/PR threshold values based on the mean motion perception threshold, determined in the absence of a pursuit signal. The comparison of the ideal and obtained M/PR thresholds show that observer thresholds are near the ideal with pursuit velocities between 5 and 18 deg/sec. The M/PR threshold values at the lowest pursuit velocities (1.3 and 2.3 deg/sec) are much higher than the ideal values. There also appears to be a slight increase in the M/PR threshold values at 25.0 deg/sec compared to the ideal, even though the M/PR threshold value is lower than is found at many other pursuit velocities. This issue will be addressed in more detail in section 4.

Figure 4.

Obtained and ideal thresholds. Observer threshold values are plotted against pursuit velocities. The upper line (squares) plots obtained thresholds. Bars denote standard errors. The bottom line (triangles) represents idealized thresholds, if dθ were equal to the extrapolated lowest motion threshold, 0.13 deg/sec.

Similar to the directional asymmetry seen with the dθ threshold values (Figure 3), Figure 5 shows the mean M/PR threshold values for the two different pursuit directions. Though the pattern of results is the same for both directions, M/PR thresholds were higher for stimuli moving in the leftward (TN) direction than for stimuli moving in the rightward (NT) direction.

Figure 5.

Thresholds split by pursuit direction. Thresholds are plotted for NT (white squares) and TN (dark squares) pursuit directions, for all pursuit velocity conditions. Bars denote standard errors.

A 2 (pursuit direction: NT and TN) × 8 (pursuit velocity: 1.3-25.0 deg/sec) ANOVA confirmed that there were significant main effects of pursuit direction (F(1, 94) = 8.04, p < 0.01), and pursuit velocity (F(7, 94) = 43.05, p = 0.00). The direction × velocity interaction was also significant (F(7,94) = 2.47, p = 0.02). Thresholds for stimuli moving in the NT direction were significantly higher than those moving in the TN direction at the two lowest velocities (i.e., 1.3 and 2.3 deg/sec), but not at the other five velocities. Pairwise comparisons (Bonferroni corrected) revealed significant differences between thresholds (collapsed across pursuit direction) at different pursuit speeds. Thresholds at 1.3 deg/sec were significantly higher than thresholds at all other pursuit velocities (all p < 0.05). Thresholds at 2.3 deg/sec were also higher than thresholds at faster pursuit velocities (i.e., 5.0 deg/sec and higher; all p = 0.00). None of the other threshold values were significantly different from one another at any other pursuit velocity (5.0-25.0 deg/sec; all p > 0.05).

In addition to depth thresholds, we also analyzed the factors affecting pursuit gains to determine observers’ pursuit accuracy, and to evaluate how dα signals influence depth thresholds. Recall that we first collected 44 trials from all seven observers, and collected 484 additional trials from four of the observers. Table 1 shows average pursuit gains for all 11 velocities, for all observers who completed the initial trials. Table 2 shows pursuit gains for the four observers who completed additional trials. Though standard errors decreased with additional trials (see Tables 1 and 2), it is apparent in Figure 6 that gains were very similar across the two data sets. Because variability was lower when more trials were completed, it is this set of data that were used in the following statistical test; however, results from either set of data support our conclusions.

Table 1.

Mean Gains for Observers in Initial Pursuit Trials

Observers Initial		0.61	1.22	1.83	3.05	4.26	5.48	7.31	10.63	14.88	19.13	25.51
1	L	0.53	0.97	1.16	0.89	1.17	0.6	0.71	1.09	1.05	0.67	0.44
	R	1.24	1.11	1.37	1.2	1.2	1.1	0.76	1.11	0.81	0.68	0.52
2	L	1.11	1.08	1.00	1.13	1.23	1.3	1.14	1.02	0.85	0.68	0.71
	R	0.72	0.84	1.02	0.85	1.14	1.08	1.08	1.05	0.94	0.82	0.62
3	L	0.99	1.16	0.63	1.04	0.88	0.89	0.96	1.00	1.09	0.95	0.85
	R	1.35	0.86	0.97	1.22	1.1	1.33	1.09	1.03	0.98	0.94	0.77
4	L	0.8	1.10	0.96	1.10	1.00	1.03	1.05	1.08	1.11	1.02	0.87
	R	0.94	1.13	1.08	1.24	1.13	1.03	1.12	1.15	1.10	0.95	0.93
5	L	0.24	0.78	0.55	0.85	0.95	1.13	1.13	1.04	1.07	0.99	0.94
	R	0.84	1.13	1.18	1.27	1.38	1.07	0.95	1.04	1.00	0.97	0.75
6	L	0.81	1.18	1.04	1.04	1.09	0.88	0.85	1.08	1.06	0.9	0.78
	R	0.63	0.84	0.98	1.01	1.30	1.13	1.05	1.03	1.00	0.99	0.81
7	L	1.94	1.12	1.43	1.05		0.95	0.82	0.62	0.57	0.93	0.68
	R	2.73	1.30	0.84	1.45	1.20	1.51	1.47	0.77	0.86	0.85	0.48
Mean	L(SE)	0.92 (0.21)	1.06 (0.05)	0.97 (0.11)	1.02 (0.04)	1.05 (0.06)	0.97 (0.08)	0.95 (0.06)	0.99 (0.06)	0.97 (0.07)	0.88 (0.05)	0.75 (0.06)
	R(SE)	1.21 (0.27)	1.03 (0.07)	1.06 (0.06)	1.18 (0.07)	1.21 (0.04)	1.18 (0.07)	1.07 (0.08)	1.03 (0.05)	0.96 (0.04)	0.89 (0.04)	0.70 (0.07)
	Overall(SE)	1.06 (0.24)	1.04 (0.06)	1.01 (0.09)	1.10 (0.06)	1.13 (0.05)	1.07 (0.08)	1.01 (0.08)	1.01 (0.06)	0.96 (0.06)	0.88 (0.05)	0.73 (0.06)

Table 2.

Mean Gains for Observers in Additional Pursuit Trials

Observers Addn'la		0.61	1.22	1.83	3.05	4.26	5.48	7.31	10.63	14.88	19.13	25.51
2	L	0.96	0.91	1.04	1.12	1.12	1.11	1.10	1.05	1.02	0.89	0.81
	R	0.64	0.97	0.97	0.94	1.04	1.00	0.93	1.03	0.95	0.87	0.69
5	L	0.89	1.10	0.97	1.01	1.05	1.07	1.03	1.09	1.14	1.09	0.91
	R	0.96	1.03	0.96	1.05	1.06	1.07	0.95	1.00	1.06	1.00	0.78
6	L	0.77	0.88	1.03	1.02	1.08	0.98	0.95	1.04	0.97	0.88	0.77
	R	1.14	1.17	1.04	1.19	1.09	1.16	1.15	0.99	1.02	0.98	0.83
7	L	1.23	1.23	1.07	1.16	1.01	1.07	0.98	0.96	1.01	0.96	0.85
	R	1.25	0.94	0.91	1.19	1.14	1.19	1.07	1.00	0.93	0.87	0.74
Mean	L(SE)	0.96 (0.10)	1.03 (0.08)	1.03 (0.02)	1.08 (0.04)	1.07 (0.02)	1.06 (0.03)	1.02 (0.03)	1.04 (0.03)	1.04 (0.04)	0.96 (0.05)	0.84 (0.03)
	R(SE)	1.00 (0.14)	1.03 (0.05)	0.97 (0.03)	1.10 (0.06)	1.09 (0.02)	1.11 (0.04)	1.03 (0.05)	1.01 (0.01)	0.99 (0.03)	0.93 (0.03)	0.76 (0.03)
	Overall(SE)	0.98 (0.10)	1.03 (0.03)	1.00 (0.02)	1.09 (0.04)	1.08 (0.00)	1.08 (0.02)	1.02 (0.01)	1.03 (0.02)	1.10 (0.03)	0.94 (0.04)	0.80 (0.02)

^aAddn'l = Additional

Figure 6.

Mean gains by pursuit direction for both initial and additional trials, for all pursuit velocities. Mean gains (eye velocity/target velocity) are plotted as a function of pursuit velocity. White data points represent the data for the initial set of trials (i.e., 44 trials). Black data points represent the data for the additional set of trials (i.e., 484 trials). Squares represent stimuli moving in the NT pursuit direction, and triangles represent stimuli moving in the TN pursuit direction. Bars denote standard errors.

A 2 (direction: left [TN] and right [NT]) × 11 (pursuit velocities: 0.61-25.51 deg/sec) ANOVA revealed a main effect of velocity (F(10,88) = 4.88, p = 0.00). Gains at 25.51 deg/sec were significantly lower than gains at all velocities but 19.13 deg/sec (p = 0.38). No other differences in gains were significant (all p > 0.05). There was no significant effect of pursuit direction on gains.

4. Discussion

Motion parallax thresholds found here are as low as possible given the observer's limits in the ability to detect retinal image movement. However, this is only true for the range of pursuit velocities between about 5 - 20 deg/sec. Above and below this range, MP thresholds diverge from this motion detection limit, and then appear to be affected by changes in the observer's pursuit ability.

These results are similar to the results of Graham et al. (1948) and Zegers (1948), who found that MP thresholds were constant for pursuit velocities between ~ 5 - 20 deg/sec. Zegers also investigated thresholds for pursuit velocities higher than 20 deg/sec, and found an increase in MP thresholds with increasing pursuit velocity. In the current study, notice that pursuit gain was less than unity (i.e., approximately 0.73) at a pursuit velocity of 25.5 deg/sec. The increase in thresholds at 25 deg/sec pursuit may be explained by the drop in pursuit accuracy. Pursuit accuracy (measured as gain) begins to decline at velocities higher than 20 deg/sec (Leigh & Zee, 1983; Lisberger, Morris, & Tyschen, 1987). It is likely, then, that higher thresholds in the 25 deg/sec condition are due to an inaccurate dα signal.

In addition to directly limiting thresholds at high pursuit velocities, pursuit eye movements may also limit the ability to optimally use dθ signals to determine MP depth thresholds at low velocities. Notice that below pursuit velocity of ~ 5 deg/sec the average pursuit gains are marked by increased variability. This is likely contributing to the greater than ideal MP threshold at these lower pursuit velocities.

Finally, the use of the M/PR allows these MP thresholds to be compared typical binocular disparity thresholds. That is, these M/PR thresholds can be converted into d_MP, depth depicted in the underlying stimulus geometry, through the relationship given in equation 1. At a viewing distance of 196 cm, an M/PR of 0.01 corresponds to a depth of 1.96 cm. This viewing geometry (d = 2 cm, f = 196 cm) would produce a retinal disparity of about 1.1 arcmin. Considering the MP stimulus was translating laterally, and viewed for only 800 ms, we believe these MP thresholds compare quite well with binocular disparity thresholds.

The M/PR provides a reliable and simple way to quantify depth thresholds, and the current study provides an important extension of previous findings. Past research has described lower MP depth thresholds in terms of equivalent disparity (ED) (Ono & Ujike, 2005). ED is given by the distance-squared law for MP, which equates perceived depth from MP with depth from BD (Nawrot, 2003b; Rogers & Graham, 1982):

$$d_M=({D_M}^2\ast \mu )∕t$$ (2)

where d_M is the stimulus depth, D_M is viewing distance, μ is equal to the magnitude of disparity generated by stimulus translation, and t is the magnitude of observer (head) translation. Essentially, this equation describes the magnitude of depth from MP as determined by the amount of stimulus movement that occurs for a given magnitude of head movement (usually 6.2-6.5 cm, representing interocular distance).

Equivalent disparity serves as one starting point for quantifying depth from MP, and is especially useful for directly comparing depth percepts from BD and MP. However, when describing depth thresholds, ED may be too coarse an approximation of depth (Nawrot & Stroyan, 2009). Using ED to quantify thresholds does not take into account dα signals, precluding a role for pursuit velocity in depth from MP. It is also important to note that MP is dynamic, but ED variables are static. ED also does not describe the underlying mechanisms of MP. ED is therefore a less appropriate quantification of MP depth thresholds than the M/PR, which explicitly accounts for dα signals and the dynamic nature and underlying mechanisms of depth from MP.

Another important consideration when examining results of depth threshold studies is that many studies employ head movements to generate the extra-retinal signal necessary for unambiguous depth from MP (see, e.g., Ono & Ujike, 2005; Ujike & Ono, 2001). When a head translation is made, two different types of eye movements are generated—the translational vestibulo-ocular response (tVOR) and smooth pursuit (Leigh & Zee, 1983; Miles & Businetti, 1992). Both types of eye movements help an observer to maintain fixation on a point in space as the observer (or the scene) translates, but it is only the smooth pursuit signal that contributes to the generation of depth perception from MP (Nawrot & Joyce, 2006). The tVOR signal, which is unnecessary for depth perception from MP, must be partitioned from the total head translation signal in order to estimate the pursuit signal, which provides the dα signal for the M/PR. The relative contribution of pursuit in maintaining fixation during head translation has been estimated to be approximately 0.5 (Liao et al., 2008; Nawrot & Joyce, 2006; Ramat & Zee, 2003).

We used this estimate of the pursuit component during translation to estimate thresholds from Ujike and Ono's (2001) work, in order to make direct comparisons with the results of the current study. First we adapted their data into log-log coordinates (see Figure 7, adapted from their Figure 4). Next, we converted their data into M/PRs. First, head velocity was converted to pursuit velocity. Using a pursuit component of head movements at 0.57 (based on Nawrot & Joyce, 2006), Ujike and Ono's (2001) head velocities ranging from 2-60 deg/sec corresponded to pursuit velocities of 1.14- 25.65 deg/sec. Then the minimum amount of relative image velocity for each pursuit velocity was estimated from results depicted in Figure 7, and M/PRs were generated by taking the ratio of estimated relative image motion to pursuit velocity.

Figure 7.

Data from Ujike and Ono (2001). Results adapted from Ujike and Ono's study and re-plotted in log-log coordinates. Relative image velocity is shown as a function of head velocity.

Figure 8 shows Ujike & Ono's (2001) original data, re-plotted as M/PRs. Compare this figure to our data plotted in Figure 4, and it becomes apparent that the pattern of results is very similar. Ujike and Ono's results show a relative stabilization of threshold values at approximately 8.0 deg/sec and higher pursuit velocities, with increasing thresholds at lower pursuit velocities. Though the pattern of results is very similar, their estimated threshold values are much lower than ours. This may be due to some information that was lost in the recalculation of their data into M/PRs. Ujike and Ono also allowed unlimited viewing time, whereas we used stimulus durations of only 800 ms, which could contribute to higher thresholds in the current study.

Figure 8.

Data estimated from Ujike and Ono (2001). From Ujike and Ono's results, we estimated observer M/PR threshold values. Thresholds are plotted as a function of estimated pursuit velocity. Compare this figure to the obtained line in Figure 2.

To summarize, the lower limit of depth perception from MP is determined by an interaction of image motion and smooth pursuit mechanisms. Changes in thresholds with different pursuit velocities are due to the limits of dθ and dα signals, both independently and in interaction with each other. This is apparent in the relative stability of thresholds across moderate pursuit velocities, as well as the higher thresholds associated with low and high pursuit velocities. A direct comparison between the findings of the current study and past research (Graham et al., 1948; Ujike & Ono, 2001; Zegers, 1948) reveals very similar patterns of results. The M/PR provides a very useful way of quantifying and describing lower depth thresholds, because it takes both motion and pursuit mechanisms into account; other methods, such as using ED, cannot explain changes in thresholds with stimulus and pursuit velocity.

Highlights.

• Motion parallax thresholds are stable at moderate pursuit velocities

• Thresholds at moderate pursuit velocities are limited by motion processing

• Non-optimal pursuit signals limit thresholds at high and low pursuit velocities

• Motion parallax thresholds are comparable to binocular disparity thresholds