Draft version

Published version: Journal of Experimental Psychology: Human Perception and Performance,

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Frank H. Durgin
Department of Psychology
Swarthmore College
500 College Avenue
Swarthmore, PA 19081-1397
PHONE: (215) 328-8678
FAX: (215) 328-7814

Abstract

The accuracy of depth judgments based upon binocular disparity or relative motion (motion parallax and object rotation) was compared in two experiments. A third experiment on stereoscopic depth constancy (the scaling of disparity information with distance) is also reported. In the first experiment, depth judgments were recorded for computer simulations of cones specified by binocular disparity, motion parallax, or stereokinesis (illusory structure from motion). In the second experiment, judgments were recorded for real cones in a structured environment, with depth information from binocular disparity, motion parallax, or object rotation. In both experiments, judgments from binocular disparity information were quite accurate, but judgments based upon geometrically equivalent or more robust motion information reflect poor recovery of quantitative depth information. In the third experiment, stereoscopic depth constancy was demonstrated for distances of 1 to 3 m using real objects in a well-illuminated, structured viewing environment in which monocular depth cues (e.g., shading) were minimized.

Comparing Depth From Binocular Disparity to Depth from Motion

It has been pointed out that the geometrical information that supports the perception of depth from binocular disparity is actually less determinate than that which supports the recovery of structure from object rotation or motion parallax (Richards, 1985). Stationary binocular stereopsis provides two 2-D perspectives on a 3-D object, whereas object rotation or motion parallax provide a continuous sampling of 2-D perspectives in the form of relative motions. The latter situations, which entail having information obtained from more than two views, can theoretically provide a more accurate (Euclidean) geometric recovery of the distal object's 3-D configuration, so long as object rigidity is assumed. However, this analysis makes other assumptions about the kinds of information available to an observer that are probably not warranted in practice. The studies here presented show that, within the range of parameters investigated for real and simulated objects, quantitative depth judgments based upon information from object rotation and motion parallax are much less accurate than those based upon binocular disparity.

The geometry of disparity and relative motion information

When an object is rotated about an axis other than the line of sight, the relative motions of features upon the object can specify the 3-D structure of the object to monocular vision (see the Appendix for a mathematical analysis). A similar recovery of structure can be obtained if the object and the viewer undergo an angular displacement with respect to one another. Although the geometry of displacement differs from that of rotation, in both cases the object is observed from more than one angle. If the observer maintains fixation on the object during angular displacement, the two cases are practically identical for small rotations/displacements. Note that it is the angular rotation of the object with respect to the viewer that is crucial; a simple translation toward or away from a monocular viewer does not provide geometric depth information(Note 1). Recovery of structure from object rotation is often referred to as the kinetic depth effect (KDE, Wallach & O'Connell, 1953; cf. also Braunstein, 1962). Depth information specified by viewer or object translation is referred to as motion parallax. Because both KDE and motion parallax with fixation may be shown to involve similar geometrical analyses, we will refer to both as structure from motion (SFM)(Note 2).

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Figure 1. Geometric similarity between disparity, motion parallax and object rotation. The same two views (D) are provided by (A) binocular disparity (two simultaneous views) (B) motion parallax (two successive views with a single displaced eye), and (C) object rotation (two monocular views of a rotating object ). If a single fixating eye is displaced along a circular arc with a radius equal to the distance to the object, the geometry of the proximal stimulus is similar to KDE (rotation at a fixed distance). Such a curved path is illustrated in Panel A, as is the straight motion parallax path. The line of sight to the center of the base is shown as fixed here, but any stationary point of fixation could be used.

The geometry of binocular disparity is similar to that of the two SFM cases. Viewing an object with two eyes provides views at two different angles to the object. Assuming central fixation, the two views of binocular disparity are equivalent to (a) a single eye displaced the interocular distance, or (b) two static frames of object rotation. These geometrical equivalences are illustrated in Figure 1, which shows how three points on an object bear the same relationship in the two represented views of (A) binocular disparity (B) motion parallax with fixation, and (C) object rotation. If the eye is fixating the object in the motion parallax case (i.e., tracking) then the two views (D) are identical to those of rotation and binocular disparity. Note that our discussion here is geometrical rather than psychological. For example, Simpson (1993) has reviewed the evidence that velocity rather than displacement information is used in the perceptual recovery of structure from motion, and other characterizations of the optical information available in these situations are possible (e.g., Lappin, 1990). Included in Panel A are the paths a single eye would take to generate KDE-like rotation (curved path) or motion parallax (straight, dashed line). The differences between the intermediate frames of continuous motion parallax and object rotation are fairly slight for small rotations and translations, and, as we will argue, unimportant in the recovery of depth. The three points chosen here are not special. The same equivalence of geometry applies to every point on the object.

It is well known that a single pair of frames is insufficient to determine metric shape, unless the transformation between the two is known. In Figure 1D the same relative displacement (or velocity) of the central point with respect to the others could arise from a shallow object with a large change of angular perspective or from a deep object with a small change in angular perspective. Disparity information, in itself, cannot distinguish(Note 3) between a near, shallow object and a far, deep object, because the angle of virtual rotation in the binocular disparity case is greater for nearer objects. In the motion cases, the angular rotation is not perfectly correlated with viewing distance, but the same ambiguity about the depth of the object would remain given only two frames of relative displacement (or relative velocity) information: It could be a deep object undergoing a small rotation or a shallow object undergoing a large rotation. The ambiguity we wish to emphasize concerns the shape of the object, not the scaling factor associated with distance. Independent of stereo or motion, if you only have two views, you need to know the angle between them. If this angle can be recovered, then so can the depth relative to the 2-D projected size of the object.

There are lots of reasons to believe, as suggested by Richards (1985), that depth information from motion is superior to that from disparity. Static stereopsis is inherently a two-frame problem. On the other hand, with motion you have a continuous sequence of frames. As we shall see, the presence of this additional information is a necessary condition for recovering angular rotation information in a monocular SFM display. Furthermore, because interocular distance is fairly small, at a fixation distance of 1 m, the angular rotation of an object provided by binocular disparity is less than 4 deg, whereas motion parallax information, such as provided by moving around in an environment, can easily exceed that. However, a closer examination of the problem will indicate that neither of these advantages is sufficient to produce accurate recovery of the angle of rotation for SFM with small rotations (e.g., less than about 15 deg). In the Appendix we derive equations for determining object shape from relative velocity and from disparity information. Although velocity information is similar to disparity information, the disparity equation requires only knowledge of fixation distance to determine shape and size, whereas the motion cases require information about angular rotation to determine shape (scaled depth), and distance information to determine size.

Recovering depth from SFM displays

As argued above, displacement (or instantaneous velocity) information is insufficient to provide quantitative depth information in the absence of information about the object's quantity of rotation or angular velocity. To make this more precise, consider a rigid object centered at the origin and rotating about the Y axis, as shown in Figure 2.

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Assuming orthographic projection, an arbitrary point will have image -coordinate Rcos(L) and image velocity = -Rsin(L) = -Z . That is, the depth of every point is given by its image velocity scaled by the inverse of the angular velocity. Thus unless is known or other information is available, depth can be known only up to a multiplicative constant.

Quantitative depth can be recovered if additional information is available. Differentiating gives the image acceleration = -Rcos(L) = - . Since is known, acceleration determines up to a sign reversal. The sign ambiguity can be resolved by resorting to perspective analysis, or by heuristics based on assuming opaque objects. In any event, velocity plus acceleration is sufficient to provide quantitative estimates of object shape.

An alternative to using acceleration information is to observe changes in relative position over a non-infinitesimal interval, and make use of some additional information. Consider two points with a common Z coordinate = (,,Z) and = (,,Z). (Such points are identifiable because their instantaneous velocities are the same.) After a rotation through an angle L, the X coordinates of the points become = cos(L) + Z sin(L) and = cos(L) + Z sin(L) and the change in distance (foreshortening) between the two points is - = (-)cos(L). In their analysis of the stereokinetic effect (SKE; Musatti, 1924), Proffitt, Rock, Hecht, and Schubert (1992) refer to the common (Z sin(L)) component of displacement as the between-contour motion, and call the change in relative position ( cos(L) - cos(L)) the within-contour motion. The former contains all the information about depth, but that information is confounded with the rotation angle L. The latter (i.e. the foreshortening) can in theory be used to specify L and permit quantitative depth judgments.

Although velocity plus either acceleration or foreshortening suffices to determine the angle of rotation of an SFM stimulus, there is substantial evidence that this information is not always used by humans for recovery of metric depth information. For example, humans are quite bad at perceiving and judging acceleration (Gottsdanker, Frick & Lockard, 1961; Simpson, 1994). Moreover, Caudek and Proffitt (1993, 1994) compared depth judgments for SFM stimuli to those obtained with SKE, which differs only in that foreshortening information is absent. They found that, for small effective rotations, judgments were similar and that the presence of foreshortening information in the SFM cases had little effect on perceived shape. This result is consistent with the idea (Todd & Bressan, 1990) that human shape judgments typically do not make effective use of higher order temporal relations such as acceleration, or comparisons of more than two views of the scene.

The case can be restated in terms of pragmatic considerations: For an angular rotation of 15 deg, the foreshortening of the distance between any two points initially parallel to the projection plane will amount to a decrease in length by 3.5%. Distinguishing a total angular rotation of 10 deg from an angular rotation of 5 deg involves distinguishing a compression factor of 0.985 from a compression factor of 0.996. But whereas these compression factors differ by only 1%, the scaled object depth implied by these two angles (for a given velocity gradient) would differ by a factor of two. We expect that it would be quite difficult to accurately discriminate this magnitude of compression information. It therefore seems that for small object rotations, or over small increments of rotation, the expected sensitivity of the motion system to quantity of rotation, and therefore quantitative depth, should not be very good.

Caudek and Proffitt (1993, 1994) have argued that perceived depth in such situations is determined more by heuristics and biases than by an appropriate geometry. Specifically, they reported (1) a bias to see SFM-specified objects as about as deep as they are wide and (2) a heuristic that objects with greater relative motions were deeper.

Before turning to the case of binocular disparity, a comment should be made about the angular information theoretically available in motion parallax. If an observer is viewing a translating object and knows that it is translating without rotating, then the angle of object rotation is given by the visual angle subtended by the translation. However, motion parallax produced by object translation suffers ambiguity on the grounds that, insofar as the accurate recovery of small angular rotations is difficult, it is pragmatically impossible for an observer to distinguish between an object that is translating and rotating and an object that is translating only. When the viewer is moving and the object is not, there are so many degrees of freedom of observer motion that the angle of rotation may again be difficult to determine visually, though the observer is quite aware of the moment-to-moment egocentric position of the object. Thus, for example, concomitant object rotation is sometimes reported in motion parallax displays, with some (appropriate) reduction in depth (Ono, Rivest, & Ono, 1986). There are a number of reports that depth from motion parallax and depth from object rotation are processed similarly, even when the motion parallax is produced by self motion (Braunstein, Liter, & Tittle, 1993; Caudek & Proffitt, 1993, 1994; Loomis & Beall, 1993), though others report some advantage to viewer-produced motion parallax (Ono & Steinbach, 1990). We believe that for small rotations angular information is not accurately recovered in either case.

Recovering depth from disparity

We have seen that disparity information, in itself, is inadequate to recover depth. Because depth specified by disparity varies as the square of distance, the same disparity information could specify a near, shallow object or a far, deep object (Richards, 1985). However, because interocular distance is a fixed quantity, the angle of virtual rotation is fully specified in the case of binocular disparity by knowing the distance to the point of zero disparity. Therefore, as shown in the Appendix, if precise information about the distance to the object is available then disparity information can be converted into absolute depth perception. Accommodation, the adjustment of the eye's lens necessary to focus images, and vergence, the rotation of the eyes for fixation in depth, have both been established as effective, if imperfect, indicators of distance, which are yoked in active vision (c.f., Owens, 1987, for a review). Moreover, there is normally an abundance of other sources of information about egocentric distance to objects including perspective, familiar size, etc. Although using such distance information may seem redundant (if distance can be precisely known, so can depth), consider that the task of identifying fixation distance is different than the task of perceiving object shape. The visual system can accomplish the latter using disparity information if a single value of fixation distance is given. Thus, although the angle of virtual object rotation available to binocular viewing at typical distances is small, the precision and accuracy of depth perception is limited by the precision and accuracy of distance perception. In fact, it is well established that perceived depth from disparity does depend upon perceived distance (Ono & Comerford, 1977), a fact ironically illustrated by the following:
One observation which has been (mistakenly) taken as support for the idea that SFM is treated as visually superior to binocular stereopsis is the well-replicated finding that stereoscopic judgments of object depth appear to be "recalibrated" after viewing the rotation of a 3-D wire object through a telestereoscope, an apparatus which optically alters interocular separation and thus disparity (Wallach, Moore, & Davidson, 1963). The interpretation originally offered for this effect was that the disparity system was being recalibrated to a new interocular distance based upon information from the SFM system, and the intended implication was that SFM was more reliable than stereopsis. However, the use of the telestereoscope incidentally introduced a mismatch between vergence information and actual optical distance (which drives accommodation). The aftereffect can actually be shown to depend on the recalibration of the yoking between the vergence and accommodative systems: It can be produced both in the absence of stereopsis and in the absence of motion, but is not produced by the presence of (tele)stereopsis and motion when artificial pupils are used to reduce blur-driven accommodation (Fischer & Ebenholtz, 1986). In other words, contrary to prior claims(Note 4), the aftereffect depended upon the visual system using (distorted) distance information provided by accommodation in its derivation of depth from disparity.

Comparing the perception of depth from disparity and depth from motion

We have argued that the instantaneous geometric information provided by disparity, motion parallax, and object rotation are roughly equivalent: A map of the disparities produced by an object viewed binocularly is similar to a velocity map of that object either rotating or translating through the same angle (with fixation). Although the recovery of veridical depth is possible if more than two frames of motion are used, we believe that there are practical difficulties in extracting such information and that empirical evidence suggests that these difficulties are insurmountable for small rotations.

Proposals that motion information is superior to disparity information are further complicated by the finding that the two extreme frames of a SFM sequence are reported to produce the same subjective depth as a more continuous multi-frame case and that affine structure, rather than Euclidean, is readily recovered from motion (Todd & Bressan, 1990). Because two-frame motion stimuli suffer the same theoretical inadequacies as static stereo displays discussed above, a natural hypothesis is that information from disparity and motion might be processed by the same algorithms. Indeed, certain similarities between the systems have been noted (e.g., Graham & Rogers, 1982; Kontsevich, 1994; Rogers & Graham, 1982). However, one might predict that the binocular system would have some advantage in deriving object depth since interocular displacement is a fixed quantity within the system. Rotation magnitude is less easily determined when observing object rotations, and the information specifying it apparently is not used for small rotations (Caudek & Proffitt, 1993, 1994).

If the angle of object rotation is not used in recovering structure from motion, but distance information is used in recovering depth from disparity, then judgments of object depth based upon disparity information ought to be superior to those based upon recovery of structure from motion for small rotations. The primary purpose of present studies was to assess this possibility by directly comparing the perceived depth of objects presented with either binocular disparity or monocular motion information. Both computer simulations of objects (Experiment 1) and real objects (Experiment 2) were used. Because of a recent demonstration of a failure of stereoscopic depth constancy (Johnston, 1991) with stereograms viewed in a dimly lit environment, a secondary goal, accomplished in Experiment 3, was to demonstrate stereoscopic depth constancy over a range of distances from 1 to 3 m with real objects in a well-structured viewing environment.

Experiment 1

In the first experiment, comparisons were made between depth perceived from monocular motion or geometrically equivalent binocular disparity information presented by computer simulation. The objects simulated were cones which varied in depth along their principal axes(Note 5). The cones were represented by five evenly spaced circular contours. In two conditions of the experiment, the depth of the cones was specified either by disparity information or by motion parallax information provided by cone translation.

A third condition was run in which stereokinetic cones were presented which were geometrically matched to those of the other two conditions and which appeared as rotating cones. Proffitt, et al. (1992) have shown that the stereokinetic phenomenon may be defined as the subset of information available in a KDE display when contour foreshortening is absent. True cone rotation was not simulated because the appropriate foreshortening for a 2.5 deg rotation in each direction is less than 0.5% of the contour diameter, which was close to the resolution of the monitor for even the largest circular contour. This condition was run for comparison with motion parallax, in which the appropriate foreshortening is accomplished, but by the change in angle of regard of the observer.

Method

Subjects.

The subjects were 30 University of Virginia undergraduates who were able to identify simple forms in random dot stereograms. Fifteen men and 15 women were divided equally among three experimental conditions. Four additional subjects were excluded from the study either because they failed to perceive depth in a random dot stereogram or because of experimenter error.

Apparatus.

The displays were produced on a Sun 3/60 workstation with a color monitor and were viewed through a stereoviewer built with first-surface mirrors. The left and right eyes received images from the left and right halves of the screen respectively, but the images were optically rotated(Note 6) such that the apparent screens were 22.9 cm x 14.6 cm (900 x 576 pixels), and were viewed at an optical distance of 72 cm. The mirrors were set so that the vergence angle of the eyes would match the optical distance to the screen for the typical observer. In the monocular conditions, stimuli were presented only to the right eye, and the left channel of the viewer was occluded.

Stimuli.

Cones of nine different depths were simulated in each of three conditions, binocular disparity (BD), motion parallax (MP) and stereokinesis (SKE). The cones were represented by five white circular contours presented against a black background. The base (largest circle) of each cone was 6 cm (4.8 deg) in diameter, and the simulated tip-to-base depths were 2.4, 3.0, 3.6, 4.5, 6.0, 9.0, 12, 15, and 18 cm . In the BD condition, the simulated stationary cones pointed directly at the observer. An interpupillary distance of 6.2 cm was assumed(Note 7), and thus each eye received an image of a cone seen along lines of sight 2.46 deg from the axis of the cone. The two motion conditions were designed to match the geometrical information provided in the disparity condition, but were presented monocularly. In the MP condition, the simulated cones moved continuously from left to right and back. The total displacement of the base was 6.2 cm, or 2.46 deg in either direction, with a speed of 4.13 cm/sec. Thus, at either extreme, the projected image was geometrically similar to that presented to one of the eyes in the disparity condition. Likewise, in the SKE condition, motion of the contours corresponded to a (KDE) cone rotation of 2.46 deg in each direction but without any foreshortening of contours. Because of the absence of foreshortening, SKE cones do not have a true simulated height. We did not attempt to simulate foreshortening information (necessary to specify true KDE) because of the resolution limits of the monitor.

A mouse-adjustable icon was simultaneously presented on the screen. The icon began as a vertical line and could be adjusted by the subject to form an isosceles triangle with the original line as the base. The icon represented a side view of the cone that the subject could use to indicate the perceived depth (shape) of the object. The base of the triangle was equal to the diameter of the base of the cone, and the cone's five contour lines were represented as lines on the triangular icon for clarity. Note that correct adjustment of the icon is a test of the perceived depth-to-base ratio and does not require accurate 2-D size perception.

Procedure.

Subjects were pseudo-randomly assigned to one of the three conditions. Each subject was required to identify a letter presented as a random-dot stereogram through the stereo viewer. Subjects were then instructed that they would be viewing simulated cones through the viewing box and were to adjust the mouse-controlled triangular icon on the screen to indicate the apparent depth (shape) of the cones. For subjects in the monocular viewing conditions, the left channel of the viewer was occluded to prevent light from reaching the left eye.

Four blocks of nine pseudo-randomly ordered stimulus trials were presented to the subject. Each of the nine cone depths was presented once per block. The first block was considered practice, and the data were not analyzed. Subjects were not informed about the number of distinct stimuli, nor the range of sizes, nor were the experimental blocks distinguished for the subject. On each trial, the subject had unlimited time to view the stimulus and adjust the mouse-controlled triangular icon. The icon and the cone were both present on the screen until the subject pressed a mouse button to indicate a satisfactory match between the icon and the cone. Depth judgments were recorded in pixels (40 pixels/cm) and converted to cm. Several subjects in the binocular disparity condition reported difficulty in fusing the tips of the deepest cones. This is consistent with the findings of Ogle (1952, 1953) who showed that there is a limited range of fusible disparities beyond which exists a range of "patent stereopsis" in which some depth is perceived despite a failure of binocular fusion.

Results

Analysis of means.

The mean judged depths for each cone in each condition are plotted in Figure 3. It is evident that subjects in the Binocular Disparity condition are fairly accurate at judging the size of the cones, while the judgments of the other two groups do not reflect much sensitivity to the geometrically defined depth. Between subject variability is indicated by the error bars in Figure 3, which represent the standard error of the mean. A 3 x 9 x 3 (Condition x Simulated Depth x Block) mixed design repeated measures ANOVA revealed a reliable main effect of Condition, F(2,27) = 6.06, p<.01, and of Simulated Depth, F(8,20)=21.2, p<.001. Block was not reliable, F(2,26)=1.15. Most important to our hypothesis, the interaction between Condition and Simulated Depth was reliable, Wilk's F(16,40) = 3.99, p<.01.

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Figure 3. Results of Experiment 1. Mean perceived depth is plotted against simulated depth for each viewing condition. (Depth is expressed as a percentage of the base's diameter, which was 6 cm.) For the SKE condition, simulated depth is defined assuming a rotation of 4.92 deg to match the other two conditions. The major diagonal dashed line represents accurate performance. Error bars represent standard errors of the mean (i.e., between-observers variability).

Individual regression slopes were computed for each subject. An ANOVA of Slope by Condition revealed a reliable effect of Condition, F(2,27) = 58.5, p<.001. Planned comparisons revealed that the BD slopes (M = 0.90) differed reliably from the MP slopes (M = 0.26) and from SKE slopes (M = 0.17), but that SKE and MP slopes did not differ from each other, (REGWQ, p < .05). SKE and MP slopes did differ from zero, indicating sensitivity to depth differences, t(9) = 5.45, p < .001, t(9) = 3.24, p < .001. Although it is evident that depth judgments fall short for the deepest cones, the mean slope of the BD group did not differ reliably from 1, t(9) = 1.76, n.s.

Within-subject variability.

How consistent were each subjects' repeated judgments of each stimulus? For purposes of comparison with other experiments this analysis was performed by first converting all judgments into equivalent retinal-disparities (or absolute differences in retinal displacement -- which is proportional to velocity if the subject maintains fixation on the object) expressed in minutes of arc. For each subject in each condition, the standard deviation was computed from the three judgments of each cone depth. This value was divided by the mean for that cone depth in that condition to produce an estimate of within-subject variability(Note 8). Because the distribution of these scores were skewed, median values (rather than means) are reported in Table 1. It is evident from the table that variability is higher in the motion parallax condition than in the SKE condition. This is consistent with the fact that, although the relative motions presented in the two monocular motion conditions are approximately the same, these are added to a pedestal velocity of translation in the MP condition. Variability appears lowest in the binocular disparity condition.

Median Within-Subject Variability as a Function of Condition and Equivalent Retinal Disparity in Experiment 1

Retinal disparity or displacement (minutes of arc)

Condition

10

13

16

20

27

42

59

78

98

Binocular Disparity	0.14	0.13	0.13	0.13	0.13	0.11	0.13	0.07	0.10
Stereokinesis	0.20	0.16	0.17	0.16	0.12	0.16	0.09	0.14	0.12
Motion Parallax	0.23	0.35	0.23	0.22	0.22	0.27	0.17	0.23	0.16

Note. Values are based on standard deviations of individual subjects' judgments divided by group cell means. Three judgments per cell per subject; ten subjects per cell.

Discussion

The similarity of depth judgments from motion parallax and SKE displays is consistent with prior research (Caudek & Proffitt, 1993), in which it was determined that foreshortening information which distinguished motion parallax from SKE was not used for small rotations/displacements. Caudek and Proffitt suggested that perceived depth was mediated by a compactness assumption. This effect is illustrated in Figure 3 by the bias of the motion-based judgments toward a depth equal to the diameter of the base. Note that depth judgments equaling the diameter of the base are indicated by the 100% value in Figure 3, and that depth from motion judgments did not depart much from this default value. According to Caudek and Proffitt, the discrimination of depths may be mediated by a heuristic that greater relative motion signals greater depth. This heuristic would fit the results of the current situation. The important novel result is the direct evidence that geometrically equivalent depth information does not lead to the same quantitative perception when presented through motion parallax as when presented through binocular disparity.

Considered pragmatically, the disadvantage of motion parallax information is not surprising. The information specifying the angular rotation of the objects in the two monocular conditions is either left out entirely (as in SKE) or potentially ambiguous (in MP). Although the cones in the MP condition are simulated as translating, they are, for practical purposes, indistinguishable from simulations of differently-sized cones which are translating and rotating simultaneously. What should the visual system "conclude"? The recovery of form from relative motion information seems to be coupled with a default assumption regarding the objects' depths, such as the compactness assumption proposed by Caudek and Proffitt (1993). On the other hand, so long as distance is known to a reasonable precision, disparity information is quite determinate. The two frames which determine depth from disparity information are simultaneous and therefore cannot normally be confounded with object motion.

Experiment 2

In Experiment 2 we used real cones in a brightly-lit, fully-structured environment. There were two SFM conditions and a static binocular viewing condition. In the MP condition, the subject moved his or her head from side to side. Not only does this reduce the ambiguity of the object's actual lack of rotational motion, but optical flow from the surround may help to specify the quantity of head motion to greater precision. This might afford the MP condition an advantage over the second SFM condition, which consisted of a simple rotation of the object which was matched to the quantity of virtual rotation in the MP condition. Except for perspective projection, the object rotation condition is similar to a KDE stimulus. However the ambiguities of rotational angle are essentially the same as in the SKE condition of the first experiment, because the foreshortening information provided by the very slight rotation used is negligible. As in Experiment 1, a static binocular disparity condition was used to assess the accuracy of depth from stereopsis.

Method

Subjects.

The subjects were 36 University of Virginia undergraduates who were able to identify simple forms in random dot stereograms. Eighteen men and 18 women were divided equally among three experimental conditions. Two additional subjects were excluded from the study because they failed to perceive depth in a random dot stereogram.

Apparatus.

Four wooden cones were used. Each was painted flat black. The tip and four equally spaced thin circular contours were painted white. All cones were 10 cm in diameter at the base. The depths of the cones were 5, 10, 15, and 20 cm. The cones screwed into a vertical metal rod hidden behind a black cloth. The black cloth extended 40 cm to the left and right of the cone center. The rest of the room was left unoccluded so as to provide a structured viewing environment. The cones were illuminated by a flood light behind and slightly above the subject's head and a fill light placed below and in front of the subjects to minimize shading information on the cones.

The cones could be manually rotated by the experimenter through a known angular displacement. Subjects' heads were restrained by a sliding chin rest which enabled side to side motion for the MP condition, but was fixed stationary for the other two conditions. The chin rest had cheek rests on the side to prevent head rotation. Depth judgments were recorded with a Macintosh SE using a mouse controlled icon similar to that used in Experiment 1.

Stimulus Geometry.

In order to minimize the influence of accommodative depth cues which might be used to discriminate differences in the absolute distance of the tip and base, the cones were viewed from a distance of 2 m (to the base). This is nearly three times the distance used in Experiment 1. Pilot studies revealed that limiting viewer displacement in the motion parallax condition to interocular distance produced little sensation of depth at this distance. As a more generous test of motion parallax, we had subjects move through a total of 25 cm, (four times the interocular distance). Thus the observer moved 3.6 deg to the right and left of the base of the cone in the motion parallax condition. In the object rotation condition, the cones were rotated 3.6 degrees to the left and to the right, which is matched to the motion parallax condition. The angle of convergence was approximately 0.9 deg for each eye for the binocular disparity group.

Procedure. As with Experiment 1, a mixed design was used. Subjects were randomly assigned to one of three conditions: Binocular disparity (BD), cone rotation (CR), or viewer-movement-induced motion parallax (MP). Within each condition, subjects made two judgments of each of the four stimulus cones. The order of presentation was broken into two blocks of four randomly ordered trials.

Subjects were told that they would be viewing real wooden cones under special viewing conditions and were instructed in the use of the mouse-adjustable icon used to register perceived depth of the cones. Subjects were advised to pay attention to the shape rather than to the absolute size of the cone when adjusting the computer icon. Subjects were not informed of the number of distinct stimuli, nor of the range of sizes.

For the monocular conditions (CR and MP) subjects wore an eyepatch on whichever eye they chose. The chinrest was adjusted so that subject's exposed eye was in line with the principal axis of the cone when the cone (CR condition) or the eye (MP condition) were at the center of their excursions. The subjects' heads remained in the chinrest throughout the procedure. The adjustable icon appeared on a screen in front of the subject, below the line of vision to the cone, and the subjects could freely look back and forth between the cone and the icon. In the MP condition, subjects were encouraged to slide their heads back and forth quite rapidly to enhance the perception of 3-D structure. Since subjects would stop sliding to adjust the icon, they were encouraged to alternate between sliding their heads and adjusting the icon.

A curtain was lowered between the subject and the cone apparatus between trials while the experimenter changed cones. Each subject made eight judgments, and was then assessed for stereo depth perception with a random dot stereogram viewed through a hand held stereoscope. Some subjects reported experiencing apparent object rotation in the motion parallax condition.

Results and Discussion

The mean judged heights for each cone in each condition are shown in Figure 4. As in Experiment 1, subjects in the Binocular Disparity condition were quite accurate at judging the depth of the cones. The judgments of the other two groups reflect much less sensitivity to the geometrically defined height of the stimulus information. A 3 x 4 (Condition x Depth) mixed design repeated measures ANOVA revealed a highly reliable main effect of Depth, F(3,31) = 64.6, p <.001 and, more importantly, a highly reliable interaction between Condition and Depth, Wilk's F(6,62) = 6.4, p < .001. There was no main effect of Condition, which indicates there is no overall bias to see greater depth when both eyes are open, for example.

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Figure 4. Results of Experiment 2. Mean perceived depth is plotted against actual cone depth for each viewing condition. (depth is expressed as a percentage of the base's diameter, which was 10 cm.). The major diagonal dashed line represents accurate performance. Error bars represent standard errors of the mean.

As in Experiment 1, individual regression slopes were computed for each subject. An ANOVA of Slope by Condition revealed a highly reliable difference between conditions F(2,33) = 17.9, p<.01. Planned comparisons between groups revealed that slopes of the BD group (M = 1.07), the MP group (M = 0.60) and the CR group (M = 0.31) all differed reliably from each other (REGWQ, p < .05). All slopes differed reliably from zero, only the slope of the binocular disparity group did not differ reliably from 1, t(11) = .729, n.s.

Median scores of within-subject variability are shown in Table 2 as a function of equivalent retinal disparity. The scores were calculated as in Experiment 1. Note that because these estimates are based on only two judgments per cell per subject, they underestimate true variability by a factor of about 0.8; see Footnote 8. The retinal disparities viewed in the binocular condition were less than those of Experiment 1, and there is a notable increase in the variability for depth judgments of the smallest disparities (shallowest cone). Variability in the motion parallax condition (with equivalent disparities or relative displacements that are four time larger) is comparable to that of the binocular disparity condition, with a mean of 0.12 in each case.

Median Within-Subject Variability as a Function of Condition and Equivalent Retinal Disparity in Experiment 2

Retinal Disparity or Displacement (arcmin)

Condition

2.73

5.61

8.64

11.8

11.0

22.5

34.7

47.5

Binocular Disparity	0.18	0.13	0.10	0.06
Cone Rotation					0.12	0.16	0.15	0.13
Motion Parallax					0.16	0.12	0.10	0.11

Note. Values are based on standard deviations for individual subjects' judgments divided by group cell means. Two judgments per cell per subject; twelve subjects per cell.

There are two main results: On the one hand, the superiority of binocular disparity was replicated, but on the other, a clear superiority was demonstrated for the motion parallax condition over the cone rotation condition. Thus, in a well-illuminated, structured environment, it appears that the active self-motion of an observer produces depth judgments which are superior in accuracy to that of equivalent object rotation, but remain inferior to static binocular viewing. The relatively good performance of observers in the monocular conditions may in part be due to monocular depth cues such as accommodation, differential luminance reflection, and other information not available in the computer simulations. Using the same cones, but different illumination conditions, Caudek and Proffitt (1994) found that observers could discriminate the relative depths of these cones in a static monocular viewing condition. Also, the angular displacements and object rotations were somewhat greater than in Experiment 1. It is probable that improved information about the angle of object displacement was available in the MP condition. However, it is also possible that the improved performance of the motion parallax condition is due, at least in part, to a reduction of the ambiguity about object rotation: Some subjects reported that some objects seemed to rotate, so they assumed they were taller than they "looked". We find no evidence to support the notion that depth from motion is equal to or superior to depth from disparity. On the contrary, perceived depth in the binocular disparity situation appears excellent in the present viewing conditions, whereas depth from motion appears to tend toward the default depth assignment of the base's diameter that was identified by Caudek and Proffitt (1993).

Is there stereoscopic depth constancy?

Although the present findings support our claim that binocular stereopsis can support more accurate depth recovery than can structure from motion with geometrically similar displays, there exist reports of the failure of depth constancy in the literature that would seem to contradict our conclusions regarding the accuracy of depth from stereopsis generally (e.g., Tyler, 1983; Collett, Schwarz, and Sobel, 1991). Perhaps the clearest case of the failure of depth constancy is that demonstrated by Johnston (1991). She presented observers with random dot stereograms which simulated elliptical cylinders with their principal axis horizontal in the picture plane. The subjects' task was to judge whether a presented cylinder appeared "squashed" or "stretched" in depth (relative to an imagined circular cylinder). Such data can be interpreted as a measure of a perceived depth to base ratio. The task was performed at three viewing distances, using five cylinder heights (vertical diameters). Her results were straightforward: Perceived depth was exaggerated for the nearest viewing distance and grossly underestimated for the farthest. Such results are consistent with an inadequate use of distance information to scale disparity information. Indeed, Johnston determined that her results were entirely consistent with the interpretation that subjects used a default presumed viewing distance of about 80 cm with a change in (perceived) distance corresponding to 26%(Note 9) of the actual change in distance from that default.

Johnston's methodology and findings are of particular interest because she was able to show that subjects did not seem to use purely visual (retinal) sources of information to perceive the shape of her smoothly curved surfaces. Rogers and Cagnello (1989) and Koenderink and Van Doorn (1976) had both suggested analyses of visually available information that could produce accurate depth from stereopsis for smoothly curved surfaces without requiring knowledge about viewing distance. Apparently, observers did not use these kinds of theoretically-available, purely visual sources of information.

Johnston's findings are consistent with the claim that stereoscopic depth judgments are scaled by information about fixation depth, but suggest that information about egocentric viewing distance is quite poor, at least under the experimental conditions used. Foley (1980) reviewed evidence that the perception of egocentric distance could be derived from vergence information and concluded that though vergence information was probably used, egocentric distance was not typically recovered accurately. In Experiment 1 of the present paper, our fixation distance (72 cm) was coincidentally quite close to the distance that appears to be the default in Johnston's data. It therefore should come as no surprise that we find such accurate depth judgments. However, our fixation distance in Experiment 2 (2 m) was similar to the longest fixation distance used by Johnston, and yet we still find apparently veridical stereoscopic perception of depth, whereas she found a marked underestimation of depth (failure of depth constancy). Why should this difference arise?

Our stimuli in Experiment 2 differ from those of Johnston (1991) in several ways. We used real cones specified by clearly demarcated contours presented in a well-lit, structured environment without a fixation target, whereas she used random dot stereograms of cylinders presented in a darkened environment with a fixation target. Assuming our result at 2 m is not an artifact, it is reasonable to suppose that the crucial differences between the situations have to do with the perception of distance rather than the recovery of disparity information itself, because it is inadequate recovery of distance information, not disparity information, which seems to underlie the failure of stereoscopic depth perception in Johnston's experiments. Thus, we suspect that the source of the difference between Johnston's results and our own is the experimental environment.

There is a great deal of evidence suggesting that distance perception is faulty in low illumination or in the absence of a surrounding context (e.g., Owens & Leibowitz, 1980). Gogel (1969b) and Gogel and Tietz (1973) have shown that in dim light most observers tend to underestimate the distance of far objects and overestimate the distance of near objects, just as Johnston's results suggest. This phenomenon is referred to as the "Specific Distance Tendency" wherein objects in reduced viewing conditions tend to be perceived as being at a default distance (usually about 2 m, according to Gogel, 1982; cf. also Foley, 1980). Owens and Leibowitz (1976) have shown that oculomotor adjustments to a binocularly viewed target in a dark environment show systematic errors which are broadly consistent with a specific distance tendency. Owens and Leibowitz (1980) measured dark vergence and dark focus of 60 subjects and found mean vergence and accommodative states appropriate to egocentric distances of 1.16 m and 0.76 m, respectively. Note that these distances are quite close to the default found in Johnston's work. It is therefore possible that the failure of depth constancy found in Johnston's experiment, and perhaps in portions of the distance perception literature, generally, is due to the low level of ambient illumination that is typically used to provide reduced viewing conditions. Distance perception may critically depend on viewing conditions that are more normal. Thus, although Johnston's results represent a crucial refutation of certain more elaborate models of depth from disparity (i.e., Koenderink & Van Doorn, 1976; Rogers & Cagnello, 1989; but cf. Rogers & Bradshaw, 1993), as she herself argues, they are entirely consistent with the thesis that depth from disparity is scaled by information about viewing distance.

Experiment 3

To demonstrate that stereoscopic depth constancy can indeed exist in a non-reduced viewing situation like that employed in Experiment 2, it is necessary to systematically vary viewing distance. The present experiment was a partial replication of the binocular disparity condition of Experiment 2 with a systematic variation of viewing distance from 1 to 3 m by 0.5 m increments. Static monocular control conditions were also tested at 1 and 2 m. A failure of depth constancy would be indicated by changes in both the slope between judged and actual depth and by absolute depth values at different distances. Because the proximal disparity produced by the same distal object decreases with greater viewing distance, if distance information is not taken adequately into account, judgments of depth from disparity should decrease as viewing distance increases.

Method

Subjects.

The subjects were 56 University of Virginia undergraduates divided evenly among seven experimental conditions. Three additional subjects were excluded from the study because of failure to follow instructions or failure to perceive depth from disparity.

Design.

A mixed design was again used. Subjects in binocular conditions were assigned to one of five viewing distances (1.0, 1.5, 2.0, 2.5, or 3.0 m, measured to the base of the cones). There were also two static monocular conditions in which subjects viewed cones from a distance of either 1.0 or 2.0 m. Each subject made single depth judgments (side view triangles) of each of four cones presented in random order in each of three blocks. From a subject's point of view, there were simply 12 trials, without break. The first block was considered practice and was not analyzed.

Apparatus.

The room used for testing was not the same as that used in Experiment 2, but the apparatus was otherwise quite similar. The same four wooden cones were used, and the same black cloth extended 40 cm to the left and right of the cone center. As before, the rest of the room was left unoccluded so as to provide a structured viewing environment, and the cones were illuminated by a flood light behind and slightly above the subject's head and a fill light placed below and in front of the subjects to minimize shading information on the cones. Subjects' heads were positioned in a chin rest. Depth judgments were recorded with a Macintosh SE using a triangular mouse-controlled icon similar to those used in the previous Experiments.

Procedure.

Subjects were told that they would be viewing real wooden cones under special viewing conditions and were instructed in the use of the mouse-adjustable icon used to register the depth of the cones. As in previous experiments, no information was provided to the subjects about the number of distinct stimuli nor about the range of sizes. They were told to represent how each cone would look from the side, but were advised to pay attention to the shape rather than to the absolute size of the cone when adjusting the computer icon. The adjustable icon appeared on a screen in front of the subject, below the line of vision to the cone, and the subjects could freely look back and forth between the cone and the icon. Both the cones and the observers were stationary in all conditions. Subjects in the monocular condition wore an eye patch over one eye. Each subject made a total of twelve judgments, of which the first four were considered practice. A barrier was raised between the subject and the cone apparatus between trials while the experimenter changed cones.

Results

The mean judged heights for each cone in each of the seven viewing conditions (5 binocular and 2 monocular) are shown in Figure 5. Individual regression slopes were computed for each subject. An ANOVA of slope by Condition revealed a highly reliable main effect of viewing condition, F(6, 49) = 31.2, p < .0001. Planned comparisons revealed no differences within the binocular or monocular conditions, but indicated that the mean slopes in the monocular conditions (M = .020) were significantly lower than those in the binocular conditions (M = 1.33), REGWQ, p < .05. The mean slope for the binocular conditions was reliably greater than 1, t(39) = 5.35, p < .0001. The mean slope of the monocular conditions did not differ reliably from 0, t(15) = 0.356, n.s. The monocular and binocular conditions were considered separately in further analyses.

Monocular conditions.

In our static monocular conditions, many subjects commented that they saw no definite depth, nor even depth order. The judgments they made are therefore to be understood as "best bet" guesses. A 2 (Viewing Distance) x 4 (Cone Depth) x 2 (Block) repeated measures ANOVA of the monocular conditions revealed a reliable interaction of Block and Viewing Distance, as well as reliable main effects of Block, F(1,14) = 8.94, p < .01, and of Cone Depth, F(3, 12) = 4.59, p < .05. Because of the interaction of viewing distance and block, 4 x 2 (Cone Depth x Block) repeated measures ANOVAs were performed on data from each of the two conditions individually. For the 1 m monocular condition, there was no effect of Block, F(1,7) = 0.13, n.s., or of Cone Depth, F(3,5) = 1.19, n.s. However, for the 2 m monocular condition, the depth judgments in the last block (M = 13.1 cm) were significantly larger than those in the first analyzed block (M = 9.4 cm), F(1,7) = 19.1, p < .01. There was also a marginal effect of Cone Depth, F(3,5) = 5.00, p = .058, suggesting some sensitivity to different depths. The mean judged depths for the four cones in the monocular 2 m condition (smallest to deepest actual cone) were 10.5, 11.3, 11.8, and 12.7 cm (recall that the cones were actually 5, 10, 15, and 20 cm in depth and had a base diameter of 10 cm). The depth judgments in these monocular conditions suggest that when there is very little monocular information available to specify object depth in our viewing situation, subjects will use a judgmental default consistent with a compactness assumption.

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Figure 5. Results of Experiment 3. Mean perceived depth of real cones viewed either binocularly (Panel A) or monocularly (Panel B) as a function of actual depth and viewing distance. (Depth is expressed as a percentage of the base's diameter, which was 10 cm.) In Panel A, standard errors are shown for the 1 m and 3 m conditions. In Panel B, standard errors are shown for the 2 m condition.

Binocular conditions.

A 5 (Viewing Distance) x 4 (Cone Depth) x 2 (Block) repeated measures ANOVA of the binocular conditions revealed a highly reliable main effect of Cone Depth, F(3, 33) = 181.0, p < .0001. Apparently because of the lower depth judgments in the 1.5 m binocular condition, there was also a marginal main effect of Viewing Distance in the binocular conditions, F(4, 35) = 2.38, p = .07. The mean judged depth in the 1.5 m condition was 10.7 cm, while the means in the 1.0, 2.0, 2.5, and 3.0 m conditions were 15.9, 14.7, 14.9 and 14.0 cm, respectively(Note 10).

The within-subject variability between the two adjustments made for each cone was calculated as in previous experiments. Because perceived distance may be expected to remain constant for each observer, within-subject variability is expected to be modulated primarily by the quantity of disparity information available, which covaries with depth and viewing distance. It can be seen in Figure 6 that the variability is greatest for displays in which the maximum disparity present is less than about 2 arcmin.

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Figure 6. Median within-subject error in Experiment 3 as a function of retinal disparity. Within-subject error was computed by taking the root mean square error for each subject's judgments of each cone and dividing by the group mean. Each plot point represents the median of 8 such ratios (corrected, here, by a factor of 0.8; see Footnote 8). Because error is computed separately for each subject, between-subject variability in perceived viewing distance is not represented. As expected, variability is greatest for extremely small disparities. Note that the abscissa represents the maximum retinal disparity in each stimulus. Four points are shown for each viewing distance, corresponding to the four cones used in the experiment.

Discussion

Using real objects in a well-illuminated, structured viewing environment we tested whether adequate distance information would be provided to support stereoscopic depth constancy. Indeed, the results are indicative of such constancy, for they do not reflect any systematic distortions of depth with changes in viewing distance from 1 to 3 m, such as those found by Johnston (1991) in a reduced viewing environment. Thus, the claim from Experiment 2 that depth from disparity information can be scaled by viewing distance has received important confirmation.

The study of depth constancy is not new, but the basis for its sometime failure is poorly understood. Johnston (1991) argued that her experiments indicated that the cues of accommodation and convergence were inadequate to support depth constancy. However, as reviewed above, there is some dispute in the literature regarding oculomotor adjustments and the perception of distance in dim light (e.g., Foley, 1980; Owens & Leibowitz, 1980). Johnston took care to establish that the vergence angle through the stereoviewer was correct for the distance, and she used a crosshair fixation device to ensure that subjects were accommodating at the right distance. Given the elegance and care with which her experiment was carried out, it may seem unreasonable to doubt the general validity of its conclusion. However, the techniques used by Johnston to establish the failure of stereoscopic depth constancy, and presumably of distance perception, entailed conditions (dim illumination) where vergence and accommodation typically fail (Leibowitz, Hennessy, & Owens, 1975; Owens & Leibowitz, 1980), and might therefore, even when correct, receive little weight in the evaluation of distance. It should not be concluded from her experiment that there is no depth constancy, nor even that oculomotor cues are entirely ineffective, for she found evidence of some distance compensation. Indeed, the effects of oculomotor cues on perceived stereoscopic depth have also been documented in a number of studies by Wallach and colleagues (O'Leary & Wallach, 1980; Wallach & Zuckerman, 1963) and other investigators (e.g., Collett et al., 1991; Cumming, Johnston, & Parker, 1991), though there are no reports of full stereoscopic depth constancy from oculomotor cues alone.

In general, perfect stereoscopic depth constancy depends critically on the veridical perception of egocentric distance. Using disparity afterimages, Cormack (1984) has demonstrated that stereoscopic depth constancy holds quite well for far viewing distances (e.g., 2000 m) if one takes into account predictable biases and failures of distance perception outlined by Gilinsky (1951). O'Leary and Wallach (1980) have shown that linear perspective cues, even when put in conflict with otherwise effective (but deceptive) oculomotor cues to distance, can affect perceived stereoscopic depth in a manner consistent with distance scaling. Moreover, Collett et al. (1991) have demonstrated that relative size effects, which affect distance perception, also affect disparity scaling. In short, sources of visual and oculomotor information which affect perceived distance seem also to influence the scaling of binocular disparity information(Note 11).

It appears that subjects did use distance information to scale depth from disparity in the present experiment. However, there are two important features of the data which suggest that depth from stereopsis was imperfect: (1) the data show evidence of an overestimation of depth for the deeper cones, and (2) there is evident variability in the depth judgments between observers. One probable source of error arguably external to the perception of depth might involve individual differences and/or difficulties in transforming 3-D perceptions into 2-D side-view responses. One could easily have a metric scale of depth along the line of sight without having a very good means of mentally transforming distance along one axis to distance along another. An even more pernicious source of variability and bias, however, may have arisen from subjects's interpretations of the experimental instructions (Brunswick, 1956; Carlson, 1977; Leibowitz & Harvey, 1969). Carlson (1977) reviewed evidence that instructions to report on the objective size of an object in a size-constancy experiment, for example, will tend to produce over-constancy, a kind of over-compensation for distance, whereas instructions to report apparent size may result in a closer approximation to constancy. Because viewing distance was varied between subjects in the present experiment, we would expect to find no "overconstancy" with respect to viewing distance, and none was found. However, the fact that the mean slopes in the binocular conditions are uniformly greater than 1 is consistent with a view that subjects may be similarly compensating for what they perceive to be a reduced viewing condition (static viewing). In other words, the subjects may have exaggerated differences in perceived depth in their responses.

A geometry of visual space?

Studies of distance perception clearly implicate a compressive scale of distance over large distances (e.g., Gilinsky, 1951; Luneburg, 1947), though the apparent compression in near space (less than 2 or 3 meters) may be relatively small in non-reduced conditions. In an outdoor full-cue viewing environment, Loomis, Da Silva, Fujita, and Fukusima (1992) have recently replicated the basic finding that perceived spatial intervals in depth are increasingly compressed relative to perceived frontal intervals as viewing distance increases from 4 to 12 meters, but also demonstrated that performance on a blind walking task (following visual inspection of the goal) is linearly related to distance (and accurate) for these same distances. Wagner (1985; cf. also Toye, 1986) examined the geometry of visual space in a somewhat larger outdoor environment and concluded that the amount of compression of perceived space (i.e. of depth relative to frontal distances) was constant over the distances he sampled (5-70 meters) and equal to about 0.5 (depth intervals are perceived as half as large as equivalent frontal intervals viewed at the same distance). Nonetheless, in comparing his findings with those of other investigators, Wagner concludes that the "geometry of visual space itself appears to be a function of stimulus conditions" and that "the visual world approaches the Euclidean ideal of veridical perception as the quantity and quality of perceptual information increases" (1985, p. 493; cf. also Baird, 1970; Suppes, 1977).

The scaling of binocular disparity information requires accurate egocentric distance, but does not otherwise entail any particular "geometry of visual space" as assessed by depth interval studies. Because the physical world is (for most behaviorally-relevant purposes) Euclidean, one would hope that the visual system would not reject Euclidean interpretations, if only it could easily get at them, but there is ample evidence that our perceptions are frequently biased by view-point dependent information. In reduced environments there is a well-demonstrated failure of accurate distance perception, even for near objects, and a concomitant failure of disparity scaling (Foley, 1980). We have suggested that the perception of near distances will approach veridicality in well-illuminated, fully structured environments (cf. also Johansson, 1973). We note that even in good illumination, there are reports of failures in the perception of near depth intervals. For example, Baird and Biersdorf (1967; cf. also Todd & Norman, 1994) found that the length of a 20.3 x 1.25 cm rectangular strip of posterboard viewed at 2 meters distance in a well-lit visual alley was compressed by a factor of about 0.85 relative to its appearance at a distance of 60 cm(Note 12).

In contrast even to this fairly slight compression observed by Baird and Biersdorf (1967) our results demonstrate no failure of depth constancy whatsoever. We can only speculate about the basis for this difference, but we note several differences between our stimuli and procedure and theirs: (1) Our subjects made shape judgments about volumetric solids whereas Baird and Biersdorf compared lengths of rectangular strips. (2) Our surrounding experimental setting was more elaborately structured than their visual alley and may have provided better peripheral information about viewing distance (e.g., differential perspective, Rogers & Bradshaw, 1993; Tyler, 1991). (3) Because the relevant depth axes of our stimuli were all aligned directly with the subject's cyclopean line of sight, conflicting size cues (i.e., from changes in angle of regard) were not confounded with viewing distance. (4) Our subjects were not individually asked to compare sizes at different viewing distances (distance was manipulated between subjects).

In summary, our concern has not been with assessing the geometry of visual space (if such a thing exists) but rather with examining how well subjects can recover the metric shape of objects viewed binocularly in near space. Thus, subjects were not required to explicitly compare spatial intervals, but rather to report on 3-D shapes. Moreover, we used real solid objects which were textured with widely spaced contours in a manner that reduces any binocular correspondence difficulties, and we had subjects view these objects in a well-lit, structured environment. The methods we have used differed from traditional constancy experiments in that each subject viewed the objects at only a single distance and different distances were compared between subjects. This strategy was used to avoid problems of recognizing the objects, but may have had the advantage of reducing attention to distance itself. We found that performance at our task was quite good and have suggested that this was probably due to access to accurate distance information from oculomotor cues as well as optical distance cues such as linear perspective and, possibly, differential perspective.

General Discussion

In the experiments presented here, judgments of object depth based on binocular disparity information were found to be substantially more accurate than judgments based on monocular motion information. This was true both for contour-specified computer simulations and for real objects viewed in a structured environment. The superiority of binocular disparity was maintained when the geometric information available from motion was equivalent or theoretically superior to that available from disparity. These findings are consistent with other evidence that information about angle of rotation in SFM displays is not recovered for small rotations (Caudek & Proffitt, 1993, 1994; Loomis & Beall, 1993). In Experiment 2, an advantage was also found for viewer induced motion parallax over object rotation in a situation where the local visual information was essentially equivalent.

Our findings are interpretable within a view of perception as a heuristic process (Braunstein, 1976). We would argue that the derivation of a single definite percept from the plethora of information sources available should proceed according to pragmatic considerations. For example, we suppose that the accurate perception of depth from disparity is possible because the visual system can be calibrated to reliably determine the distance to the point of zero disparity from vergence, accommodation, and other cues in normal viewing environments. There is evidence that oculomotor sources of information are fairly accurate within a range of 25-200 cm (Leibowitz, Shina & Hennessy, 1972), though they may fail in dim illumination. Obviously, distance determination from oculomotor cues will deteriorate with increasing distance, but within the range of space normally relevant to object manipulation, binocular disparity should be quite reliable. On the other hand, the crucial parameters required to make SFM accurately determine object depth are often difficult to obtain, and are likely to be less precise in any case.

There are a number of models which suggest that SFM might generate a set of structures all related by a single parameter--essentially an affine structure (Aloimonos & Brown, 1989; Bennett, Hoffman, Nicola, & Prakash, 1989; Huang & Lee, 1989; Koenderink & van Doorn, 1981). To derive a definite object depth from such models requires either the overt specification of the free parameter, or some other constraint upon the visual interpretation. From our point of view, however, the apparent success of affine derivations of structure (e.g., Todd & Bressan, 1990) may simply result from the fact that it is typically impossible to precisely specify crucial parameters of the Euclidean equations (see Appendix) from within the local visual flow field: The relative velocities or displacements necessary to specify it are too difficult to discriminate. For example, Eagle and Blake (1994) found that when thresholds for relevant visual information are matched, performance is as good for Euclidean tasks as for affine ones. In cases of reduced angular rotation or translation, the perceptual system should only accept an affine (or weaker) interpretation of motion information because the visual system cannot reliably extract more. The findings of Todd and his colleagues (Norman & Todd, 1993; Todd & Bressan, 1990; Todd & Norman, 1991) are therefore consistent with our argument.

Norman and Todd (1993) have shown that instantaneous stretching of a SFM-specified object along the line of sight is essentially undetectable, which is entirely consistent with the view that the angle of rotation is not easily recovered and that depth is often assigned by default assumptions. Pollick (1994) provides evidence suggesting that these "unperceived" mutations of form may often instead be perceived as changes in angular velocity of rotation -- a Euclidean interpretation approximately consistent with the 2-D projections involved. Stretching in other directions can also be undetectable, however, suggesting that there is a fair amount of tolerance in the system. For example, SKE stimuli simulate a different kind of stretching which is nonetheless perceived as rigid (Proffitt et al., 1992). On the other hand, Loomis and Eby (1988) have shown that simulations of rigid elongated objects appear to stretch and contract (with their 2-D projection) as they rotate, and more recently, Caudek and Proffitt (1994) have shown that a simulation of a rotating wire form appears most rigid if it actually stretches and contracts along an axis rotating with the object so that its projected 2-D size remains fairly constant. All of these findings suggest that there are limits to the amount of depth the visual system is willing to recover from motion, and that (perceived) violations of rigidity may be primarily determined by marked changes in the 2-D envelope in which the stimulus is projected.

This last finding points toward an explanation of the specific depth assignments that are found in SFM research. The relative depth assignments produced in the current experiments reflect an appropriate discrimination of differences: Displays simulating deeper objects are judged to simulate deeper objects. Displays of shallower objects are judged to be shallower. However, such discriminations appear to be driven by differences in the 2-D displacement of near object features relative to the base because they also occurred with illusory depth in the case of SKE (cf. also Caudek & Proffitt, 1993, 1994). Included in these depth judgments, however, is a central tendency which is not explained by models of geometric SFM. The depth of the shallowest objects is overestimated based upon motion information while that of the deepest ones is underestimated. Caudek and Proffitt have attempted to explain this phenomenon in terms of a compactness assumption. A compactness assumption represents a default assumption that the particular view one has of an object is randomly selected among those possible. This particular assumption should come into play when depth is insufficiently determined. According to this view, in the absence of other information, the best estimate of the depth of a solid object is provided by its apparent width (width should be used preferentially to height because objects are often oriented relative to the gravity-specified vertical). We believe that this heuristic may be regarded as one of many within a biological perceptual system which can take into account both the content and the reliability of the information available to it.

The SFM stimuli used in our experiments involved small angles of rotation, and we have argued that for such small angles, the foreshortening (or acceleration) information available to specify the angle of rotation to an observer is understandably difficult to recover (cf. Simpson, 1994). This is not to say that angular information is never used. For larger angles of rotation (>15 deg) there is clear evidence that compression information affects judgments of depth in the appropriate direction (Braunstein et al, 1993; Caudek & Proffitt, 1994; Proffitt et al., 1992). For example, Braunstein et al. (1993) have shown that when additional compression information was added to a motion parallax display (indicating rotations of 20 deg and 28.4 deg out of the picture plane) the apparent depth of a monocularly viewed horizontal dihedral angle was decreased. This is consistent with an increase in perceived object rotation.

Moreover, there is reason to believe that there is better information about the effective angle in self-produced motion parallax when an enriched viewing environment is available. In Experiment 2, our motion parallax condition demonstrated a greater sensitivity to depth than did our equivalent object rotation. We believe that this difference can be understood as arising from richer sources of information about effective object rotation in the motion parallax condition. This information could be given by sensitivity to the change in angle of regard, which might be supported by optic flow information in well-illuminated, structured environment.

A convergent interpretation of how depth from motion parallax might be scaled more accurately when supporting information is given has been put forth by Ono et al. (1986). They provided an analysis of motion parallax (defined as differential projected motions produced by self-motion) as an analog of binocular disparity in which they expressed the quantity of parallax-induced "disparity" relative to a unit of head motion which was equal to interocular distance. From this analysis, they argued that viewing distance must be taken into account, in the same manner as in stereopsis, to retrieve depth from motion parallax. To test this conjecture they used random dot motion parallax displays which were yoked to head motion, like those of Rogers and Graham (1982), but the displays were viewed from several distances in a well-illuminated environment. In one study their stimuli simulated vertical sinusoidal forms with matched retinal characteristics of size, peak-to-peak horizontal distance, and movement-induced parallax at each of two distances, 40 and 80 cm. With a doubling of distance, equivalent parallax (like equivalent disparity) signals a change in depth by a factor of roughly four, whereas a failure of depth constancy would predict no change in perceived depth. They found a mean change in perceived depth by a factor of about 3.5, consistent with nearly complete depth constancy from motion parallax. This result is consistent with the current analysis of information about the effective angular rotation of objects: At a greater viewing distance, the same horizontal motion of a viewer produces less (effective) angular rotation of the object. Thus, an observer in Ono et al.'s situation who successfully extracted information about ego-relative object velocity (angular rotation) would appear to implicitly have taken distance into account(Note 13). At present, Ono et al.'s account in terms of distance and our own in terms of angle appear to be notational variants, but it is clear from their results, and is strongly suggested by our Experiment 2, that information necessary to specify the effective angular displacement of objects can indeed be used in the recovery of depth from motion parallax.

The heuristic view of perception has implications for understanding the interaction of various kinds of depth information. Recent research has attempted to examine the quantitative combination of depth information from different sources including structure from motion and binocular disparity (Bulthoff & Mallot, 1988; Clark & Yuille, 1990; Dosher, Sperling, & Wurst, 1986; Johnston, Cumming, & Parker, 1993; Landy, Maloney, & Young, 1991; Rogers & Collett, 1989; Tittle & Braunstein, 1991, 1993). Tittle and Braunstein, for example, found that motion facilitates depth from disparity. Although this is partly an incidental facilitation in helping to solve the stereo correspondence problem (a problem which is most pronounced for transparent random-dot-specified surfaces of the kind used for their research), they argue that differential velocity is an important factor. The present studies suggest that modeling the combination of cues involved in perceiving depth may involve including inherent default biases such as the compactness assumption, and specific distance tendencies. Moreover, the result of having multiple kinds of depth information is unlikely to be adequately modeled by a combination of independent quantitative depth estimates from different sources. We would argue that, much as disparity information must be combined with distance information to produce depth, the SFM system does not provide, in itself, a definite depth estimate except in combination with other information which includes (somewhat ineffective) default assumptions. Tittle and Braunstein (1991) report data on the combination of different levels of disparity- and KDE-specified depth which are quite consistent with the findings of this paper: The primary determinant of perceived depth (Euclidean shape) is binocular disparity, and the motion information seems to additively increment the perceived depth in a manner consistent with the SKE heuristic of Caudek and Proffitt (1993, 1994).

In general, efforts to study depth-cue combination quantitatively must recognize that disparity information, per se, does not provide metric depth information without taking distance into account. Insofar as different sources of distance information can be used, a comprehensive study of depth cue combination involving binocular disparity would probably need to explicitly manipulate the kinds and quality of distance information available. The manner in which disparity information combines with other kinds of cues to 3-D shape may depend upon the reliability of the distance information necessary to support stereoscopic depth perception. Under reduced viewing conditions, with a static observer, that reliability may be low.

With respect to investigations involving stereoscopic depth (and motion parallax), our findings suggest that computer simulations in a dimly lit or otherwise unstructured environment may represent a very limited scenario for investigations of metric form perception (for a noteworthy exception, see Lappin & Love, 1992). Although experiments in reduced environments can help to adjudicate between theories that do not suggest a role for a supporting environment, a full theory of stereoscopic depth perception seems likely to depend on the distance information that may be available in well-illuminated, fully-structured environments. Proffitt and Kaiser (1986) have pointed out a number of potential limitations introduced by computer simulation, generally, and recommend that natural objects also be used, when possible, to provide convergent information with computer simulation. Indeed, there has recently been a report (Buckley & Frisby, 1993) of a situation in which stereoscopic depth perceptions involving real objects differed from those involving stereograms. Although results from studies employing reduced viewing conditions and computer simulated stimuli are not, in themselves, invalid, they may be limited, and the present study demonstrates the usefulness of having additional data involving not only real objects but also a well-illuminated, structured environment. We feel that not only the "reality" of the target stimuli themselves, but also that of the environment in which they are presented is an important factor to consider in investigations of stereoscopic depth perception by itself or in combination with other information about depth.
A new complementarity of structure from motion and binocular stereopsis has recently been proposed by Olson (1991). Although some stereoscopic depth can be recovered even when disparities are so large that fusion fails (Ogle, 1952), the range of retinal disparities which can be precisely discriminated is quite limited (McKee, Levi, & Bowne, 1990). Thus disparity information for near fixations is most informative within a fairly small region about the fixation plane (Ogle, 1953; Westheimer & McKee, 1978). Although the proportional depth of field from small disparities increases with distance, the accuracy of depth from stereopsis for fixation distances of more than a few meters may be limited by the precision or accuracy of distance perception(Note 14). On the other hand, motion parallax can function over the entire visual field and at least recover a rough sense of global structure as the observer moves through the environment (e.g., Cutting, 1986). Thus, the two systems appear to have different domains of maximum utility (cf. also McKee et al., 1990). We believe that this view of the complementarity of motion and disparity information has promise for guiding future research.

Conclusion

The accurate perception of depth from binocular disparity information entails the use of accurate distance information, which, in non-reduced circumstances, may be precise enough to specify object depth within the range of distances normally available for human manipulation. With computer simulations and with real objects, we found that depth judgments based upon static binocular viewing were quite accurate in this range. Conversely, the recovery of depth from relative motions produced by object rotation or motion parallax entails obtaining angular displacement information that may be difficult to determine precisely in many instances. Our studies confirm that depth is not accurately recovered even in the case of motion parallax, where angular displacement information can be fairly well specified. For small rotations, depth magnitude perceived from relative motions appears to be mediated by heuristic assumptions and biases (Caudek & Proffitt, 1993, 1994). We conclude that for near objects in a fully-structured environment, the recovery of depth from static binocular viewing is normally well determined, while depth from motion is poorly determined for small rotations and angular displacements.

Appendix

Footnotes

1. This assumes orthographic projection. Under perspective, translation toward the target does provide depth information. However, the information is unstable around the focus of expansion and hence is not likely to be useful.(back to text)

2. KDE is typically simulated by orthographic projection which is of theoretical significance because it eliminates perspective information. "SFM" is often reserved for orthographic projection (usually KDE) so as to distinguish motion information, per se, from the perspective information available from motion parallax. We use the term "SFM" for both motion parallax and KDE (with or without orthographic projection, in the latter case). Our usage is primarily a matter of expository convenience, but we note that there is evidence that the underlying perceptual processing is the same in either case (Braunstein, Liter & Tittle, 1993). (back to text)

3. Vertical disparities, which can theoretically be used to solve for metric depth, are present in stereoscopic images, but this information is weak at points near fixation. Cumming, Johnston, and Parker (1991) have argued that vertical disparities or differential perspective in stereoscopic images are not used. Rogers and Bradshaw (1993) have provided evidence that this information can be effective for very large displays (~80û), but we will consider smaller displays here. (back to text)

4. Wallach et al. had dismissed the notion that depth distortion was due to accommodative misperception of distance because they found no evidence of a concomitant size distortion such as would be produced by misperceived distance. In fact, the expected magnitude of the size distortion would be much smaller than the depth distortion and apparently went undetected. (back to text)

5. One might be concerned that these frontal views are "degenerate" as described by Todd and Norman (1991), who argued that SFM information would fail to discriminate between shapes which differ only in affine stretching along the line of sight. However this is precisely the point. If the cones were shown at oblique angles, then different length cones would be readily discriminable from each other as affine objects along the line of sight: That is, they would project different families of metric structures. The purpose of the experiment was to test for accuracy of quantitative object depth along the line of sight, not affine shape discrimination. Note that if the observer may assume (e.g., from the experimental context) that the 3-D form being simulated is symmetrical about some axis, then the affine structure derived from the motion of a three-quarter view, for example, could theoretically be constrained to a single metric interpretation, because only one member of the projective family will be symmetrical in metric space. (back to text)

6. The optics of image rotation may be best illustrated with right angle prisms. Two such prisms, placed face to face at a 90û angle, are used in binoculars to rotate the image (inverted by the objective lens) by 180û. In fact, any amount of rotation may be produced by setting the relative orientation of the two prisms to half the desired rotation, (e.g., a 90û image rotation is produced by a relative prism orientation of 45û). The same image rotation was accomplished with two appropriately oriented mirrors in the present apparatus. (back to text)

7. Many authors assume an interpupillary separation of 6.5 cm. According to the Handbook of Human Factors (Salvendy, 1987), the median interpupillary separation is about 6.27 cm (fixating on infinity). At nearer fixations, our value should therefore be closer to the median than is the traditional 6.5 cm, though the difference in perceived depth should only be on the order of 5%. (back to text)

8. Because we were working from only three data points for each subject in each cell, it was necessary to ascertain whether our variability estimates were biased. To accomplish this we ran simulations in which 1000 sets of three scores were sampled from a normal distribution with a mean of 100 and a standard deviation of 10, 20, 30 or 40. We calculated the sample standard deviation score divided by the mean for each of the 1000 samples. The resulting scores were roughly normally distributed, but underestimated the standard deviation of the population by a factor of about 0.9. Specifically, the means were: 0.089, 0.18, 0.27, and 0.38. This should be taken in to account when evaluating our reported errors. (N.B. For samples with a cell N of 2, the simulation estimates were 0.081, 0.16, 0.25, and 0.35, requiring a correction by about 0.8). (back to text)

9. Johnston's figure is probably artificially low because she used images with matched retinal sizes and textures. Stimulus size itself (among similar stimuli) is a cue to distance which interacts with oculomotor cues (Collett et al., 1991). Foley (1980, 1985) suggests a gain of 0.5. (back to text)

10. To determine whether these marginal differences would replicate, a follow-up experiment was conducted in which the 1.5 and 2.0 meter conditions were repeated with 8 subjects each. The marginal differences did not replicate. The mean depth for new subjects in the 1.5 meter condition was 13.2 cm, which did not differ reliably from the 12.0 cm mean of the new subjects in the 2.0 meter condition, t(15) = 1.18, n.s. In addition, the mean slope between judged and actual depth at 1.5 m (1.30) did not differ reliably from that at 2 m (1.14), t(15) = 0.915, n.s. (back to text)

11. O'Leary and Wallach (1980) also reported that the deceptive manipulation of the distance cue of familiar size (a shrunken dollar bill) affected perceived stereoscopic depth in a manner predicted by constancy. However, their own data and that of other investigators (Predebon, 1993) indicate that only linear-scaling (not quadratic) occurs, as if perceived depth is scaled proportionally to judged size, and indeed that the effect of familiar size on distance perception itself (as against distance judgment) is fairly inconsequential (Gogel, 1969a; Gogel & Da Silva, 1987; Predebon, 1992). (back to text)

12. Baird and Biersdorf (1967) used a well-illuminated visual alley with a black floor and green walls in which they presented strips of white posterboard which were 1.25 cm wide and varied in length. The strips were viewed binocularly from one end of the alley, and size matches were made between standard strips (20.3 cm in length) presented at several distances (from 61 cm to 549 cm) and comparison strips presented at the extreme distances. The authors were the observers and they tried to make "objective" matches for pairs of strips presented frontally or lying flat on the floor of the visual alley (46 cm below eye level). Let us consider the comparison of the near strip (presented at 61 cm) with an intermediate strip (presented upright at 204 cm, flat at 206 cm). For the frontally viewed strips, Baird and Biersdorf found slight overconstancy (consistent with "objective" instructions): A strip at 61 cm distance needed to be 20.9 cm, on average, to appear equal to one of 20.3 cm presented at 204 cm. On the other hand, underconstancy was observed for the strips laid flat: A flat strip at 61 cm needed to be only 18.0 cm to appear equal in length to one of 20.3 cm viewed at 206 cm (compression by 0.89). Moreover, even at 309 cm an object of 20.3 cm was matched to one of 18.3 cm viewed at 61 cm (compression by 0.90). If the overconstancy factor of frontal size is taken into account, these results can be interpreted as evidence of roughly an 0.85 compression of a depth interval along the surface of the alley with a roughly three-fold increase in absolute viewing distance. Although, depth and frontal views are not truly segregated because of the elevation of the eye above the surface, this amount of compression is much less than that observed at greater viewing distances (e.g., Wagner, 1985). (back to text)

13. Rivest, Ono, and Saida (1989) have demonstrated that depth judgments from observer-produced motion parallax displays are affected by information about apparent distance. However, their results are even more consistent with the recovery of angle information: (1) They did not find any effect of manipulating subjects' vergence angle. (2) They did find differences in perceived depth from motion parallax for displays presented alongside dollar bills of which the size was manipulated (á la O'Leary and Wallach, 1980), however the depth differences were also quantitatively consistent with simple size scaling (i.e., were roughly linear rather than quadratic). (3) The only condition in which the change in judged depth was quadratically related to a change in apparent distance, was when apparent distance to a display was manipulated using an induction screen placed in front of the display and the change in angular position of the center of the motion parallax display (through the induction aperture) was made consistent with the intended "false" distance. In this case the change in angular regard was consistent with the nearer "false" distance. It therefore seems that angular information, rather than distance, per se, may be most important for scaling depth from motion parallax. (back to text)

14. This statement must be qualified by consideration of results (e.g., Loomis et al., 1992) demonstrating fairly precise egocentric distance information available for blind walking to targets up to 12 meters distant. (back to text)

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Appendix

In this appendix we explore the relationship between depth and optical flow in kinetic depth and motion parallax, and compare it to the relationship between disparity and depth in stereopsis. We assume visual fixation in each case as follows: In the kinetic depth case, the target object rotates around the fixation point, and in motion parallax the observer translates while keeping the gaze fixed on the target. In stereopsis, the optic axes of the eyes converge at a point on the target. The analysis will show that recovering depth in the two motion cases requires information that is more difficult to obtain than the information required to recover depth from stereo. The derivations are based on those of Olson (1991) and Ballard and Ozcandarli (1988).

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Figure A-1 shows the imaging situation for the two motion cases. We assume perspective projection with the focal point of the imaging system at the origin. The image plane is parallel to the X-Y plane and passes through (0,0,f), so that a world point (X,Y,Z) projects to (x,y) = f(X/Z, Y/Z) in the image plane. For the case of motion parallax, the observer translates along the X axis at a speed of , maintaining fixation on point F. This induces an instantaneous rotation of object points about F with angular velocity = /ZF. For the kinetic depth case, the observer is stationary ( = 0) and the target rotates about F with a constant speed of .
Consider object point P at a distance r from the fixation point. P can be expressed as (Xp,Yp,Zp) = (-r cosL, Yp, Zf - r sinL), so the x coordinate of its image plane projection is given by

Differentiating with respect to time gives

Solving for Z and combining with the expressions for P yields the depth:

For the motion parallax case this can be written in terms of the observer's velocity:

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The case of stereopsis under binocular fixation is essentially a discrete version of the observer translation case. Figure A-2 shows the assumed geometry. In this situation it can be shown (Olson, 1991) that the depth of an arbitrary world point is given by

When the gaze direction is forward (qL Å qR), for the region of central vision the above expression can be well approximated by

Å

The depth equations for the three cases are similar in form, and all require knowledge of the fixation depth ZF in order to determine absolute depth. For reasonably close fixations ZF can be estimated from accommodation and (in the case of stereopsis) vergence angle. Aside from ZF, the stereo depth equation depends only on quantities that can be measured in the images (disparity and position of feature points) and on intrinsic properties of the imaging system (baseline and focal length). The motion equations, on the other hand, require an additional extrinsic parameter, the angular velocity . For the motion parallax case this is, in principle, computable from the observer's translational velocity and the fixation depth, or simply from the angular velocity required to track the fixation point. For the kinetic depth case, the depth is underdetermined unless the angle of rotation can be extracted. In the Introduction, we show how angular rotation information could be derived from foreshortening information, although for small rotations or small increments of rotation this information is poorly specified.
In summary, all of the parameters required to determine depth from stereoscopic disparity are readily available. In the case of motion parallax, an additional parameter (the angular velocity) must be estimated, and it must be assumed that the target does not have a concomitant rotation. If the target moves (as in the kinetic depth case), depth is underdetermined. The fact that recovering structure from motion requires more assumptions and has an additional free parameter suggests that obtaining a complete 3-D solution in the motion cases may not be worth the effort. This might explain why, in the experiments described in this paper, shape judgments based on stereo information proved far more accurate than judgments based on either kinetic depth or motion parallax.

Author Notes

This research was supported by National Aeronautics and Space Administration Grants NAG2-812 and NCA2-708 to Dennis R. Proffitt. Helpful comments on earlier drafts of this paper were provided by J. Todd, J. Lappin, L. Kontsevich, and F. Pollick. We thank Jonathan Midgett and Melissa Salva for assisting in the collection of data. Steve Jacquot programmed the displays for Experiment 1 and helped the first author design the stereoviewer, which was constructed, as were the mechanical apparati of Experiments 2 and 3, by Ron Simmons.

Created by Richa Jain, rjain1@swarthmore.edu