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Methods  |   December 2013
The dichoptiscope: An instrument for investigating cues to motion in depth
Author Affiliations
  • Ian P. Howard
    Centre for Vision Research, York University, Toronto, Canada
  • Kazuho Fukuda
    Department of Information Processing, Tokyo Institute of Technology, Yokohama, Japan
    fukuda@ip.titech.ac.jp
  • Robert S. Allison
    Centre for Vision Research, Department of Computer Science & Engineering, York University, Toronto, Canada
    allison@cse.yorku.cahttp://percept.eecs.yorku.ca/
Journal of Vision December 2013, Vol.13, 1. doi:10.1167/13.14.1
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      Ian P. Howard, Kazuho Fukuda, Robert S. Allison; The dichoptiscope: An instrument for investigating cues to motion in depth. Journal of Vision 2013;13(14):1. doi: 10.1167/13.14.1.

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Abstract
Abstract
Abstract:

Abstract  A stereoscope displays 2-D images with binocular disparities (stereograms), which fuse to form a 3-D stereoscopic object. But a stereoscopic object creates a conflict between vergence and accommodation. Also, motion in depth of a stereoscopic object simulated solely from change in target vergence produces anomalous motion parallax and anomalous changes in perspective. We describe a new instrument, which overcomes these problems. We call it the dichoptiscope. It resembles a mirror stereoscope, but instead of stereograms, it displays identical 2-D or 3-D physical objects to each eye. When a pair of the physical, monocular objects is fused, they create a dichoptic object that is visually identical to a real object. There is no conflict between vergence and accommodation, and motion parallax is normal. When the monocular objects move in real depth, the dichoptic object also moves in depth. The instrument allows the experimenter to control independently each of several cues to motion in depth. These cues include changes in the size of the images, changes in the vergence of the eyes, changes in binocular disparity within the moving object, and changes in the relative disparity between the moving object and a stationary object.

Introduction
The Oxford English Dictionary defines a stereoscope as “a device in which two pictures of the same object taken at slightly different angles are viewed together, creating an impression of depth and solidity.” Such pictures are known as stereograms. We will refer to the three-dimensional object seen in a stereoscope as a “stereoscopic object.” A stereoscopic object differs from a real object in the following ways: 
  1.  
    Although the eyes can converge on parts of a stereoscopic object at different distances, they cannot accommodate on different parts of the object, because stereograms are two-dimensional. Convergence and accommodation are therefore dissociated when one views a stereoscopic object.
  2.  
    Motion of the head produces motion parallax between near and far parts of a real object. This relative motion provides depth information. Motion parallax is not produced in a stereoscopic object. Instead, the object appears to hold a fixed orientation relative to the head, just as a real object does when motion of the head does not produce motion parallax.
  3.  
    When a real object moves in depth, the disparity between near and far parts of the object changes as an inverse function of the square of distance. The disparity in a stereoscopic object changes only as an inverse function of viewing distance.
  4.  
    When a real object moves in depth, perspective relationships between the images of near and far parts of the object change. These changes occur in each eye. They do not occur when stereograms are moved in depth.
  5.  
    When a real object moves in depth, there is a change in each eye in the way near parts occlude far parts. There are no such changes when stereograms are moved in depth. Also, when a real object moves in depth, there is a change in interocular differences in the way near parts occlude far parts. In other words, there is a change in da Vinci stereopsis. This change does not occur in a stereoscopic object.
All these limitations can be overcome by using a pair of identical physical objects as stimuli rather than stereograms. But an instrument displaying identical physical objects to the two eyes is not a stereoscope. We call such an instrument a “dichoptiscope.” We call two identical objects displayed in a dichoptiscope, one to each eye, a pair of “monocular objects.” The monocular objects may be two-dimensional or three-dimensional.1 We call the object formed by fusing two monocular objects a “dichoptic object.” A dichoptic object is visually indistinguishable from each of the binocularly-viewed monocular objects (the proximal stimulus is identical). One can converge and accommodate on different parts of a dichoptic object, and head motion produces normal motion parallax. A dichoptic object contains all the perspective of the monocular objects. When the monocular objects move in depth along the two visual axes, the dichoptic object also moves in depth. Changes in the perspective and disparity of the moving dichoptic object are identical to the changes in a real moving object. However, a dichoptic object differs from a real object in one crucial respect: A real object is a single distal stimulus, whereas the two monocular objects comprising a dichoptic object are distinct distal stimuli. This means that cues to distance in a stationary dichoptic object can be independently manipulated. Also, we can manipulate how these cues to distance change as a dichoptic object moves in depth. The cues to motion in depth that the instrument is designed to manipulate are covered in the following sections. 
Cues to motion in depth
Changing image size
The images of an object change in size as the object moves in depth. We will refer to changing image size as “looming.” 
Parallax
Parallax is the relative motion of the images of stimuli at different distances produced by motion of the head or by motion of the stimuli relative to the head. Parallax includes changing accretion or deletion of a far surface by an opaque near surface. 
Changing accommodation and image focus
Changing accommodation refers to changes in the focal length of the lens as a fixated object moves in depth. Changing image focus refers to the changing blur of the image of an object as it moves out of the plane in which the eye is accommodated. 
Changing absolute disparity
Absolute binocular disparity refers to the disparity between the images of a point without reference to the disparity of any other points. When the eyes are converged symmetrically on a point at infinity, the angular horizontal disparity, θ, between the images of a point at distance D for an interocular distance a is given by θ = 2 arctan(a/2D). The same function defines the angle of vergence when the eyes converge on the object. Convergence or divergence of the eyes produces a corresponding change in the absolute disparity of all points in the field of view. Also, a change in the absolute disparity of the two images in a stereoscope or in the dichoptiscope tends to induce a corresponding change in vergence. 
Changing internal relative disparity
Internal relative disparity refers to the relative disparities between parts of an object. Consider a short rod on the cyclopean line of sight extending out from a point midway between the eyes. As the rod approaches, the horizontal disparity between its near and far ends (its internal disparity) increases approximately as an inverse function of the square of the distance of the rod from the eyes. Now consider a textured surface lying in a frontal plane. The horizontal disparities over the surface depend on the curvature of the horopter at the distance of the surface and on the distances of points from the median plane. As the surface recedes, the horopter flattens and the horizontal disparities diminish. The vertical disparities over a frontal surface depend on the distance of the surface and on the horizontal and vertical positions of points on the surface. Figure 1 shows an example of these internal disparities. The pattern of horizontal and vertical disparities produced by points on a frontal surface in the center of the visual field is uniquely determined by the distance of the surface (Howard & Rogers, 2012, p. 381). 
Figure 1
 
An illustration of the binocular disparities produced by a rectangular grid in a frontal plane. The left panel illustrates the keystone distortion pattern produced in the left (blue) and right (red) images, respectively. The right panel shows disparity vectors between corresponding grid corners in the left and right images.
Figure 1
 
An illustration of the binocular disparities produced by a rectangular grid in a frontal plane. The left panel illustrates the keystone distortion pattern produced in the left (blue) and right (red) images, respectively. The right panel shows disparity vectors between corresponding grid corners in the left and right images.
Changing external relative disparity
External relative disparity refers to the disparity between the images of a moving object relative to the disparity of a stationary object. The relative external disparity between a point moving in depth and a stationary point increases as the depth interval between the two points increases. 
Changing interocular differences of occlusion
The extent to which a near surface occludes a far surface differs between the two eyes. This forms the basis of Da Vinci stereopsis (Howard & Rogers, 2012, p. 267). As an object moves in depth, interocular differences in occlusion change both within the moving object and between the moving object and a stationary object. 
Traditional control of the cues to motion in depth
In traditional stereoscopes, the disparity between the half images is controlled by the separation between the left and right eye images. Hering's (1879) haploscope and synoptophore (major amblyoscope; Stanworth, 1958) are more flexible in that vergence can be controlled without changing internal disparity. Sometimes accommodation can be stimulated in these instruments by sliding the targets along the haploscope arms or by introducing lenses. Other cues must be simulated in the images themselves, most conveniently with computer graphics. Computer-graphic displays offer considerable flexibility and continually improving quality but fall short of the fidelity of the real world. Computer graphics on a 2-D display also cannot simulate cues to accommodation, and several devices have been developed to provide accommodation in computer displays by providing a volumetric display using time multiplexing, layering, or other techniques. These displays have been an active area of research for several decades, but few of them have been used in vision science. Notable exceptions are custom multilayer (Akeley, Watt, Girshick, & Banks, 2004; Eagle, Paige, Sucharov, & Rogers, 1999; Paige, Neil, & Sucharov, 1998) and variable-lens (Love et al., 2009; Shibata et al., 2005) systems that have been used to study depth-cue conflict. These displays provide a way to approximate the blur cues to accommodation by using a small number of depth intervals. In contrast, real physical motion of the stimulus (Rushton & Duke, 2009; Welchman, Tuck, & Harris, 2004) provides for high-fidelity stimuli but does not allow for the dissociation of depth cues that is possible in the dichoptiscope. 
General structure of the dichoptiscope
Figure 2 is a photograph of the dichoptiscope, and Figure 3 is a diagrammatic plan view. The monocular objects, indicated by green and blue triangles, are mounted on frames on opposite sides of the two orthogonal mirrors, arranged as in a Wheatstone stereoscope. The orientation and position of each monocular object on its frame can be adjusted by a gimbal and two orthogonal threads. These adjustments are used to align the two monocular objects so that they fuse into a single dichoptic object when viewed through the mirrors. Each frame is mounted on a stimulus base that is mounted on a carriage, as shown in Figure 4. The frame can be rotated about its midvertical axis on the base, and the base can be rotated about the same axis on the carriage. We will see that these rotations vary the internal binocular disparity of the dichoptic object. 
Figure 2
 
A photograph of the apparatus. The stimulus in this case is a textured 2-D plane printed on transparent acetate sheets and backlit by electroluminescent panels. In general, front-lit 3-D or 2-D solid objects on a matte background or backlit transparent 3-D or 2-D objects can be used.
Figure 2
 
A photograph of the apparatus. The stimulus in this case is a textured 2-D plane printed on transparent acetate sheets and backlit by electroluminescent panels. In general, front-lit 3-D or 2-D solid objects on a matte background or backlit transparent 3-D or 2-D objects can be used.
Figure 3
 
Plan of the dichoptiscope. In this setting, the tracks are aligned with the apex of the mirrors so that vergence changes normally as the dichoptic object moves along the midline. Also, the rods are aligned with the apex of the mirrors so that the internal disparity of the dichoptic object changes correctly. The shaded area on the right indicates the detail shown in Figure 4.
Figure 3
 
Plan of the dichoptiscope. In this setting, the tracks are aligned with the apex of the mirrors so that vergence changes normally as the dichoptic object moves along the midline. Also, the rods are aligned with the apex of the mirrors so that the internal disparity of the dichoptic object changes correctly. The shaded area on the right indicates the detail shown in Figure 4.
Figure 4
 
Detailed structure of the mounting of the right-eye monocular object (shaded area in Figure 3). The linear motion along the track, the linear motion perpendicular to the track, and the rotation of the object on the carriage can all be independently and dynamically controlled, allowing for independent control of physical distance, vergence demand, internal disparity, and external relative disparity. For details, see figure annotations and the text.
Figure 4
 
Detailed structure of the mounting of the right-eye monocular object (shaded area in Figure 3). The linear motion along the track, the linear motion perpendicular to the track, and the rotation of the object on the carriage can all be independently and dynamically controlled, allowing for independent control of physical distance, vergence demand, internal disparity, and external relative disparity. For details, see figure annotations and the text.
The two carriages are mounted on tracks that extend from the observer on either side. The carriages with their monocular objects can be moved equally in opposite directions along the tracks. When the tracks are aligned with the apex of the mirrors, as in Figure 3, the dichoptic object moves in depth along the observer's midline, as in Movie 1. Its movement in depth is visually identical to the motion of a real object. The tracks can be rotated equally in opposite directions about vertical pivots. It will be assumed that each pivot is 50 cm from an eye. We will see that rotation of the tracks controls the way the eyes converge or diverge as the dichoptic object moves in depth. 
Instead of being moved along the main track, each monocular object can be moved along a lateral track orthogonal to the main track. This lateral motion changes the absolute disparity of the dichoptic object while the object remains at a constant distance from the eyes. A computer-controlled servomotor moves the two carriages at the same velocity along the main tracks or along the lateral tracks through a distance determined by the positions of microswitches on the tracks. 
 
Movie 1.
 
The motion of the dichoptic object when the tracks and rods are set as in Figure 3. The back-and-and forth movement of the left and right monocular objects along the physical tracks can be seen to the left and right of the subject, respectively. When viewed through the mirrors, this object pair forms a dichoptic image in front of the subject moving in depth. The dotted lines show the visual axes of the two eyes if they track the object. The movement in depth of the dichoptic object is visually identical to the movement of a real object.
A horizontal steel rod passes through a linear bearing in the center of each stimulus base. The bearing keeps the rod orthogonal to the base. The end of each rod near the observer is mounted on a vertical rod pivot. Rotation of the rods on the pivots causes the monocular objects to rotate in opposite directions, which changes the internal disparity of the dichoptic object. 
Any pair of identical 2-D or 3-D objects can be placed in the dichoptiscope. Solid objects placed on a matte background and carefully front lit can be used. We, however, typically mount our monocular objects on electroluminescent panels in order to perform experiments in a dark room. These panels produce very even white light with no pixels, and they can be varied in luminance. We compose textured surfaces of 2-D or 3-D objects on a computer and print them on transparent acetate sheets. The texture elements are transparent on a black opaque background, and the unused area of the electroluminescent panels is covered with black velvet. Masking of the unused portion of the panel and all other surfaces in view with black velvet, combined with modest intensity of back illumination of the stimuli, ensures that nothing but the light through the transparent texture elements is visible. Flat surfaces of the printed acetate sheets are simply placed on the luminous panels. Depth steps are made from two superimposed or juxtaposed sheets separated in depth. Surfaces curved in depth, cones, and cylinders are made by forming acetate sheets round a template. More complex patterns can be formed by 3-D printing with transparent materials. 
Control of cues to motion in depth by the dichoptiscope
We will now describe how sources of information about motion in depth are controlled in the dichoptiscope. Figure 3 shows that the mirrors reflect the monocular objects, the tracks, the rods, and the rod pivots into a virtual mirror space extending out from the observer. The observer perceives only the dichoptic object. Its visual features are determined by the locations and motions of the reflections of the tracks and rods in the mirror space. Therefore, the following figures and movies show only objects in the mirror space. The real objects are on each side of the observer. 
Looming
The images of the dichoptic object loom naturally when the dichoptic object moves along the main track. Looming is eliminated when the carriages are held at a fixed distance from the eyes. With looming absent, changes in absolute disparity (vergence) can be set to any value by moving the monocular objects along the lateral tracks. Also, with looming absent, changes in internal disparity can be set to any value by setting the rod pivots to different positions. When the rods are removed, as in Movie 2, there is no looming, but internal disparity changes in the normal way with changes in vergence. 
 
Movie 2.
 
When the stimuli move laterally at a fixed distance, there is no looming, but vergence changes. The monocular objects are actually located to the left and right of the subject (not shown) but are combined through the mirrors to form the superimposed dichoptic view shown (mirror space). The dotted lines show the visual axes of the two eyes when they track the object.
Looming can be isolated as the only cue indicating nonzero motion in depth by closing one eye or by aligning each track and each rod with the eye on the same side, as shown in Movie 3
 
Movie 3.
 
When the tracks and rods are aligned with the eyes, there is looming but no change in vergence demand or internal disparity. The monocular objects are actually located to the left and right of the subject (not shown) but are combined through the mirrors to form the superimposed dichoptic view shown. The dotted lines showing the visual axes of the two eyes is superimposed with the rods aligned with the eyes.
Note that in this case the looming cue is produced by changes in physical and optical distance, and therefore accommodation demand follows the looming stimulus. Accommodation and blur are relatively weak but still potentially useful depth cues and could signal motion-in-depth (Nguyen, Howard, & Allison, 2005; Watt, Akeley, Ernst, & Banks, 2005). Furthermore, while vergence demand does not change in this condition, accommodative convergence (Donders, 1864) could be elicited with monocular viewing. Accommodation could be eliminated or weakened as factor in the apparatus, when appropriate, through observation of stimuli through a pinhole aperture. 
Parallax
Parallax can be introduced by viewing a dichoptic object moving in depth with respect to a stationary object or a ground plane seen through semisilvered mirrors. Parallax can also be introduced by side-to-side motion of the observer's head. 
Absolute horizontal disparity (vergence)
When the tracks are parallel and aligned with the apex of the mirrors, the reflected tracks are superimposed on the observer's midline, as shown in Figure 3. Therefore, the dichoptic object moves along the midline and its images change in absolute horizontal disparity, just like those of a real object. If the dichoptic object is binocularly fixated, the eyes converge so as to cancel the change in disparity, just as they do when a real object approaches along the midline. 
Now consider what happens when each track is rotated about its pivot 50 cm from the eye so that each reflected track is aligned with the visual axis of an eye, as shown in Movie 3. Each monocular object moves along a visual axis, so there is no change in absolute disparity and the eyes must remain converged at 50 cm to maintain a fused image. Therefore, changes in vergence demand are eliminated. If the rods are also aligned with the eyes, as in Movie 3, internal disparity remains constant. It the rods pivot in the midline, disparity changes normally. 
When the tracks are rotated to a larger angle, the change in absolute disparity and the attendant change in vergence are reversed. The observer must diverge when the dichoptic object approaches and converge when it retreats, as shown in Movie 4
 
Movie 4.
 
When the tracks are rotated beyond the visual axes, vergence is reversed. Looming and internal disparity change normally. As in Movie 2, only the unfolded dichoptic view is shown. The dotted lines show the visual axes of the two eyes when they track the object, and the point labeled “convergence” indicates the changing vergence demand.
When each track is rotated toward the opposite eye, the change in absolute disparity is greater than normal and the eyes must converge or diverge more than they do when a real object moves in depth, as shown in Movie 5. Thus, the change in absolute disparity of the moving dichoptic object, and the attendant change in vergence, may be set to any value without changing other cues to motion in depth. 
 
Movie 5.
 
When the tracks are rotated toward the midline, vergence changes more than normal. Looming and internal disparity change normally. As in Movie 2, only the unfolded dichoptic view is shown. The dotted lines show the visual axes of the two eyes when they track the object, and the point labeled “convergence” indicates the changing vergence demand.
Figure 5 shows the geometry of the dichoptiscope under this manipulation of the angle of the main tracks to control vergence demand. The figure shows the position of the right eye and its mirror-space view of the apparatus; the geometry for the left eye and stimulus can be obtained by symmetry. When the eyes converge at the object's base position and the left and right side of the apparatus are set with symmetrical angles relative to the midline, vergence angle θverg (we define angles as positive when counterclockwise here) and the distance Dverg defined by this vergence angle can be defined in terms of the dichoptic object's position by the following equations:  and  Here, a/2 represents the position of right eye's nodal point. It can be seen that the vergence demand specified by Dverg is the physical distance of the object, Ybase, attenuated or amplified by a function of its lateral movement, Xbase. As Ybase and Xbase are coupled due to their common motion, this modulation produces the increased or decreased vergence demand. 
Figure 5
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the main tracks are pivoted symmetrically to vary the relationship between physical distance and vergence demand. The geometric construction shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The long rectangle shaded light gray is a main track, which can be rotated around the track pivot, which in turn can be set at a desired distance from the observer. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 5
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the main tracks are pivoted symmetrically to vary the relationship between physical distance and vergence demand. The geometric construction shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The long rectangle shaded light gray is a main track, which can be rotated around the track pivot, which in turn can be set at a desired distance from the observer. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
The position of the dichoptic object, (Xbase, Ybase), as a function of translation along the track and track rotation is defined by the following equations:  and  Here, Yt-pivot and θtrack represent the pivot distance and angle of the main track, respectively. Both of these parameters can be set manually. Tbase represents the position on the main track, which can be measured by reading the scale on the track. Replacing Tbase with Tswitch(far) or Tswitch(near), the position of far/near microswitches on the main track, determines the object position at maximum or minimum distance in a motion-in-depth trajectory. 
As shown in Movie 2, vergence demand can also be controlled in the absence of any looming by moving the stimuli in opposite directions along the lateral tracks for the two eyes. Figure 6 represents the geometry of the object motion along the lateral tracks for the right eye in this situation. Vergence demand angle, θverg, and the corresponding vergence-demand distance, Dverg, are also defined by Equations 1 and 2. The position of object base is defined by Ybase and Xbase. By construction, these are  the constant distance of lateral-track origin from the observer, and  the varying position of the dichoptic object on the lateral track. Substituting Equations 5 and 6 into Equations 1 and 2, we obtain  and  The maximum and minimum vergence distance for a given motion trajectory can be calculated by replacing Tobject with Tswitch(right) or Tswitch(left), the position of right/left microswitches on the lateral track. 
Figure 6
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the lateral tracks are used to stimulate vergence demand without looming. The diagram shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The longer, light-gray rectangle is the lateral track for the right eye. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 6
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the lateral tracks are used to stimulate vergence demand without looming. The diagram shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The longer, light-gray rectangle is the lateral track for the right eye. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Internal relative disparity
The near and far parts of a real stationary object produce images with binocular disparity because the eyes see the object from different vantage points. Therefore, the internal disparities in a dichoptic object can be changed by simply rotating the monocular objects in opposite directions about their midvertical axes. In the dichoptiscope, this rotation is produced by the rods that pass through linear bearings in the stimulus frames. When both rod pivots are under the apex of the mirrors, as in Figure 7a, the reflections of the rods are superimposed on the midline, and the internal disparity of the dichoptic object is the same as that of each monocular object. When each rod pivot is so positioned that the reflection of the rod is aligned with the eye, as shown in Figure 5b, the two monocular objects have the same orientation with respect to the visual axes. The images in the two eyes therefore fall on geometric corresponding points, and internal disparities are eliminated. When each rod pivot is on the temporal side of each eye, the internal disparity is reversed. When each rod pivot is moved toward the opposite eye, the internal disparity is increased. Thus by the turn of a knob that sets both pivots to different positions, the internal disparity of the dichoptic object can be made larger or smaller than normal or reversed without affecting other cues to depth. 
Figure 7
 
The locations of the rod pivots determine how internal disparity changes as the dichoptic object moves along the midline. (a) Disparity changes normally because the rods are aligned with the midline. The dichoptic object appears like a real object approaching along the midline. (b) Disparity remains constant because the rods are aligned with the visual axes, which keeps the orientation of the stimuli with respect to the visual axes constant.
Figure 7
 
The locations of the rod pivots determine how internal disparity changes as the dichoptic object moves along the midline. (a) Disparity changes normally because the rods are aligned with the midline. The dichoptic object appears like a real object approaching along the midline. (b) Disparity remains constant because the rods are aligned with the visual axes, which keeps the orientation of the stimuli with respect to the visual axes constant.
Now consider an object moving in depth along the midline. The angle of slant that the object makes with respect to each visual axis changes. In effect, the images in the two eyes rotate about vertical axes in opposite directions. This produces a change in internal disparity as an object moves in depth. Disparity changes approximately in inverse proportion to the square of viewing distance. In the dichoptiscope, the rate at which internal disparity changes is controlled by rotating each monocular object while it moves along its track. The rotation is produced by rotation of the rods on their pivots. Assume that the eyes are converged on the center of the dichoptic object. When both rod pivots are under the apex of the mirrors, as in Figure 7a, the change in internal disparity is the same as that produced by a real object moving along the midline. When the reflected image of each rod is aligned with the eye on the same side, as in Figure 7b, each monocular object moves along the visual axis of an eye. Therefore, internal disparity remains constant. The value of the unchanging internal disparity depends on the initial orientation of each monocular object relative to the carriage. Through moving the rod pivots to different locations, the change in internal disparity as the stimulus moves in depth can be normal, greater than normal, less than normal, or reversed. 
Figures 8 and 9 show the right-eye geometry for the relationship between the action of the eye rods and the internal disparity while the dichoptic object moves along the main track and the lateral track, respectively. Internal disparity is determined by the angle of slant that the object makes with respect to each visual axis, as described in the previous paragraph. This angle, which we call θobj-vis, is calculated by the following simple equation:  Here, θobj represents the angle of slant that the object makes with respect to the eye rod, which is a constant since the eye rod is attached to the base with linear bearings. The variable θrod represents the angle that the rod makes relative to the y-axis; that value can be calculated by the following equation (except that θrod is constant at 0 when the eye rods are not in use):  Here, Xrod is the position of the eye-rod pivot, while (Xbase, Ybase) represents the position of the dichoptic object and is calculated by Equations 3 and 4 for motion on the main track (Figure 8) or Equations 5 and 6 for motion on the lateral track (Figure 9). 
Figure 8
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the main track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The main track, object base, and microswitches are drawn as in Figure 5. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves in depth along the main track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 8
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the main track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The main track, object base, and microswitches are drawn as in Figure 5. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves in depth along the main track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 9
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the lateral track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The lateral track, object base, and microswitches are drawn as in Figure 6. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves along the lateral track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 9
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the lateral track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The lateral track, object base, and microswitches are drawn as in Figure 6. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves along the lateral track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
The vergence angle, θverg, is given in Equation 1. Note that when the eye rods are pivoted under the observer's eye (Xrod = a/2), the rotation due to the eye rods cancels the change in internal disparity due to vergence leaving internal disparity constant, since   
External relative disparity
As an object approaches, the disparity in its images changes with respect to the images of a stationary object. At any instant, the external relative disparity is proportional to the distance in depth between the moving object and the stationary object. In the dichoptiscope, external disparity is provided by placing a stationary object in front of the observer so that it is seen through semisilvered mirrors. The dichoptic object is therefore superimposed on the stationary object. As the dichoptic object moves in depth, it appears to move in depth relative to the stationary object. External disparity may also be provided by a textured surface in any orientation, including a ground plane. The dichoptic object can be made to move through, past, or over a stationary surface. 
The instrument described here allows an experimenter to independently control many visual cues to motion in depth. But much simpler versions of the apparatus could be built to investigate one or a small number of cues. 
Acknowledgments
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. 
Commercial relationships: none. 
Corresponding author: Robert S. Allison. 
Email: allison@cse.yorku.ca. 
Address: Center for Vision Research, York University, Toronto, Ontario, Canada. 
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Footnotes
1  Note that there is flexibility in the choice of the physical objects, which can be identical physical props, 2-D- or 3-D-printed objects, or a variety of electronic displays. While typically identical, they do not need to be (or they could change over time, in the case of electronic displays), allowing for additional flexibility.
Figure 1
 
An illustration of the binocular disparities produced by a rectangular grid in a frontal plane. The left panel illustrates the keystone distortion pattern produced in the left (blue) and right (red) images, respectively. The right panel shows disparity vectors between corresponding grid corners in the left and right images.
Figure 1
 
An illustration of the binocular disparities produced by a rectangular grid in a frontal plane. The left panel illustrates the keystone distortion pattern produced in the left (blue) and right (red) images, respectively. The right panel shows disparity vectors between corresponding grid corners in the left and right images.
Figure 2
 
A photograph of the apparatus. The stimulus in this case is a textured 2-D plane printed on transparent acetate sheets and backlit by electroluminescent panels. In general, front-lit 3-D or 2-D solid objects on a matte background or backlit transparent 3-D or 2-D objects can be used.
Figure 2
 
A photograph of the apparatus. The stimulus in this case is a textured 2-D plane printed on transparent acetate sheets and backlit by electroluminescent panels. In general, front-lit 3-D or 2-D solid objects on a matte background or backlit transparent 3-D or 2-D objects can be used.
Figure 3
 
Plan of the dichoptiscope. In this setting, the tracks are aligned with the apex of the mirrors so that vergence changes normally as the dichoptic object moves along the midline. Also, the rods are aligned with the apex of the mirrors so that the internal disparity of the dichoptic object changes correctly. The shaded area on the right indicates the detail shown in Figure 4.
Figure 3
 
Plan of the dichoptiscope. In this setting, the tracks are aligned with the apex of the mirrors so that vergence changes normally as the dichoptic object moves along the midline. Also, the rods are aligned with the apex of the mirrors so that the internal disparity of the dichoptic object changes correctly. The shaded area on the right indicates the detail shown in Figure 4.
Figure 4
 
Detailed structure of the mounting of the right-eye monocular object (shaded area in Figure 3). The linear motion along the track, the linear motion perpendicular to the track, and the rotation of the object on the carriage can all be independently and dynamically controlled, allowing for independent control of physical distance, vergence demand, internal disparity, and external relative disparity. For details, see figure annotations and the text.
Figure 4
 
Detailed structure of the mounting of the right-eye monocular object (shaded area in Figure 3). The linear motion along the track, the linear motion perpendicular to the track, and the rotation of the object on the carriage can all be independently and dynamically controlled, allowing for independent control of physical distance, vergence demand, internal disparity, and external relative disparity. For details, see figure annotations and the text.
Figure 5
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the main tracks are pivoted symmetrically to vary the relationship between physical distance and vergence demand. The geometric construction shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The long rectangle shaded light gray is a main track, which can be rotated around the track pivot, which in turn can be set at a desired distance from the observer. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 5
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the main tracks are pivoted symmetrically to vary the relationship between physical distance and vergence demand. The geometric construction shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The long rectangle shaded light gray is a main track, which can be rotated around the track pivot, which in turn can be set at a desired distance from the observer. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 6
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the lateral tracks are used to stimulate vergence demand without looming. The diagram shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The longer, light-gray rectangle is the lateral track for the right eye. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 6
 
Right-eye mirror-space representation of the geometry of a motion trajectory when the lateral tracks are used to stimulate vergence demand without looming. The diagram shows the stimulus position and vergence angle produced at an arbitrary point in the trajectory, assuming the eyes track the stimulus. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The longer, light-gray rectangle is the lateral track for the right eye. The short rectangle painted dark gray is the object base, which can mount a real 2-D or 3-D object as a dichoptic stimulus. The equations in the text are derived for the origin of an object coordinate frame attached to the object base; the visual axis of each eye, in this case the right, is assumed to pass through this origin (green dashed line).
Figure 7
 
The locations of the rod pivots determine how internal disparity changes as the dichoptic object moves along the midline. (a) Disparity changes normally because the rods are aligned with the midline. The dichoptic object appears like a real object approaching along the midline. (b) Disparity remains constant because the rods are aligned with the visual axes, which keeps the orientation of the stimuli with respect to the visual axes constant.
Figure 7
 
The locations of the rod pivots determine how internal disparity changes as the dichoptic object moves along the midline. (a) Disparity changes normally because the rods are aligned with the midline. The dichoptic object appears like a real object approaching along the midline. (b) Disparity remains constant because the rods are aligned with the visual axes, which keeps the orientation of the stimuli with respect to the visual axes constant.
Figure 8
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the main track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The main track, object base, and microswitches are drawn as in Figure 5. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves in depth along the main track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 8
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the main track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The main track, object base, and microswitches are drawn as in Figure 5. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves in depth along the main track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 9
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the lateral track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The lateral track, object base, and microswitches are drawn as in Figure 6. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves along the lateral track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
Figure 9
 
Right-eye mirror-space representation of the geometry of the control of internal disparity during action of the lateral track. The apparatus for the right eye shown here is actually located to the right of the subject but is combined with the left eye's view through the mirror to form a superimposed dichoptic view. The lateral track, object base, and microswitches are drawn as in Figure 6. A real object mounted on the object base is represented by the red triangle, which maintains a constant angle with respect to the eye rod while the object moves along the lateral track. The internal disparity is determined by the angle that the dichoptic object makes with respect to the visual axis.
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