Larry F. Hodges Geometric Considerations for College of Computing-0280 Graphics, Visualization Stereoscopic Virtual Environments & Usability Center Atlanta, Georgia 30332 Elizabeth Thorpe Davis of School Psychology-0170 Abstract Graphics, Visualization & Center Usability We examine the relationship among the different geometries implicit in a stereoscopic Institute of Georgia Technology virtual environment. In particular, we examine in detail the relationship of retinal dispar- Atlanta, 30332 Georgia ity, fixation point, binocular visual direction, and screen parallax. We introduce the con- cept of a volumetric spatial unit called a stereoscopic voxel. Due to the shape of stereo- scopic voxels, apparent depth of points in space may be affected by their horizontal placement. Downloaded from http://direct.mit.edu/pvar/article-pdf/2/1/34/1622602/pres.1993.2.1.34.pdf by guest on 26 September 2021 I Introduction The display component of the most common implementations of virtual environments provides the user with a visual image that incorporates stereopsis as a visual cue. Examples include head-mounted displays (Teitel, 1990), time- multiplexed CRT-based displays (Deering, 1992), and time-multiplexed pro- jection systems (Cruz-Neira, Sandin, DeFanti, Kenyon, & Hart, 1992). It is not always recognized, however, that the characteristics of a stereoscopic image can be very different from that of a monoscopic perspective image. The visual impression given by stereoscopic images are very sensitive to the geometry of the visual system of a user, the geometry ofthe display environment, and the modeling geometry assumed in the computation of the scene. In this paper we present a tutorial whose purpose is to review these basic geometries and to ana- lyze their relationship to each other in a stereoscopic virtual environment. In Section 2 we review the geometry of binocular vision and retinal disparity. In Section 3 we discuss modeling geometry and its relationship to retinal dis- parity. Section 4 describes the effects of the discrete nature of display geometry and the distortions caused by optical and tracking artifacts. In Section 5 we re- visit our approximate model of the visual system geometry and discuss its limi- tations. 2 Visual System Geometry Stereopsis results from the two slightly different views of the external world that our laterally displaced eyes receive (e.g., Schor, 1987; Tyler, 1983). This difference is quantified in terms of retinal disparity. If both eyes are fixated on a point, f1, in space, then an image of f 1 is focused at corresponding points in the center of the fovea1 of each eye. Another point, f2, at a different spatial Presence, Volume 2, Number I, Winter 1993 1. The fovea is the part of the human retina that possesses the best spatial resolution or visual © 1993 The Massachusetts Institute of Technology acuity. 34 PRESENCE: VOLUME 2, NUMBER I Hodges and Davis 35 location, would be imaged at points in each eye that may not be the same distance from the fovea. This difference in distance is the retinal disparity. Normally we measure this distance as the sum of the angles ôi + Ô2 as shown in Figure 1, where angles measured from the center of fo- vea toward the outside of each eye are negative. In the example shown in Figure 1, ôi has a positive value, ô2 has a negative value, and their sum (8i + ô2) is positive. The retinal disparity that results from these two differ- ent views can provide information about the distance or Left Eye Right Eye Downloaded from http://direct.mit.edu/pvar/article-pdf/2/1/34/1622602/pres.1993.2.1.34.pdf by guest on 26 September 2021 depth of an object as well as about the shape of an ob- Retinal = Si + 52 ject. Stereoacuity is the smallest depth that can be de- disparity tected based on retinal disparity. In some humans, under Figure I. Retinal disparity. optimal stimulus conditions, stereoacuities of 5" or less can be obtained (Westheimer & McKee, 1980; McKee, stereoacuities of more than 5" are more 1983); however, where a is the angle of convergence, Di is the distance common 8c Shor 8c (Davis, King, Anoskey, 1992; from the midpoint of the interocular axis to the fixated Wood, 1983). point, f1, and i is the interocular distance (Arditi, 1986; Graham, 1965).2 (See Fig. 2.) Notice that the angle of 2.1 Binocular Visual Direction convergence, a, is inversely related to the distance, D\, of the fixated from the observer; this inverse rela- Visual direction is the location of point perceived spatial tion is nonlinear. an object relative to the observer. Usually, it is measured For another point in space, f2, located at a distance in terms of azimuth (left and of the of fixa- right point D2, the of convergence is ß Note that and of elevation and below the of angle (see Fig. 2). tion) (above point since fixation). Often, the binocular visual direction of an ob- ject is not the same as the monocular visual direction of a + a + c + b + d= 180 either can this at a eye. (You verify yourself by looking and very close object first with one eye, then with the other eye, then with both eyes. Notice how the apparent spa- ß + c + d = 180 tial location of the object changes.) Hering proposed then that binocular visual direction will lie midway between the directions of the monocular others have re- a ß = (-a) + (-b) = (8, + 82) images; - ported that the binocular visual direction will lie some- The difference in vergence (a ß) is angles — equiva- where between the left and right monocular visual direc- lent to the retinal disparity between the two points in tions, but not necessarily midway (e.g., Tyler, 1983). space, measured in units of visual angle. This retinal dis- parity is monotonically related to the depth between the 2.2 Convergence Angles and Retinal Disparities 2. For asymmetric convergence of the two eyes, the formula for the For symmetric convergence of the two eyes on a angle of convergence is basically the same as that shown for symmetric The difference is that D now the fixated in space, f1, the of convergence is convergence. represents perpendicular point angle distance from the interocular axis to the frontoparallel plane that inter- defined as sects the asymmetrically converged point of fixation. This interpreta- tion of the convergence angle formula for asymmetric convergence is a = 2 arctan(¿/2D!) not exact, but it is a good approximation. 36 PRESENCE: VOLUME 2, NUMBER I Dh * Downloaded from http://direct.mit.edu/pvar/article-pdf/2/1/34/1622602/pres.1993.2.1.34.pdf by guest on 26 September 2021 d2 Figure 2. Convergence angles. Figure 3. Vieth-Mueller circle. two points in space (i.e., at a constant distance, D\, a The Vieth-Mueller circle is a theoretical horopter de- larger depth corresponds to a larger retinal disparity); fined only in terms of geometric considerations. This this monotonie relationship is a nonlinear one. horopter is a circle in the horizontal plane that intersects If an object is closer than the fixation point, the retinal each eye at the first nodal point of the eye's lens system disparity will be a negative value. This is known as a (Gulick & Lawson, 1974; Ogle, 1968). (See Fig. 3.) crossed disparity because the two eyes must cross to fixate This circle defines a locus of points with zero disparity. the closer object. Conversely, if an object is farther than However, in devising this theoretical horopter it is as- the fixation point, the retinal disparity will be a positive sumed that the eyes are perfect spheres with perfectly value. This is known as uncrossed disparity because the spherical optics and that the eyes rotate about axes that two eyes must uncross to fixate the farther object. An pass only through their first optical nodal points (e.g., object located at the fixation point or whose image falls Arditi, 1986; Bishop, 1985, Gulick 6k Lawson, 1974). on corresponding points in the two retinae has a zero None of these assumptions is strictly true. Thus, when disparity. one compares the Vieth-Mueller circle to any empiri- cally determined horopter there is a discrepancy between the theoretical and With few 2.3 Horopters empirical horopters. excep- tions (Deering, 1992) the eye geometry used for image Corresponding points on the two retinae are de- calculations for stereoscopic virtual environments as- fined as being the same vertical and horizontal distance sume a visual model consistent with the Veith-Mueller from the center of the fovea in each eye (e.g., Tyler, circle. We will address the results of this assumption in 1983; Arditi, 1986; Davis & Hodges, 1994). When the Section 5. two eyes binocularly fixate on a given point in space, Horopters usually describe a locus of points that there is a locus of points in space that fall on correspond- should result in zero disparity. Stereopsis, however, occurs ing points in the two retinae. This locus of points is the when there is a nonzero disparity that gives rise to the horopter, a term originally used by Aguilonius in 1613. percept of depth. That is, an object or visual stimulus The horopter can be defined either theoretically or em- can appear closer or farther than the horopter for crossed pirically. and uncrossed disparity, respectively. Hodges and Davis 37 Display Screen Left eye position Downloaded from http://direct.mit.edu/pvar/article-pdf/2/1/34/1622602/pres.1993.2.1.34.pdf by guest on 26 September 2021 Right eye position Object with negative parallax Figure 4. Screen parallax. 3 Modeling Geometry The amount of screen parallax in this case may be computed with respect to the geometric model of the For any type of stereoscopic display, we are model- scene as what would be seen each if the were ing by eye image = p i(D d)/D projected onto a screen or window being viewed by an - observer (Hodges, 1992).