Ординатура / Офтальмология / Английские материалы / Seeing_De Valois_2000

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FIGURE 3 Changes in horopter curvature produced by nonuniform magnification of corresponding points in the temporal retinas relative to those in the nasal retinas. Nonuniform magnification of corresponding points of the nasal hemi-retinae flatten the horopter relative to the Vieth-Muller circle. The absolute curvature varies with viewing distance and becomes flatter as distance increases from concave at near (c) to convex at far (a). At an intermediate distance (b), the horopter coincides with the fronto-parallel plane when the Hering–Hillebrand deviation (H) equals the interpupillary distance/the viewing distance (Abathic Distance). All three empirical curves correspond to the same Hering–Hille- brand deviation, and the apparent curvature changes result from spatial geometry. (From BINOCULAR VISION AND STEREOPSIS by Ian P. Howard and Brian J. Rogers, Copyright © 1995 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc., and the authors.)

The horopter can also be represented analytically by a plot of the ratio of longitudinal angles subtended by empirical horopter points (Right/Left) on the y axis as a function of retinal eccentricity of the image in the right eye (Figure 4). Changes in curvature from a circle to an ellipse result from nonuniform magni- fication of retinal points in one eye. If the empirical horopter is flatter than the theoretical horopter, corresponding retinal points are more distant from the fovea on the nasal than temporal hemi retina (see Figure 3). A tilt of the horopter around a vertical axis results from a uniformly closer spacing of corresponding points to the fovea in one eye than the other eye (uniform magnification e ect) (see Figure 2).

D. The Vertical Horopter

The theoretical vertical point-horopter for a finite viewing distance is limited by the locus of points in space where homologous visual directions will intersect real

Analytical plot of horopter data. The tangent of the angular subtense of the right eye image (tan or) is plotted on the x axis, and the tangent ratio of the angle of subtense of the two images

(R) for each point along the empirical horopter is plotted on the y axis. H is the slope of the resulting linear function, and it quantifies horopter curvature (Hering–Hillebrand deviation), where a slope of zero equals the curvature of the Vieth-Muller circle.

objects and is described by a vertical line that passes through the fixation point in the midsagittal plane (Figure 5). Eccentric points in tertiary gaze (points with both azimuth and elevation) lie closer to one eye than the other eye. Because they are imaged at di erent vertical eccentricities from the two foveas, tertiary points cannot be imaged on theoretically corresponding retinal points. However, all points at an infinite viewing distance can be imaged on homologous retinal regions, and at this viewing distance the vertical horopter becomes a plane.

The empirical vertical horopter is declinated in comparison to the theoretical horopter (Figure 6). Helmholtz (1909) reasoned this was because of a horizontal shear of the two retinae, which causes a real vertical plane to appear inclinated or a real horizontal plane, such as the ground, to lie close to the empirical vertical horopter.

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The theoretical vertical horopter. Corresponding visual lines from the midvertical meridians of the two eyes intersect in a vertical line in the median plane of the head. With near convergence, visual lines from any other pair of corresponding vertical meridians do not intersect, since the image of any eccentric vertical line in one eye is larger than the image of that line in the other eye. With parallel convergence, the horopter is a plane at infinity, orthogonal to the visual axes. (From BINOCULAR VISION AND STEREOPSIS by Ian P. Howard and Brian J. Rogers, Copyright © 1995 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc., and the authors.)

E. Coordinate Systems for Binocular Disparity

All coordinate systems for binocular disparity use the same theoretical horopter as a reference for zero retinal image disparity. Targets not lying on the horopter subtend nonzero disparities, and the magnitudes of these nonzero disparities depend on the coordinate system used to describe them. Howard and Rogers (1995) describe five coordinate systems (their Table 2.1), and these correspond to the oculomotor coordinate systems of Polar, Helmholtz, Fick, Harms, and Hess (Schor, Maxwell, & Stevenson, 1994). Howard and Rogers and most vision scientists use the Harms system. The di erent coordinate systems will quantify vertical and horizontal disparity equally for targets at optical infinity, as all points subtend zero disparity at this distance, but their disparity will change at finite viewing distances.This is because the visual axes and retinal projection screens are not parallel or coplanar at near viewing distances associated with convergences of the two eyes. Each of the five coordinate systems may be suitable for only one aspect of vision, and apparently none of them is adequate to describe binocular disparity as processed by the visual system.

Corresponding points can be described in terms of Cartesian coordinates of constant azimuth and elevation. Contours in space that appear at identical constant horizontal eccentricities or constant heights by the two eyes are, by Hering’s definition of identical visual directions, imaged on binocular corresponding points. The theoretical retinal regions that correspond to constant azimuth and elevation can be described with epipolar geometry. Retinal division lines are retinal loci that give rise to the sense of constant elevation or azimuth. Regions in space that lie along the perspective projection of these retinal division lines are derived with transformations from the spherical retinal space to isopters in a planar Euclidean space. Isopters are contours on a tangent screen that appear to be straight and horizontal (or vertical).The relationship between the retinal division lines and isopters is a perspective projection through a projection center. The choice of retinal division lines (major or minor circles) and projection center (nodal point or bulbcenter) determines the coordinate system (Figure 7) (Tschermak-Seysenegge, 1952). The theoretical case is simplified by assuming the projection point and nodal point of the eye both lie at the center of curvature of the retinal sphere (bulbcenter). As shown in Figure 7, there are several families of theoretical retinal division lines. There are great circles, such as in the Harms system, which resemble lines of longitude on the globe (Schor et al., 1994).Their projections all pass through the bulbcenter as planes and form straight-line isopters on a tangent screen. There could also be a family of

R

FIGURE 6 The empirical vertical horopter. For near convergence, the vertical horopter is a straight line inclined top away in the median plane of the head. The dotted lines on the retinas indicate the corresponding vertical meridians. The solid lines indicate the true vertical meridian. (From BINOCULAR VISION AND STEREOPSIS by Ian P. Howard and Brian J. Rogers, Copyright © 1995 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc., and the authors.)

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FIGURE 7 Epipolar geometry. Projection of great circles retinal division lines through the nodal point intersect the planar surfaces in space as straight lines. Projection of minor circles scribed on the retina through the nodal point intersect the planar surface in space as Parabolas. The minor circle retinal division lines closely approximate the retinal loci of perceived constant elevation isopters on the planar surface in space. The projections through the nodal point represent a transformation from spherical coordinates in the eye to Euclidean coordinates in space. (Reprinted from Ophthalmic and Physiological Optics, 14, Schor, C. M., Maxwell, J., & Stevenson, S. B. (1994). Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze. 279–286, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom.)

minor circles, such as in the Hess coordinate system, which resembles lines of latitude on the globe (Schor et al., 1994). Their projections also pass through the bulbcenter as conic sections and form parabolic isopters on a tangent screen. None of these necessarily represents the set of retinal division lines used by the visual system, because retinal division lines, by definition, must produce a sense of equal elevation or equal azimuth. As will be discussed below, empirical measures of iso-ele- vation indicate that the projection point actually lies in front of the nodal point, such that it produces barrel distortions of the visual field (Liu & Schor, 1997).

The tangent plane projections represent a map of constant or iso-elevation and iso-azimuth contours. The only viewing distance at which projection maps of the two eyes can be superimposed is at optical infinity. At finite viewing distances, the eyes converge and the retinal projection screens are no longer parallel or coplanar such that a single tangent plane will make di erent angles with the axes of the two eyes. Because the coordinate systems describing binocular disparity are retinal based, their origins (the lines of sight) move with the eyes so that they project relative to

the direction of the visual axis. In eye movements, the reference systems are headcentric at the primary position of gaze so that the coordinate systems do not change during convergence. The consequence of having a retinal-based coordinate system is that the projected tangent planes of the two eyes are not coplanar in convergence. This problem is solved by mapping both eyes onto a common tangent plane that is parallel to the face (fronto-parallel plane), which results in trapezoidal distortion of the projected isopters (Liu & Schor, 1997).

When the fronto-parallel maps of the two eyes isopters are superimposed, they only match along the primary vertical and horizontal meridians, and they are disparate elsewhere. Consequently, the theoretical horopter is not a surface but rather it is made up of a horizontal (V-M) circle and vertical line in the midsagittal plane (Figure 5). All coordinate systems yield the same pure horizontal or vertical disparity along these primary meridians, but they will compute di erent magnitudes of combined vertical and horizontal disparities in tertiary field locations.

The separations of the two eye’s iso-elevation and iso-azimuth isopters describe vertical and horizontal misalignment of visual directions originating from corresponding retinal division lines (binocular discrepancy). Binocular discrepancy is distinguished from binocular disparity, which refers to the binocular misalignment of retinal images of any object in space from corresponding retinal points. Binocular discrepancy provides the objective reference points of each eye in the fronto-par- allel plane for zero binocular disparity (Liu & Schor, 1997). Binocular discrepancy and disparity are related, as the discrepancy map also describes the pattern of binocular disparities subtended by points lying in the fronto-parallel plane.

F. Monocular Spatial Distortions and the Empirical

Binocular Disparity Map

The mismatch of the projection fields of iso-elevation and iso-azimuth isopters at near viewing distances results in a binocular discrepancy map or field on the frontoparallel plane. An empirically derived map needs to take into account the horizontal shear of the two monocular image spaces, described by Helmholtz, as well as other monocular spatial distortions. For example, horizontal contours that are viewed eccentrically above or below the point of fixation are seen as bowed or convex relative to the primary horizontal meridian (barrel distortion). Horizontal lines must be adjusted to a pincushion distortion to appear straight (Helmholtz, 1909). The retinal division lines corresponding to the pincushion adjustments needed to make horizontal lines appear at a constant or iso-elevation resembles the minor circles of the Hess coordinate system (Liu & Schor, 1997). These retinal division lines have been used to predict an empirical binocular discrepancy map for finite viewing distances.

The discrepancy map is derived from the combination of horizontal shear with empirically measured iso-elevation and iso-azimuth retinal division lines.Their projection from the two eyes onto isopters in a common fronto-parallel plane centered

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Binocular discrepancy map of isoelevation and iso-azimuth lines. Projection of the minor circle retinal division lines of the two eyes form intersecting iso-elevation and iso-azimuth grid in space. The grid is distorted by the horizontal shear of the retina that causes declination of the empirical vertical horopter shown in Figure 6. The grid is a complex pattern of horizontal and vertical disparities that are greater in the lower than upper field as a result of the horizontal shear of the two retinae. (Reprinted from Liu, L. & Schor, C. M. Monocular spatial distortions and binocular correspondence.

Journal of the Optical Society of America: Vision Science, 7, 1740–55, Copyright © 1998. With kind permission from the Optical Society of America, and the authors.)

along the midsagittal axis produces a complex pattern of vertical discrepancies that are greater in the lower than upper visual field, as is illustrated in Figure 8 (Liu & Schor, 1998). Because the pincushion distortions vary dramatically between observers (Liu & Schor, 1977), it is impossible to generalize these results and predict the vertical discrepancy map for individuals and the magnitude of vertical disparities that are quantified relative to this map.

The general pattern of the binocular discrepancy distribution is the direct consequence of the basic monocular distortions (horizontal shear and vertical trapezoid) and therefore should be valid for most cases. However, the quantitative details of the map may vary among observers because Liu and Schor (1997) have demonstrated that di erent observers have di erent amounts of monocular pincushion distortion. It is important to realize the idiosyncrasy of binocular correspondence when designing experiments that involve manipulating large disparities in the periphery. Several theoretical models suggested that vertical disparity may carry

information that is necessary for scaling stereoscopic depth (Gillam & Lawergren, 1983; Mayhew & Longuet-Higgins, 1982). Because vertical disparity is much smaller in magnitude compared to horizontal disparity, the empirical verification of the above theoretical speculations usually involves manipulating vertical disparities in the far periphery (Cumming, Johnston, & Parker, 1991; Rogers & Bradshaw, 1993). In such studies, accurate knowledge about the status of vertical correspondence across the visual field becomes critical because the discrepancy between geometric stimulus correspondence and empirical retinal correspondence may be large enough in the periphery to actually a ect the amount of vertical disparity delivered by the stimulus.

III. BINOCULAR SENSORY FUSION

A. Panum’s Fusional Areas

Binocular correspondence is described above as though there was an exact point- to-point linkage between the two retinal images. However, Panum (1858) observed that corresponding points on the two retinae are not points at all, but regions. He reported that in the vicinity of the fovea, the fusional system would accept two disparate contours as corresponding if they fell within a radius of 0.026 mm of corresponding retinal loci. This radius corresponds to a visual angle of approximatelydegrees. The consequence of Panum’s area is that there is a range of disparities that yield binocular singleness. The horizontal extent of Panum’s area gives the horopter a thickness in space (Figure 9). The vertical extent also contributes to single binocular vision when targets are viewed eccentrically in tertiary directions of gaze. Spatial geometry dictates that because these tertiary targets are closer to one eye than the other, they form unequal image sizes in the two eyes which subtend di erent heights and accordingly vertical disparities. Binocular sensory fusion of these targets is a consequence of Panum’s areas.

Panum’s area functions as a bu er zone to eliminate diplopia for small disparities near the horopter. Stereopsis could exist without singleness, but the double images near the fixation plane would be a distraction. The depth of focus of the human eye serves a similar function. Objects that are nearly conjugate to the retina appear as clear as objects focused precisely on the retina. The bu er for the optics of the eye is much larger than the bu er for binocular fusion. The depth of focus of the eye is approximately 0.75 diopters. Panum’s area can be expressed in equivalent units and is only 0.08-meter angles or approximately one-tenth the magnitude of the depth of focus. Thus we are more tolerant of focus errors than we are of convergence errors.

Both accommodation and stereo-fusion are sensitive to di erences in distance within their respective bu er zones. Accommodations can respond to defocus that is below our threshold for blur detection (Kotulak & Schor, 1986), and the eyes can sense small depth intervals ( 10 arc sec) and converge in response to disparities (3 arc min) (Riggs & Niehl, 1960) that are smaller than Panum’s area. Thus Panum’s

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Panum’s area gives the horopter volume. The lines depicting the theoretical horizontal and vertical horopters shown in Figures 1 and 5 become volumes when measured empirically with a singleness horopter, where a range of retinal points in one eye appear fused with a single point in the contralateral eye (Panum’s area). (From BINOCULAR VISION AND STEREOPSIS by Ian P. Howard and Brian J. Rogers, Copyright © 1995 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc., and the authors.)

area is not a threshold or limit-to-depth sensitivity, nor is it limited by the minimal noise or instability of the binocular vergence system. However, it does allow the persistence of single binocular vision in the presence of constant changes in retinal image disparity caused by various oculomotor disturbances. For example, considerable errors of binocular alignment ( 15 arc min) may occur during eye tracking of dynamic depth produced either by object motion or by head and body movements (Steinman & Collewijn, 1980).

B. Allelotropia

The comparison between the depth of focus and Panum’s area suggests that single binocular vision is simply a threshold or resolution limit of visual direction discrimination (Le Grand, 1953). Although this is partially true, as will be described below, Panum’s area also provides a combined percept in 3-D space that can be di erent in shape than either of its monocular components. The averaging of monocular shapes and directions is termed allelotropia. For example, it is possible to fuse two horizontal lines curved in opposite directions and perceive a straight line

binocularly. There are limits to how dissimilar the two monocular images can be to support fusion, and when these limits are exceeded, binocular rivalry suppression occurs in which only one eye perceives in a given visual direction at one time. Clearly, fusion is not a simple summation process or a suppression process (Blake & Camisa, 1978) or one of rapid alternate viewing as proposed by Verhoe (1935). The combination of the two retinal images follows many of the rules of binocular visual direction as described by Hering (Ono, 1979).

Allelotropia is not restricted to fused images. Dichoptically viewed stimuli subtending large disparities, beyond the fusion limit, are perceived in separate directions. The separation of these diplopic images is less than what is predicted by their physical disparity (Rose & Blake, 1988), demonstrating that visual directions can di er under monocular and binocular viewing conditions even when fusion is absent. This observation suggests that allelotropia may be an independent process from binocular sensory fusion.

C. Spatial Constraints

The range of singleness or size of Panum’s area and its shape are not a constant. It is not a static zone of fixed dimension; it varies with a wide variety of parameters. The classical description of the fusional area, as described by Panum, is an ellipse with the long axis in the horizontal meridian (Mitchell, 1966; Ogle & Prangen, 1953; Panum, 1858). The horizontal radius is degree, whereas the vertical radius is degree. The elliptical shape indicates that we have a greater tolerance for horizontal than vertical disparities. Perhaps this is because vergence fluctuations associated with accommodation are mainly horizontal, and because the range of horizontal disparities presented in a natural environment is far greater for vertical disparities.

D. Spatial Frequency

Both the shape and size of Panum’s area vary. Panum’s area increases in size as the spatial frequency of the fusion target decreases (Schor, Wood, & Ogawa, 1984a) (Figure 10). The horizontal extent increases with spatial frequencies below 2.5 cpd. Panum’s fusional area, centered about the fovea, has a range from 10–400-arc min as spatial frequency is decreased to 0.075 cpd (cycles per degree). At these lower spatial frequencies, the fusion range approaches the upper disparity limit for static stereopsis. The variation of the horizontal fusion range may be interpreted as two subcomponents of the fusion mechanism that process the position and phase of disparity. The fusion limit is determined by the least sensitive of the two forms of disparity. The fusion range at high spatial frequencies can be attributed to a 10-arc min positional limit, and the range at low spatial frequencies can be attributed to a 90 phase disparity. At high spatial frequencies, the 90 phase limit corresponds to a smaller angle than the 10-arc min positional disparity, and consequently fusion is