Stereoscopic Information

Prepublication version. Frontiers of Psychology, section on Perceptual Science, in a collection of articles on “the Venetian blind effect” (edited by William W. Stine): Frontiers in Psychology, Perception Science, http://journal.frontiersin.org/Journal/10.3389/fpsyg.2014.00870/full,

What is Binocular Disparity? Joseph S. Lappin Vanderbilt University

Abstract importance of this problem were highlighted What are the geometric primitives of by Julesz’s elegant experiments with binocular disparity? The Venetian blind effect random-dot stereograms. These random and other converging lines of evidence indicate texture patterns contain large numbers of that stereoscopic derives from identical elements with countless potential disparities of higher-order structure in images of binocular correspondences and disparities. surfaces. Image structure entails spatial Evidently, the corresponding image features variations of intensity, texture, and motion, cannot be individual texture elements. jointly structured by observed surfaces. The Cooperative visual interactions among local spatial structure of binocular disparity texture elements on smooth surfaces seem corresponds to the spatial structure of surfaces. necessary for , as Julesz and Marr Independent spatial coordinates are not necessary for stereoscopic vision. Stereopsis is & Poggio emphasized. Research continues highly sensitive to structural disparities on the visual processes that yield associated with local surface shape. Disparate correspondence (Blake & Wilson, 2011). positions on retinal anatomy are neither Beyond the , necessary nor sufficient for stereopsis. however, binocular disparity involves a

representation of spatial structure. Spatial

positions of corresponding image features 1. Introduction: Spatial information. are often represented in relation to Stereoscopic vision provides important hypothetical anatomically defined retinal information about the spatial structure of coordinates; and disparity is represented as the surrounding world. The two eyes offer a binocular difference in these coordinates. largely similar optical images but from By definition, these retinal coordinates are slightly different vantage points. The independent of optical image structure. resulting small disparities between the two This spatial representation is testable, monocular images constitute visually however, with plausible alternative important information not available in either hypotheses. The present article reviews image alone. The binocular visual system is evidence about the spatial structure of extraordinarily sensitive to this stereoscopic binocular disparity. Articles by Lappin & information. Craft (1997, 2000) and Lappin, Norman, & But what, exactly, is binocular disparity? Phillips (2011) are also relevant. The issue is not terminology, but the input As discussed by Lappin et al. (2011), two information. Identifying the input is psychophysical criteria for identifying necessary for determining how that input is information for vision are resolution and processed. invariance. Resolution involves precision of One aspect of this problem is the discrimination, limited by variability. In “correspondence problem” — to identify short, what do the two eyes see best? corresponding spatial elements in the two Information and geometric structure are also monocular images (Julesz, 1960, 1971; Marr defined by invariance — by the groups of & Poggio, 1976, 1979). The nature and transformations of observational conditions Stereoscopic Information

(e.g., viewing position and illumination) relative to an independent reference frame or under which they remain invariant. Such topologically, relative to the surrounding invariance is experimentally testable. image structure. Examples of both approaches are common in vision science. 2. Image intensities and visual space. The concept of binocular disparity often involves the intuitive concept of space as 2.1. The Venetian blind effect. Several phenomena motivate reexamination of independent of the objects and patterns it binocular disparity. One motivation is the contains. Intuitively, retinal anatomy might “Venetian blind effect” (VBE, for short) — provide such spatial coordinates. where dichoptic intensity differences of Alternatively, the topology of spatial vertical gratings with non-disparate edges relations at a given point may be described produce a perceived change in 3D surface in several ways. Topological parameters slant. Apparently, spatial disparity is not include (a) complexity (number of points or necessary. regions), (b) dimensionality, and (c) scale Cibis & Haber (1951), Ogle (1962), and (size of neighborhood). Howard & Rogers (2002) suggest that the A familiar topological description is VBE requires no revision of theories of Fourier analysis. The Fourier power stereopsis: Monocular intensity patterns spectrum involves correlations between may affect spatial position signals — image contrasts at pairs of points. The because light scattering or nonlinear visual Fourier phase spectrum specifies relative signaling may affect spatial disparity. positions of various wavelengths, involving relations among triples of points (Yellott, Extensive studies by Stine and colleagues (Dobias & Stine, 2012; Filley, 1993). The phase spectrum is essential to Khutoryansky, Dobias, & Stine, 2011; Hetley most aspects of visible image structure, & Stine, 2011), however, clearly demonstrate including stereopsis (Blake & Wilson, 2011; that the VBE derives from disparate DeAngelis, Ohzawa, & Freeman, 1995; intensities not spatial positions. Disparate Piotrowski & Campbell, 1982; Smallman & intensities and edge positions have additive McLeod, 1994). The power and phase effects on perceived depth; and the two spectra are translation-invariant. Neither disparities can cancel each other. requires retinal coordinates. The VBE is also consistent with other Another topological description is based experimental evidence that disparities in on differential geometry. Koenderink & van surface highlights and shading contribute to Doorn are chiefly responsible for developing perception of 3D structure (Bülthoff & the differential geometry of image structure Mallot, 1988; Nefs, 2008; Norman, Todd, & (e.g., 1976, 1992a, 1992b, 1997; Phillips, 1995; Todd, Norman, Koenderink, & Koenderink, 1986, 1990). Kappers, 1997; Vuong, Domini, & Caudek, The spatial structure of image intensity 2006). Surface structure affects binocular provides visible information about variations disparities in both space and shading. The in surface orientation relative to both VBE is one of several lines of evidence that viewing and illumination directions. vision uses both dimensions of information. Countless illustrations are found in literatures on image shading in photography, 2.2. Image intensities & spatial positions painting, computer vision, and vision science co-vary. Monocular image structure involves (e.g., Koenderink & van Doorn, 2004). spatial variations of intensity. Regardless of Evidently, the VBE also illustrates such one’s representation of the physical effects. dimensions, space and intensity are not visually independent. The VBE shows that retinal position disparity is not necessary for stereopsis. The spatial position of a given optical Other experiments reviewed below show that feature (e.g., edge) can be represented

2 Stereoscopic Information disparate retinal positions are also optical information can be represented in insufficient. many ways. Few representations suffice for stereoscopic perception, however. 2.3. Perceived surface slant is imprecise. Stereopsis is not necessary for perceiving Perceived depth in the VBE seems smaller, a 3D world, but visual experience is much less compelling, and less reliable than that clearer with stereopsis than without it. from disparate spatial positions. Differences in perception with and without Is stereopsis simply insensitive to stereopsis are subjectively profound, as intensity disparities? Actually, binocular described by Oliver Sacks (“Stereo Sue”, in vision seems quite sensitive to dichoptic The mind’s eye, 2010) and Bruce Bridgeman contrast differences; and these contrast (http://www.bbc.com/future/story/201207 differences affect perceived spatial positions 19-awoken-from-a-2d-world). in binocularly fused images (Ding & Moreover, stereopsis greatly improves Sperling, 2006). spatial acuity. Acuity thresholds for One source of variable perceived surface binocularly disparate relative positions are slant in the VBE is that dichoptic intensity about 25% of those for the same patterns differences have two complementary without disparity (Berry, 1948; Lappin & perceptual effects — on binocular brightness Craft, 2000; Westheimer & McKee, 1979). as well as depth rotation (Hetley & Stine, What, then, is the structure of 2011). Hetley & Stine found that the relative stereoscopic perception? Is depth a magnitudes of these two effects varied perceptually created third dimension? That between observers and conditions, but the is a common intuition, but not the only combined effect was relatively constant. possibility. Another limitation of the VBE is that Alternatively, stereoscopic space and surface slant is not reliably perceived depth may derive from visible relations anyway — from binocular disparity, among objects. Several hypotheses are structure-from-motion, image shading, possible about the primitive visual topology texture, or other information. This perceptual limitation is hardly surprising: of perceived space. Image information about surface orientation Experimental research indicates that necessarily depends on the observer’s surface shape is an elementary visual viewing position. Experimental evidence property. From traditional perspectives, this about the imprecision of stereoscopic slant conclusion is very counter-intuitive. Higher- perception is reviewed below (Section 3.3). order object structures would seem to derive from simpler visual cues. 3. Stereoscopic depth perception. Contemporary understanding of the To identify input information for visual role of surfaces and surface shape is stereopsis, one can work backwards from due chiefly to Koenderink and van Doorn perceptual output to optical input: What (e.g., 1992a, b, 1997; Koenderink, 1990). structure of binocular disparity is necessary Basic theoretical results include: (1) and sufficient for perceiving environmental Environmental object surfaces and their structures in depth? retinal images are both 2-dimensional This strategy exemplifies means-end manifolds, described at any point by spatial analysis (Simon, 1996) and Gibson’s (1966) derivatives in two principal orthogonal method in “The senses considered as directions. (2) The differential structures of perceptual systems”. This method is environmental surfaces and the binocular common in engineering, but it differs from disparity fields of their images are starting with presumed retinal input. A approximately isomorphic. (3) Image difficulty with the conventional input-first information about local surface shape is nd approach is that binocular disparity and given by the 2 -order differential structure

3 Stereoscopic Information of the image fields of binocular disparity and 3.2. Perceived depth differences are motion , which specify the ratio of imprecise. An alternative hypothesis is that minimum and maximum curvature at each stereopsis provides perception of depth position. (4) 2nd-order image information differences between pairs of points. about local surface shape can be estimated The retinal separation between two directly without first estimating lower-order points and associated binocular disparity is properties such as depth or surface invariant with the locus of fixation. But the orientation. (5) Variations in local surface relation between pair-wise image disparity shape are invariant with depth, slant, and and physical depth difference still depends curvedness. on distance of the objects from the observer. Before examining experimental evidence, When viewing distance, D, is large relative to consider alternative hypotheses about the inter-ocular separation, I, then for a perceived absolute and relative depths. given disparity (in pair-wise separation), ∂, the corresponding depth difference, ∆d, 3.1. Absolute depths of individual points increases approximately with the square of are visually undefined. The simplest spatial the viewing distance: primitive is an individual point. Spatial ∆d ≈ (D2/I) ∂ (1) positions and binocular disparities of points might be visually defined by retinal anatomy. This strong influence of viewing distance is a This is a common intuitive conception. fundamental limitation of pair-wise disparities. As expected, perceived depth Nevertheless, a single point is generally differences are unreliable. recognized as stereoscopically ambiguous without a reference point at fixation (Howard Studies by McKee, Levi, & Bowne (1990) & Rogers, 2002). and Norman et al. (2008) found that perceived depth differences between two Binocular alignment of the two retinal objects were imprecise, as quantified by coordinate systems is problematic, however, large Weber fractions. McKee et al. found because alignment varies substantially with thresholds for stereoscopic depth differences the direction and distance of gaze — see about 3-5 times higher than those for Howard & Rogers (1995, 2002). Alignment is monocular separations of the same stimuli. also perturbed by disparate eye-movements Norman et al. found similar imprecision, (Collewijn & Erkelens, 1990; Ferman, with Weber fractions (coefficient of variation Collewijn, Jansen, & van den Berg, 1987; = SD/M) ~22%. In contrast, Weber fractions Steinman, Levinson, Collewijn, & van der for simply detecting depth are less than Steen, 1985). 0.5% (e.g., Lappin & Craft, 1997, 2000). Despite these misalignments, the perceived 3D structure of the world usually 3.3. Stereoscopic surface slant is appears constant under changes in gaze imprecise. Koenderink & van Doorn (1976; direction and distance. This perceptual Koenderink, 1986) showed that surface slant stability conflicts with the hypothesis that affects the ‘deformation’ component of the stereoscopic depth derives from retinal 1st-order spatial derivatives of the binocular positions. Moreover, stereoacuity thresholds disparity field — involving disparate shapes for relative position are robust under of triangular surface patches. The disparate motions of the monocular images deformation component is invariant with (Lappin & Craft, 1997, 2000; Steinman et image translation, expansion, and rotation, al., 1985; van Ee & Erkelens, 1996; but it varies with viewing direction and Westheimer & McKee, 1978). Thus, distance (see Howard & Rogers, 2002, chap. stereoscopic depth cannot derive from 21). Accordingly, perceived surface slant is disparities in retinal positions of individual ambiguous. points. Slant detection is also anisotropic,

4 Stereoscopic Information because the eyes are horizontally separated, Lappin et al., 2011; Perotti, Todd, Lappin, & with more sensitivity to vertical than Phillips, 1998; Todd, 2004; Todd, horizontal disparity gradients (Gillam & Koenderink, van Doorn, & Kappers, 1996; Ryan, 1992; Rogers & Graham, 1983). Todd, Norman, Koenderink, & Kappers, 1997; van Damme & van de Grind, 1993). The predictable unreliability of slant discriminations has been found Norman et al. (1991) found accurate experimentally (e.g., Todd, Tittle, & Norman, perception of surface smoothness. Random- 1995). Current evidence is limited, however: dot triangle-wave surfaces, discontinuous at Judgmental reliability is often not reported; their extrema, were discriminated from very viewing distance and context are often similar smooth surfaces (fundamental + 3rd constant; and disparity gradients usually co- harmonic of the triangle-wave) with slight vary with texture gradients and other curvature at the extrema. Smoothness information. discriminations were more accurate than detections of the differences in Fourier power Experiments by Norman and colleagues spectra. Thus, stereoscopic perception (2006, 2009) found that stereopsis adds very yielded curved surfaces (2nd-order structure), little to the limited precision of slant not depths or slants. estimates based on texture, relative motion, and shading. Surfaces in both studies were Shape discriminations are more reliable seen at a constant distance; and judgments than and independent of perceived depth would have been less precise with varied differences (Perotti et al., 1998; Todd, 2004; viewing distances. Todd, Koenderink, et al., 1996; Todd, Norman, et al. 1997; van Damme & van de Steep surface slants may be difficult to Grind, 1993). Smooth surface shape, discriminate or even detect when disparity therefore, is a fundamental visual property, changes too much in too small an area. not derived from perceived depths or slants. Filippini & Banks (2009) evaluated stereoscopic detection of large depth 4. Binocular disparity. gradients, using random-dot saw-tooth surfaces in noise. Signal/noise thresholds What does stereoscopic perception tell us for surface detection rose rapidly for about binocular disparity, the input disparity/separation ratios greater than 1.0, information for stereopsis? as predicted by cross-correlation models. 4.1. Disparity involves image structure. Other experiments, however, have found The first principle is that stereoscopic input that depth changes on smooth surfaces are involves disparate image structures, not more visible than predicted by a cross- disparate retinal positions. Stereoscopic correlation model. Allenmark & Read (2010) hyperacuity (resolution finer than the eye’s found that large depth changes were as photoreceptor density, point spread visible on smooth sine-wave surfaces as on function, and diffraction limit) is robust square-waves. Norman, Lappin, & Zucker under random perturbations of retinal image (1991) found very accurate discriminations positions in each eye (Sections 3.1 & 3.4). of surface smoothness, exceeding Thus, monocular spatial positions are predictions of cross-correlation or other visually defined relative to the surrounding linear models. image.

3.4. Surface shape is a perceptual 4.2. Disparity involves surface shape. primitive. Human observers can Stereoscopic vision is directly sensitive to the discriminate very small variations in surface shapes of environmental surfaces (Section shape — with greater precision than for 3.4). Surface shape is discriminated more discriminations of depth or slant, and reliably than seemingly simpler properties; invariant under random perturbations of and hyperacuity for surface shape is depth and slant (e.g., Lappin & Craft, 2000; maintained under random perturbations of

5 Stereoscopic Information lower-order disparities associated with magnitudes of the two principal curvatures relative depth and slant (Lappin & Craft, (horizontal and vertical in this illustration). 2000; Norman et al., 1991; Perotti et al., These patterns exemplify the qualitative 1998). possibilities for smooth surfaces. Stereoscopic perception of surface shape Figure 2 demonstrates the robust visual is possible because of structural correspondences between environmental surfaces and binocular disparities — involving 2nd-order spatial derivatives (Koenderink & van Doorn, 1992a; Lappin & Craft, 2000; Lappin et al. 2011; Todd, 2004).

4.3. Disparity of 2nd-order image structure. The “2nd-order differential structure” of binocular disparity is simpler than it might first seem. The relevant structure is just the radial symmetry of the neighborhood around every local image point. The disparate binocular images of a surface differ by a deformation of this symmetry. The qualitative form of this local image deformation corresponds to the local surface shape, invariant with the observer’s viewing position.

Fig. 1. Schematic forms of image deformations produced by rotating the viewpoint of a circular surface patch around its central vertical axis. Rotation direction Fig. 2. Stereo illustrations of perceived shape and concavity vs. convexity are ambiguous. from binocular disparity, invariant under global The shapes, from the left, are planar (0 image transformations by 2D rotation and curvature), parabolic (0 curvature in one shear. Shape and shading are random and axis), parabolic, elliptic (with the same sign of mutually independent. Top: undistorted stereo, curvature in both axes), and hyperbolic with right image rotated in depth around the (opposite signs of curvature in the two axes). vertical axis by about 5º. Center: right image (Illustration from Lappin & Craft, 2000, Fig. rotated about 7º. Bottom: right image expanded 3, p. 14. Copyright 2000 by the American and compressed by about 7% in orthogonal Psychological Association.) axes (‘pure shear’). The left image is identical in all three pairs. (Illustration from Lappin, Figure 1 illustrates these image Norman, & Phillips, 2011, Fig. 10, p. 2368. deformations for each of the possible surface Copyright 2011 by the Psychonomic Society.) shapes. As may be seen, these stereo deformations correspond, from left to right, to local images of a plane, horizontal sensitivity to smooth variations in these local cylinder, vertical cylinder, ellipsoid, and structural disparities in images of randomly saddle — as specified by the relative shaped surfaces. Image information about

6 Stereoscopic Information local surface shape is preserved under significant global disparity changes References produced by rotating, dilating, or shearing the image plane — as illustrated by the Allenmark, F. & Read, J.C.A. (2010). middle and lower panels. Like most random- Detectability of sine- versus square-wave element stereograms, the random intensities disparity gratings: A challenge for current in these patterns are independent of surface models of depth perception. Journal of shape and binocular disparity; but here Vision, 10(8):17, 1-16. depths and intensities both vary smoothly, Berry, R.N. (1948). Quantitative relations without sharp edges. Unlike most natural among vernier, real depth, and stereoscopic images, shading is unrelated to surface depth acuities. Journal of Experimental shape; and the intensities are not disparate. Psychology, 38, 708-721. 4.4. Binocular disparity in the Venetian Blake, R. & Wilson, H. (2011). Binocular blind effect. The VBE involves perceived vision. Vision Research, 51, 754-770. rotation of vertical bars. Such planar Bülthoff, H.H. & Mallot, H.A. (1988). rotations ordinarily produce bilaterally Integration of depth modules: stereo and symmetric dilation or compression of shading. Journal of the Optical Society of horizontal scale, as seen at the left of Figure America A, 5(10), 1749-1758. 1. Changing the horizontal distribution of relative intensities may have similar effects Cibis, P.A. & Haber, H. (1951). Ansiopia on visual neurons responsive to the left-right and perception of space. Journal of the balance of surrounding stimulation. Optical Society of America, 41(10), 676-683. Perceived rotation seems a plausible and Collewijn, H. & Erkelens, C.J. (1990). understandable result of this image Binocular eye movements and the perception disparity. of depth. In Kowler, E. (Ed.). Eye movements As Dobias and Stine (2012) note, the and their role in visual and cognitive explanation for the direction of perceived processes (pp. 213-261). Amsterdam: rotation is not immediately obvious. Image Elsevier. shading from reflective surfaces depends on DeAngelis, G.C., Ohzawa, I., & Freeman, illumination direction as well as surface R.D. (1995). Neuronal mechanisms orientation. For special cases, however, with underlying stereopsis: how do simple cells in Lambertian shading (equal scattering in all the visual cortex encode binocular disparity? directions), radiant surfaces, and surfaces Perception, 24, 3-31. illuminated from behind, image intensity is greater when the surface is perpendicular to Ding, J. & Sperling, G. (2006). A gain- the viewing direction. Thus, the surface control theory of binocular combination. orientation may plausibly appear more Proceedings of the National Academy of perpendicular (and thus expanded) toward Sciences, 103(4), 1141-1146. the eye with greater relative intensity or Dobias, J.J. & Stine, W.W. (2012). contrast. Temporal dynamics of the Venetian blind In general, stereoscopically perceived effect. Vision Research, 60, 79-94. surfaces derive from binocular disparities of Ferman, L., Collewijn, H., Jansen, T.C., higher-order image structures. For the visual & Van den Berg, A.V. (1987). Human gaze system, spatial position and intensity are stability in the horizontal, vertical and correlated dimensions. Relative spatial torsional direction during voluntary head positions involve relative intensities. Both movements, evaluated with a three- are structured by surfaces, and both dimensional scleral induction coil technique. constitute information about surface Vision Research, 27(5), 811-828. structure, not depth as such. Filley, E.T., Khutoryansky, N., Dobias,

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