Kent A. Stevens

The visual interpretation of surface contours

This article examines the computational problems underlying the 3-D interpretation of surface contours. A surface contour is the image of a curve across a physical surface, such as the edge of a shadow cast across a surface, a gloss contour, wrinkle, seam, or pigmentation marking. Surface contours by and large are not as restricted as occluding contours and therefore pose a more difficult interpretation problem. Nonetheless, we are adept at perceiving a definite 3-D surface from even simple line drawings (e.g., graphical depictions of continuous functions of two variables). The solution of a specific surface shape comes by assuming that the physical curves are particularly restricted in their geometric relationship to the underlying surface. These geometric restrictions are examined.

Computation of locally parallel structure

A Moiré-like effect can be observed in dot patterns consisting of two superimposed copies of a random dot pattern where one copy has been expanded, translated, or rotated. One perceives in these patterns a structure that is locally parallel. Our ability to perceive this structure is shown by experiment to be limited by the local geometry of the pattern, independent of the overall structure or the dot density. A simple representation of locally parallel structure is proposed, and it is found to be computable by a noniterative, parallel algorithm. An implementation of this algorithm is demonstrated. Its performance parallels that observed experimentally, providing a potential explanation for human performance. Advantages are discussed for the early description of locally parallel structure in the course of visual processing.

The information content of texture gradients

Texture gradients are systematic variations in projected surface texture. It is expected that image variables such as texture density or size carry information about the surface orientation and distance. This article reexamines the information content of texture gradients and reviews the relevant psychophysics. Slant is shown to be difficult to infer from a texture gradient, but tilt (the direction of slant) is easily and reliably determined. Regarding distance, a new texture measure is introduced that has a simple geometric definition, and from which distance can be readily computed. Variables affecting the precision and the accuracy of the distance computation are discussed.

Integrating stereopsis with monocular interpretations of planar surfaces

Experiments are reported that involved spatial judgments of planar surfaces that had contradictory stereo and monocular information. Tasks included comparing the relative depths of two points on the depicted surface and judging the surfaces apparent spatial orientation. It was found that for planar surfaces the 3D perception was dominated by the monocular interpretation, despite the strongly contradictory stereo information. We propose that stereo information is effectively integrated only where the surface exhibits curvature features or edge discontinuities, i.e. where the second spatial derivatives of disparity are nonzero. Planar surfaces induce constant gradients of disparity and are thus effectively featureless to stereopsis. Further observations are reported regarding nonplanar surfaces, where contradictory monocular information can still be effectively rivalrous with that suggested stereoscopically.

Slant-tilt: The visual encoding of surface orientation

A specific form for the internal representation of local surface orientation is proposed, which is similar to Gibsons (1950) “amount and direction of slant”. Slant amount is usually quantifed by the angle σ between the surface normal and the line of sight (0°≦σ≦90°). Slant direction corresponds to the direction of the gradient of distance from the viewer to the surface, and may be defined by the image direction τ to which the surface normal would project (0°≦τ≦360°). Since the direction of slant is specified by the tilt of the projected surface normal, it is referred to as surface tilt (Stevens, 1979; Marr, 1982). The two degrees of freedom of orientation are therefore quantified by slant, an angle measured perpendicular to the image plane, and tilt, an angle measured in the image plane. The slanttilt form provides several computational advantages relative to some other proposals and is consistent with various psychological phenomena. Slant might be encoded by various means, e.g. by the cosine of the angle, by the tangent, or linearly by the angle itself. Experimental results are reported that suggest that slant is encoded by an internal parameter that varies linearly with slant angle, with resolution of roughly one part in 100. Thus we propose that surface orientation is encoded in human vision by two quantities, one varying linearly with slant angle, the other varying linearly with tilt angle.

The Analogy between Stereo Depth and Brightness

Apparent depth in stereograms exhibits various simultaneous-contrast and induction effects analogous to those reported in the luminance domain. This behavior suggests that stereo depth, like brightness, is reconstructed, ie recovered from higher-order spatial derivatives or differences of the original signal. The extent to which depth is analogous to brightness is examined. There are similarities in terms of contrast effects but dissimilarities in terms of the lateral inhibition effects traditionally attributed to underlying spatial-differentiation operators.

Surface tilt (the direction of slant): a neglected psychophysical variable.

Surface slant (the angle between the line of sight and the surface normal) is an important psychophysical variable. However, slant angle captures only one of the two degrees of freedom of surface orientation, the other being thedirection of slant. Slant direction, measured in the image plane, coincides with the direction of the gradient of distance from viewer to surface and, equivalently, with the direction the surface normal would point if projected onto the image plane. Since slant direction may be quantified by the tilt of the projected normal (which ranges over 360 deg in the frontal plane), it is referred to here assurface tilt. (Note that slant angle is measured perpendicular to the image plane, whereas tilt angle is measured in the image plane.) Compared with slant angle’s popularity as a psychophysical variable, the attention paid to surface tilt seems undeservedly scant. Experiments that demonstrate a technique for measuring apparent surface tilt are reported. The experimental stimuli were oblique crosses and parallelograms, which suggest oriented planes in 3-D. The apparent tilt of the plane might be probed by orienting a needle in 3-D so as to appear normal, projecting the normal onto the image plane, and measuring its direction (e.g., relative to the horizontal). It is shown to be preferable, however, to merely rotate a line segment in 2-D, superimposed on the display, until it appears normal to the perceived surface. The apparent surface tilt recorded in these experiments corresponded closely to that predicted by assuming the 3-D configurations consist of equal-length lines and perpendicular intersections.

The Relation Between Proximity and Brightness Similarity in Dot Patterns

The Gestalt studies demonstrated the tendency to visually organize dots on the basis of similarity, proximity, and global properties such as closure, good continuation, and symmetry. The particular organization imposed on a collection of dots is thus determined by many factors, some local, some global. We discuss computational reasons for expecting the initial stages of grouping to be achieved by processes with purely local support. In the case of dot patterns, the expectation is that neighboring dots are grouped as a function of proximity and similarity of contrast, by processes that are independent of the overall organization and the various global factors. We describe experiments that suggest a purely local relationship between proximity and brightness similarity in perceptual grouping.

Probing depth in monocular images

It is generally expected that depth (distance) is the internal representational primitive that corresponds to much of the perception of 3D. We tested this assumption in monocular surface stimuli that are devoid of distance information (due to orthographic projection and the chosen surface shape, with perspective projection used as a control) and yet are vividly three-dimensional. Slant judgments were found to be in close correspondence with the actual geometric slant of the stimuli; the spatial orientation of the surfaces was perceived accurately. The apparent depth in these stimuli was then tested by superimposing a stereo depth probe over the monocular surface. In both the perspective and orthographic projection the gradient of perceived depth, measured by matching the apparent depth of the stereo probe with that of the monocular surface at a series of locations, was substantial. The experiments demonstrate that in orthographic projection the visual system can compute from local surface orientation a depth quantity that is commensurate with the relative depth derived from stereo disparity. The depth data suggests that, at least in the near field, the zero value for relative depth lies at the same absolute depth as the stereo horopter (locus of zero stereo disparity). Relative to this zero value, the depth-from-slant computation seems to provide an estimate of distance information that is independent of the absolute distance to the surface.

Detecting structure by symbolic constructions on tokens

Geometric organization is readily detected in discrete textures such as dot patterns. A common proposal is that orientation-tuned receptive field mechanisms provide the local orientation information from which the global organizations emerge. Alternatively, the local orientation might be attributed to grouping constructions between adjacent tokens, each representing the position of a dot and its attributes such as color, size, and contrast. Geometric organization would then emerge by grouping operations on selected tokens that are similar, adjacent, and aligned. It is the ability to group on the basis of similarity that most strongly differentiates this from the energy-summating receptive field approach. Using dot patterns with rivalrous organization, we demonstrate grouping phenomena that are difficult to attribute to a broad class of energy summation detectors operating in the spatial frequency domain, which we therefore attribute to perceptual groupings on tokens. We discuss the computational differences between feature detection and structure detection, and suggest that orientation-tuned receptive field mechanisms, while appropriate for the former task, have little application to the latter.