We used quadratic shapes in several psychophysical shape-from-stereo tasks. The shapes were elegantly represented in a 2-D parameter space by the scale-independent shape index and the scale-dependent curuedness. Using random-dot stereograms to depict the surfaces, we found that the shape of hyperbolic surfaces is slightly more difficult to recognize than the shape of elliptic surfaces. We found that curvedness (and indirectly, scale) has little or no influence on shape recognition.Wheatstone (1838) was the first to observe that a dichoptic presentation of two projected images (differing slightly in the viewpoint) could induce a depth percept. This important finding reveals the ability of the human visual system to make depth-related judgments using disparity information. Generally, it is thought that stereopsis stems from the direct calculation of depth from disparities. This is a very logical assumption, because this is the inverse of the method that we normally use to generate stereograms. Recent research (Brookes & Stevens, 1989a, 1989bRogers & Cagenello, 1989; Brookes, 1987, 1988) suggests that depth reconstruction of surfaces is done indirectly through surface shape descriptors such as curvatures and discontinuities of disparity fields. This indicates that shape recognition is not simply a matter of calculating some kind of depth map (Gibson, 1950). Thus, we might expect shape recognition to depend on the type of shape that is seen. Most shape-from-stereo research is done with a collection of rather arbitrary shapes (Uttal, 1987;Uttal, Davis, Welke, & Kakarala, 1988) or a very restricted set of shapes, for instance, cylinders (Johnston, 1991, Rogers & Cagenello, 1989. Here, we present a more systematic approach to the research of recognition of shape with stereo.
Shape DefinitionsFirst of all, we need a more concise definition of "shape." Clearly, only a few objects can be identified by a descriptive name. To describe the infinitely large family of arbitrary shapes requires some kind of restriction and we will therefore consider local surface patches only.Position and attitude of a local surface patch do not contribute to its shape, because they are dependent on the observer-object geometry only and are not an intrinsic
71property of the patch. Therefore, we choose our frame ofreference so as to get rid of these terms. We place the origin at the fixation point and align the z-axis parallel to the surface normal. With these coordinates, we can describe a general patch as a Taylor series expansion:I . z = -(ax 2 +by 2) + hIgher order terms.( 1) 2 An infinite number of normal sections (the intersection of a plane containing the surface normal and the surface itself) can be drawn through the origin on the patch. Together, they totally define the patch. The curvature in a point of such a normal section is given by:where the x-axis is chosen in the plane of the normal section and tangent to the surface (zx denotes derivation of z with respect to x). Euler (see, e. g., AIeksandrov, Kolmogorov, & Lavrent'ev, 196...