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Unlike the Metelli monochrome transparencies, when overlays and their backgrounds have chromatic content, the inferred surface colors may not always be physically realizable, and are in some sense “imaginary.” In these cases, the inferred chromatic transmittance or reflectance of the overlay lies outside the RGB spectral boundaries. Using the classical Metelli configuration, we demonstrate this illusion and briefly explore some of its attributes. Some observer differences in perceiving transparencies are also highlighted. These results show that the perception of transparency is much more complex than conventionally envisioned.

Phenomenal transparency is commonly studied by using a stimulus configuration introduced by Metelli: a bipartite patch, divided into equal left and right halves is overlaid with a smaller, concentric bipartite patch, divided along the same line. Observers are instructed to report either a transparent patch over an opaque bipartite field or a mosaic of four opaque patches. We show theoretically and empirically that these are only two of five generic perceptual categories, namely, transparent patch, transparent annulus (hole), mosaic, partial transparency, and multiple transparency (ambiguous) cases. Thus Gestalt factors complicate the interpretation "phenomenal transparency." We propose a framework that avoids this complication. There is excellent agreement between predictions and results.

We investigate the ability of human observers to judge the direction of illumination from image texture. Photographs of 61 real surfaces were used, taken from the Columbia-Utrecht Reflectance and Texture (Curet) database (http:/www.cs.columbia.edu/CAVE/curet). All samples were normally viewed but obliquely illuminated, the elevation of the source being 22.5 degrees, 45.0 degrees, or 67.5 degrees. The illumination was with a collimated, parallel beam. Stimuli were presented in random orientation, and observers had to judge both the elevation and the azimuth of the source. Observers judged the azimuth within approximately 15 degrees, except for the fact that they committed random (with approximately 50% probability) sign flips (180 degrees flips). Connected with this finding is the fact that observers judged the illumination to be from above rather than below in the overwhelming majority of cases, despite the fact that each case occurred with equal probability. The elevation of the illumination can be judged to some extent but is not far above chance level. The data are in good agreement with a simple model that bases the estimate of illumination direction on the second-order statistics of local luminance gradients. This locates the locus of the probable mechanism very early in the visual stream.

General surface scattering is characterized through the bidirectional reflection distribution function (BRDF). The BRDF is a function of the directions of the incident and remitted beams and thus depends on four parameters. Under very general assumptions one shows that the BRDF is invariant under interchange of the incident and remitted beams, the so-called Helmholtz reciprocity. For isotropic surfaces the BRDF depends only on the absolute value of the difference between the azimuths of the incident and remitted beams. Since these exhaust the symmetries, the BRDF is a very complicated function. For many applications it would be advantageous to be able to summarize empirical data or to smooth and/or interpolate (often even extrapolate) BRDF data. We present a principled way to do this, exactly respecting the symmetry properties.

We study the illuminance texture due to three-dimensional surface mesostructure. In the conventional rendering algorithms various methods of approximate photometry are used whose domains of validity have been ill defined. We introduce the necessary photometric concepts and do a comparative study of rendering algorithms. In the computer graphics/vision literature the physically distinct causes of illuminance variation, namely, the obliqueness of surface elements and the effects of vignetting of extended sources, are thoroughly confused. This leads to serious errors when expressions are used in contexts outside their original realm. The treatment of the texture that is due to the illumination of surface mesostructure is an example. Erroneous results are obtained when simple expressions for the illuminance of convex bodies are interpreted in terms of equivalent sources and when these again are used in calculations involving different geometrical structures. We classify current methods in terms of the approximations (often implicitly) made and compare results for a large class of applications: that of surface mesostructure on convex objects. Large differences among the various approximative methods are found. We discuss simple methods that might be used to produce more realistic results at reasonable computational cost.

We consider the qualitative structure of the illuminance distribution that is due to the illumination of a generic smooth surface under the following simplifying conditions: The surface bidirectional reflection distribution function is a constant (that is, it is Lambertian and with constant albedo), the surface is homogeneously illuminated, and vignetting and interreflection effects are absent. In that case the critical points of the illuminance distribution are determined by the local third-order differential structure of the surface, that is, by the gradients of its principal curvatures. We find the classes of local surface structure that may yield various types of critical points under variation of the direction of incidence. The parabolic points of the surface play a distinguished role. Two distinct generic types of parabolic points exist that behave qualitatively differently with respect to illumination. Finally, we consider various relaxations of the rather restrictive initial constraints.

We investigate the structure of the second-order spatial differential structure of optic flow for the case of a rigid surface in parallel projection. We disregard such problems as that of noise and that of how to obtain optic flow from image flow in the first place. The focus of the investigation is on the essential geometric structures. It is shown that the second-order structure is conveniently represented as a tensor that allows one to obtain the second-order directional derivative of the flow for arbitrary directions in the field of view. This second-order directional derivative in an arbitrary direction turns out to be a vector in the epipolar direction with a magnitude that is proportional to the normal curvature of the surface in the direction of derivation. Our analysis involves only affine concepts and constructions; metric concepts do not enter into the analysis. The second-order structure immediately reveals the projection of Dupin’s indicatrix, which captures important affine aspects of shape. When the second-order structure is combined with the first-order differential structure and after the introduction of a metric, only one degree of ambiguity is left: The rate of turn about an axis in the frontoparallel plane can be traded against the height of relief. Finally we construct hypothetical neural implementations for the second-order flow analysis. A simple, center–surround organization suffices, although one may not pool over orientations.

The movement parallax field due to the translation of an observer relative to a plane surface is studied in an infinitesimal neighborhood of a visual direction. The parallax field is decomposed into elementary transformations: a translation, a rigid rotation, a similarity, and a deformation. A topologically invariant classification based on critical-point analysis is also obtained. It is shown that the field is either that of a node or that of a saddle point. Numerical results for a general case are offered as illustration. We discuss the relevance of the local, as opposed to the global structure of the parallax field for visual perception and the visual space sense.

It is shown that there exist classes of useful partial solutions to what has become known as the structure from motion problem, that is, the problem of finding the three-dimensional shape as well as the movement in space of objects on the basis of their optical projections in several distinct phases of the movement. The class of solutions treated in this paper is quite different from the usual treatment in that the rigidity condition is relaxed so as to include the important class of bending deformations. For instance, we show a case in which important aspects of shape are obtained for a configuration of seven points, no four of which move in a rigid way, and which can be obtained from only two views. (This violates all requirements of the structure from motion theorem.) We discuss an algorithm that obtains the three-dimensional configuration of a polyhedral vertex up to a relief transformation (a similar transformation to the one that maps object to image space in paraxial optics), in the presence of arbitrary flections of the edges, from the projection and the instantaneous-movement parallax. Such a solution can, for instance, be used to predict side views of the vertex, as is demonstrated for a specific example. The solution can be used in an iterative fashion on triangulations to solve the case of extended curved surfaces suffering arbitrary bendings. The class of bending deformations encompasses all deformations that conserve distances along the surface (but not necessarily through space); thus stretchings are excluded. Practical examples include, e.g., deformations of flexible but nonstretchable shells. Actual numerical examples illustrate the practicability of the method.

A new phenomenon is reported: stroboscopically displayed moving spatial noise patterns are subjectively described as a superposition of a smoothly moving noise pattern and a somewhat irregular grating. The periodicity of the grating depends on the displacement of the noise pattern between two successive stroboscopic presentations. The detection threshold for the ghost grating is similar to that for stationary sine-wave gratings. The visual system must integrate the stimulus for at least 0.5--1 s in order for such an illusory grating to appear. Yet the subjectively smoothly moving noise pattern looks sharp.

The problem of interreflections for Lambertian surfaces of arbitrary shape and with varying reflectance is of interest for many practical applications. We present a general method to approach this problem. We define photometric modes that are uncoupled in the sense that each mode may be assigned a (pseudo) reflectance and that interreflections among modes vanish. Then the problem is formally identical with that of a convex body, in which interreflections are of no importance. The photometric modes depend on the shape of the body. In many practical cases one or a few modes dominate, and the reflected radiance depends more on the shape of the body (the dominant mode) than on the precise irradiance distribution. A few examples are treated explicitly. The redistribution of radiation described by the modes is treated by means of the net vector flux and the space density of radiation. Knowledge of these fields for the dominant mode yields considerable intuitive insight in the physical situation and provides the means to estimate the effects of painting part of the surface or of the introduction of screens.