Complexity, Confusion, and Perceptual Grouping. Part I: The Curve-like Representation

Intermediate-level vision is central to form perception, and we outline an approach to intermediate-level segmentation based on complexity analysis. We focus on the problem of edge detection, and how edge elements might be grouped together. This is typical because, once the local structure is established, the transition to global structure must be effected and context is critical. To illustrate, consider an edge element inferred from an unknown image. Is this local edge part of a long curve, or part of a texture? If the former, which is the next element along the curve? If the latter, is the texture like a hair pattern, in which nearby elements are oriented similarly, or like a spaghetti pattern, in which they are not? Are there other natural possibilities? Such questions raise issues of dimensionality, since curves are 1-D and textures are 2-D, and also of complexity. Working toward a measure of representational complexity for vision, in this first of a pair of papers we develop a foundation based on geometric measure theory. The main result concerns the distribution of tangents in space and in orientation, which serves as a formal basis for the concrete measure of representational complexity developed in the companion paper.