Topological Reconstruction of a Smooth Manifold-Solid from Its Occluding Contour

Abstract

This paper describes a simple construction for building a combinatorial model of a smooth manifold-solid from a labeled-figure representing its occluding contour. The motivation is twofold. First, deriving the combinatorial model is an essential intermediate step in the visual reconstruction of solid-shape from image contours. A description of solid-shape consists of a metric and a topological component. Both are necessary: the metric component specifies how the topological component is embedded in three-dimensional space. The paneling construction described in this paper is a procedure for generating the topological component from a labeled-figure representing the occluding contour. Second, the existence of this construction establishes the sufficiency of a labeling scheme for line-drawings of smooth solid-objects originally proposed by Huffman (1971). By sufficiency, it is meant that every set of closed plane-curves satisfying this labeling scheme is shown to correspond to a generic view of a manifold-solid. Together with the Whitney theorem (Whitney, 1955), this confirms that Huffman's labeling scheme correctly distinguishes possible from impossible smooth solid-objects.