Toward Non-Parametric Digital Shape Representation and Recovery

TL;DR: It is shown that if an object is parallel regular, very few digital boundary patterns are realizable and that each such digital neighborhood has a well-defined geometric interpretation with respect to tangent direction, and described how a generalized torus can be recovered from a single intensity image using digital non-parametric models and methods.

Abstract: In this paper, the authors argue that shape representation and recovery can be accomplished using digital non-parametric models and methods. First, we describe some recent results in digital topology and digital geometry. It is shown, for example, that for compact, twice-differentiable objects, which we refer to as parallel regular, a digitization resolution always exists that preserves topology arid that a topology-preserving digitization resolution r always preserves the qualitative differential geometry of the object surface. Moreover, it is shown that if an object is parallel regular, very few digital boundary patterns are realizable and that each such digital neighborhood has a well-defined geometric interpretation with respect to tangent direction. We then define the set of digital boundary curves as either an adjacency graph or a grammar, and show how to recursively enumerate the set of such curves. Finally, we describe how a generalized torus can be recovered from a single intensity image using digital non-parametric models and methods.