Chapter 1 analysis and synthesis of smooth shapes

Abstract

Smooth shapes are defined as connected regularized point sets whose boundary is “smooth”. We distinguish, in the structure of an object, between a “main structure” and the overlying “details” or “textures” and we propose a complementarity principle for the description of shape: “Shape demands description at different levels of detail: global descriptions require low detail whereas local descriptions require higher detail. The relative emphasis depends on the application”. We suggest that the two entities can be separated in terms of frequency analysis, the low frequency component being associated with the main structure, and the high frequency component with details/textures. The representation of shape that is proposed consists of two complementary components: (i) the “skeletal representation” of the main structure, which uses the concept of Medial Axis Transform, and (ii) the “zero-crossing representation” of the details/textures which extends to the contours of shapes the theorem by Logan already applied to problems of vision. The two representations are discussed with regard to problems of analysis and synthesis.