Continuous skeletons from discrete objects

Abstract: "The Medial Axis Transform (MAT) was defined by Blum in the sixties as an alternate description of the shape of an object. Since then, its potential applicability in a wide range of engineering domains has been acknowledged. However, this potential has never quite been realized, except recently in two dimensions. One reason is the difficulty in defining algorithms for finding the MAT, especially in three dimensions. Another reason is the lack of incentive for modeling designs directly in MAT's. Given this impasse, some lateral thinking appears to be in order. Perhaps the MAT per se is not the only skeleton which can be used. Are there other, more easily derived skeletons, which share the properties of the MAT which are of interest in engineering design? In this work, we identify a set of properties of the MAT which, we argue, are of primary interest. Briefly, these properties are dimensional reduction (in the sense of having no interior), topological equivalence, and invertibility. For the restricted class of discrete objects, we define an algorithm for identifying a point set, generally called a skeleton, which shares these properties with the MAT. The algorithm will be defined for two dimensions, and a proof will be outlined. The true focus of our work, however, lies in the extension of this to three dimensions. We present ideas on how the 2D algorithm can be extended to 3D objects, and also present a line of argument which should extend the proofs of the 2D algorithm to three dimensions.