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For over 30 years, Blum's Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with form, but it is one that is notoriously difficult to apply in a robust and stable way. It is well documented how a tiny change to an object's boundary can cause a large change in its MAT. There has also been great difficulty in using the MAT to decompose an object into a hierarchy of parts reflecting the natural parts-hierarchy that we perceive. This paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts. A small change in an object's boundary corresponds to a small change in its substance but may involve a large change in its connection information. The problems with Blum's MAT are generated because it does not explicitly represent this dichotomy of information. To use the Blum MAT to it's full potential, this paper presents a method for separating the substance and connection information of an object. This provides a natural parts-hierarchy while eliminating instabilities due to small boundary changes. The method also allows for graded, fuzzy classifications of object parts to match the ambiguity in human perception of many objects. [ABSTRACT FROM AUTHOR]

Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from the Blum medial axis that considerably overcame the weakness. Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges (Ogniewicz, 1993; Székely, 1996; Näf, 1996), shock loci of reaction-diffusion equations (Kimia et al., 1995; Siddiqi and Kimia, 1996), and height ridges of medialness (cores) (Fritsch et al., 1994; Morse et al., 1993; Pizer et al., 1998). These are different from the Blum medial axis, and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary, and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries. These mathematical properties and computational abilities are laid out and compared and contrasted in this paper. [ABSTRACT FROM AUTHOR]