Burning the medial axis

Medial axis is a classical shape descriptor that is widely used in computer graphics, computer vision, and pattern recognition. Defined elegantly as the locus of points with multiple nearest neighbors on the object boundary, the medial axis preserves both the structure and topology of the object in a compact form - A geometry that has one lower dimension than the object itself. In many applications, medial geometry at even lower dimensions are desirable. For example, the medial curve of a 3D object is useful for deformable shape matching and character animation. The medial point of an object is useful for object alignment and tracking. Although numerous heuristic approaches have been developed for computing medial curves and points of a 3D object, there has been little progress in developing a sound mathematical definition of these lower-dimensional medial geometry. To the best of our knowledge, the only definition of the medial curve of a 3D object was proposed in [Dey and Sun 2006]. However, their definition is quite different from that of the medial axis, and the defined medial curve is not guaranteed to preserve the topology of the object, which is a key property of the medial axis.