Hierarchical Discrete Medial Axis: application to fast discrete object visualization and analysis

In digital geometry, the study of a multiresolution representation of digital objects has been carried out (using an homotopic thinning) [8]. This hierarchical structure does not change the medial axis (i.e. the balls radii) of the shape on the first stage, and provides the medial axis pruning. In this PhD, we would like to develop a hierarchical structure which, as in the continuous case, is flexible with respect to the reversibility of the construction: for instance, one possibility is to propose a ball merging process. The discrete power diagram will most certainly be of great help in this process, especially to study the relations between balls. In computational geometry, the power diagram is a generalization of the Voronoi diagram where each site is associated to a given radius [3]. In the general case, the cells of this diagram are convex polygons and this tool is widely used for ball interaction computation and surface reconstruction [4, 2, 1]. Concerning the sphere tree structure, the simplification processes use extensively this diagram in order to locally decide medial axis simplifications. In [7], we proposed a study of the relations between the discrete medial axis and the discrete power diagram. In order to have a hierarchical representation of the discrete medial axis, the first step consists in precisely analyzing this diagram structure. Compared to the continuous structure used in computational geometry, the discrete approach benefits from the fact that information loss can be computed exactly (Hamming-like distance) when geometrical approximations are done. This work will be carried out with a special interest for shape analysis and fast visualization of discrete objects based on the sphere-tree structure.