Robustness Issues in Surface Reconstruction

The piecewise linear reconstruction of a surface from a sample is a well studied problem in computer graphics and computational geometry. A popular class of reconstruction algorithms lter a subset of triangles of the three dimensional Delaunay triangulation of the sample and subsequently extract a manifold from the ltered triangles. Here we report on robustness issues that turned out to be crucial in implementations.

Delaunay Triangulations Approximate Anchor Hulls

Recent results establish that a subset of the Voronoi diagram of a point set that is sampled from the smooth boundary of a shape approximates the medial axis. The corresponding question for the dual Delaunay triangulation is not addressed in the literature. We show that, for two dimensional shapes, the Delaunay triangulation approximates a specific structure which we call anchor hulls. Since similar shapes have similar anchor hulls, they provide an useful tool in matching shapes. We demonstrate that our approximation result is useful in this application. 1

Recursive geometry of the flow complex and topology of the flow complex filtration

AbstractThe flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in Rk. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have found applications in surface reconstruction, shape matching, and molecular modeling. In this article we give an algorithm for computing the flow complex of weighted points in any dimension. The algorithm reflects the recursive structure of the flow complex. On the basis of the algorithm we establish a topological similarity between flow shapes and the nerve of a corresponding ball set, namely homotopy equivalence

Shape Dimension and Approximation from Samples

There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to. Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory