Analysis of Curve Reconstruction by Meshless Parameterization

This paper proposes and analyzes a method called meshless parameterization for reconstructing curves from unordered point samples. The method solves a linear system of equations based on convex combinations so as to map the sampled points into corresponding parameter values, whose natural ordering provides the ordering of the points. Using the theory of M-matrices, we derive natural conditions on the point sample which guarantee the correct ordering. A sufficient condition is that the underlying curve be tangent-continuous and free of self-intersections and that the sample is dense enough.