Defining, contouring, and visualizing scalar functions on point-sampled surfaces

Abstract

This paper addresses the definition, contouring, and visualization of scalar functions on unorganized point sets, which are sampled from a surface in 3D space; the proposed framework builds on moving least-squares techniques and implicit modeling. Given a scalar function f : P → R , defined on a point set P , the idea behind our approach is to exploit the local connectivity structure of the k -nearest neighbor graph of P and mimic the contouring of scalar functions defined on triangle meshes. Moving least-squares and implicit modeling techniques are used to extend f from P to the surface M underlying P . To this end, we compute an analytical approximation f ̃ of f that allows us to provide an exact differential analysis of f ̃ , draw its iso-contours, visualize its behavior on and around M , and approximate its critical points. We also compare moving least-squares and implicit techniques for the definition of the scalar function underlying f and discuss their numerical stability and approximation accuracy. Finally, the proposed framework is a starting point to extend those processing techniques that build on the analysis of scalar functions on 2-manifold surfaces to point sets.