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

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) over tilde of f that allows us to provide an exact differential analysis of (f) over tilde, 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. (c) 2010 Elsevier Ltd. All rights reserved.