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→Rf:P→R, defined on a point set PP, the idea behind our approach is to exploit the local connectivity structure of the kk-nearest neighbor graph of PP and mimic the contouring of scalar functions defined on triangle meshes. Moving least-squares and implicit modeling techniques are used to extend ff from PP to the surface MM underlying PP. To this end, we compute an analytical approximation f̃ of ff that allows us to provide an exact differential analysis of f̃, draw its iso-contours, visualize its behavior on and around MM, and approximate its critical points. We also compare moving least-squares and implicit techniques for the definition of the scalar function underlying ff 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.

Research highlights
► This paper addresses the definition, contouring, and visualization of scalar functions on unorganized point sets.
► The proposed framework builds on moving least-squares techniques and implicit modeling.
► 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.