Visualizing Point Cloud Classifiers by Curvature Smoothing Supplementary Materials

A Curvature approximation proof Theorem 1. Let pi ∈R3 be a point in point cloud. Let H = {x : 〈x,n〉+D = 0,x ∈R3},n ∈ R3, ||n|| = 1 be the plane fitted to the neighbors of pi. Let hi be the projection of pi on H. Assuming pi’s neighbors distribute evenly and densely on a ring surrounding hi, then the curvature normal at pi can be approximated by the expression 1 2k2 (hi− pi), where k is the distance from pi to any of its neighbor.