Computing geodesies on point clouds

Abstract

Given two points on an object represented by point cloud, we first obtain an approximately shortest path between the two points as an initial active curve by using Dijkstra's algorithm, then we use square distance minimization method to compute the active curve iteratively to be the geodesic between the two points on the point cloud. We define the objective function to be constraints of the distance between the active curve and the point cloud as well as the arc length of the active curve, and then minimize the objective function step by step by relocating the control points of the active curve until a convergence result is achieved. The method avoids triangulating or reconstructing the point cloud to be a surface model so that it is practicable for dealing with the point cloud with a huge number of scattered points.