Geodesics on Point Clouds

Hongchuan Yu,Zheng Jiao,Jian J Zhang

doi:10.1155/2014/860136

Hongchuan Yu, Zheng Jiao + Show 1 more

Open Access

https://doi.org/10.1155/2014/860136

Copy DOI

Abstract

We present a novel framework to compute geodesics on implicit surfaces and point clouds. Our framework consists of three parts, particle based approximate geodesics on implicit surfaces, Cartesian grid based approximate geodesics on point clouds, and geodesic correction. The first two parts can effectively generate approximate geodesics on implicit surfaces and point clouds, respectively. By introducing the geodesic curvature flow, the third part produces smooth and accurate geodesic solutions. Differing from most of the existing methods, our algorithms can converge to a given tolerance. The presented computational framework is suitable for arbitrary implicit hypersurfaces or point clouds with high genus or high curvature.

Highlights

Geodesics play an important role in differential geometry
We have presented a novel framework for computing geodesics on implicit surfaces and point clouds
The main feature of this work is its ability to produce smooth and accurate geodesics on the surface. This is due to the introduction of geodesic curvature flow in our framework

Summary

Introduction

Geodesics play an important role in differential geometry. The applications range from finite element computation to computer aided geometric design, from computer animation to robotic navigation, and from brain flattening and warping in computational neuroscience to machine learning on manifolds. One of the main challenges is that all computation must be carried out on a set of scattered points rather than parameterized surfaces or meshes. We will tackle this challenge and concentrate on two scenarios: implicit surfaces and point clouds, which are the common forms used for representing point set surfaces. The presented geodesic computation framework works well for implicit surfaces and point clouds and can be applied to high dimensional datasets. Our framework consists of three parts, approximate geodesics on implicit surfaces, approximate geodesics on point clouds, and geodesic correction. Point set surfaces might be given in an implicit form in many applications [1,2,3,4,5,6], it is impractical to generate an implicit surface for every point cloud

Objectives

Methods

Conclusion