Computing Smooth Quasi-geodesic Distance Field (QGDF) with Quadratic Programming

Abstract

Computing geodesic distances on polyhedral surfaces is an important task in digital geometry processing. Speed and accuracy are two commonly-used measurements of evaluating a discrete geodesic algorithm. In applications, such as parametrization and shape analysis, a smooth distance field is often preferred over the exact, non-smooth geodesic distance field. We use the term Quasi-geodesic Distance Field (QGDF) to denote a smooth scalar field that is as close as possible to an exact geodesic distance field. In this paper, we formulate the problem of computing QGDF into a standard quadratic programming (QP) problem which maintains a trade-off between accuracy and smoothness. The proposed QP formulation is also flexible in that it can be naturally extended to point clouds and tetrahedral meshes, and support various user-specified constraints. We demonstrate the effectiveness of QGDF in defect-tolerant distances and symmetry-constrained distances.