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

Highlights

• Our method is able to seek a trade-off between accuracy and smoothness.

• It can be easily extended to point clouds and tetrahedral meshes.

• It supports 3D models with density functions or anisotropic metric tensor fields.

• We present two applications: defect tolerant and symmetry driven geodesics.

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.

Introduction

Fast and accurate estimation of geodesic distances is a fundamental research topic in computer graphics and computational geometry. It is central to various applications ranging from surface sampling/parametrization/skinning [1], [2], [3] to shape segmentation/registration/edit [4], [5].

There is a large body of literature [6], [7], [8], [9], [10], [11], [12] on computing a geodesic distance field on a polygonal mesh surface. Some of them [6], [7], [8], [9] are exact while some [10], [11], [12] are approximate. Speed and accuracy are often viewed as a tradeoff for users to select a favorite geodesic algorithm for their specific application occasions. But researchers find that accuracy should not be an absolute evaluation criterion because an exact geodesic distance field is generally continuous but not smooth, which is unfavorable to many applications, e.g., defining an intrinsic shape signature and calculating a local parametrization. Therefore, in recent years, there is a trend toward taking the smoothness requirement into consideration when one evaluates a geodesic algorithm. For example, commute distance [13], diffusion distance [14], biharmonic distance [15] and heat diffusion induced distance [11] are typical examples. However, the resulting distance fields, in spite of being smooth, may be far from the exact geodesic distance fields, especially around the “ridge” points (a point is said to be a ridge point if there exist two different shortest paths); See Fig. 1. The consideration of smoothness brings a new question: how to make a balance between accuracy and smoothness?

In this paper, we use the term Quasi-geodesic Distance Field (QGDF) to denote the distance field we expect. To be exactly, the desirable features of QGDF include two aspects: (a) QGDF is a smooth scalar field on the given surface, and (b) QGDF is sufficiently close to the exact geodesic distance field. Users are allowed to use a unique parameter to balance accuracy and smoothness. Our idea is inspired by an interesting observation: the solution to the shortest path problem on graphs can be found by a linear programming [16]. Based on this, we formulate the problem of computing a geodesic distance field from a pure optimization perspective. Given a mesh $\mathbf{M}=(\mathbf{V},\mathbf{E},\mathbf{F})$ and a specific vertex $v_i\in \mathbf{V}$ as the source, we compute the unknown distances ${\left\{d_i\right\}}_{i=1}^n$ via quadratic programming, which maximizes $\sum d_i$ while minimizes the Laplacian energy of ${\left\{d_i\right\}}_{i=1}^n$, subject to the triangle inequalities for each vertex triple contributed by a triangle $f_i\in \mathbf{F}$. In implementation, we use a parameter to control the balance between the two terms that compose the objective function. The quadratic programming based formulation can be solved efficiently by many mature optimization solvers. Experimental results show that the timing cost increases linearly with the number of vertices.

Furthermore, our formulation is very flexible. First, it can be extended to various surface representations including point clouds and tetrahedral meshes. Second, it is able to support various user-specified constraints, which is a big benefit for us to solve many variants of the standard geodesic problem. The nice feature motivates us to apply the proposed algorithm in two applications including defect tolerant geodesic distance fields and symmetry-constrained distance fields, which cannot be easily handled by the known approaches, to our best knowledge.