A numerical framework for elastic surface matching, comparison, and interpolation

Abstract

Surface comparison and matching is a challenging problem in computer vision. While reparametrization-invariant Sobolev metrics provide meaningful elastic distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult. Square root normal fields (SRNF) considerably simplify the computation of certain elastic distances between parametrized surfaces. Yet they leave open the issue of finding optimal reparametrizations, which induce elastic distances between unparametrized surfaces. This issue has concentrated much effort in recent years and led to the development of several numerical frameworks. In this paper, we take an alternative approach which bypasses the direct estimation of reparametrizations: we relax the geodesic boundary constraint using an auxiliary parametrization-blind varifold fidelity metric. This reformulation has several notable benefits. By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns. Moreover, the problem lends itself to a coarse-to-fine multi-resolution implementation, which makes the algorithm scalable to large meshes. Furthermore, this approach extends readily to higher-order feature maps such as square root curvature fields and is also able to include surface textures in the matching problem. We demonstrate these advantages on several examples, synthetic and real.

arXiv comment21 pages, 11 figures, 1 table, 3 algorithms. Forthcoming in the International Journal of Computer Vision

References in corpus (13)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• The varifold representation of non-oriented shapes for diffeomorphic registration
• Sobolev metrics on shape space of surfaces
• Sobolev--type metrics in the space of curves
• Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces
• Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves
• Almost local metrics on shape space of hypersurfaces in n-space
• Why Use Sobolev Metrics on the Space of Curves
• On Approximating the Riemannian 1-Center
• Gauge Invariant Framework for Shape Analysis of Surfaces
• Functional Currents : a new mathematical tool to model and analyse functional shapes
• Fractional Sobolev metrics on spaces of immersions

Cited by in corpus (2)

• Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework
• Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies