Inexact Elastic Shape Matching in the Square Root Normal Field Framework

Abstract

This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel framework, which has several potential advantages over previous works. First, our variational formulation allows us to minimize over reparametrizations without discretizing the reparametrization group. Second, the objective function and gradient are easy to implement and efficient to evaluate numerically. Third, the initial and target surface may have different samplings and even different topologies. Fourth, texture can be incorporated as additional information in the matching term similarly to the fshape framework. We demonstrate the usefulness of this approach with several numerical examples of curves and surfaces.2010 Mathematics Subject Classification. 68U05, 49Q10, 58D10.

Intrinsic Riemannian metrics on spaces of curves: theory and computation

“…[25,23,21,49,32] among other references. But it can also be applied in the context of elastic metric matching, as recent works such as [1,6] have shown. In this section, we will assume that curves are immersed in the Euclidean space R d .…”

“…and leads, after discretization, to a simple minimization problem over the vertices of the deformed curve. This formulation is for instance implemented in [6]. Note that this could also apply in principle to other simplifying transforms associated to different choices of elastic parameters.…”

An inexact matching approach for the comparison of plane curves with general elastic metrics

“…Our approach provides a robust way to deal with geodesic distance computations in the presence of noise and perturbations. Moreover, we expect that this approach could be extended to the more challenging situation of immersed surfaces: this has so far only been touched upon for some very specific choice of metric in [6]. Another promising avenue, which is the subject of ongoing work by the authors, is to leverage the flexibility of the varifold representation for modelling and estimating weight functions defined on the shapes.…”

Closed surfaces with different shapes that are indistinguishable by the SRNF

“…It is a fascinating mathematical problem to distinguish shapes of different surfaces, and to quantify how different these shapes are from each other. One promising candidate for solving this problem has been the SRNF (square root normal field) method introduced by Jermyn et al in [5], It has been the subject of several subsequent publications, including [4], [6], and [2]. Given an oriented surface M with a Riemannian metric, this method introduces a pseudometric on the space Imm(M, R The purpose of this paper is to give examples demonstrating that this pseudometric fails to be a metric on shape space for every domain M .…”