Elastic Shape Matching of Parameterized Surfaces Using Square Root Normal Fields

Ian H. Jermyn¹, Sebastian Kurtek², Eric Klassen³, and Anuj Srivastava⁴

¹Department of Mathematical Sciences, Durham University, Durham, England

²Department of Statistics, The Ohio State University, Columbus, Ohio, USA

³Department of Mathematics, Florida State University, Tallahassee, Florida, USA

⁴Department of Statistics, Florida State University, Tallahassee, Florida, USA

Abstract. In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods.

LNCS 7576, p. 804 ff.

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