Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching

Abstract

An efficient algorithm for isometry-invariant matching of surfaces is presented. The key idea is computing the minimum-distortion mapping between two surfaces. For this purpose, we introduce the generalized multidimensional scaling, a computationally efficient continuous optimization algorithm for finding the least distortion embedding of one surface into another. The generalized multidimensional scaling algorithm allows for both full and partial surface matching. As an example, it is applied to the problem of expression-invariant three-dimensional face recognition.Gromov-Hausdorff distance ͉ isometric embedding ͉ iterative-closestpoint ͉ partial embedding T he problem of comparison between deformable surfaces has recently gained a lot of interest in pattern recognition, especially in the fields of three-dimensional (3D) data analysis and synthesis. It arises, for example, in medical imaging and 3D face recognition (1). The fundamental question is how to efficiently yet accurately quantify the similarity between a given reference surface ("model") and some other surface ("probe"), usually a bent version of the model. We would like to capture the distinction of the model and the probe intrinsic properties associated with the metric structure of the surface, while ignoring the extrinsic properties that describe the way the surface is immersed into the ambient space and that often change while the surface bends. A deformation that preserves the intrinsic structure of the surface is called an ''isometry.'' Our goal is thus to define a computable isometry-invariant measure of similarity between surfaces. Often, the problem is even more complicated, when having the probe only partially available (see Fig. 1). We refer to the latter setting as ''partial surface matching'' and to the case where the whole probe is available as ''full surface matching.'' One of the earliest attempts of isometry-invariant surface matching is the classical ''iterative closest point'' (ICP) algorithm (2). It addresses a particular case of the partial matching problem where only rigid (Euclidean) isometries are allowed. An efficient method for the construction of near-isometry-invariant representations of surfaces [called ''canonical forms'' (CFs)] based on Euclidean embeddings was presented in ref. 3, as a generalization of ref. 4. This approach used a multidimensional scaling (MDS) algorithm (5). MDS is closely related to dimensionality reduction (6, 7) and can be performed in a computationally efficient manner. Euclidean embeddings are used in theoretical computer science for representing metric spaces usually arising from geometry of graphs (8).Recently, Mémoli and Sapiro (9) used a discrete probabilistic approximation of the Gromov-Hausdorff (GH) distance (10) to compare surfaces up to isometries. Their method is based on evaluating all of the permutations between two sets of surface samples, which is computationally expensive. Moreover, the GH distance is inappropriate for partial matching.One of the main contributions of this work is t...

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