The goal of our work is to develop an efficient, automatic algorithm for discovering point correspondences between surfaces that are approximately and/or partially isometric. Our approach is based on three observations. First, isometries are a subset of the Möbius group, which has low-dimensionality – six degrees of freedom for topological spheres, and three for topological discs. Second, computing the Möbius transformation that interpolates any three points can be computed in closed-form after a mid-edge flattening to the complex plane. Third, deviations from isometry can be modeled by a transportation-type distance between corresponding points in that plane. Motivated by these observations, we have developed a Möbius Voting algorithm that iteratively: 1) samples a triplet of three random points from each of two point sets, 2) uses the Möbius transformations defined by those triplets to map both point sets into a canonical coordinate frame on the complex plane, and 3) produces “votes” for predicted correspondences between the mutually closest points with magnitude representing their estimated deviation from isometry. The result of this process is a fuzzy correspondence matrix, which is converted to a permutation matrix with simple matrix operations and output as a discrete set of point correspondences with confidence values. The main advantage of this algorithm is that it can find intrinsic point correspondences in cases of extreme deformation. During experiments with a variety of data sets, we find that it is able to find dozens of point correspondences between different object types in different poses fully automatically.

We present an efficient computational framework for isometry-invariant comparison of smooth surfaces. We formulate the Gromov-Hausdorff distance as a multidimensional scaling (MDS)-like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples. For isometry-invariant comparison of surfaces in the case of partially missing data, we present the partial embedding distance, which is computed using a similar scheme. The main idea is finding a minimum-distortion mapping from one surface to another, while considering only relevant geodesic distances. We discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency.

(will be inserted by the editor) A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching

Abstract—An algorithm is proposed for 3D face recognition in the presence of varied facial expressions. It is based on combining the match scores from matching multiple overlapping regions around the nose. Experimental results are presented using the largest database employed to date in 3D face recognition studies, over 4,000 scans of 449 subjects. Results show substantial improvement over matching the shape of a single larger frontal face region. This is the first approach to use multiple overlapping regions around the nose to handle the problem of expression variation. Index Terms—Biometrics, face recognition, three-dimensional face, facial expression. 1

We present a geometric framework for automatically finding intrinsic correspondence between three-dimensional nonrigid objects. We model object deformation as near isometries and find the correspondence as the minimum-distortion mapping. A generalization of multidimensional scaling is used as the numerical core of our approach. As a result, we obtain the possibility to manipulate the extrinsic geometry and the texture of the objects as vectors in a linear space. We demonstrate our method on the problems of expression-invariant texture mapping onto an animated three-dimensional face, expression exaggeration, morphing between faces, and virtual body painting.

Recent studies on three-dimensional face recognition proposed to model facial expressions as isometries of the facial surface. Based on this model, expression-invariant signatures of the face were constructed by means of approximate isometric embedding into flat spaces. Here, we apply a new method for measuring isometry-invariant similarity between faces by embedding one facial surface into another. We demonstrate that our approach has several significant advantages, one of which is the ability to handle partially missing data. Promising face recognition results are obtained in numerical experiments even when the facial surfaces are severely occluded.

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This paper explores the problem of similarity criteria between nonrigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the object and the latter to how it is laid out in the Euclidean space. Both criteria have their advantages and disadvantages: extrinsic similarity is sensitive to nonrigid deformations, while intrinsic similarity is sensitive to topological noise. In this paper, we approach the problem from the perspective of metric geometry. We show that by unifying the extrinsic and intrinsic similarity criteria, it is possible to obtain a stronger topology-invariant similarity, suitable for comparing deformed shapes with different topology. We construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a set-valued distance. Numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail.

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In this paper, we propose a high-order graph matching formulation to address non-rigid surface matching. The singleton terms capture the geometric and appearance similarities (e.g., curvature and texture) while the high-order terms model the intrinsic embedding energy. The novelty of this paper includes: 1) casting 3D surface registration into a graph matching problem that combines both geometric and appearance similarities and intrinsic embedding information, 2) the first implementation of high-order graph matching algorithm that solves a non-convex optimization problem, and 3) an efficient two-stage optimization approach to constrain the search space for dense surface registration. Our method is validated through a series of experiments demonstrating its accuracy and efficiency, notably in challenging cases of large and/or non-isometric deformations, or meshes that are partially occluded. 1.