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A GromovHausdorff framework with diffusion geometry for topologicallyrobust nonrigid shape matching
 IMA PREPRINT SERIES# 2240
, 2009
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ShapeGoogle: geometric words and expressions for invariant shape retrieval
, 2010
"... The computer vision and pattern recognition communities have recently witnessed a surge of featurebased methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words ” and treat them using text search approaches following the ..."
Abstract

Cited by 33 (5 self)
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The computer vision and pattern recognition communities have recently witnessed a surge of featurebased methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words ” and treat them using text search approaches following the “bag of features ” paradigm. In this paper, we explore analogous approaches in the 3D world applied to the problem of nonrigid shape retrieval in large databases. Using multiscale diffusion heat kernels as “geometric words”, we construct compact and informative shape descriptors by means of the “bag of features ” approach. We also show that considering pairs of “geometric words ” (“geometric expressions”) allows creating spatiallysensitive bags of features with better discriminativity. Finally, adopting metric learning approaches, we show that shapes can be efficiently represented as binary codes. Our approach achieves stateoftheart results on the SHREC 2010 largescale shape retrieval benchmark.
Spectral GromovWasserstein Distances for Shape Matching
"... We introduce a spectral notion of distance between shapes and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our constr ..."
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Cited by 16 (1 self)
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We introduce a spectral notion of distance between shapes and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our construction is similar to the recently proposed GromovWasserstein distance, but rather than viewing shapes merely as metric spaces, we define our distance via the comparison of heat kernels. This allows us to relate our distance to previously proposed spectral invariants used for shape comparison, such as the spectrum of the LaplaceBeltrami operator. In addition, the heat kernel provides a natural notion of scale, which is useful for multiscale shape comparison. We also prove a hierarchy of lower bounds for our distance, which provide increasing discriminative power at the cost of increase in computational complexity. 1.
SpectralDriven IsometryInvariant Matching of 3D Shapes
, 2009
"... This paper presents a matching method for 3D shapes, which comprises a new technique for surface sampling and two algorithms for matching 3D shapes based on pointbased statistical shape descriptors. Our sampling technique is based on critical points of the eigenfunctions related to the smaller eige ..."
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Cited by 13 (1 self)
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This paper presents a matching method for 3D shapes, which comprises a new technique for surface sampling and two algorithms for matching 3D shapes based on pointbased statistical shape descriptors. Our sampling technique is based on critical points of the eigenfunctions related to the smaller eigenvalues of the LaplaceBeltrami operator. These critical points are invariant to isometries and are used as anchor points of a sampling technique, which extends the farthest point sampling by using statistical criteria for controlling the density and number of reference points. Once a set of reference points has been computed, for each of them we construct a pointbased statistical descriptor (PSSD, for short) of the input surface. This descriptor incorporates an approximation of the geodesic shape distribution and other geometric information describing the surface at that point. Then, the dissimilarity between two surfaces is computed by comparing the corresponding sets of PSSDs with bipartite graph matching or measuring the L1distance between the reordered feature vectors of a proximity graph. Here, the reordering is given by the Fiedler vector of a Laplacian matrix
Gromovhausdorff distances in Euclidean spaces
 In Proc. Computer Vision and Pattern Recognition (CVPR
"... The purpose of this paper is to study the relationship between measures of dissimilarity between shapes in Euclidean space. We first concentrate on the pair GromovHausdorff distance (GH) versus Hausdorff distance under the action of Euclidean isometries (EH). Then, we (1) show they are comparable i ..."
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Cited by 12 (6 self)
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The purpose of this paper is to study the relationship between measures of dissimilarity between shapes in Euclidean space. We first concentrate on the pair GromovHausdorff distance (GH) versus Hausdorff distance under the action of Euclidean isometries (EH). Then, we (1) show they are comparable in a precise sense that is not the linear behaviour one would expect and (2) explain the source of this phenomenon via explicit constructions. Finally, (3) by conveniently modifying the expression for the GH distance, we recover the EH distance. This allows us to uncover a connection that links the problem of computing GH and EH and the family of Euclidean Distance Matrix completion problems. The second pair of dissimilarity notions we study is the so called LpGromovHausdorff distance versus the Earth Mover’s distance under the action of Euclidean isometries. We obtain results about comparability in this situation as well. 1.
GromovHausdorff Stable Signatures for Shapes Using Persistence
, 2009
"... We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on thepersistencediagramsofsuitablefiltrationsbuilton topofthesespaces.Weprovethestabilityofoursignatures under GromovHausdorff perturbations of the spaces. We also extend these results ..."
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Cited by 12 (2 self)
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We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on thepersistencediagramsofsuitablefiltrationsbuilton topofthesespaces.Weprovethestabilityofoursignatures under GromovHausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are wellsuited for the study of unstructured point cloud data, which we illustrate through an application in shape classification.
Matching shapes by eigendecomposition of the laplacebeltrami operator
, 2010
"... We present a method for detecting correspondences between nonrigid shapes, that utilizes surface descriptors based on the eigenfunctions of the LaplaceBeltrami operator. We use clusters of probable matched descriptors to resolve the sign ambiguity in matching the eigenfunctions. We then define a m ..."
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Cited by 12 (6 self)
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We present a method for detecting correspondences between nonrigid shapes, that utilizes surface descriptors based on the eigenfunctions of the LaplaceBeltrami operator. We use clusters of probable matched descriptors to resolve the sign ambiguity in matching the eigenfunctions. We then define a matching cost that measures both the descriptor similarity, and the similarity between corresponding geodesic distances measured on the two shapes. We seek for correspondence by minimizing the above cost. The resulting combinatorial problem is then reduced to the problem of matching a small number of feature points using quadratic integer programming. 1.
Isometryinvariant matching of point set surfaces
 In Proc. of the Eurographics workshop on 3D object retrieval
, 2008
"... Shape deformations preserving the intrinsic properties of a surface are called isometries. An isometry deforms a surface without tearing or stretching it, and preserves geodesic distances. We present a technique for matching point set surfaces, which is invariant with respect to isometries. A set of ..."
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Cited by 10 (1 self)
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Shape deformations preserving the intrinsic properties of a surface are called isometries. An isometry deforms a surface without tearing or stretching it, and preserves geodesic distances. We present a technique for matching point set surfaces, which is invariant with respect to isometries. A set of reference points, evenly distributed on the point set surface, is sampled by farthest point sampling. The geodesic distance between reference points is normalized and stored in a geodesic distance matrix. Each row of the matrix yields a histogram of its elements. The set of histograms of the rows of a distance matrix is taken as a descriptor of the shape of the surface. The dissimilarity between two point set surfaces is computed by matching the corresponding sets of histograms with bipartite graph matching. This is an effective method for classifying and recognizing objects deformed with isometric transformations, e.g., nonrigid and articulated objects in different postures.
Hierarchical matching of nonrigid shapes
 International Conference on Scale Space and Variational Methods in Computer Vision (SSVM’11
, 2011
"... Abstract. Detecting similarity between nonrigid shapes is one of the fundamental problems in computer vision. While rigid alignment can be parameterized using a small number of unknowns representing rotations, reflections and translations, nonrigid alignment does not have this advantage. The major ..."
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Cited by 2 (2 self)
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Abstract. Detecting similarity between nonrigid shapes is one of the fundamental problems in computer vision. While rigid alignment can be parameterized using a small number of unknowns representing rotations, reflections and translations, nonrigid alignment does not have this advantage. The majority of the methods addressing this problem boil down to a minimization of a distortion measure. The complexity of a matching process is exponential by nature, but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth twomanifolds. Here we model shapes using both local and global structures, and provide a hierarchical framework for the quadratic matching problem.
Metric Approaches to Invariant Shape Similarity
, 2009
"... Nonrigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance ..."
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Nonrigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance is understanding which properties of the shape are attributed to deformations and which are invariant, i.e., remain unchanged. This chapter presents an approach to nonrigid shapes from the point of view of metric geometry. Modeling shapes as metric spaces, one can pose the problem of shape similarity as the similarity of metric spaces and harness tools from theoretical metric geometry for the computation of such a similarity. 1