Journal ArticleDOI
Gromov–Wasserstein Distances and the Metric Approach to Object Matching
TLDR
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers.Abstract:
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.read more
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References
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Proceedings ArticleDOI
Spectral Gromov-Wasserstein distances for shape matching
TL;DR: It is shown that the spectral notion of distance between shapes 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.
Proceedings ArticleDOI
Gromov-Hausdorff distances in Euclidean spaces
TL;DR: A connection is uncovered that links the problem of computing GH and EH and the family of Euclidean Distance Matrix completion problems and the pair are comparable in a precise sense that is not the linear behaviour one would expect.
Proceedings ArticleDOI
Symmetries of non-rigid shapes
TL;DR: This paper poses the problem of finding intrinsic symmetries of non-rigid shapes and proposes an efficient method for their computation.
Journal ArticleDOI
Size homotopy groups for computation of natural size distances
TL;DR: For every manifold M endowed with a structure described by a function from M to the vector space R k, a parametric family of groups, called size homotopy groups, is introduced and studied in this paper.
Journal Article
Geodesic object representation and recognition
A. Ben Hamza,Hamid Krim +1 more
TL;DR: A shape signature that captures the intrinsic geometric structure of 3D objects is described, based on a global geodesic distance defined on the object surface, and takes the form of a kernel density estimate.