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A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching

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Abstract

In this paper, the problem of non-rigid shape recognition is studied from the perspective of metric geometry. In particular, we explore the applicability of diffusion distances within the Gromov-Hausdorff framework. While the traditionally used geodesic distance exploits the shortest path between points on the surface, the diffusion distance averages all paths connecting the points. The diffusion distance constitutes an intrinsic metric which is robust, in particular, to topological changes. Such changes in the form of shortcuts, holes, and missing data may be a result of natural non-rigid deformations as well as acquisition and representation noise due to inaccurate surface construction. The presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points, its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non-rigid shapes.

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Authors and Affiliations

  1. Department of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel

    Alexander M. Bronstein, Michael M. Bronstein & Ron Kimmel

  2. Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, USA

    Mona Mahmoudi & Guillermo Sapiro

Authors
  1. Alexander M. Bronstein
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  2. Michael M. Bronstein
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  3. Ron Kimmel
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  4. Mona Mahmoudi
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  5. Guillermo Sapiro
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Corresponding author

Correspondence to Guillermo Sapiro.

Additional information

This paper is dedicated to Prof. Gromov in the occasion of receiving the Abel Prize in May 2009.

This work is partially supported by NSF, ONR, NGA, ARO, DARPA, NIH, and by the Israel Science Foundation (ISF grant No. 623/08).

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Bronstein, A.M., Bronstein, M.M., Kimmel, R. et al. A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching. Int J Comput Vis 89, 266–286 (2010). https://doi.org/10.1007/s11263-009-0301-6

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