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Equi-affine Invariant Geometry for Shape Analysis

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Abstract

Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tessellation. We show that in some shape analysis tasks, equi-affine-invariant intrinsic geometries often outperform their Euclidean-based counterparts. We further explore the potential of this metric in facial anthropometry of newborns. We show that intrinsic properties of this homogeneous group are better captured using the equi-affine metric.

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Acknowledgements

This research was supported by European Community’s FP7-ERC program, grant agreement no. 267414. This research was mostly done while D.R. was a PhD candidate in the Technion, Israel.

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Authors and Affiliations

  1. Media Lab, Massachusetts Institute of Technology (MIT), Cambridge, USA

    Dan Raviv

  2. School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel

    Alexander M. Bronstein

  3. Institute of Computational Science, Faculty of Informatics, Università della Svizzera Italiana, Lugano, Switzerland

    Michael M. Bronstein

  4. Department of Neonatology, Carmel Medical Center and Faculty of Medicine, Technion, Haifa, Israel

    Dan Waisman

  5. Dept. of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel

    Nir Sochen

  6. Dept. of Computer Science, Technion, Haifa, Israel

    Ron Kimmel

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  1. Dan Raviv
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Correspondence to Dan Raviv.

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Raviv, D., Bronstein, A.M., Bronstein, M.M. et al. Equi-affine Invariant Geometry for Shape Analysis. J Math Imaging Vis 50, 144–163 (2014). https://doi.org/10.1007/s10851-013-0467-y

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