Affine Invariant Geometry for Non-rigid Shapes

Abstract

Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity—by measuring the lack of isometry, deal with similarity or scaling—by multiplying the Euclidean arc-length by the Gaussian curvature, and manage equi-affine transformations—by resorting to the special affine arc-length definition in classical equi-affine differential geometry. Here, we propose a computational framework that is invariant to the full affine group of transformations (similarity and equi-affine). Thus, by construction, it can handle non-rigid shapes. Technically, we add the similarity invariant property to an equi-affine invariant one and establish an affine invariant pseudo-metric. As an example, we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison.

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Acknowledgments

We thank the editor and the reviewers for their valuable comments that helped us improve the presentation and writeup of the paper. This research was supported by the Office of Naval Research (ONR) award number N00014-12-1-0517 and by Israel Science Foundation (ISF) Grant Number 1031/12.

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Correspondence to Dan Raviv.

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Communicated by C. Schnörr.

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Raviv, D., Kimmel, R. Affine Invariant Geometry for Non-rigid Shapes. Int J Comput Vis 111, 1–11 (2015). https://doi.org/10.1007/s11263-014-0728-2

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Keywords

  • Heat Kernel
  • Gaussian Curvature
  • Affine Transformation
  • Scale Invariant Feature Transform
  • Beltrami Operator