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Hamilton-Jacobi Skeletons

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Abstract

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grassfire flow and measure the average outward flux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the flux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

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Authors and Affiliations

  1. School of Computer Science & Center for Intelligent Machines, McGill University, Montreal, QC, H3A 2A7, Canada

    Kaleem Siddiqi & Sylvain Bouix

  2. Department of Electrical & Computer Engineering and Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA, 30082-0250, USA

    Allen Tannenbaum

  3. Departments of Computer Science and Electrical Engineering and Center for Computational Vision & Control, Yale University, New Haven, CT, 06520-8285, USA

    Steven W. Zucker

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  1. Kaleem Siddiqi
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  4. Steven W. Zucker
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Siddiqi, K., Bouix, S., Tannenbaum, A. et al. Hamilton-Jacobi Skeletons. International Journal of Computer Vision 48, 215–231 (2002). https://doi.org/10.1023/A:1016376116653

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