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Geodesic Distance and Curves Through Isotropic and Anisotropic Heat Equations on Images and Surfaces

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Abstract

This paper proposes a method to extract geodesic distance and geodesic curves using heat diffusion. The method is based on Varadhan’s formula that helps to obtain a numerical approximation of geodesic distance according to metrics based on different heat flows. The heat equation can be utilized by regarding an image or a surface as a medium for heat diffusion and letting the user set at least one source point in the domain. Both isotropic and anisotropic diffusions are considered here to obtain geodesics according to their respective metrics. (1) In the part of the paper where we deal with the isotropic case, we use gray-level intensity to compute the conductivity, i.e., those pixels with gray-levels similar to the source point would have higher conductivity. The model of Perona and Malik, which inhibits heat from diffusing out of homogeneous regions, is also used for geodesic computations in this paper. The two methods are combined and used for more complicated cases. We can also use the norm of the gradient of an image as the feature in the Perona and Malik model to make the heat diffuse along boundaries and edges. (2) For the anisotropic case, we use different eigenvectors and eigenvalues to compose the diffusion tensors to concentrate heat flow along chosen directions. Furthermore, to automate the process of extracting geodesic lines, we propose two automatic methods: a new voting method and a key point method, which are both especially designed for the heat-based method. Our algorithms are tested on synthetic and real images as well as on a mesh. The results are very promising and demonstrate the robustness of the algorithms.

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Acknowledgments

We would like to thank the anonymous reviewers, as well as Aristide-Oswald Bartet, for their useful comments that allowed us to improve this paper. Our Special thanks to Dr. Vivek Kaul and Prof.Anthony Yezzi who made a complete reading of our paper in order to check for correct English and helped for this revision. Also, many thanks to Jean-Marie Mirebeau, Dario Prandi, and Gabriel Peyré for fruitful discussions.

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  1. CEREMADE, UMR CNRS 7534, Université Paris Dauphine, PSL, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Fang Yang & Laurent D. Cohen

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  1. Fang Yang
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  2. Laurent D. Cohen
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Correspondence to Fang Yang.

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Yang, F., Cohen, L.D. Geodesic Distance and Curves Through Isotropic and Anisotropic Heat Equations on Images and Surfaces. J Math Imaging Vis 55, 210–228 (2016). https://doi.org/10.1007/s10851-015-0621-9

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