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On the Link between Gaussian Homotopy Continuation and Convex Envelopes

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Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8932))

Abstract

The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simplified problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or implicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progressively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification.

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  1. Computer Science and Artificial Intelligence Lab. (CSAIL), Massachusetts Institute of Technology (MIT), USA

    Hossein Mobahi & John W. Fisher III

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  1. Hossein Mobahi
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  2. John W. Fisher III
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Editors and Affiliations

  1. Department of Mathematics, University of Bergen, 7800, Bergen, Norway

    Xue-Cheng Tai

  2. Department of Mathematics, University of California, Los Angeles, CA, USA

    Egil Bae

  3. The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, S.A.R.

    Tony F. Chan

  4. Telemark University College, Postboks 203, 3901, Porsgrunn, Norway

    Marius Lysaker

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Mobahi, H., Fisher, J.W. (2015). On the Link between Gaussian Homotopy Continuation and Convex Envelopes. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-14612-6_4

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