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Denoising Tensors via Lie Group Flows

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Book cover Variational, Geometric, and Level Set Methods in Computer Vision (VLSM 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3752))

Abstract

The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to matrix Lie groups. We formulate the problem of these SO(N) flows in terms of the principal chiral model (PCM) action. This action is defined over a Lie group manifold. By minimizing the PCM action with respect to the group element, we obtain the equations of motion for the group element (or the corresponding connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows to regularize in particular the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N). This type of regularization preserves their properties (i.e., the orthogonality and the determinant). A special numerical scheme that preserves the Lie group structure is used. However, these flows regularize the tensor field isotropically and therefore discontinuities are not preserved. We modify the functional and thereby the gradient descent PDEs in order to obtain an anisotropic tensor field regularization. We demonstrate our formalism with various examples.

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Authors and Affiliations

  1. Department of Applied Mathematics, Tel-Aviv university, Ramat-Aviv, Tel-Aviv, 69978, Israel

    Y. Gur & N. Sochen

Authors
  1. Y. Gur
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  2. N. Sochen
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Editors and Affiliations

  1. MAS - Ecole Centrale Paris, Grande Voie des Vignes, 92295, Chatenay-Malabry, France

    Nikos Paragios

  2. I.N.R.I.A, 2004 route des lucioles,, 06902, Sophia-Antipolis, France

    Olivier Faugeras

  3. Department of Mathematics, UCLA,  

    Tony Chan

  4. Image and Pattern Analysis Group, Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany

    Christoph Schnörr

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Gur, Y., Sochen, N. (2005). Denoising Tensors via Lie Group Flows. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds) Variational, Geometric, and Level Set Methods in Computer Vision. VLSM 2005. Lecture Notes in Computer Science, vol 3752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11567646_2

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  • DOI: https://doi.org/10.1007/11567646_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29348-4

  • Online ISBN: 978-3-540-32109-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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