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Partial Differential Equations for Interpolation and Compression of Surfaces

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

Abstract

Partial differential equations (PDEs) have recently shown to be very promising for image interpolation and compression. Inspired from this work, we present a PDE based approach to interpolation of surfaces from scattered point sets using the geometric diffusion equation. Triangulated surfaces are considered in the discrete setting, and the geometric diffusion equation is discretized by the finite element method directly on the triangular mesh. Furthermore, a PDE based method for lossy compression of triangulated surfaces is presented. The idea is to store only a few relevant vertex coordinates in the encoding step. In the decoding step, the remaining vertices are reconstructed by solving the geometric diffusion equation. Finally, two modified reconstruction methods are proposed that are shown to improve the compression quality for both images and surfaces. These reconstruction methods approximate instead of interpolating, and have links to Hopscotch methods for the numerical solution of PDEs. Experiments are presented illustrating that results of high quality can be obtained using simple geometric diffusion without any information on surface normals.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Norway

    Egil Bae

  2. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building E1.1, 66041, Saarbrücken, Germany

    Joachim Weickert

Authors
  1. Egil Bae
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  2. Joachim Weickert
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

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Bae, E., Weickert, J. (2010). Partial Differential Equations for Interpolation and Compression of Surfaces. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_1

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  • DOI: https://doi.org/10.1007/978-3-642-11620-9_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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