Journal of Mathematical Imaging and Vision

, Volume 27, Issue 1, pp 5–27 | Cite as

Level Lines Selection with Variational Models for Segmentation and Encoding

  • Coloma Ballester
  • Vicent Caselles
  • Laura Igual
  • Luis Garrido


This paper discusses the interest of the Tree of Shapes of an image as a region oriented image representation. The Tree of Shapes offers a compact and structured representation of the family of level lines of an image. This representation has been used for many processing tasks such as filtering, registration, or shape analysis. In this paper we show how this representation can be used for segmentation, rate distortion optimization, and encoding. We address the problem of segmentation and rate distortion optimization using Guigues algorithm on a hierarchy of partitions constructed using the simplified Mumford-Shah multiscale energy. To segment an image, we minimize the simplified Mumford-Shah energy functional on the set of partitions represented in this hierarchy. The rate distortion problem is also solved in this hierarchy of partitions. In the case of encoding, we propose a variational model to select a family of level lines of a gray level image in order to obtain a minimal description of it. Our energy functional represents the cost in bits of encoding the selected level lines while controlling the maximum error of the reconstructed image. In this case, a greedy algorithm is used to minimize the corresponding functional. Some experiments are displayed.


mathematical morphology tree structure segmentation rate distortion morphological encoding minimal description length 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Coloma Ballester
    • 1
  • Vicent Caselles
    • 1
  • Laura Igual
    • 1
  • Luis Garrido
    • 1
  1. 1.Dept. de TecnologiaUniversitat Pompeu-FabraBarcelonaSpain

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