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Level Set Methods for Structural Inversion and Image Reconstruction

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Handbook of Mathematical Methods in Imaging

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Abstract

In this chapter, an introduction is given into the use of level set techniques for inverse problems and image reconstruction. Several approaches are presented which have been developed and proposed in the literature since the publication of the original (and seminal) paper by F. Santosa in 1996 on this topic. The emphasis of this chapter, however, is not so much on providing an exhaustive overview of all ideas developed so far, but on the goal of outlining the general idea of structural inversion by level sets, which means the reconstruction of complicated images with interfaces from indirectly measured data. As case studies, recent results (in 2D) from microwave breast screening, history matching in reservoir engineering, and crack detection are presented in order to demonstrate the general ideas outlined in this chapter on practically relevant and instructive examples. Various references and suggestions for further research are given as well.

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Acknowledgments

OD thanks Diego Álvarez, Natalia Irishina, Miguel Moscoso and Rossmary Villegas for their collaboration on the exciting topic of level set methods in image reconstruction, and for providing figures which have been included in this chapter. He thanks the Spanish Ministerio de Educacion y Ciencia (Grants FIS2004-22546-E and FIS2007-62673), the European Union (Grant FP6-503259), the French CNRS and Univ. Paris Sud 11, and the Research Councils UK for their support of some of the work which has been presented in this chapter. DL thanks Jean Cea for having introduced him to the fascinating world of shape optimal design, Fadil Santosa for his contribution to his understanding of the linkage between shape optimal design and level set evolutions, and Jean-Paul Zolésio for his precious help on both topics, plus his many insights on topological derivatives.

Authors
  1. Oliver Dorn
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  2. Dominique Lesselier
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Editors and Affiliations

  1. Computational Science Center, University of Vienna, Nordbergstrasse 15, Vienna, Austria

    Otmar Scherzer

  2. RICAM, Austrian Academy of Sciences, Linz, Austria

    Otmar Scherzer

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Dorn, O., Lesselier, D. (2011). Level Set Methods for Structural Inversion and Image Reconstruction. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_10

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