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Inverse Problems and Imaging (IPI)
 

A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

Pages: 137 - 166, Volume 5, Issue 1, February 2011      doi:10.3934/ipi.2011.5.137

 
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Esther Klann - Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraβe 69, A-4040 Linz, Austria (email)
Ronny Ramlau - Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraβe 69, A-4040 Linz, Austria (email)
Wolfgang Ring - Institut für Mathematik, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria (email)

Abstract: This paper presents a level-set based approach for the simultaneous reconstruction and segmentation of the activity as well as the density distribution from tomography data gathered by an integrated SPECT/CT scanner.
   Activity and density distributions are modeled as piecewise constant functions. The segmenting contours and the corresponding function values of both the activity and the density distribution are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and -- for fixed contours -- over the spaces of piecewise constant density and activity distributions which may be discontinuous across their corresponding contours. For the latter step a Newton method is used to solve the nonlinear optimality system. Shape sensitivity calculus is used to find a descent direction for the cost functional with respect to the geometrical variables which leads to an update formula for the contours in the level-set framework. A heuristic approach for the insertion of new components for the activity as well as the density function is used. The method is tested for synthetic data with different noise levels.

Keywords:  Level set method, shape sensitivity analysis, tomography, active contours, Mumford-Shah functional, inverse problems.
Mathematics Subject Classification:  34K29, 44A12, 49M05, 65K10, 92C50, 94A08.

Received: March 2009;      Revised: February 2010;      Available Online: February 2011.

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