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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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# A Mumford–Shah-type approach to simultaneous reconstruction and segmentation for emission tomography problems with Poisson statistics

Esther Klann
• Corresponding author
• Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany;and Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraße 69, 4040, Linz, Austria
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• Other articles by this author:
/ Ronny Ramlau
• Radon Institute for Computational and Applied Mathematics (RICAM) and Johannes Kepler University Linz, Industrial Mathematics Institute and Doctoral Program “Computational Mathematics”, Altenbergerstraße 69, 4040 Linz, Austria
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• Other articles by this author:
/ Peng Sun
• Johannes Kepler University Linz, Doctoral Program “Computational Mathematics”, Altenbergerstraße 69,4040 Linz, Austria
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Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/jiip-2016-0077

## Abstract

We propose a variational model to simultaneous reconstruction and segmentation in emission tomography. As in the original Mumford–Shah model [27] we use the contour length as penalty term to preserve edge information whereas a different data fidelity term is used to measure the information discrimination between the computed tomography data of the reconstructed object and the observed (or simulated) data. As data fidelity term we use the Kullback–Leibler divergence which originates from the Poisson distribution present in emission tomography. In this paper we focus on piecewise constant reconstructions which is a reasonable assumption in medical imaging. The segmenting contour as well as the corresponding reconstructions are found as minimizers of a Mumford–Shah-type functional over the space of piecewise constant functions. The numerical scheme is implemented by evolving the level-set surface according to the shape derivative of the functional. The method is validated for simulated data with different levels of noise.

MSC 2010: 44A12; 47A52; 65K10; 92C55; 94A08

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Accepted: 2016-12-17

Published Online: 2017-01-12

Published in Print: 2017-08-01

The work of E. Klann has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 600209 (TU Berlin – IPODI) as well as from the Austrian Science Funds (FWF) under grant T 529-N18 and the MPNS COST Action MP1207. The work of R. Ramlau and P. Sun is supported by DK (Doktoratskolleg) program “Computational Mathematics” (W1214) granted by Austrian Science Funds (FWF). The work of P. Sun was also supported by CSC-FWF joint program.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 4, Pages 521–542, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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© 2017 Walter de Gruyter GmbH, Berlin/Boston.