Geometric reconstruction in  bioluminescence tomography

Tim Kreutzmann; Andreas Rieder

doi:10.3934/ipi.2014.8.173

Article Contents

2014, Volume 8, Issue 1: 173-197. Doi: 10.3934/ipi.2014.8.173

This issue Previous Article Convergence rates for Kaczmarz-type regularization methods Next Article Heat source identification based on $l_1$ constrained minimization

Geometric reconstruction in bioluminescence tomography

Tim Kreutzmann^1, and
Andreas Rieder^1,

1.
Fakultät für Mathematik, Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie (KIT), D-76128 Karlsruhe

Received: February 29, 2012

Revised: June 30, 2013

Published: January 2014

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Abstract

In bioluminescence tomography the location as well as the radiation intensity of a photon source (marked cell clusters) inside an organism have to be determined given the outside photon count. This inverse source problem is ill-posed: it suffers not only from strong instability but also from non-uniqueness. To cope with these difficulties the source is modeled as a linear combination of indicator functions of measurable domains leading to a nonlinear operator equation. The solution process is stabilized by a Tikhonov like functional which penalizes the perimeter of the domains. For the resulting minimization problem existence of a minimizer, stability, and regularization property are shown. Moreover, an approximate variational principle is developed based on the calculated domain derivatives which states that there exist smooth almost stationary points of the Tikhonov like functional near to any of its minimizers. This is a crucial property from a numerical point of view as it allows to approximate the searched-for domain by smooth domains. Based on the theoretical findings numerical schemes are proposed and tested for star-shaped sources in 2D: computational experiments illustrate performance and limitations of the considered approach.

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Mathematics Subject Classification: Primary: 65J15, 65J20.

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