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Variational Methods

In Imaging and Geometric Control

Ed. by Bergounioux, Maïtine / Peyré, Gabriel / Schnörr, Christoph / Caillau, Jean-Baptiste / Haberkorn, Thomas

Series:Radon Series on Computational and Applied Mathematics 18

eBook (PDF)
Publication Date:
January 2017
Copyright year:
2017
ISBN
978-3-11-043039-4
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6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

Beretta, Elena / Muszkieta, Monika / Naetar, Wolf / Scherzer, Otmar

Abstract

We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x), from a single measurement of the absorbed energy E(x) = μ(x)u(x), where u satisfies the elliptic partial differential equation

−∇ ⋅ (D(x)∇u(x)) + μ(x)u(x) =0 in Ω ⊂ ℝN .

This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in [31], we show that when the coefficients μ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of μ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a Mumford-Shah-like functional, which we implement numerically and test on simulated two-dimensional data.

Citation Information

Variational Methods

In Imaging and Geometric Control

Edited by Bergounioux, Maïtine / Peyré, Gabriel / Schnörr, Christoph / Caillau, Jean-Baptiste / Haberkorn, Thomas

De Gruyter

2016

Pages: 202–224

ISBN (Online): 9783110430394

DOI (Chapter): https://doi.org/10.1515/9783110430394-006

DOI (Book): https://doi.org/10.1515/9783110430394

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