In Imaging and Geometric Control
Ed. by Bergounioux, Maïtine / Peyré, Gabriel / Schnörr, Christoph / Caillau, Jean-Baptiste / Haberkorn, Thomas
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6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
Beretta, Elena / Muszkieta, Monika / Naetar, Wolf / Scherzer, Otmar
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 , 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.
Elena Beretta, Monika Muszkieta, Wolf Naetar, Otmar Scherzer (2016). 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients. In Maitine Bergounioux, Gabriel Peyré, Christoph Schnörr, Jean-Baptiste Caillau, Thomas Haberkorn (Eds.), Variational Methods: In Imaging and Geometric Control (pp. 202–224). Berlin, Boston: De Gruyter. https://doi.org/10.1515/9783110430394-006
Book DOI: https://doi.org/10.1515/9783110430394
Online ISBN: 9783110430394© 2016 Walter de Gruyter GmbH, Berlin/Munich/Boston