A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

Elena Beretta, Monika Muszkieta, Wolf Naetar, Otmar Scherzer

Research output: Chapter in Book/Report/Conference proceedingChapter


We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x),from asingle measurement of the absorbed energy E(x)= μ(x)u(x),where u satisfies the elliptic partial differential equation -Δ • (D(x)Δu(x)) + μ(x)u(x) = 0in ω cRN . 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.

Original languageEnglish (US)
Title of host publicationVariational Methods
Subtitle of host publicationIn Imaging and Geometric Control
PublisherDe Gruyter
Number of pages23
ISBN (Electronic)9783110430394
ISBN (Print)9783110439236
StatePublished - Jan 11 2017


  • Inverse problems
  • Mathematical imaging
  • Mumford-Shah functional
  • Quantitative photoacoustic tomography

ASJC Scopus subject areas

  • General Mathematics
  • General Computer Science
  • General Engineering


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