## Abstract

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 language | English (US) |
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Title of host publication | Variational Methods |

Subtitle of host publication | In Imaging and Geometric Control |

Publisher | De Gruyter |

Pages | 202-224 |

Number of pages | 23 |

ISBN (Electronic) | 9783110430394 |

ISBN (Print) | 9783110439236 |

State | Published - Jan 11 2017 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- General Computer Science
- General Engineering