A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

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.