Inverse diffusion theory of photoacoustics

By Guillaume Bal, Gunther Uhlmann. 2009

182 citations recorded by the source corpus.

This paper analyzes the reconstruction of diffusion and absorption parameters in an elliptic equation from knowledge of internal data. In the application of photoacoustics, the internal data are the amount of thermal energy deposited by high frequency radiation propagating inside a domain of interest. These data are obtained by solving an inverse wave equation, which is well studied in the literature. We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determines two constitutive parameters in diffusion and Schrödinger equations. Stability of the reconstruction is guaranteed under additional geometric constraints of strict convexity. No geometric constraints are necessary when 2n internal data for well-chosen boundary conditions are available, where n is spatial dimension. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.

Subfields: Biomedical Engineering, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/8fa52f490e9b3ac1b958bdcaf6cab7156b2d28eb

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In Biomedical Engineering

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

Inverse problems: seeing the unseen (2014, methodology): “CGO solutions have been used in these hybrid methods in [15–18,42,110].”
Quantitative spectroscopic photoacoustic imaging: a review. (2012, unknown): “Rn δDðx 0Þ∇G0ðx; x 0Þ · ∇Φ0ðx 0Þdx 0: (44)”
The Calderón inverse problem for isotropic quasilinear conductivities (2021, background): “1 is known, see [2], [13, Proposition 2], [35], [31], we shall present here a very simple proof of it for the convenience of the reader.”
Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions (2011, background): “We know that for γ and c with the aforementioned regularity and for |ρ| sufficiently large, then ψρ is of order |ρ|−1 in C(2)(X̄) [7].”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Such problems have applications in photoacoustic tomography [4, 3] and have been studied recently in the Bayesian inference setting in [41].”