Analysis of the Linearized Problem of Quantitative Photoacoustic Tomography

Abstract

Quantitative image reconstruction in photoacoustic tomography requires the solution of a coupled physics inverse problem involving light transport and acoustic wave propagation. In this paper we address this issue employing the radiative transfer equation as accurate model for light transport. As main theoretical results, we derive several stability and uniqueness results for the linearized inverse problem. We consider the case of single illumination as well as the case of multiple illuminations assuming full or partial data. The numerical solution of the linearized problem is much less costly than the solution of the non-linear problem. We present numerical simulations supporting the stability results for the linearized problem and demonstrate that the linearized problem already gives accurate quantitative results.

Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography

“…In the following, we discuss the methods that we have outlined in the previous section. (19): We assume that the scattering coefficient is known and we restrict ourself to reconstructing the absorption coefficient. Then, the proximal gradient and proximal stochastic gradient algorithm, respectively, read…”

“…With the abbreviation M(µ) := θ · ∇ x + µ a + µ s (I − K), the RTE can be written in compact form M(µ)Φ = q, where µ = (µ a , µ s ) is the unknown parameter pair. In the case of exact data, the multi-source problem in QPAT (19) then can be reformulated as the problem of finding the tuple z := (µ, (…”

“…Therefore, the computational effort for the proximal stochastic gradient method is approximately 3/4 of the proximal gradient algorithm using full gradients. For the algorithms based on the standard formulation (19), calculating adaptive step sizes similar to (51) is time-consuming as this requires another evaluation of the forward operator F i and therefore another solution of the RTE. Therefore, we simply use a constant step size rule; in our numerical experiments, it turned out that s k = 0.5 is a suitable choice.…”

“…Flowcharts of stochastic gradient algorithms for QPAT proposed in this paper. Left: algorithm based on the standard formulation (19). Right: algorithm based on the novel MULL formulation (35).…”

“…The initial pressure distribution only provides qualitative information about the tissue-relevant parameters, as it is the product of the optical absorption coefficient and the spatially varying optical intensity, which again indirectly depends on the tissue parameters. Quantitative photoacoustic tomography (QPAT) addresses this issue and aims at quantitatively estimating the tissue parameters by supplementing the inversion of the acoustic wave equation with an inverse problem for light propagation (see, for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). Basic principles of PAT.…”