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On the optimal choice of the illumination function in photoacoustic tomography

Phuoc-Truong Huynh, Barbara Kaltenbacher

TL;DR

This work tackles photoacoustic tomography in lossy media via a time-fractional damping model, formulating the inverse problem as identifying a spatial source $a(x)$ from surface measurements. The authors develop a Bayesian MAP framework, derive an explicit adjoint for the forward operator $\mathcal W_i$, and study how the laser illumination function $i(t)$ can be optimally chosen using $A$-optimality under Gaussian priors and noise. A projection-based finite-dimensional scheme is introduced to efficiently approximate the posterior trace and to ensure convergence of the OED problem, with theoretical results on existence and stability. Numerical experiments illustrate the impact of the damping parameter $\alpha$, the choice of priors, and the benefit of optimized illumination in improving reconstruction quality, while also providing a mapping from simulation to physical units for practical interpretation.

Abstract

This work studies the inverse problem of photoacoustic tomography (more precisely, the acoustic subproblem) as the identification of a space-dependent source parameter. The model consists of a wave equation involving a time-fractional damping term to account for power law frequency dependence of the attenuation, as relevant in ultrasonics. We solve the inverse problem in a Bayesian framework using a Maximum A Posteriori (MAP) estimate, and for this purpose derive an explicit expression for the adjoint operator. On top of this, we consider optimization of the choice of the laser excitation function, which is the time-dependent part of the source in this model, to enhance the reconstruction result. The method employs the $A$-optimality criterion for Bayesian optimal experimental design with Gaussian prior and Gaussian noise. To efficiently approximate the cost functional, we introduce an approximation scheme based on projection onto finite-dimensional subspaces. Finally, we present numerical results that illustrate the theory.

On the optimal choice of the illumination function in photoacoustic tomography

TL;DR

This work tackles photoacoustic tomography in lossy media via a time-fractional damping model, formulating the inverse problem as identifying a spatial source from surface measurements. The authors develop a Bayesian MAP framework, derive an explicit adjoint for the forward operator , and study how the laser illumination function can be optimally chosen using -optimality under Gaussian priors and noise. A projection-based finite-dimensional scheme is introduced to efficiently approximate the posterior trace and to ensure convergence of the OED problem, with theoretical results on existence and stability. Numerical experiments illustrate the impact of the damping parameter , the choice of priors, and the benefit of optimized illumination in improving reconstruction quality, while also providing a mapping from simulation to physical units for practical interpretation.

Abstract

This work studies the inverse problem of photoacoustic tomography (more precisely, the acoustic subproblem) as the identification of a space-dependent source parameter. The model consists of a wave equation involving a time-fractional damping term to account for power law frequency dependence of the attenuation, as relevant in ultrasonics. We solve the inverse problem in a Bayesian framework using a Maximum A Posteriori (MAP) estimate, and for this purpose derive an explicit expression for the adjoint operator. On top of this, we consider optimization of the choice of the laser excitation function, which is the time-dependent part of the source in this model, to enhance the reconstruction result. The method employs the -optimality criterion for Bayesian optimal experimental design with Gaussian prior and Gaussian noise. To efficiently approximate the cost functional, we introduce an approximation scheme based on projection onto finite-dimensional subspaces. Finally, we present numerical results that illustrate the theory.

Paper Structure

This paper contains 22 sections, 15 theorems, 114 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Fractional derivatives and fractional Sobolev spaces
  4. Trace-class and Hilbert-Schmidt operators
  5. Setting of the photoacoustic tomography problem
  6. Forward and adjoint problems
  7. Bayesian inverse problem
  8. Optimal design of illumination function
  9. Optimality criterion
  10. Projection scheme
  11. Finite-dimensional discretization of OED
  12. Discretization for the Bayesian inverse problem
  13. Evaluation of the discretized optimal design functional and its gradient
  14. Numerical results
  15. Setting of the problem
  16. ...and 7 more sections

Key Result

Proposition 3.2

For $\alpha\in(0,1]$, $s\in[0,1]$ and every $f \in H^{s(1-\alpha)/2}(0,T;H^{-s}(\Omega))$, there exists a unique weak solution of eq:wave_equation, that is, of eq:wave_equation_weak. Furthermore, the solution map is linear and bounded, where

Figures (10)

  • Figure 1: Snapshots of the state variable for $\alpha = 0.3$
  • Figure 2: Snapshots of the state variable for $\alpha = 0.8$
  • Figure 3: Ground truth (left) and reconstruction results for $\alpha = 0.3$ (middle) and $\alpha = 0.8$ (right) with $I_0 = 0.5 \cdot 10^2$, obtained using the bi-Laplacian prior
  • Figure 4: Reconstruction results for $\alpha = 0.3$ (middle) and $\alpha = 0.8$ (right) with $I_0 = 2\cdot 10^2$, obtained using the bi-Laplacian prior
  • Figure 5: Reconstruction results with bi-Laplacian prior (middle) and Ornstein-Uhlenbeck prior (right) with $I_0 = 0.5 \cdot 10^2$ and $\alpha = 0.3$
  • ...and 5 more figures

Theorems & Definitions (28)

  • Definition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • Remark 3.5: Injectivity of $\mathcal{W}_{i}$
  • Lemma 3.6
  • Corollary 3.7
  • Proposition 3.8
  • proof
  • ...and 18 more