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A high contrast and resolution reconstruction algorithm in quantitative photoacoustic tomography

Anwesa Dey, Alfio Borzi, Souvik Roy

TL;DR

This work addresses recovering the diffusion coefficient $D$ and absorption coefficient $\sigma_a$ in quantitative photoacoustic tomography (QPAT) from measured initial pressure data by formulating a regularized PDE-constrained optimization that includes a sparsity-promoting term on $\sigma$ and a Kubelka–Munk prior linking $D$ and $\sigma_a$ via $D\approx\bar{D}=\frac{1}{3c(\sigma+\sigma_b)}$. The inverse problem is solved with a fast sequential quadratic Hamiltonian (SQH) algorithm based on the Pontryagin maximum principle, which yields a robust, grid-search-based, pointwise update of the coefficients and adaptive penalty updates to ensure monotonic reduction of the objective. Theoretical and numerical results demonstrate high-contrast, high-resolution reconstructions across diverse phantoms (disk, heart–lung, Shepp–Logan) and noise levels, validating the method’s stability and effectiveness. The contribution provides a practical framework for biomedical QPAT with potential impact on accurate tissue characterization and cancer detection under near-infrared illumination, leveraging nonsmooth PDE optimization and PMP-derived optimality conditions.

Abstract

A framework for reconstruction of optical diffusion and absorption coefficients in quantitative photoacoustic tomography is presented. This framework is based on a Tikhonov-type functional with a regularization term promoting sparsity of the absorption coefficient and a prior involving a Kubelka-Munk absorption-diffusion relation that allows to obtain superior reconstructions. The reconstruction problem is formulated as the minimization of this functional subject to the differential constraint given by a photon-propagation model. The solution of this problem is obtained by a fast and robust sequential quadratic hamiltonian algorithm based on the Pontryagin maximum principle. Results of several numerical experiments demonstrate that the proposed computational strategy is able to obtain reconstructions of the optical coefficients with high contrast and resolution for a wide variety of objects.

A high contrast and resolution reconstruction algorithm in quantitative photoacoustic tomography

TL;DR

This work addresses recovering the diffusion coefficient and absorption coefficient in quantitative photoacoustic tomography (QPAT) from measured initial pressure data by formulating a regularized PDE-constrained optimization that includes a sparsity-promoting term on and a Kubelka–Munk prior linking and via . The inverse problem is solved with a fast sequential quadratic Hamiltonian (SQH) algorithm based on the Pontryagin maximum principle, which yields a robust, grid-search-based, pointwise update of the coefficients and adaptive penalty updates to ensure monotonic reduction of the objective. Theoretical and numerical results demonstrate high-contrast, high-resolution reconstructions across diverse phantoms (disk, heart–lung, Shepp–Logan) and noise levels, validating the method’s stability and effectiveness. The contribution provides a practical framework for biomedical QPAT with potential impact on accurate tissue characterization and cancer detection under near-infrared illumination, leveraging nonsmooth PDE optimization and PMP-derived optimality conditions.

Abstract

A framework for reconstruction of optical diffusion and absorption coefficients in quantitative photoacoustic tomography is presented. This framework is based on a Tikhonov-type functional with a regularization term promoting sparsity of the absorption coefficient and a prior involving a Kubelka-Munk absorption-diffusion relation that allows to obtain superior reconstructions. The reconstruction problem is formulated as the minimization of this functional subject to the differential constraint given by a photon-propagation model. The solution of this problem is obtained by a fast and robust sequential quadratic hamiltonian algorithm based on the Pontryagin maximum principle. Results of several numerical experiments demonstrate that the proposed computational strategy is able to obtain reconstructions of the optical coefficients with high contrast and resolution for a wide variety of objects.
Paper Structure (7 sections, 4 theorems, 48 equations, 7 figures, 6 tables, 1 algorithm)

This paper contains 7 sections, 4 theorems, 48 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formulation of a QPAT optimization problem
  3. Analysis of the QPAT inverse problem
  4. The SQH algorithm for solving the QPAT inverse problem
  5. Numerical results
  6. Conclusion
  7. Proof of Theorem \ref{['sqh2']}

Key Result

Proposition 1

Let $D\in D_{ad}$, $\sigma\in \Sigma_{ad}$. Then there exists a unique solution $u \in H^1(\Omega)$ to eq:DA.

Figures (7)

  • Figure 1: Test Case 1: Exact $D$ and $\sigma$ represented by disk phantoms.
  • Figure 2: Test Case 1: The reconstructed disk phantoms with different choices of the values of the regularization weights.
  • Figure 3: Test Case 1: The reconstructed disk phantoms using the weights $\alpha=1,\xi_1=0.01,\xi_2=20,\gamma=0.01$.
  • Figure 4: Test Case 2: The actual and the reconstructed phantoms
  • Figure 5: Test Case 3: The actual and the reconstructed phantoms
  • ...and 2 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • Theorem 4.1
  • proof