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Recovery guarantees for compressed sensing photoacoustic tomography

Alessandro Felisi

TL;DR

This work addresses stable recovery in photoacoustic tomography (PAT) when data are available only on a partial boundary. By imposing a wavelet-sparse prior on the initial pressure $u_0$ and leveraging a general compressed sensing-inverse problems framework, it proves Lipschitz-stable recovery from $m$ random local-average measurements on small detectors, provided $m$ scales as $m\gtrsim j_0 s$ up to log factors and the detectors satisfy a diameter bound $\mathrm{diam}_{\mathbb{R}^3}(E_i)\le\mu 2^{-j_0}$ with truncation error controlled. The reconstruction is obtained by solving an $\ell^1$-minimization over a truncated wavelet subspace $\mathcal{M}_{\le j_0}$, and the result accommodates noise of level $\beta$ and a sparsity defect. The analysis combines forward-map stability (via a quasi-diagonalization bound), a balancing property, and coherence bounds of tensorized wavelets, together with a trace identity on spheres and Huygens’ principle; it further extends to general sensing patterns. Practically, the findings provide the first rigorous theoretical guarantees for PSAT-like recovery in PAT under partial data and randomized detectors, enabling efficient and provable reconstruction from compressed boundary measurements.

Abstract

Photoacoustic tomography is an emerging medical imaging technology whose primary aim is to map the high-contrast optical properties of biological tissues by leveraging high-resolution ultrasound measurements. Mathematically, this can be framed as an inverse source problem for the wave equation over a specific domain. In this work, for the first time, it is shown how, by assuming signal sparsity, it is possible to establish rigorous stable recovery guarantees when the data collection is given by spatial averages restricted to a limited portion of the boundary. Our framework encompasses many approaches that have been considered in the literature. The result is a consequence of a general framework for subsampled inverse problems developed in previous works and refined stability estimates for an inverse problem for the wave equation with surface measurements.

Recovery guarantees for compressed sensing photoacoustic tomography

TL;DR

This work addresses stable recovery in photoacoustic tomography (PAT) when data are available only on a partial boundary. By imposing a wavelet-sparse prior on the initial pressure and leveraging a general compressed sensing-inverse problems framework, it proves Lipschitz-stable recovery from random local-average measurements on small detectors, provided scales as up to log factors and the detectors satisfy a diameter bound with truncation error controlled. The reconstruction is obtained by solving an -minimization over a truncated wavelet subspace , and the result accommodates noise of level and a sparsity defect. The analysis combines forward-map stability (via a quasi-diagonalization bound), a balancing property, and coherence bounds of tensorized wavelets, together with a trace identity on spheres and Huygens’ principle; it further extends to general sensing patterns. Practically, the findings provide the first rigorous theoretical guarantees for PSAT-like recovery in PAT under partial data and randomized detectors, enabling efficient and provable reconstruction from compressed boundary measurements.

Abstract

Photoacoustic tomography is an emerging medical imaging technology whose primary aim is to map the high-contrast optical properties of biological tissues by leveraging high-resolution ultrasound measurements. Mathematically, this can be framed as an inverse source problem for the wave equation over a specific domain. In this work, for the first time, it is shown how, by assuming signal sparsity, it is possible to establish rigorous stable recovery guarantees when the data collection is given by spatial averages restricted to a limited portion of the boundary. Our framework encompasses many approaches that have been considered in the literature. The result is a consequence of a general framework for subsampled inverse problems developed in previous works and refined stability estimates for an inverse problem for the wave equation with surface measurements.

Paper Structure

This paper contains 18 sections, 10 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical model and stability issues
  3. Main contribution
  4. Comparison with the literature
  5. Main elements in the analysis
  6. Structure of the paper
  7. Setting and main result
  8. Wavelet dictionary and sparsity
  9. PDE model for PAT
  10. Detectors and measurements
  11. Main result
  12. General sensing patterns
  13. Proofs
  14. Proof of Proposition \ref{['lem:L2_stability']}
  15. Proof of Proposition \ref{['prop:ass_detectors']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $U$ be the linear operator defined by mapping the initial datum $u_0$ to the restriction to $\Sigma \times [0, T]$ of the corresponding solution of the wave equation with constant velocity $c$. Suppose that $(K,\Sigma)$ does not satisfy the visibility condition. Then there do not exist constants $\mu, \delta, C > 0$ and $s_0, s_1 \geq 0

Figures (2)

  • Figure 1: Example of failure of the visibility condition. (a) The set $K$ and the acquisition surface $\Sigma$. (b) A nonzero element $\xi$ and the associated line bundle that does not intersect a neighbourhood $\Omega_{\Sigma}$ of $\Sigma$. (c) An open set $\Omega_K$ contained in the intersection of $K$ and the line bundle.
  • Figure 2: The sets in the proof of Proposition \ref{['prop:fond_prop2']}. In red, the set $\partial B_t(x)\cap C_{j,n}$; in yellow, the set $\partial B_t(x)\cap Q_{j,n}$.

Theorems & Definitions (18)

  • Definition : nguyen2011
  • Theorem : nguyen2011
  • Theorem
  • Remark 1.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • ...and 8 more