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Full Field Inversion of the Attenuated Wave Equation: Theory and Numerical Inversion

Ngoc Do, Markus Haltmeier, Richard Kowar, Linh V. Nguyen, Robert Nuster

TL;DR

This work addresses the inverse problem of full-field photoacoustic tomography with damped, spatially varying sound speed by showing uniqueness and stability for recovering the initial data from single-time measurements $u(\cdot,T)|_{\Omega^c}$ and by developing an exact time-reversal approach. It introduces a one-step reconstruction framework that bypasses separate inversion of the X-ray transform, employing iterative and variational regularization to recover the source directly from data $\mathbf{X}\mathbf{W}f$, and demonstrates this with numerical tests in full-angle and limited-angle scenarios. Theoretical results hinge on a forward operator $\mathbf{W}_T$, an explicit adjoint via time reversal, and a visibility-driven stability estimate, while the numerics implement a 2D discretization with k-space solvers, Radon transforms, and TV or quadratic regularization. Overall, the paper bridges theory and computation for full-field PAT, showing reliable reconstructions under noise and partial data and highlighting TV-regularization and time-reversal-based methods as effective in practice.

Abstract

Standard photoacoustic tomography (PAT) provides data that consist of time-dependent signals governed by the wave equation, which are measured on an observation surface. In contrast, the measured data from the recently invented full-field PAT is the Radon transform of the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known about the full-field PAT problem. In this paper, we study full-field photoacoustic tomography with spatially variable sound speed and spatially variable damping. In particular, we prove the uniqueness and stability of the associated single-time full-field wave inversion problem and develop algorithms for its numerical inversion using iterative and variational regularization methods. Numerical simulations are presented for both full-angle and limited-angle data cases

Full Field Inversion of the Attenuated Wave Equation: Theory and Numerical Inversion

TL;DR

This work addresses the inverse problem of full-field photoacoustic tomography with damped, spatially varying sound speed by showing uniqueness and stability for recovering the initial data from single-time measurements and by developing an exact time-reversal approach. It introduces a one-step reconstruction framework that bypasses separate inversion of the X-ray transform, employing iterative and variational regularization to recover the source directly from data , and demonstrates this with numerical tests in full-angle and limited-angle scenarios. Theoretical results hinge on a forward operator , an explicit adjoint via time reversal, and a visibility-driven stability estimate, while the numerics implement a 2D discretization with k-space solvers, Radon transforms, and TV or quadratic regularization. Overall, the paper bridges theory and computation for full-field PAT, showing reliable reconstructions under noise and partial data and highlighting TV-regularization and time-reversal-based methods as effective in practice.

Abstract

Standard photoacoustic tomography (PAT) provides data that consist of time-dependent signals governed by the wave equation, which are measured on an observation surface. In contrast, the measured data from the recently invented full-field PAT is the Radon transform of the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known about the full-field PAT problem. In this paper, we study full-field photoacoustic tomography with spatially variable sound speed and spatially variable damping. In particular, we prove the uniqueness and stability of the associated single-time full-field wave inversion problem and develop algorithms for its numerical inversion using iterative and variational regularization methods. Numerical simulations are presented for both full-angle and limited-angle data cases
Paper Structure (22 sections, 7 theorems, 38 equations, 3 figures, 2 algorithms)

This paper contains 22 sections, 7 theorems, 38 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Full field PAT
  3. Main results
  4. Theory
  5. Uniqueness
  6. Stability
  7. Iterative time-reversal
  8. Numerical inversion
  9. Problem description
  10. Regularized inversion
  11. Iterative regularization:
  12. Variational regularization:
  13. Adjoint operator
  14. Numerical details
  15. Wave operator:
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $u$ be the solution of eq:wave1-eq:wave3 with initial data $\mathbf{f} \in \mathbb{X}(\Omega)$.

Figures (3)

  • Figure 1: Simulation setup: Phantom $f$ (top left), sound speed $c$ (top middle), attenuation $a$ (top right), final time pressure $\mathop{\mathrm{\mathbf{W}}}\nolimits f = p(\cdot, T)$ (bottom left), full angle data $\mathop{\mathrm{\mathbf{X}}}\nolimits \mathop{\mathrm{\mathbf{W}}}\nolimits f$ (bottom middle), limited angle data $\mathop{\mathrm{\mathbf{M}}}\nolimits_I \mathop{\mathrm{\mathbf{X}}}\nolimits \mathop{\mathrm{\mathbf{W}}}\nolimits f$ (bottom right).
  • Figure 2: Results for full angular data where the top row shows the optimal reconstructions and the bottom row the relative reconstruction errors depending on the iteration index. Left: CGNE iterative regularization. Middle: FBS for quadratic regularization. Right: CP for TV-regularization.
  • Figure 3: Results for limited angular data where the top row shows the optimal reconstructions and the bottom row the relative reconstruction errors depending on the iteration index. Left: CGNE iterative regularization. Middle: FBS for quadratic regularization. Right: CP for TV-regularization

Theorems & Definitions (16)

  • Lemma 2.1: Auxiliary uniqueness results
  • proof
  • Theorem 2.2: Uniqueness of full field wave inversion
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6: Stability of full field wave inversion
  • proof
  • ...and 6 more