Photoacoustic image reconstruction from full field data in heterogeneous media

TL;DR

This work analyzes full-field photoacoustic tomography (FFD-PAT) with variable sound speed by formulating the forward model and exploring both two-step and novel one-step reconstruction strategies. It shows that, for constant speed, full data allow stable, unique recovery via invertible Radon/ wave operators, while variable speed necessitates a sequence of transforms; the authors then propose preconditioned one-step Landweber and variational methods to directly recover the PA source from full-field projections, addressing smoothing inherent in the X-ray transform. Numerical results in 2D demonstrate that the one-step approach yields competitive, sometimes superior, reconstructions and reduces artifacts compared to the two-step method, especially when speed variations are significant. The work highlights both practical improvements and open theoretical questions for limited data and 3D extensions, suggesting directions for future research in robust, scalable FFD-PAT inversion.

Abstract

We consider image reconstruction in full-field photoacoustic tomography, where 2D projections of the full 3D acoustic pressure distribution at a given time T>0 are collected. We discuss existing results on the stability and uniqueness of the resulting image reconstruction problem and review existing reconstruction algorithms. Open challenges are also mentioned. Additionally, we introduce novel one-step reconstruction methods allowing for a variable speed of sound. We apply preconditioned iterative and variational regularization methods to the one-step formulation. Numerical results using the one-step formulation are presented, together with a comparison with the previous two-step approach for full-field photoacoustic tomography

Figures (3)

• Figure 1: Phantom and sound speed. Left: PA source to be reconstructed. Right: Used trapping sound speed profile taken from qian2011new.
• Figure 2: Data simulation. Top left: Pressure $\mathop{\mathrm{\mathbf{W}}}\nolimits_T h$ at time $T= 2$. Top right: Noisy X-ray transform $\mathop{\mathrm{\mathbf{X}}}\nolimits \mathop{\mathrm{\mathbf{W}}}\nolimits_T h + \mathrm{noise}$. Bottom right: Noisy data $G = \chi_ D \mathop{\mathrm{\mathbf{X}}}\nolimits \mathop{\mathrm{\mathbf{W}}}\nolimits_T h + \mathrm{noise}$ used as input for the one-step methods. Bottom left: Recovered final time pressure $\mathop{\mathrm{\mathbf{X}}}\nolimits^{-1} G \simeq \mathop{\mathrm{\mathbf{W}}}\nolimits_T h$ used as input for the second step of the two-step method. The white circles indicate the imaging domain $D_1$ and the stripe inside the white lines is the missing region $[-1,1]$.
• Figure 3: Reconstruction results. Top left: Relative $L^2$-reconstruction error with the one-step method. Top right: Reconstruction with one-step methods after 60 iterations. Center left: Relative $L^2$-reconstruction error with the two-step method. Center right: Reconstruction with two-step methods after 60 iterations. Bottom left: Relative $L^2$-reconstruction error assuming constant sound speed for the reconstruction algorithm. Bottom right: Reconstruction assuming constant sound speed after 60 iterations.