Analysis of Iterative Methods in Photoacoustic Tomography with Variable Sound Speed

TL;DR

This paper improves iterative reconstruction methods for photoacoustic tomography by proposing modifications to the adjoint operator, leading to faster convergence and better error estimates, supported by theoretical analysis and numerical experiments.

Contribution

It introduces new modifications to the adjoint operator in iterative methods, enhancing convergence speed and stability in photoacoustic image reconstruction.

Findings

Nesterov's method and CG converge faster than Landweber and time reversal.

The proposed methods achieve linear convergence in $L^2$ and $H^1$ norms.

Numerical results confirm improved performance in full and limited view data.

Abstract

In this article, we revisit iterative methods for solving the inverse problem of photoacoustic tomography in free space. Recently, there have been interesting developments on explicit formulations of the adjoint operator, demonstrating that iterative methods are an attractive choice for photoacoustic image reconstruction. In this work, we propose several modifications of current formulations of the adjoint operator which help speed up the convergence and yield improved error estimates. We establish a stability analysis and show that, with our choices of the adjoint operator, the iterative methods can achieve a linear rate of convergence, in the $L^2$-norm as well as in the $H^1$-norm. In addition, we analyze the normal operator from the microlocal analysis point of view. This gives insight into the convergence speed of the iterative methods and choosing proper weights for the mapping spaces. Finally, we present numerical results using various iterative reconstruction methods for full as well as limited view data. Our results demonstrate that Nesterov's fast gradient and the CG methods converge faster than Landweber's and iterative time reversal methods in the visible as well as the invisible case.