[Paper Review] Iterative Methods for Photoacoustic Tomography with Variable Sound Speed

Markus Haltmeier, Linh V. Nguyen|arXiv (Cornell University)|Nov 22, 2016

Photoacoustic and Ultrasonic Imaging4 citations

TL;DR

This paper proposes modified adjoint operators for iterative photoacoustic tomography reconstruction with variable sound speed, accelerating convergence and improving error estimates. It establishes linear convergence rates in $L^2$ and $H^1$ norms under visibility conditions and demonstrates superior performance of Nesterov's and conjugate gradient methods over Landweber and iterative time reversal in both visible and invisible cases.

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 could be an attraktive 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, Landweber's and the CG methods can achieve a linear rate of convergence either in $L^2$ or $H^1$ norm under the visibility condition. In addition, we analyze the normal operator from the microlocal analysis point of view. This helps us to have more insight into the convergence speed of the iterative methods as well as choosing proper weights for the mapping spaces. Finally, we present numerical results using various iterative reconstruction methods for trapping as well as non-trapping sound speed. 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.

Motivation & Objective

• To improve convergence speed and error estimates in iterative photoacoustic tomography reconstruction with variable sound speed.
• To develop modified formulations of the adjoint operator that enhance performance of iterative solvers.
• To establish theoretical convergence rates for Landweber and conjugate gradient methods in $L^2$ and $H^1$ norms under visibility conditions.
• To analyze the normal operator using microlocal analysis for deeper insight into convergence behavior and optimal weighting in function spaces.
• To evaluate and compare numerical performance of iterative methods across trapping and non-trapping sound speed configurations.

Proposed method

• Derives modified adjoint operators based on explicit formulations to accelerate convergence in iterative reconstruction.
• Applies stability analysis to prove linear convergence rates in $L^2$ and $H^1$ norms under the visibility condition.
• Performs microlocal analysis of the normal operator to understand singularities and guide choice of function space weights.
• Employs Landweber, conjugate gradient (CG), Nesterov's fast gradient, and iterative time reversal methods for numerical evaluation.
• Uses variable sound speed models to test performance in both trapping and non-trapping configurations.
• Introduces problem-specific weighting in mapping spaces informed by microlocal insights to improve reconstruction quality.

Experimental results

Research questions

• RQ1 How can the adjoint operator in photoacoustic tomography be modified to improve convergence speed and error estimates with variable sound speed?
• RQ2 What is the theoretical convergence rate of Landweber and CG methods under the visibility condition in $L^2$ and $H^1$ norms?
• RQ3 How does microlocal analysis of the normal operator inform the choice of function space weights for iterative solvers?
• RQ4 How do iterative methods compare in performance when reconstructing images with trapping versus non-trapping sound speed profiles?
• RQ5 What is the relative convergence speed of Nesterov's fast gradient and CG methods compared to Landweber and iterative time reversal in visible and invisible cases?

Key findings

• The proposed modified adjoint operators enable Landweber's and conjugate gradient methods to achieve a linear rate of convergence in both $L^2$ and $H^1$ norms under the visibility condition.
• Nesterov's fast gradient and conjugate gradient methods converge faster than Landweber's and iterative time reversal methods in both visible and invisible cases.
• Microlocal analysis of the normal operator provides insight into the structure of singularities, guiding optimal weighting in function spaces.
• The stability analysis confirms robust convergence behavior for the modified iterative schemes under the visibility condition.
• Numerical results demonstrate improved reconstruction quality and faster convergence with the proposed adjoint formulations across diverse sound speed models.