Nonlinearity parameter imaging in the frequency domain

TL;DR

This paper develops a frequency domain approach for imaging nonlinearity parameters in ultrasound wave equations, introducing a reconstruction algorithm and proving convergence of a regularized Newton method.

Contribution

It transforms the nonlinear wave equation problem into a coupled Helmholtz system and proposes a novel reconstruction algorithm with convergence analysis.

Findings

Successful implementation of the reconstruction algorithm.

Proof of convergence for the regularized Newton method.

Validation through numerical tests.

Abstract

Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where the Westervelt equation gets replaced by a coupled system of Helmholtz equations with quadratic nonlinearities. For the case of the to-be-determined nonlinearity coefficient being a characteristic function of an unknown, not necessarily connected domain $D$, we devise and test a reconstruction algorithm based on weighted point source approximations combined with Newton's method. In a more abstract setting, convergence of a regularised Newton type method for this inverse problem is proven by verifying a range invariance condition of the forward operator and establishing injectivity of its linearisation.