Nonlinear ultrasound imaging modeled by a Westervelt equation

TL;DR

This paper addresses nonlinear ultrasound imaging modeled by the Westervelt equation, demonstrating unique recovery of nonlinearity coefficients and proposing an inversion algorithm with numerical results in the frequency domain.

Contribution

It introduces a novel method for uniquely recovering the nonlinearity coefficient from boundary measurements using second order linearization and Gaussian beams.

Findings

Successful numerical implementation of the inversion algorithm.

Proof of unique recovery of the nonlinearity coefficient.

Reduction of the inverse problem to geodesic ray transform inversion.

Abstract

We consider the ultrasound imaging problem governed by a nonlinear wave equation of Westervelt type with variable wave speed. We show that the coefficient of nonlinearity can be recovered uniquely from knowledge of the Dirichlet-to-Neumann map. Our proof is based on a second order linearization and the use of Gaussian beam solutions to reduce the problem to the inversion of a weighted geodesic ray transform. We propose an inversion algorithm and report the results of a numerical implementation to solve the nonlinear ultrasound imaging problem in a transmission setting in the frequency domain.