Time reversal in thermoacoustic tomography—an error estimate

Abstract

In thermoacoustic tomography an object is irradiated by a short electromagnetic pulse and the absorbed energy causes a thermoelastic expansion. This expansion leads to a pressure wave propagating through the object. The goal of thermoacoustic tomography is the recovery of the initial pressure inside the object from measurements of the pressure wave made on a surface surrounding the object. The time reversal method can be used for approximating the initial pressure when the sound speed inside the object is variable (non-trapping as well as trapping). This article presents error estimates for the time reversal method in the cases of variable, non-trapping sound speeds. Numerical examples for non-trapping as well as for trapping sound speeds are provided.

Artifact Trapping During Time Reversal Photoacoustic Imaging for Acoustically Heterogeneous Media

“…Recently, time reversal photoacoustic imaging has been described as the "least restrictive" imaging algorithm on the basis that it relies on fewer assumptions than many other image reconstruction algorithms [6], [13]. For example, it is applicable to closed measurement surfaces of any shape, is immune to acoustic sources outside the measurement surface, and makes no assumptions about the initial time rate of change of the pressure (or equivalently the initial acoustic particle velocity).…”

“…For example, it is applicable to closed measurement surfaces of any shape, is immune to acoustic sources outside the measurement surface, and makes no assumptions about the initial time rate of change of the pressure (or equivalently the initial acoustic particle velocity). It has also been shown to work reasonably well even when certain assumptions fail, e.g., with a heterogeneous sound speed distribution, and in two dimensions [6], [13]. In addition to these advantages, there is no axiomatic constraint on the efficiency of the algorithm; it is as fast as the numerical propagation code on which it is based.…”

Reconstruction of Microwave Absorption Properties in Heterogeneous Tissue for Microwave-Induced Thermo-Acoustic Tomography

“…Hristova et al [6][7][8] applied time reversal mirror (TRM) to reconstruct the image under the assumption that the sound velocity (SV) distribution of heterogeneous media is known. However in engineering the SV distribution of the heterogeneous medium is usually unknown.…”

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…Namely, since the solution p of (9) has compactly supported initial data, its energy is decaying inside any bounded domain, in particular inside Ω (see Section 3.1.2 and [34,67] and references therein about local energy decay). On the other hand, if there is non-uniqueness, there exists a non-zero f such that g(y, t) = 0 for all y ∈ S and t. This means that we can add homogeneous Dirichlet boundary conditions p | S = 0 to (9).…”

“…Under the non-trapping condition, as it is shown in (11) (see [34,129,130]), the time decay is exponential in odd dimensions, but only algebraic in even-dimensions. Although, in order to obtain theoretically exact reconstruction, one would have to start the time reversal at T = ∞, numerical experiments (e.g., [68]) and theoretical estimates [67] show that in practice it is sufficient to start at the values of T when the signal becomes small enough, and to approximate the unknown value of p(x, T ) by zero (a more sophisticated cut-off is used in [123], which leads to an equation with a contraction operator). This works [52,68] even in 2D (where decay is the slowest) and in inhomogeneous media.…”