Thermoacoustic tomography with variable sound speed

Abstract

We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T ] so that all signals issued from the domain leave it after time T . In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary.

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…In this case, stability follows by solving the wave equation in reverse time starting from t = ∞, as it is done in [3]. In fact, Lipschitz stability in this case holds for any observation time exceeding T (Ω) (see [123], where microlocal analysis is used to prove this result).…”

“…We will try to give the reader a feeling of the general state of affairs with stability, referring to the literature (e.g., [5,68,78,103,123]) for further exact details.…”

“…In the case of a variable speed of sound, there still are uniqueness theorems for partial data [123,124], e.g.…”

“…Parametrix reconstructions can be either accepted as approximate images, or used as starting points for iterative algorithms. See [123] for a recent discussion of parametrices.…”

“…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.…”