Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media

Abstract

The paper starts with a comparative discussion of features and limitations of the three types of recent approaches to reconstruction in thermoacoustic/photoacoustic tomography: backprojection formulae, eigenfunction expansions and time reversal. The latter method happens to be the least restrictive. It is then considered in more detail, e.g. its relation to trapping properties of the medium. The time reversal method is exact only in the case of a constant sound speed in odd dimension, due to validity of the Huygens' principle. The next best case is of non-trapping speed in odd dimensions. The authors provide 2D examples and discuss the features of numerical reconstructions for constant and variable (both non-trapping and trapping) speeds, showing that this technique works surprisingly well even under the most unfavorable circumstances (variable, and even trapping sound speed in 2D). In particular, a 'limited view' effect due to trapping is observed and explained. Finally, an initial consideration of the problem of sound speed recovery is also provided.

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…(e.g., [68,140,141]) If the speed is constant, a point x 0 is in the visible region, if and only if any line passing through x 0 intersects at least once the acquisition surface S (and thus a detector location). The reader can now see an important difference between the acoustically homogeneous and inhomogeneous media.…”

“…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.…”

“…However, trapping speed does cause instability [68]. Indeed, since some of the rays are trapped inside Ω, the information about some singularities never reaches S (no matter for how long one collects the data), and thus, as it is shown in [91], the reconstruction is not even Hölder stable, and the singular values have super-algebraic decay.…”

“…in breast imaging, the medium is assumed to be acoustically homogeneous, this is not a good assumption in many other cases. It has been observed (e.g., [68,71]) that replacing even slightly varying speed of sound with its average value might significantly distort the image; not only the numerical values, but also the shapes of interfaces between the tissues will be reconstructed incorrectly. Thus, the question of estimating c(x) correctly becomes important.…”

“…Direct usage of range conditions attempted in [68] has lead only to extremely weak and not practically useful results so far. A revealing relation to the transmission eigenvalue problem well known in inverse problems (see [29] for the survey) was recently discovered by D. Finch.…”