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.

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.

The article addresses the possibility of robust detection of geometrically small, low emission sources on a significantly stronger background. This problem is important for homeland security. A technique of detecting such sources using Compton type cameras is developed, which is shown on numerical examples to have high sensitivity and specificity and also allows to assign confidence probabilities of the detection. 2D case is considered in detail

For a graph G and a, b ∈ V (G), the shortest path reconfiguration graph of G with respect to a and b is denoted by S(G, a, b). The vertex set of S(G, a, b) is the set of all shortest paths between a and b in G. Two vertices in V (S(G, a, b)) are adjacent, if their corresponding paths in G differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We also show that the shortest path graph of a grid graph is an induced subgraph of a lattice.