No abstract

The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography.

We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo-and photo-acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in R n surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.

An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo-and photoacoustic tomography. Closed-form inversion formulae are currently known only for the case when the centers of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly knownsuch as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centers of the integration spheres) lie on a surface of a cube. This algorithm reconsrtucts 3-D images thousands times faster than backprojection-type methods.

We present a new reconstruction algorithm for single-photon emission computed tomography. The algorithm is based on the Novikov explicit inversion formula for the attenuated Radon transform with non-uniform attenuation. Our reconstruction technique can be viewed as a generalization of both the filtered backprojection algorithm and the Tretiak-Metz algorithm. We test the performance of the present algorithm in a variety of numerical experiments. Our numerical examples show that the algorithm is capable of accurate image reconstruction even in the case of strongly non-uniform attenuation coefficient, similar to that occurring in a human thorax.