Recovering a function from its spherical mean values in two and three dimensions

The problem of determining a function from a subset of its spherical means has a rich history in pure and applied mathematics. Its connection with photoacoustic imaging has already been made abundantly clear in earlier chapters in this volume: when the sound speed νs in a medium is constant the pressure at time t and point r is expressed in terms of spherical means of the pressure, and its time derivative, at any earlier time t′.We begin this chapter with a review of the connections between spherical means and the several equations which arise in the time or frequency domains in the analysis of photoacoustic imaging and show how the inverse problem of photoacoustic imaging can be formulated as the problem of recovering a function from some collection of its spherical means. In §2 we discuss the uniqueness problem of when a function is determined by a collection of its spherical means, and when the initial value of a wave equation is determined by the value of the solution on a subset of the spatial domain over a given time interval. In §3 we specialize to the question of actually recovering (or approximately recovering) a function supported in a region D in space (or in the plane) from its spherical means with centers on the boundary of D for fairly general regions D. In connection with this topic, it is convenient to also discuss some related work on the characterization of the range of the spherical mean transform, and of related wave equations. (Several other chapters in this volume have some discussion of the range: see the chapter by S. Patch and the chapter by M. Agranovsky, P. Kuchment, and L. Kunyansky.) In §4, we specialize to the case when the family of spherical means are centered on sets with simple geometry, and present some of the filtered back-projection formulas that have been found. This section is largely independent of the preceding two, and so the reader interested in explicit formulas can jump directly from the introduction. We shall confine the discussion of this chapter to two and three dimensions, since these are relevant to photoacoustic tomography. Many of the results do have higher dimensional generalizations, and the interested reader may pursue these in the original articles.