ABSTRACT
The problem of determining a function from a subset of its spherical means has a rich history in pure and applied mathematics. This chapter reviews connections between spherical means and the several equations which arise in the time or frequency domains in the analysis of photoacoustic imaging. It focuses on the recovery of a function supported in a region D in space from its spherical means with centers on the boundary of D for fairly general regions D. The chapter discusses some related work on the characterization of the range of the spherical mean transform, and of related wave equations. It explains the special case where the family of spherical means is over spheres is centered on sets with simple geometry, and presents some of the filtered back-projection formulas that have been found. When the domain is a ball, there are convenient formulas of filtered back-projection type in all dimensions.
