Hybrid imaging modalities and the range of the spherical means transform

The last two decades saw proliferation of novel coupled-physics imaging modalities. A variety of sensitive but safe and inexpensive medical imaging methods has been  developed, that hold great promise for the breast imaging for cancer, detection of ischemia, hemorrhaging, blood clots, etc. These imaging techniques work by combining  high-resolution ultrasound with electromagnetic fields that are highly sensitive to the features of interest. In the first part of my talk I will overview the most interesting of these modalities, with emphasis on the underlying mathematics.

Inevitably, the introduction of the new techniques has posed a variety of new and exciting inverse problems. In particular, a prominent role in this field is played by the spherical means operator, that maps a function into a collection of integrals over spheres with centers lying on a given surface. The problem I will discuss is the description of the range of this operator. As it happens, all of the classical results are suboptimal, in that they use twice the amount of needed data. I will present a novel range description, that overcomes this issue.

The talk will be accessible to graduate students.

(This is a joint work with Peter Kuchment, Texas A&M University)