Mathematics

Let M be a differentiable manifold and N another manifold which parametrizes a smoothly varying family of submanifolds of M. Integral geometry, in its modern formulation, studies the mapping which carries a geometric object on M such as a function, form, or vector field to the object on N obtained by "integration"
over the given family of submanifolds. Several specific cases are important in medical imaging. The best known is the x-ray transform which integrates over straight lines in Euclidean space and is the mathematical model of the CAT scan. Others occur in emission tomography, Doppler
ultrasound, and thermoacoustic tomography. I will introduce these imaging
modalities and discuss the extent to which the canonical questions of
uniqueness, inversion and range characterization have been answered in these
specific cases.