Imaging and Inverse Problems of Partial Differential Equations
by
Frank Natterer
Univerität Münster

The "mother" of imaging methods is computerized tomography (CT). From a mathematical point of view CT is just an application of Radon's 1907 inversion formula for the Radon transform. So, the mathematics of imaging is usually considered as part of integral geometry in the Gelfand-Helgason sense, i.e. reconstructing functions from integrals over manifolds of lower dimension.

Recently imaging has got a different slant. Virtually all imaging problems can be formulated as inverse problems of PDE's, i. e. estimating coefficients of PDE's from boundary measurements of the solution. In CT the relevant differential equation is the transport equation, in optical imaging the diffusion equation, in acoustic imaging the wave equation. Integral geometry comes in only because integral operators are often approximate solution operators to these equations.

In the talk we describe the progresses and unifications that have been achieved recently by this broader view of imaging.

Date received: March 24, 2005