Generalized transforms of Radon type and their applications

Abstract

These notes represent an extended version of the contents of the third lecture delivered at the AMS Short Course "Radon Transform and Applications to Inverse Problems" in Atlanta in January 2005. They contain a brief description of properties of some generalized Radon transforms arising in inverse problems. Here by generalized Radon transforms we mean transforms that involve integrations over curved surfaces and/or weighted integrations. Such transformations arise in many areas, e.g. in Single Photon Emission Tomography (SPECT), Electrical Impedance Tomography (EIT) thermoacoustic Tomography (TAT), and other areas. 1 Numerous other reasons to study this transform are known, e.g. Radar and Sonar imaging, approximation theory, PDEs, potential theory, complex analysis, etc. [2, 93]. Although in dimensions higher than two one should probably use the word "spherical" rather than "circular," we will use for simplicity the latter. This should not create any confusion.

Book Review: Integral geometry and Radon transforms

“…The model in single photon emission tomography [52] involves a Radon line transform with a measure that is not group invariant. Models in radar [56], sonar, and thermoacoustic tomography [1] involve circular, elliptical, or spherical mean transforms and their generalizations, and researchers use harmonic and microlocal analysis, PDEs, and group theory to develop reconstructions methods and properties.…”

On some inversion formulas for Riesz potentials and k-plane transforms

“…The latter contains application of Theorem A to approximate and explicit inversion of the k-plane transform arising in integral geometry and tomography; see [1,5,6,7,8,9,10,11,14,15,16,21,19] and references therein. We recall that C 0 (R n ) denotes the space of continuous functions on R n vanishing at infinity.…”

Photoacoustic Tomography

“…One needs to mention that in the case of the constant sound speed, the reconstruction task can be described as an integral geometry problem dealing with a spherical mean operator [2,22] (see also [9,13,15,16,21,23,28,29,34] and references therein for integral geometric methods). This relation with integral geometry all but disappears for non-constant speed.…”