 
Summary: SIAM J. MATH. ANAL. c 2006 Society for Industrial and Applied Mathematics
Vol. 38, No. 2, pp. 681692
A RANGE DESCRIPTION FOR THE PLANAR CIRCULAR RADON
TRANSFORM
GAIK AMBARTSOUMIAN AND PETER KUCHMENT
Abstract. The transform considered in the paper integrates a function supported in the unit
disk on the plane over all circles centered at the boundary of this disk. Such a circular Radon
transform arises in several contemporary imaging techniques, as well as in other applications. As is
common for transforms of Radon type, its range has infinite codimension in standard function spaces.
Range descriptions for such transforms are known to be very important for computed tomography
for instance, when dealing with incomplete data, error correction, and other issues. A complete range
description for the circular Radon transform is obtained. Range conditions include the recently found
set of momenttype conditions, which happens to be incomplete, as well as other conditions that have
less standard form. In order to explain the procedure better, a similar (nonstandard) treatment of
the range conditions is described first for the usual Radon transform on the plane.
Key words. thermoacoustic tomography, Radon transform, range description
AMS subject classifications. 44A12, 92C55, 65R32
DOI. 10.1137/050637492
1. Introduction. The following "circular" Radon transform, which is the main
object of study in this article, arises in several applications, including the newly
