IJPAM: Volume 49, No. 3 (2008)

ON A CLASS OF GENERALIZED RADON TRANSFORMS
AND ITS ROLE IN IMAGING SCIENCE

T.T. Truong$^1$, M.K. Nguyen$^2$
$^1$Laboratoire de Physique Théorique et Modélisation
University of Cergy-Pontoise (LPTM)
CNRS UMR 8089, Boulevard du Port
Cergy-Pontoise Cedex, 95302, FRANCE
e-mail: truong@u-cergy.fr
$^2$Equipes Traitement de l'Image et du Signal
University of Cergy-Pontoise/ENSEA (ETIS)
CNRS UMR 8051, Boulevard du Port
Cergy-Pontoise Cedex, 95302, FRANCE
e-mail: mai.nguyen-verger@u-cergy.fr


Abstract.Integral transforms based on geometrical objects, i.e. the so-called generalized Radon transforms, play a key role in integral geometry in the sense of I.M. Gelfand. In this work, we discuss the properties of a newly established class of Conical Radon Transforms (CRT), which are defined on circular cones having fixed axis direction and variable opening angle. In particular, we describe its inversion process, i.e. the recovery of an unknown function from the set of its integrals on cone surfaces, or its conical projections. This transform is the basis for a new gamma-ray emission imaging principle, which works with Compton scattered radiation and offers the remarkable advantage of functioning with a fixed detector instead of a rotating one, as in conventional emission imaging modalities.

Received: August 14, 2008

AMS Subject Classification: 44A12, 45B05, 45Q05, 65Rxx

Key Words and Phrases: integral geometry, Radon transform, gamma-ray imaging

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 49
Issue: 3