International Journal of Biomedical Imaging 
Volume 2007 (2007), Article ID 92780, 11 pages
doi:10.1155/2007/92780
Research Article

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

T. T. Truong,1 M. K. Nguyen,2 and H. Zaidi3

 1Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, Cergy-Pontoise 95302, France
2Laboratoire Equipes de Traitement des Images et du Signal, CNRS UMR 8051, Ecole Nationale Supérieure de l' Electronique et de ses Applications, Université de Cergy-Pontoise, 6 Avenue du Ponceau, Cergy-Pontoise 95014, France
3 Division of Nuclear Medicine, Geneva University Hospital, Geneva 4 1211, Switzerland
Received 27 September 2006; Accepted 20 February 2007

Recommended by Zhaotian Zhang

Abstract

The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the two-dimensional Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties which may be relevant for active researchers in the field.