Study of generalized radon transforms and applications in compton scattering tomography

Abstract

Since the advent of the first ionizing radiation imaging devices initiated by Godfrey Newbold Hounsfield and Allan MacLeod Cormack, Nobel Prizes in 1979, the requirement for new non-invasive imaging techniques has grown. These techniques rely upon the properties of penetration in the matter of X and gamma radiation for detecting a hidden structure without destroying the illuminated environment. They are used in many fields ranging from medical imaging to non-destructive testing through. However, the techniques used so far suffer severe degradation in the quality of measurement and reconstructed images. Usually approximated by a noise, these degradations require to be compensated or corrected by collimating devices and often expensive filtering. These degradation is mainly due to scattering phenomena which may constitute up to 80% of the emitted radiation in biological tissue. In the 80's a new concept has emerged to circumvent this difficulty : the Compton scattering tomography (CST).This new approach proposes to measure the scattered radiation considering energy ranges ( 140-511 keV) where the Compton effect is the phenomenon of leading broadcast. The use of such imaging devices requires a deep understanding of the interactions between radiation and matter to propose a modeling, consistent with the measured data, which is essential to image reconstruction. In conventional imaging systems (which measure the primary radiation) the Radon transformdefined on the straight lines emerged as the natural modeling. But in Compton scattering tomography, the measured information is related to the scattering energy and thus the scattering angle. Thus the circular geometry induced by scattering phenomenon makes the classical Radon transform inadequate.In this context, it becomes necessary to provide such Radon transforms on broader geometric manifolds.The study of the Radon transform on new manifolds of curves becomes necessary to provide theoretical needs for new imaging techniques. Cormack, himself, was the first to extend the properties of the conventional Radon transform of a family of curves of the plane. Thereafter several studies have been done in order to study the Radon transform defined on different varieties of circles, spheres, broken lines ... . In 1994 S.J. Norton proposed the first modality in Compton scattering tomography modeled by a Radon transform on circular arcs, the CART1 here. In 2010, Nguyen and Truong established the inversion formula of a Radon transform on circular arcs, CART2, to model the image formation in a new modality in Compton scattering tomography. The geometry involved in the integration support of new modalities in Compton scattering tomography lead them to demonstrate the invertibility of the Radon transform defined on a family of Cormack-type curves, called C_alpha. They illustrated the inversion procedure in the case of a new transform, the CART3, modeling a new modeling of Compton scattering tomography. Based on the work of Cormack and Truong and Nguyen, we propose to establish several properties of the Radon transform on the family C_alpha especially on C1. We have thus demonstrated two inversion formulae that reconstruct the original image via its circular harmonic decomposition and itscorresponding transform. These formulae are similar to those established by Truong and Nguyen. We finally established the well-known filtered back projection and singular value decomposition in the case alpha = 1. All results established in this study provide practical problems of image reconstruction associated with these new transforms. In particular we were able to establish new inversion methods for transforms CART1,2,3 as well as numerical approaches necessary for the implementation of these transforms. All these results enable to solve problems of image formation and reconstruction related to three Compton scattering tomography modalities.In addition we propose to improve models and algorithms es