PhD thesis - Modalities of Compton Scattering Tomography: concept, modelling and associated inverse problems

PhD thesis defended on December 7th, 2021, before the comittee:

• Jérôme Mars, Professor, Institut Polytechnique de Grenoble, President
• Kacem Chehdi, Professor, Université de Rennes, Reviewer
• Voichita Maxim, Assistant Professor, INSA Lyon, Reviewer
• Fabrice Mériaudeau, Professor, Université de Bourgogne Franche Conté, Examinator
• Jean-Christophe Pesquet, Professor, Centrale Supelec, Examinator
• Maï K. Nguyen Verger, Professor, CY Cergy Paris Université, Thesis supervisor
• Laurent Dumas, Professor, Université de Versailles Saint Quentin, Thesis supervisor
• Geneviève Rollet, Assistant Professor, CY Cergy Paris Université, Thesis advisor
• Jean Avan, Senior Researcher, CY Cergy Paris Université, Guest

Context: Since the seminal works of Cormack and Hounsfield, Nobel Prize of Medicine, Computed Tomography (CT) has become one of the privileged imaging techniques to explore internal structures of matter without destroying it. Classical CT uses only transmitted radiation without any deviation from the source to the detector. However, a significant proportion of emitted photons is scattered by Compton effect inside matter. As a consequence, Compton scattering causes serious degradation, in terms of blurring and loss of contrast on reconstructions. This may lead to influence the conclusions of diagnoses. Solving the scattering problem remains an important challenge. Current CT modalities (i.e., configurations) attempt to eliminate these scattered radiations by the use of collimation or filtering allowing to select only the transmitted photons, thanks to their direction or their energy. With such equipment, only few photons arrive at the detector. Compton scattered radiation can also be considered as a useful and significant part of information, which can be used for image reconstruction. This observation is the starting point of the development of imaging systems of Compton Scattering Tomography (CST).

Contributions: The main contribution of my thesis is the proposition of a new CST modality, of circular geometry, made of a fixed source and a ring of detectors passing through the source. This CST modality, called Circular CST (CCST), conceptually surpasses the previous proposed systems since no movement of the source and detectors is required to acquire a complete set of data necessary for reconstruction.

While CT is modelled by the Radon transform (RT) on lines, the geometry of the scattered radiation leads to consider circle arcs and generalized RT on the convenient family of circle arcs. In the case of CCST, the modelling of its operation leads to a new RT on a specific family of double arcs of circles passing through a fixed point (the point source). The first result of this thesis was the inversion of this RT to solve the reconstruction problem. With collimation at detectors, two approaches for inversion have been proposed in ¹ and in ². Moreover, the second proposed algorithm has shown promising results for acquisition of both small and large objects with a double scanning configuration. We also studied CCST without collimation. In absence of an analytical inversion formula, we proposed to use a regularization method in order to have simulation results. Otherwise, we studied also the possibility to introduce a movement for detectors. This study results in the proposition of an inversion formula for the associated RT, in two and three dimensions ³. The corresponding reconstruction algorithms have proved impossible to use in practice for a CST system.

We also studied two other CST systems. The first one, introduced also during this thesis, consists of a fixed source and a detector rotating around the source. The two-dimensional design of this modality does not use collimation at detectors. Consequently, the modelling of data acquisition leads to a RT on another family of double circle arcs. As a first theoretical result, we established the analytic inversion of the RT ⁴. We also investigated its extension in three dimensions and proved the invertibility of the corresponding RT on toric surfaces⁵. The second studied modality, previously introduced by Webber, consists of a pair source detector separated by a fixed distance one from each other. The source and the detector translate simultaneously to perform data acquisition. We suggested a suitable inverse formulation for simulations ⁶.