The use of scattered radiation for tomographic reconstruction continues to be a current challenge for designing future imaging systems. In the energy range of X and gamma rays used for biomedical imaging and non- destructive evaluation, Compton scattering is the predominant effect. The one-to-one correspondence between angle and energy of scattered radiation allows the exploitation of the energy information for reconstruction and consequently the emergence of modalities of Compton Scattering Tomography (CST). For two-dimensional systems, the modelling of the imaging processes leads to new Radon transform on circular arcs according to the geometry of the modality. In this context, a new modality, named Circular Compton Scattering Tomography (CCST) has been proposed recently. This system is made of a fixed source and a ring of detectors passing through the source. The purpose of this work is to extend this modality to three dimensions. Two geometries will be proposed in this study: the first one is made of a fixed source and fixed detectors placed on a sphere passing through the source and the second one considers detectors placed on a cylinder, containing also the fixed source. These three-dimensional setups conserve the assets of CCST which include the ability of a faster scanning compared to existing systems, the convenience for small object scanning, having a fixed system (avoiding mechanical rotation) and the compactness compared to planar configurations. The modelling of image acquisition in these new cases leads to Radon transforms on spindle tori resulting of the revolution of a circle along the axis Source-Detector. Numerical simulations are carried out in this work and show the theoretical feasibility of these systems.
Characterization and interpretation of flat ancient material objects, such as those found in archaeology, paleoenvironments, paleontology, and cultural heritage, have remained a challenging task to perform by means of conventional x-ray tomography methods due to their anisotropic morphology and flattened geometry. To overcome the limitations of the mentioned methodologies for such samples, an imaging modality based on Compton scattering is proposed in this work. Classical x-ray tomography treats Compton scattering data as noise in the image formation process, while in Compton scattering tomography the conditions are set such that Compton data become the principal image contrasting agent. Under these conditions, we are able, first, to avoid relative rotations between the sample and the imaging setup, and second, to obtain three-dimensional data even when the object is supported by a dense material by exploiting backscattered photons. Mathematically this problem is addressed by means of a conical Radon transform and its inversion. The image formation process and object reconstruction model are presented. The feasibility of this methodology is supported by numerical simulations.