Circular Compton scattering tomography (CCST) where a fixed radiation source and a number of regularly spaced detectors are positioned on a fixed circular frame is recently proposed [1]. It has multiple advantages such as compact and motion-free system, possibility of combination with classic fan-beam CT (computed tomography) as a bi-imaging system, capacity of scanning both small and large objects.

In the case where the detectors are collimated to split up scattered photons coming from two opposite sides of the source-detector segment, the modelling of CCST’s data acquisition leads to a Radon transform on a family of arcs of circles passing through a fixed point (the point source). The analytical inversion of this Radon transform is derived from Cormack’s earlier works.

In the case of non-collimated detectors, the corresponding Radon transform is defined on a specific family of double circular arcs and named DCART (double circular arc Radon transform). The exact inverse formula for this new Radon transform on pair of circles is not available presently. Recently, deep learning-based techniques appear as promising alternatives to solve the ill-posed inverse problems in CT reconstruction from limited-angle and sparse-view projection data. In our work we propose a neural network architecture acting on both image and data domains. The particularity of this architecture lies in its capability to map the projection (Radon domain) to image domain at different scales of the data while extracting important image features used at reconstruction. The obtained results suggest that removing the collimator at detectors in CCST is feasible thanks to deep learning-based techniques.

Another way to by-pass the collimator at detectors is to design a CST with a single detector rotating around a fixed source. The corresponding Radon transform and its inverse are established in [2,3] but the CST is no longer a motion-free system.

[1] C. Tarpau, J. Cebeiro, M. A. Morvidone, M. K. Nguyen. A new concept of Compton scattering tomography and the development of the corresponding circular Radon transform, IEEE Transactions on Radiation and Plasma Medical Sciences (IEEE-TRPMS) 4.4: 433-440, 2020. https://doi.org/10.1109/TRPMS.2019.2943555

[2] C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet, M. A. Morvidone. Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography, IEEE Transactions on Computational Imaging (IEEE-TCI) 6: 958-967, 2020. https://doi.org/10.1109/TCI.2020.2999672

[3] J. Cebeiro, C. Tarpau, M. A. Morvidone, D. Rubio, M. K. Nguyen, On a three-dimensional Compton scattering tomography system with fixed source, Inverse Problems, Special issue on Modern Challenges in Imaging 37: 054001, 2021. https://doi.org/10.1088/1361-6420/abf0f0.

We present injectivity and microlocal analyses of a new generalized Radon transform, R, which has applications to a novel scanner design in 3-D Compton Scattering Tomography (CST), which we also introduce here. Using Fourier decomposition and Volterra equation theory, we prove that R is injective and show that the image solution is unique. Using microlocal analysis, we prove that R satisfies the Bolker condition (sometimes called the “Bolker assumption”), and we investigate the edge detection capabilities of R. This has important implications regarding the stability of inversion and the amplification of measurement noise. This paper provides the theoretical groundwork for 3-D CST using the proposed scanner design.