Analytic Inversion of a Radon Transform on Double Circular Arcs With Applications in Compton Scattering Tomography

Cécilia Tarpau (1, 2) , Javier Cebeiro (1) , Mai K. Nguyen (1) , Genevieve Rollet (3) , Marcela Morvidone (4)

1 ETIS - UMR 8051 - Equipes Traitement de l'Information et Systèmes
2 LMV - Laboratoire de Mathématiques de Versailles
3 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation
4 CEDEMA - Centro de Matematica Aplicada [San Martín]

In this work we introduce a new Radon transform which arises from a new modality of Compton Scattering Tomog-raphy (CST). This new system is made of a single detector rotating around a fixed source. Unlike some previous CST, no collimator is used at the detector. Such a system allows us to collect scattered photons coming from two opposite sides of the source-detector segment, hence the manifold of the associated Radon transform is a family of double circular arcs. As first main theoretical result, an analytic inversion formula is established for this new Radon transform. This is achieved through the formulation of the transform in terms of circular harmonic expansion satisfying the consistency conditions in Cormack's sense. Moreover, a fast and efficient numerical implementation via an alternative formulation based on Hilbert transform is carried out. Simulation results illustrate the theoretical feasibility of the new system. From a practical point of view, an uncollimated detector system considerably increases the amount of collected data, which is particularly significant in a scatter imaging system.

Domaines: Imagerie médicale; Traitement du signal et de l'image [eess.SP]

Cécilia Tarpau, Javier Cebeiro, Mai K. Nguyen, Genevieve Rollet, Marcela Morvidone. Analytic Inversion of a Radon Transform on Double Circular Arcs With Applications in Compton Scattering Tomography. IEEE Transactions on Computational Imaging, 2020, 6, pp.958-967. ⟨10.1109/TCI.2020.2999672⟩. ⟨hal-02922205⟩