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Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry

  • *Corresponding author: Cécilia Tarpau

    *Corresponding author: Cécilia Tarpau 

C. Tarpau research work is supported by grants from Région Île-de-France (in Mathematics and Innovation) 2018-2021 and LabEx MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l'Incertitude et des Interactions) (No. ANR-11-LBX-0023-01). J. Cebeiro research work is supported by a postdoctoral grant from the University of San Martín. He is also partially supported by SOARD-AFOSR (grant number FA9550-18-1-0523)

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  • In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly implemented in the Fourier domain. Simulations are carried out to illustrate the efficiency of the proposed reconstruction algorithm.

    Mathematics Subject Classification: Primary: 44A12, 65R10; Secondary: 94A08.


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  • Figure 1.  General functioning principle of a CST system. Photons are emitted by source $ S $, interact at sites $ M $, and are recorded at site $ D $. When a photon is detected carrying an energy $ E(\omega_1) $ (resp. $ E(\omega_2) $), the possible interaction sites lie on the upper (resp. lower) circle arc which subtends the angle $ (\pi-\omega_1) $ (resp. $ (\pi-\omega_2) $)

    Figure 2.  Previous proposed CST modalities. (A): Fixed source and detectors placed on a line. (B): Rotating pair source-detector diametrically opposed. (C): Rotating pair source detector. (D): Fixed source and detectors placed on a ring. (E): Detector rotating around a fixed source. (F) Source-detector translating simultaneously along two parallel lines. In all figures: The source $ S $ is represented by a red point. The detector(s) $ D $ is (are) represented by blue point(s). The $ M $, $ M_i $ or $ M'_i $ in black, are running points and examples of scattering site. An example of trajectory for a photon whose scattering site is $ M $ is shown in purple. The corresponding scattering angle is denoted $ \omega $. The object to scan is represented in grey. The red continuous curves are the examples of scanning circles arcs. For (C), (E) and (F), the dashed circles (resp. lines) represents the circular (resp. linear) paths on which move the sensors

    Figure 3.  Setup and parameterization of the CST modality proposed in [34]. The source $ S $ and the detector are respectively represented by a red and a blue point. To make the difference between the four half-arcs, $ S_1, S_3 $ and $ S_2, S_4 $ are respectively depicted in red and green. $ \Omega_{1, 2, 3, 4} $ denote the centres of the circles supporting the half-arcs $ S_{1, 2, 3, 4} $. The point $ M $ is an example of a scattering site

    Figure 4.  (A) Original object: Derenzo phantom. (B) Corresponding acquired data for $ N_{SD} = 2048 $ and $ N_r = 1024 $. A distance of one pixel is left between the upper part of the image and the detector path ($ \delta = 1 $, see 5.2.1)

    Figure 5.  Reconstruction results of the Derenzo phantom 4a for (A) $ \delta = 1 $ (NMSE = 0.0112), (B) $ \delta = 26 $ (NMSE = 0.0074) and (C) $ \delta = 51 $ (NMSE = 0.0061) pixel(s)

    Figure 6.  Evaluation of the number of source-detector positions on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for (A) $ x_{0, max} = 2N $ (NMSE = 0.0084), (B) $ 3N $ (NMSE = 0.0049) and (C) $ 4N $ NMSE = 0.0061) where $ \Delta_{x_0} = 1 $. Second row: Reconstruction results for (D) $ 0.5 $ (NMSE = 0.0046), (E) $ 1 $ (NMSE = 0.0049) and (F) $ 0.5 $ NMSE = 0.0049) detector per unit length and $ x_{0, max} = 3N $ remains constant

    Figure 7.  Evaluation of the number of scanning circles on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for (A) $ r_{max} = 2N $ (NMSE = 0.0058), (B) $ 3N $ (NMSE = 0.0040) and (C) $ 4N $ (NMSE = 0.0049) where $ \Delta_{r} = 1 $. Second row: Reconstruction results for (D) $ \Delta_{r} = 1 $ (NMSE = 0.0040), (E) $ 2 $ (NMSE = 0.0043) and (F) $ 4 $ (NMSE = 0.0043) detector per unit length and $ r_{max} = 3N $ remains constant

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