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On multiple scattering in Compton scattering tomography and its impact on fan-beam CT

  • *Corresponding author: Lorenz Kuger

    *Corresponding author: Lorenz Kuger 
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  • The recent development of energy-resolving scintillation crystals opens the way to new types of applications and imaging systems. In the context of computerized tomography (CT), it enables to use the energy as a dimension of information supplementing the source and detector positions. It is then crucial to relate the energy measurements to the properties of Compton scattering, the dominant interaction between photons and matter. An appropriate model of the spectral data leads to the concept of Compton scattering tomography (CST). Multiple-order scattering constitutes the major difficulty of CST. It is, in general, impossible to know how many times a photon was scattered before being measured. In the literature, this nature of the spectral data has often been eluded by considering only the first-order scattering in models of the spectral data. This consideration, however, does not represent the reality as second- and higher-order scattering are a substantial part of the spectral measurement. In this work, we propose to tackle this difficulty by an analysis of the spectral data in terms of modeling and mapping properties. Due to the complexity of the multiple order scattering, we model and study the second-order scattering and extend the results to the higher orders by conjecture. The study ends up with a general reconstruction strategy based on the variations of the spectral data which is illustrated by simulations on a joint CST-CT fan beam scanner. We further show how the method can be extended to high energetic polychromatic radiation sources.

    Mathematics Subject Classification: 45Q05, 65R10.


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  • Figure 1.  Illustration of the differential cross-section for the Compton effect

    Figure 2.  Geometry of the first order scattering (fig. 2a) for a human thorax phantom (fig. 2b)

    Figure 3.  Top: Illustrations of the data $ g_1 $ in 3a and $ g_1+g_2 $ in 3b. $ g_2 $ alters the structure of the measured spectrum by adding large components not seen in $ g_1 $. The plot in 3c shows the measured spectrum for a fixed detector, corresponding to the row designated by $ {\bf{d}} $ in the left two illustrations. Bottom: Reconstructions of the electron density of a thorax phantom (ground truth in 3d) using as forward model only $ \mathcal{T}_1 $ and data $ \text{Pois}(g_1) $ (3e) or $ \text{Pois}(g_1+g_2) $ (3f). Note how details of small size or lower contrast like the spine are badly imaged in the second setting

    Figure 4.  Geometry of the second order scattering

    Figure 5.  Contribution of $ g_1 $ and $ g_2 $ to the measured spectra of two small disks

    Figure 6.  For the thorax phantom and $ E_0 = 1.173 $ MeV, we depict the data of a single source-detector pair to show how the differential operator influences the data. Left: unprocessed data, right: the data after applying $ D_E^\gamma $

    Figure 7.  Scanning of the object for 8 angular views

    Figure 8.  (8a): Ground truth phantom. (8d): prior reconstruction from sparse data CT step eq. (24). (8b) and (8e): Solutions of problem eq. (25), $ \lambda = 0 $ in (8b) and $ \lambda > 0 $ tuned in (8e). (8c) and (8f): Solutions of problem eq. (26) with TV parameters $ \lambda = 0 $ (least-squares fit) in (8c) and $ \lambda > 0 $ tuned by the L-curve method in (8f)

    Figure 9.  (9a): The second phantom with an aluminium ring. (9d): prior reconstruction from sparse data CT step eq. (24). (9b) and (9e): Solutions of problem eq. (25) with $ \lambda = 0 $ (9b) and $ \lambda > 0 $ tuned (9e). (9c) and (9f): Solutions of problem eq. (26) with $ \lambda = 0 $ (least-squares fit) in (9c) and $ \lambda > 0 $ tuned in (9f)

    Figure 10.  (10a): The second phantom, now with an iron ring. (10d): prior reconstruction from sparse data CT step eq. (24). (10b) and (10e): Solutions of problem eq. (25) with $ \lambda = 0 $ in (10b) and $ \lambda > 0 $ tuned in (10e). (10c) and (10f): Solutions of problem eq. (26) with TV parameters $ \lambda = 0 $ (least-squares fit) in (10c) and $ \lambda > 0 $ tuned in (10f)

    Figure 11.  (11a): Ground truth phantom. (11b) and (11c): Solution of eq. (26) with $ \lambda = 0 $ in (11b) (least-squares fit) and $ \lambda > 0 $ tuned in (11c)

    Figure 12.  Results for the aluminium object (12a): Ground truth phantom. (12b) and (12c): Solution of eq. (26) with $ \lambda = 0 $ in (12b) (least-squares fit) and $ \lambda > 0 $ tuned in (12c)

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