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

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)