Bragg scattering tomography

Figure 1.  The portal scanner geometry. The scanned object is labelled as $ f $. The detectors are collimated to planes, and the scattering events occur along lines $ L = \{x_2 = a, x_3 = 0\} $, for some $ -1<a<1 $. The scatter from $ L $ is measured by detectors $ \mathbf{d}\in\{x_2 = 1, x_3 = \epsilon\} $, for some $ \epsilon>0 $

Figure 2.  A scattering event occurs at a scattering site $ \mathbf{x} $, for photons emitted from a source $ \mathbf{s} $ and recorded at a detector $ \mathbf{d} $. The initial photon energy is $ E $ and the scattered energy is $ E_s $. Here $ \mathbf{v} $ is the direction normal to the detector surface, displayed as a square in the $ x_1x_3 $ plane. The scattering angle is $ \omega = 2\theta $. where $ \theta $ is the Bragg angle

Figure 3.  Plot of the curves of integration for the Bragg transform for varying $ E $ and $ x_2 $. $ s_1 = 0 $ is fixed

Figure 4.  A circle with center $ \mathbf{c} = (-\sqrt{r^2-1}, 0) $, radius $ r $ is pictured. The circle intersects the dashed line at two points, whose $ x_1 $ coordinates are the solutions to (4.6)

Figure 5.  Bragg scanning modality in the $ n = 2 $ case. A square detector $ \hat{ \mathbf{d}} = ( \mathbf{s}, 1) $ is shown opposite a source $ \hat{ \mathbf{s}} = ( \mathbf{s}, -1) $, and collects photons (shown as wavy lines) scattered from points $ \hat{ \mathbf{x}} = ( \mathbf{x}, 1-|\textbf{b}-\hat{ \mathbf{s}}|) $ on the crystal plane. The crystal sample (the red plane) is placed between, and is parallel to the $ \hat{ \mathbf{s}} $-plane and $ \hat{ \mathbf{d}} $-plane. The center of the base of the cone is $ \textbf{b} $, the source opening angle is $ \beta $ (as in figure 1a), and the source width is $ w = |\textbf{b}-\hat{ \mathbf{s}}|\tan\beta $. The momentum transfer is $ q = Eq_1(| \mathbf{x}- \mathbf{s}|) = E\sin\frac{\omega}{2} $

Figure 6.  $ q_1 $ curve examples. For the decreasing curves displayed we would have to choose $ w>1 $ for the injectivity of $ \mathfrak{B}_1 $ to hold

Figure 7.  Plot of the curves of integration for the offset Bragg transform for varying $ E $ and $ \frac{E_M}{q_1(w)} $. The line $ \{q = E_m\} $ is displayed in black

Figure 8.  Invertible design regions $ S $ (the blue regions on the top row) and possible linear $ \Phi $ (the red lines on the bottom row) for varying $ \beta $. Note that the $ \epsilon $ scales of the figures are different for each $ \beta $

Figure 9.  Venetian blind detector configurations for varying source opening angles $ \beta $. We show 21 detector arrays at $ \epsilon = \Phi(x_2) $ for $ x_2\in\left\{-1+\frac{j-1}{10} : 1\leq j\leq 21\right\} $, where for each $ \beta $, $ \Phi $ is the corresponding straight line relationship of figure 8. The blue lines represent the collimation planes which intersect the tunnel at position $ T = 420(-x_2+1) $, where $ x_2 = \Phi^{-1}(\epsilon) $

Figure 10.  $ F(\cdot, Z) $ plots for varying $ Z $. The plots are normalized in $ L^{\infty} $ (by max value)

Figure 11.  Top row – 2-spherical phantom. Bottom row – 4-spherical phantom. The image values in the left column are included for visualization (e.g. to distinguish between different materials) and have no physical meaning

Figure 12.  2-spherical phantom reconstructions for all $ (x_2, c_{\text{avg}})\in\{205,410,615\}\times\{1, 10\} $, and the Ground Truth (GT) on the top row. To clarify, the GT does not vary with $ c_{\text{avg}} $ and is included for comparison

Figure 13.  4-spherical phantom reconstructions for all $ (x_2, c_{\text{avg}})\in\{205,410,615\}\times\{1, 10\} $, and the Ground Truth (GT) on the top row. To clarify, the GT does not vary with $ c_{\text{avg}} $ and is included for comparison