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Bragg scattering tomography

  • * Corresponding author: James W. Webber

    * Corresponding author: James W. Webber 
This material is supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061-ED0001
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  • Here we introduce a new forward model and imaging modality for Bragg Scattering Tomography (BST). The model we propose is based on an X-ray portal scanner with linear detector collimation, currently being developed for use in airport baggage screening. The geometry under consideration leads us to a novel two-dimensional inverse problem, where we aim to reconstruct the Bragg scattering differential cross section function from its integrals over a set of symmetric $ C^2 $ curves in the plane. The integral transform which describes the forward problem in BST is a new type of Radon transform, which we introduce and denote as the Bragg transform. We provide new injectivity results for the Bragg transform here, and describe how the conditions of our theorems can be applied to assist in the machine design of the portal scanner. Further we provide an extension of our results to $ n $-dimensions, where a generalization of the Bragg transform is introduced. Here we aim to reconstruct a real valued function on $ \mathbb{R}^{n+1} $ from its integrals over $ n $-dimensional surfaces of revolution of $ C^2 $ curves embedded in $ \mathbb{R}^{n+1} $. Injectivity proofs are provided also for the generalized Bragg transform.

    Mathematics Subject Classification: Primary: 45Axx, 44A12.

    Citation:

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  • 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

    Table 1.  Relative least square error values $ \epsilon_{\text{ls}} $ for all experiments conducted. The values in the left-hand columns give the $ x_2 $ coordinates (in mm) of the sphere centers and the equations of the scanning line profiles (as illustrated in figure 11)

    $ \epsilon_{\text{ls}} $ $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ $ \epsilon_{\text{ls}} $ $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
    $ x_2=0 $ $ 0.18 $ $ 0.58 $ $ x_2 = 0 $ $ 0.23 $ $ 0.73 $
    $ x_2=205 $ $ 0.18 $ $ 0.56 $ $ x_2 = 205 $ $ 0.22 $ $ 0.71 $
    $ x_2=615 $ $ 0.18 $ $ 0.57 $ $ x_2 = 615 $ $ 0.22 $ $ 0.71 $
    (A) 2-spherical phantom (B)4-spherical phantom
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    Table 2.  Mean $ F_1 $ score results. The values in the left-hand columns give the $ x_2 $ coordinates (in mm) of the sphere centers and the equations of the scanning line profiles (as illustrated in figure 11)

    Mean $ F_1 $ score $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ Mean $ F_1 $ score $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
    $ x_2=0 $ $ 0.90 $ $ 0.84 $ $ x_2 = 0 $ $ 0.86 $ $ 0.67 $
    $ x_2=205 $ $ 0.90 $ $ 0.90 $ $ x_2 = 205 $ $ 0.89 $ $ 0.70 $
    $ x_2=615 $ $ 0.91 $ $ 0.90 $ $ x_2 = 615 $ $ 0.87 $ $ 0.73 $
    (A) 2-spherical phantom (B) 4-spherical phantom
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    DownLoad: CSV
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