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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)

  • [1] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229.  doi: 10.1088/0266-5611/10/6/003.
    [2] A. AlmansaC. BallesterV. Caselles and G. Haro, A TV based restoration model with local constraints, Journal of Scientific Computing, 34 (2008), 209-236.  doi: 10.1007/s10915-007-9160-x.
    [3] M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker and K. Olsen, Xcom: Photon cross sections database, NIST, URL http://physics.nist.gov/xcom, Accessed: 2020-09-05.
    [4] L. BratemanA. M. Jacobs and L. T. Fitzgerald, Compton scatter axial tomography with x-rays: SCAT-CAT, Physics in Medicine and Biology, 29 (1984), 1353-1370.  doi: 10.1088/0031-9155/29/11/004.
    [5] M. Burger and S. Osher, A guide to the TV zoo, in Level Set and PDE-Based Reconstruction Methods in Imaging, Lecture Notes in Mathematics, 2013, 1–70. doi: 10.1007/978-3-319-01712-9_1.
    [6] J. CebeiroC. TarpauM. A. MorvidoneD. Rubio and M. K. Nguyen, On a three-dimensional compton scattering tomography system with fixed source, Inverse Problems, 37 (2021), 054001.  doi: 10.1088/1361-6420/abf0f0.
    [7] R. L. Clarke and G. Van Dyk, A new method for measurement of bone mineral content using both transmitted and scattered beams of gamma-rays, Physics in Medicine and Biology, 18 (1973), 532-539.  doi: 10.1088/0031-9155/18/4/005.
    [8] B. L. Evans, J. B. Martin, L. W. Burggraf and M. C. Roggemann, Nondestructive inspection using compton scatter tomography, in 1997 IEEE Nuclear Science Symposium Conference Record, vol. 1, 1997,386–390.
    [9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Springer Boston, 1984. doi: 10.1007/978-1-4684-9486-0.
    [10] A. Greenleaf and A. Seeger, Oscillatory and fourier integral operators with degenerate canonical relations, Publicacions Matematiques, (2002), 93–141. doi: 10.5565/PUBLMAT_Esco02_05.
    [11] J. Gödeke and G. Rigaud, Imaging based on compton scattering: Model-uncertainty and data-driven reconstruction methods, 2022.
    [12] B. Hahn, Reconstruction of dynamic objects with affine deformations in computerized tomography, Journal of Inverse and Ill-posed Problems, 22 (2014), 323-339.  doi: 10.1515/jip-2012-0094.
    [13] B. N. Hahn and M.-L. Kienle Garrido, An efficient reconstruction approach for a class of dynamic imaging operators, Inverse Problems, 35 (2019), 094005, 26 pp. doi: 10.1088/1361-6420/ab178b.
    [14] L. Hörmander, Fourier integral operators. I, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.
    [15] N. KanematsuT. Inaniwa and M. Nakao, Modeling of body tissues for monte carlo simulation of radiotherapy treatments planned with conventional x-ray CT systems, Physics in Medicine and Biology, 61 (2016), 5037-5050.  doi: 10.1088/0031-9155/61/13/5037.
    [16] O. Klein and Y. Nishina, Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac, Zeitschrift Für Physik, 52 (1929), 853-868.  doi: 10.1007/BF01366453.
    [17] V. P. Krishnan and E. T. Quinto, Microlocal Analysis in Tomography, 847–902, Springer New York, New York, NY, 2015.
    [18] P. KuchmentK. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems, 11 (1995), 571-589.  doi: 10.1088/0266-5611/11/3/006.
    [19] P. G. Lale, The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography, Physics in Medicine and Biology, 4 (1959), 159-167.  doi: 10.1088/0031-9155/4/2/305.
    [20] C. Leroy and P.-G. Rancoita, Principles of Radiation Interaction in Matter and Detection, World Scientific, Singapore, 2011. doi: 10.1142/8200.
    [21] F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898719284.
    [22] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898718324.
    [23] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc radon transform for compton scattering tomography, Inverse Problems, 26 (2010), 065005.  doi: 10.1088/0266-5611/26/6/065005.
    [24] S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.  doi: 10.1063/1.357668.
    [25] V. P. Palamodov, An analytic reconstruction for the compton scattering tomography in a plane, Inverse Problems, 27 (2011), 125004.  doi: 10.1088/0266-5611/27/12/125004.
    [26] G. Rigaud, 3D Compton scattering imaging with multiple scattering: Analysis by FIO and contour reconstruction, Inverse Problems, 37 (2021), Paper No. 064001, 39 pp. doi: 10.1088/1361-6420/abf22b.
    [27] G. Rigaud, Compton scattering tomography: Feature reconstruction and rotation-free modality, SIAM Journal on Imaging Sciences, 10 (2017), 2217-2249.  doi: 10.1137/17M1120105.
    [28] G. Rigaud and B. N. Hahn, 3d compton scattering imaging and contour reconstruction for a class of radon transforms, Inverse Problems, 34 (2018), 075004.  doi: 10.1088/1361-6420/aabf0b.
    [29] L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.
    [30] P. C. Shrimpton, Electron density values of various human tissues: In vitro compton scatter measurements and calculated ranges, Physics in Medicine and Biology, 26 (1981), 907-911.  doi: 10.1088/0031-9155/26/5/010.
    [31] E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine and Biology, 53 (2008), 4777-4807.  doi: 10.1088/0031-9155/53/17/021.
    [32] J. P. StonestromR. E. Alvarez and A. Macovski, A framework for spectral artifact corrections in x-ray ct, IEEE Transactions on Biomedical Engineering, BME-28 (1981), 128-141.  doi: 10.1109/TBME.1981.324786.
    [33] A. C. Tanner and I. R. Epstein, Multiple scattering in the Compton effect. I. Analytic treatment of angular distributions and total scattering probabilities, Phys. Rev. A, 13 (1976), 335-348.  doi: 10.1103/PhysRevA.13.335.
    [34] A. C. Tanner and I. R. Epstein, Multiple scattering in the Compton effect. II. Analytic and numerical treatment of energy profiles, Phys. Rev. A, 14 (1976), 313-327.  doi: 10.1103/PhysRevA.14.313.
    [35] A. C. Tanner and I. R. Epstein, Multiple scattering in the Compton effect. III. Monte Carlo calculations, Phys. Rev. A, 14 (1976), 328-340.  doi: 10.1103/PhysRevA.14.328.
    [36] C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet and L. Dumas, On the design of a cst system and its extension to a bi-imaging modality, 2020.
    [37] T. T. Truong and M. K. Nguyen, Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations, Intech, 2012. doi: 10.5772/50012.
    [38] T. T. TruongM. K. Nguyen and H. Zaidi, The mathematical foundations of 3d compton scatter emission imaging, International Journal of Biomedical Imaging, 2007 (2007), 092780.  doi: 10.1155/2007/92780.
    [39] J. WangZ. Chi and Y. Wang, Analytic reconstruction of compton scattering tomography, Journal of Applied Physics, 86 (1999), 1693-1698.  doi: 10.1063/1.370949.
    [40] J. WangT. LiH. Lu and Z. Liang, Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography, IEEE Transactions on Medical Imaging, 25 (2006), 1272-1283.  doi: 10.1109/TMI.2006.882141.
    [41] J. Webber, X-ray compton scattering tomography, Inverse Problems in Science and Engineering, 24 (2016), 1323-1346.  doi: 10.1080/17415977.2015.1104307.
    [42] J. Webber and E. L. Miller, Compton scattering tomography in translational geometries, Inverse Problems, 36 (2020), 025007.  doi: 10.1088/1361-6420/ab4a32.
    [43] J. W. Webber and E. T. Quinto, Microlocal analysis of a Compton tomography problem, SIAM J. Imaging Sci., 13 (2020), 746-774.  doi: 10.1137/19M1251035.
    [44] J. W. Webber and S. Holman, Microlocal analysis of a spindle transform, Inverse Problems & Imaging, 13 (2019), 231-261.  doi: 10.3934/ipi.2019013.
    [45] J. W. Webber and W. R. B. Lionheart, Three dimensional compton scattering tomography, Inverse Problems, 34 (2018), 084001.  doi: 10.1088/1361-6420/aac51e.
    [46] Z. ZhuK. WahidP. BabynD. CooperI. Pratt and Y. Carter, Improved compressed sensing-based algorithm for sparse-view ct image reconstruction, Computational and Mathematical Methods in Medicine, 2013 (2013), 185750.  doi: 10.1155/2013/185750.
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