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A wavelet frame constrained total generalized variation model for imaging conductivity distribution
Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry
1. | Laboratoire de Physique Théorique et Modélisation (UMR 8089), CY Cergy Paris Université, CNRS, Pontoise, 95302, France |
2. | Equipes Traitement de l'Information et Systèmes (UMR 8051), CY Cergy Paris Université, ENSEA, CNRS, Pontoise, 95302, France |
3. | Laboratoire de Mathématiques de Versailles (UMR 8100), Université de Versailles Saint-Quentin, CNRS, Versailles, 78035, France |
4. | Instituto de Tecnologías Emergentes y Ciencias Aplicadas (ITECA), UNSAM-CONICET, Escuela de Ciencia y Tecnología, Centro de Matemática Aplicada (CEDEMA), San Martín, 1650, Argentina |
In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly implemented in the Fourier domain. Simulations are carried out to illustrate the efficiency of the proposed reconstruction algorithm.
References:
[1] |
I. Ayad, C. Tarpau, M. K. Nguyen and N. S. Vu, Deep morphological network-based artifact suppression for limited-angle tomography, in Proceedings of the 25th International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV'21), Las Vegas, United States, 2021. |
[2] |
R. N. Bracewell,
Numerical transforms, Science, 248 (1990), 697-704.
doi: 10.1126/science.248.4956.697. |
[3] |
J. Cebeiro, M. K. Nguyen, M. Morvidone and A. Noumowé,
New "improved" Compton scatter tomography modality for investigative imaging of one-sided large objects, Inverse Problems in Science and Engineering, 25 (2017), 1676-1696.
doi: 10.1080/17415977.2017.1281920. |
[4] |
J. Cebeiro, M. K. Nguyen, M. Morvidone and C. Tarpau, An interior Compton Scatter Tomography, in 25th IEEE Nuclear Science Symposium and Medical Imaging Conference 2018 (IEEE NSS/MIC'18), Sydney, Australia, 2018.
doi: 10.1109/NSSMIC.2018.8824374. |
[5] |
J. Cebeiro, C. Tarpau, M. A. Morvidone, D. 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. |
[6] |
R. Clarke and G. Van Dyk, Compton-scattered gamma rays in diagnostic radiography, in Medical Radioisotope Scintigraphy. Ⅵ Proceedings of a Symposium on Medical Radioisotope Scintigraphy, 1969. |
[7] |
A. M. Cormack,
Representation of a function by its line integrals, with some radiological applications, Journal of Applied Physics, 34 (1963), 2722-2727.
|
[8] |
P. E. Cruvinel and F. A. Balogun,
Compton scattering tomography for agricultural measurements, Engenharia Agricola, 26 (2006), 151-160.
doi: 10.1590/S0100-69162006000100017. |
[9] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, McGraw-Hill Book Company, New York, 1954. |
[10] |
F. Farmer and M. P. Collins,
A new approach to the determination of anatomical cross-sections of the body by compton scattering of gamma-rays, Physics in Medicine & Biology, 16 (1971), 577.
doi: 10.1088/0031-9155/16/4/001. |
[11] |
S. Gautam, F. Hopkins, R. Klinksiek and I. Morgan,
Compton interaction tomography I. Feasibility studies for applications in earthquake engineering, IEEE Transactions on Nuclear Science, 30 (1983), 1680-1684.
doi: 10.1109/TNS.1983.4332614. |
[12] |
I. S. Gradshteyn, I. M. Ryzhik, D. Zwillinger and V. Moll, Table of Integrals, Series, and Products, 8th ed. Academic Press, Amsterdam, 2014. Available from: https://cds.cern.ch/record/1702455. |
[13] |
G. Harding and E. Harding,
Compton scatter imaging: A tool for historical exploration, Applied Radiation and Isotopes, 68 (2010), 993-1005.
doi: 10.1016/j.apradiso.2010.01.035. |
[14] |
E. M. Hussein, M. Desrosiers and E. J. Waller,
On the use of radiation scattering for the detection of landmines, Radiation Physics and Chemistry, 73 (2005), 7-19.
doi: 10.1016/j.radphyschem.2004.07.006. |
[15] |
K. C. Jones, G. Redler, A. Templeton, D. Bernard, J. V. Turian and J. C. Chu,
Characterization of Compton-scatter imaging with an analytical simulation method, Physics in Medicine & Biology, 63 (2018), 025016.
doi: 10.1016/j.tcs.2018.05.007. |
[16] |
P. Lale,
The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography, Physics in Medicine & Biology, 4 (1959), 159.
doi: 10.1088/0031-9155/4/2/305. |
[17] |
M. K. Nguyen and T. T. Truong, Imagerie par rayonnement gamma diffusé., Hermès Science, 2006. |
[18] |
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: 10.1088/0266-5611/26/9/099802. |
[19] |
S. J. Norton,
Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.
doi: 10.1063/1.357668. |
[20] |
P. G. Prado, M. K. Nguyen, L. Dumas and S. X. Cohen,
Three-dimensional imaging of flat natural and cultural heritage objects by a Compton scattering modality, Journal of Electronic Imaging, 26 (2017), 011026.
|
[21] |
J. Radon,
Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Akad. Wiss., 69 (1917), 262-277.
doi: 10.1090/psapm/027/692055. |
[22] |
G. Redler, K. C. Jones, A. Templeton, D. Bernard, J. Turian and J. C. Chu,
Compton scatter imaging: A promising modality for image guidance in lung stereotactic body radiation therapy, Medical Physics, 45 (2018), 1233-1240.
doi: 10.1002/mp.12755. |
[23] |
G. Rigaud,
Compton scattering tomography: feature reconstruction and rotation-free modality, SIAM Journal on Imaging Sciences, 10 (2017), 2217-2249.
doi: 10.1137/17M1120105. |
[24] |
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. |
[25] |
G. Rigaud, M. K. Nguyen and A. K. Louis,
Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography, Inverse Problems in Science and Engineering, 20 (2012), 809-839.
doi: 10.1080/17415977.2011.653008. |
[26] |
G. Rigaud, R. Régnier, M. K. Nguyen and H. Zaidi,
Combined modalities of Compton scattering tomography, IEEE Transactions on Nuclear Science, 60 (2013), 1570-1577.
doi: 10.1109/TNS.2013.2252022. |
[27] |
C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet and M. A. Morvidone,
Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography, IEEE Transactions on Computational Imaging, 6 (2020), 958-967.
doi: 10.1109/TCI.2020.2999672. |
[28] |
C. Tarpau and M. K. Nguyen,
Compton scattering imaging system with two scanning configurations, Journal of Electronic Imaging, 29 (2020), 130.
doi: 10.1117/1.JEI.29.1.013005. |
[29] |
C. Tarpau, J. Cebeiro, M. Morvidone and M. K. Nguyen, A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform, IEEE Transactions on Radiation and Plasma Medical Sciences, 2019.
doi: 10.1109/TRPMS.2019.2943555. |
[30] |
T. T. Truong, Function reconstruction from reflection symmetric radon data, Symmetry, 12 (2020).
doi: 10.3390/sym12060956. |
[31] |
T. Truong and M. K. Nguyen,
Compton scatter tomography in annular domains, Inverse Problems, 35 (2019), 054005.
doi: 10.1088/1361-6420/ab0b76. |
[32] |
T. T. Truong and M. K. Nguyen, Recent developments on Compton scatter tomography: Theory and numerical simulations, in Numerical Simulation-From Theory to Industry, IntechOpen, 2012. |
[33] |
T. T. Truong and M. K. Nguyen,
Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality, Inverse Problems, 27 (2011), 125001.
doi: 10.1088/0266-5611/27/12/125001. |
[34] |
J. Webber and E. L. Miller,
Compton scattering tomography in translational geometries, Inverse Problems, 36 (2020), 025007.
doi: 10.1088/1361-6420/ab4a32. |
[35] |
J. W. Webber and W. R. Lionheart,
Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), 084001.
doi: 10.1088/1361-6420/aac51e. |
[36] |
J. W. Webber and S. Holman,
Microlocal analysis of a spindle transform, Inverse Problems and Imaging, 13 (2019), 231-261.
doi: 10.3934/ipi.2019013. |
[37] |
J. W. Webber and E. T. Quinto,
Microlocal analysis of a Compton tomography problem, SIAM Journal on Imaging Sciences, 13 (2020), 746-774.
doi: 10.1137/19M1251035. |
[38] |
J. W. Webber, E. T. Quinto and E. L. Miller, A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging, 36 (2020), 074002.
doi: 10.1088/1361-6420/ab8f82. |
show all references
References:
[1] |
I. Ayad, C. Tarpau, M. K. Nguyen and N. S. Vu, Deep morphological network-based artifact suppression for limited-angle tomography, in Proceedings of the 25th International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV'21), Las Vegas, United States, 2021. |
[2] |
R. N. Bracewell,
Numerical transforms, Science, 248 (1990), 697-704.
doi: 10.1126/science.248.4956.697. |
[3] |
J. Cebeiro, M. K. Nguyen, M. Morvidone and A. Noumowé,
New "improved" Compton scatter tomography modality for investigative imaging of one-sided large objects, Inverse Problems in Science and Engineering, 25 (2017), 1676-1696.
doi: 10.1080/17415977.2017.1281920. |
[4] |
J. Cebeiro, M. K. Nguyen, M. Morvidone and C. Tarpau, An interior Compton Scatter Tomography, in 25th IEEE Nuclear Science Symposium and Medical Imaging Conference 2018 (IEEE NSS/MIC'18), Sydney, Australia, 2018.
doi: 10.1109/NSSMIC.2018.8824374. |
[5] |
J. Cebeiro, C. Tarpau, M. A. Morvidone, D. 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. |
[6] |
R. Clarke and G. Van Dyk, Compton-scattered gamma rays in diagnostic radiography, in Medical Radioisotope Scintigraphy. Ⅵ Proceedings of a Symposium on Medical Radioisotope Scintigraphy, 1969. |
[7] |
A. M. Cormack,
Representation of a function by its line integrals, with some radiological applications, Journal of Applied Physics, 34 (1963), 2722-2727.
|
[8] |
P. E. Cruvinel and F. A. Balogun,
Compton scattering tomography for agricultural measurements, Engenharia Agricola, 26 (2006), 151-160.
doi: 10.1590/S0100-69162006000100017. |
[9] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, McGraw-Hill Book Company, New York, 1954. |
[10] |
F. Farmer and M. P. Collins,
A new approach to the determination of anatomical cross-sections of the body by compton scattering of gamma-rays, Physics in Medicine & Biology, 16 (1971), 577.
doi: 10.1088/0031-9155/16/4/001. |
[11] |
S. Gautam, F. Hopkins, R. Klinksiek and I. Morgan,
Compton interaction tomography I. Feasibility studies for applications in earthquake engineering, IEEE Transactions on Nuclear Science, 30 (1983), 1680-1684.
doi: 10.1109/TNS.1983.4332614. |
[12] |
I. S. Gradshteyn, I. M. Ryzhik, D. Zwillinger and V. Moll, Table of Integrals, Series, and Products, 8th ed. Academic Press, Amsterdam, 2014. Available from: https://cds.cern.ch/record/1702455. |
[13] |
G. Harding and E. Harding,
Compton scatter imaging: A tool for historical exploration, Applied Radiation and Isotopes, 68 (2010), 993-1005.
doi: 10.1016/j.apradiso.2010.01.035. |
[14] |
E. M. Hussein, M. Desrosiers and E. J. Waller,
On the use of radiation scattering for the detection of landmines, Radiation Physics and Chemistry, 73 (2005), 7-19.
doi: 10.1016/j.radphyschem.2004.07.006. |
[15] |
K. C. Jones, G. Redler, A. Templeton, D. Bernard, J. V. Turian and J. C. Chu,
Characterization of Compton-scatter imaging with an analytical simulation method, Physics in Medicine & Biology, 63 (2018), 025016.
doi: 10.1016/j.tcs.2018.05.007. |
[16] |
P. Lale,
The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography, Physics in Medicine & Biology, 4 (1959), 159.
doi: 10.1088/0031-9155/4/2/305. |
[17] |
M. K. Nguyen and T. T. Truong, Imagerie par rayonnement gamma diffusé., Hermès Science, 2006. |
[18] |
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: 10.1088/0266-5611/26/9/099802. |
[19] |
S. J. Norton,
Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.
doi: 10.1063/1.357668. |
[20] |
P. G. Prado, M. K. Nguyen, L. Dumas and S. X. Cohen,
Three-dimensional imaging of flat natural and cultural heritage objects by a Compton scattering modality, Journal of Electronic Imaging, 26 (2017), 011026.
|
[21] |
J. Radon,
Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Akad. Wiss., 69 (1917), 262-277.
doi: 10.1090/psapm/027/692055. |
[22] |
G. Redler, K. C. Jones, A. Templeton, D. Bernard, J. Turian and J. C. Chu,
Compton scatter imaging: A promising modality for image guidance in lung stereotactic body radiation therapy, Medical Physics, 45 (2018), 1233-1240.
doi: 10.1002/mp.12755. |
[23] |
G. Rigaud,
Compton scattering tomography: feature reconstruction and rotation-free modality, SIAM Journal on Imaging Sciences, 10 (2017), 2217-2249.
doi: 10.1137/17M1120105. |
[24] |
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. |
[25] |
G. Rigaud, M. K. Nguyen and A. K. Louis,
Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography, Inverse Problems in Science and Engineering, 20 (2012), 809-839.
doi: 10.1080/17415977.2011.653008. |
[26] |
G. Rigaud, R. Régnier, M. K. Nguyen and H. Zaidi,
Combined modalities of Compton scattering tomography, IEEE Transactions on Nuclear Science, 60 (2013), 1570-1577.
doi: 10.1109/TNS.2013.2252022. |
[27] |
C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet and M. A. Morvidone,
Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography, IEEE Transactions on Computational Imaging, 6 (2020), 958-967.
doi: 10.1109/TCI.2020.2999672. |
[28] |
C. Tarpau and M. K. Nguyen,
Compton scattering imaging system with two scanning configurations, Journal of Electronic Imaging, 29 (2020), 130.
doi: 10.1117/1.JEI.29.1.013005. |
[29] |
C. Tarpau, J. Cebeiro, M. Morvidone and M. K. Nguyen, A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform, IEEE Transactions on Radiation and Plasma Medical Sciences, 2019.
doi: 10.1109/TRPMS.2019.2943555. |
[30] |
T. T. Truong, Function reconstruction from reflection symmetric radon data, Symmetry, 12 (2020).
doi: 10.3390/sym12060956. |
[31] |
T. Truong and M. K. Nguyen,
Compton scatter tomography in annular domains, Inverse Problems, 35 (2019), 054005.
doi: 10.1088/1361-6420/ab0b76. |
[32] |
T. T. Truong and M. K. Nguyen, Recent developments on Compton scatter tomography: Theory and numerical simulations, in Numerical Simulation-From Theory to Industry, IntechOpen, 2012. |
[33] |
T. T. Truong and M. K. Nguyen,
Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality, Inverse Problems, 27 (2011), 125001.
doi: 10.1088/0266-5611/27/12/125001. |
[34] |
J. Webber and E. L. Miller,
Compton scattering tomography in translational geometries, Inverse Problems, 36 (2020), 025007.
doi: 10.1088/1361-6420/ab4a32. |
[35] |
J. W. Webber and W. R. Lionheart,
Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), 084001.
doi: 10.1088/1361-6420/aac51e. |
[36] |
J. W. Webber and S. Holman,
Microlocal analysis of a spindle transform, Inverse Problems and Imaging, 13 (2019), 231-261.
doi: 10.3934/ipi.2019013. |
[37] |
J. W. Webber and E. T. Quinto,
Microlocal analysis of a Compton tomography problem, SIAM Journal on Imaging Sciences, 13 (2020), 746-774.
doi: 10.1137/19M1251035. |
[38] |
J. W. Webber, E. T. Quinto and E. L. Miller, A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging, 36 (2020), 074002.
doi: 10.1088/1361-6420/ab8f82. |







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