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August  2022, 16(4): 771-786. doi: 10.3934/ipi.2021075

Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry

Cécilia Tarpau 1,2,3,, , Javier Cebeiro 4, , Geneviève Rollet 1, , Maï K. Nguyen 2, and  Laurent Dumas 3,

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

*Corresponding author: Cécilia Tarpau

Received  June 2021 Revised  November 2021 Published  August 2022 Early access  December 2021

Fund Project: C. Tarpau research work is supported by grants from Région Île-de-France (in Mathematics and Innovation) 2018-2021 and LabEx MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l'Incertitude et des Interactions) (No. ANR-11-LBX-0023-01). J. Cebeiro research work is supported by a postdoctoral grant from the University of San Martín. He is also partially supported by SOARD-AFOSR (grant number FA9550-18-1-0523)

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.

Keywords: Analytic inversion, Compton scattering tomography, image formation, image reconstruction, radon transform on double circle arcs.
Mathematics Subject Classification: Primary: 44A12, 65R10; Secondary: 94A08.
Citation: Cécilia Tarpau, Javier Cebeiro, Geneviève Rollet, Maï K. Nguyen, Laurent Dumas. Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry. Inverse Problems and Imaging, 2022, 16 (4) : 771-786. doi: 10.3934/ipi.2021075
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.

Figure 1.  General functioning principle of a CST system. Photons are emitted by source $ S $, interact at sites $ M $, and are recorded at site $ D $. When a photon is detected carrying an energy $ E(\omega_1) $ (resp. $ E(\omega_2) $), the possible interaction sites lie on the upper (resp. lower) circle arc which subtends the angle $ (\pi-\omega_1) $ (resp. $ (\pi-\omega_2) $)
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Figure 2.  Previous proposed CST modalities. (A): Fixed source and detectors placed on a line. (B): Rotating pair source-detector diametrically opposed. (C): Rotating pair source detector. (D): Fixed source and detectors placed on a ring. (E): Detector rotating around a fixed source. (F) Source-detector translating simultaneously along two parallel lines. In all figures: The source $ S $ is represented by a red point. The detector(s) $ D $ is (are) represented by blue point(s). The $ M $, $ M_i $ or $ M'_i $ in black, are running points and examples of scattering site. An example of trajectory for a photon whose scattering site is $ M $ is shown in purple. The corresponding scattering angle is denoted $ \omega $. The object to scan is represented in grey. The red continuous curves are the examples of scanning circles arcs. For (C), (E) and (F), the dashed circles (resp. lines) represents the circular (resp. linear) paths on which move the sensors
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Figure 3.  Setup and parameterization of the CST modality proposed in [34]. The source $ S $ and the detector are respectively represented by a red and a blue point. To make the difference between the four half-arcs, $ S_1, S_3 $ and $ S_2, S_4 $ are respectively depicted in red and green. $ \Omega_{1, 2, 3, 4} $ denote the centres of the circles supporting the half-arcs $ S_{1, 2, 3, 4} $. The point $ M $ is an example of a scattering site
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Figure 4.  (A) Original object: Derenzo phantom. (B) Corresponding acquired data for $ N_{SD} = 2048 $ and $ N_r = 1024 $. A distance of one pixel is left between the upper part of the image and the detector path ($ \delta = 1 $, see 5.2.1)
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Figure 5.  Reconstruction results of the Derenzo phantom 4a for (A) $ \delta = 1 $ (NMSE = 0.0112), (B) $ \delta = 26 $ (NMSE = 0.0074) and (C) $ \delta = 51 $ (NMSE = 0.0061) pixel(s)
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Figure 6.  Evaluation of the number of source-detector positions on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for (A) $ x_{0, max} = 2N $ (NMSE = 0.0084), (B) $ 3N $ (NMSE = 0.0049) and (C) $ 4N $ NMSE = 0.0061) where $ \Delta_{x_0} = 1 $. Second row: Reconstruction results for (D) $ 0.5 $ (NMSE = 0.0046), (E) $ 1 $ (NMSE = 0.0049) and (F) $ 0.5 $ NMSE = 0.0049) detector per unit length and $ x_{0, max} = 3N $ remains constant
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Figure 7.  Evaluation of the number of scanning circles on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for (A) $ r_{max} = 2N $ (NMSE = 0.0058), (B) $ 3N $ (NMSE = 0.0040) and (C) $ 4N $ (NMSE = 0.0049) where $ \Delta_{r} = 1 $. Second row: Reconstruction results for (D) $ \Delta_{r} = 1 $ (NMSE = 0.0040), (E) $ 2 $ (NMSE = 0.0043) and (F) $ 4 $ (NMSE = 0.0043) detector per unit length and $ r_{max} = 3N $ remains constant
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