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February  2022, 16(1): 68-88. doi: 10.3934/ipi.2021041

A mathematical approach towards THz tomography for non-destructive imaging

Simon Hubmer 1,, , Alexander Ploier 2, , Ronny Ramlau 1,3, , Peter Fosodeder 4, and  Sandrine van Frank 4,

1. 

Johann Radon Institute Linz, Altenbergerstraße 69, A-4040 Linz, Austria
2. 

Doctoral Program Computational Mathematics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria
3. 

Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria
4. 

Research Center for Non-Destructive Testing GmbH (RECENDT), Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Simon Hubmer

Received  October 2020 Revised  April 2021 Published  February 2022 Early access  May 2021

In this paper, we consider the imaging problem of terahertz (THz) tomography, in particular as it appears in non-destructive testing. We derive a nonlinear mathematical model describing a full THz tomography experiment, and consider linear approximations connecting THz tomography with standard computerized tomography and the Radon transform. Based on the derived models we propose different reconstruction approaches for solving the THz tomography problem, which we then compare on experimental data obtained from THz measurements of a plastic sample.

Keywords: Terahertz tomography, tomographic imaging, non-destructive testing, radon transform, inverse and ill-posed problems.
Mathematics Subject Classification: Primary: 65J22, 78A10; Secondary: 65J20.
Citation: Simon Hubmer, Alexander Ploier, Ronny Ramlau, Peter Fosodeder, Sandrine van Frank. A mathematical approach towards THz tomography for non-destructive imaging. Inverse Problems and Imaging, 2022, 16 (1) : 68-88. doi: 10.3934/ipi.2021041
References:
[1]

E. Abraham, A. Younus, C. Aguerre, P. Desbarats and P. Mounaix, Refraction losses in terahertz computed tomography, Optics Communications, 283 (2010), 2050-2055.  doi: 10.1016/j.optcom.2010.01.013.

[2]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.

[3]

I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[4]

V. Dicken, Simultaneous Activity and Attenuation Reconstruction in Single Photon Emission Computed Tomography, a Nonlinear Ill-Posed Problem, PhD thesis, Universität Potsdam, 1998.

[5]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.  doi: 10.1088/0266-5611/15/4/307.

[6]

R. J. B. Dietz, N. Vieweg, T. Puppe, A. Zach, B. Globisch, T. Göbel, P. Leisching and M. Schell, All fiber-coupled THz-TDS system with kHz measurement rate based on electronically controlled optical sampling, Optics Letters, 39 (2014), 6482-6485.  doi: 10.1364/OL.39.006482.

[7]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.

[8]

B. Ferguson, S. Wang, D. Gray, D. Abbot and X.-C. Zhang, T-ray computed tomography, Optics Letters, 27 (2002), 1312-1314.  doi: 10.1364/OL.27.001312.

[9]

P. Fosodeder, S. Hubmer, A. Ploier, R. Ramlau, S. van Frank and C. Rankl, Phase-contrast THz-CT for non-destructive testing, Optics Express, 29 (2021), 15711-15723.  doi: 10.1364/OE.422961.

[10]

S. C. Garcea, Y. Wang and P. J. Withers, X-ray computed tomography of polymer composites, Composites Science and Technology, 156 (2018), 305-319.  doi: 10.1016/j.compscitech.2017.10.023.

[11] D. J. Griffiths, Introduction to Electrodynamics, 4th edition, Cambridge University Press, 2017.  doi: 10.1017/9781108333511.
[12]

J.-P. Guillet, B. Recur, L. Frederique, B. Bousquet, L. Canioni, I. Manek-Hönninger, P. Desbarats and P. Mounaix, Review of terahertz tomography techniques, Journal of Infrared, Millimeter, and Terahertz Waves, 35 (2014), 382-411.  doi: 10.1007/s10762-014-0057-0.

[13]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[14]

P. C. Hansen and J. Jorgensen, AIR tools Ⅱ: Algebraic iterative reconstruction methods, improved implementation, Numerical Algorithms, 79 (2018), 107-137.  doi: 10.1007/s11075-017-0430-x.

[15]

S. Hubmer and R. Ramlau, Convergence analysis of a two-point gradient method for nonlinear ill-posed problems, Inverse Problems, 33 (2017), 095004, http://stacks.iop.org/0266-5611/33/i=9/a=095004. doi: 10.1088/1361-6420/aa7ac7.

[16]

Y. Jin, G. Kim and S. Jeon, Terahertz Dielectric Properties of Polymers, Journal of the Korean Physical Society, 49 (2006), 513-517. 

[17]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, in Radon Series on Computational and Applied Mathematics, Vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[18]

B. Littau, J. Tepe, G. Schober, S. Kremling, T. Hochrein and P. Heidemeyer, Entwicklung und Evaluierung der Potenziale von Terahertz-Tomografie-Systemen, SKZ - Das Kunststoffzentrum (Eds.), Shaker Verlag, Aachen, 2016.

[19]

A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, 1989. doi: 10.1007/978-3-322-84808-6.

[20]

A. K. Louis and P. Maass, Contour Reconstruction in 3-D X-Ray CT, IEEE Transactions on Medical Imaging, 12 (1993), 764-769.  doi: 10.1109/42.251129.

[21]

S. Mukherjee and J. Federici, Study of structural defects inside natural cork by pulsed terahertz tomography, in 2011 International Conference on Infrared, Millimeter, and Terahertz Waves, (2011). doi: 10.1109/irmmw-THz.2011.6104965.

[22]

F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284.

[23]

J. Neu and C. A. Schmuttenmaer, Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS), Journal of Applied Physics, 124 (2018), 231101. doi: 10.1063/1.5047659.

[24]

D. M. Oberlin and E. M. Stein, Mapping properties of the radon transform, Indiana University Mathematics Journal, 31 (1982), 641-650.  doi: 10.1512/iumj.1982.31.31046.

[25]

B. Recur, J. P. Guillet, L. Bassel, C. Fragnol, I. Manek-Hönninger, J. Delagnes, W. Benharbone, P. Desbarats, J. Domenger and P. Mounaix, Terahertz radiation for tomographic inspection, Optical Engineering, 51 (2012), 1-8.  doi: 10.1117/1.OE.51.9.091609.

[26]

O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numerical Functional Analysis and Optimization, 17 (1996), 197-214.  doi: 10.1080/01630569608816691.

[27]

S. Sommer, M. Koch and A. A. Buda, Terahertz time-domain spectroscopy of plasticized poly(vinyl chloride), Analytical Chemistry, 90 (2018), 2409-2413.  doi: 10.1021/acs.analchem.7b04548.

[28]

J. Tepe, T. Schuster and B. Littau, A modified algebraic reconstruction technique taking refraction into account with an application in terahertz tomography, Inverse Problems in Science and Engineering, 25 (2017), 1448-1473.  doi: 10.1080/17415977.2016.1267168.

[29]

G. Trichopoulos and K. Sertel, Broadband terahertz computed tomography using a 5k-pixel real-time THz camera, Journal of Infrared, Millimeter, and Terahertz Waves, 36 (2015), 675-686.  doi: 10.1007/s10762-015-0144-x.

[30]

A. Wald and T. Schuster, Tomographic terahertz imaging using sequential subspace optimization, in New Trends in Parameter Identification for Mathematical Models, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-70824-9_14.

[31]

S. Wang and X.-C. Zhang, Pulsed terahertz tomography, Journal of Physics D: Applied Physics, 37 (2004), R1–R36. doi: 10.1088/0022-3727/37/4/R01.

[32]

M. Yahyapour, A. Jahn, K. Dutzi, T. Puppe, P. Leisching, B. Schmauss, N. Vieweg and A. Deninger, Fastest thickness measurements with a terahertz time-domain system based on electronically controlled optical sampling, Applied Sciences, 9 (2019), 1283. doi: 10.3390/app9071283.

show all references

References:
[1]

E. Abraham, A. Younus, C. Aguerre, P. Desbarats and P. Mounaix, Refraction losses in terahertz computed tomography, Optics Communications, 283 (2010), 2050-2055.  doi: 10.1016/j.optcom.2010.01.013.

[2]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.

[3]

I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[4]

V. Dicken, Simultaneous Activity and Attenuation Reconstruction in Single Photon Emission Computed Tomography, a Nonlinear Ill-Posed Problem, PhD thesis, Universität Potsdam, 1998.

[5]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.  doi: 10.1088/0266-5611/15/4/307.

[6]

R. J. B. Dietz, N. Vieweg, T. Puppe, A. Zach, B. Globisch, T. Göbel, P. Leisching and M. Schell, All fiber-coupled THz-TDS system with kHz measurement rate based on electronically controlled optical sampling, Optics Letters, 39 (2014), 6482-6485.  doi: 10.1364/OL.39.006482.

[7]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.

[8]

B. Ferguson, S. Wang, D. Gray, D. Abbot and X.-C. Zhang, T-ray computed tomography, Optics Letters, 27 (2002), 1312-1314.  doi: 10.1364/OL.27.001312.

[9]

P. Fosodeder, S. Hubmer, A. Ploier, R. Ramlau, S. van Frank and C. Rankl, Phase-contrast THz-CT for non-destructive testing, Optics Express, 29 (2021), 15711-15723.  doi: 10.1364/OE.422961.

[10]

S. C. Garcea, Y. Wang and P. J. Withers, X-ray computed tomography of polymer composites, Composites Science and Technology, 156 (2018), 305-319.  doi: 10.1016/j.compscitech.2017.10.023.

[11] D. J. Griffiths, Introduction to Electrodynamics, 4th edition, Cambridge University Press, 2017.  doi: 10.1017/9781108333511.
[12]

J.-P. Guillet, B. Recur, L. Frederique, B. Bousquet, L. Canioni, I. Manek-Hönninger, P. Desbarats and P. Mounaix, Review of terahertz tomography techniques, Journal of Infrared, Millimeter, and Terahertz Waves, 35 (2014), 382-411.  doi: 10.1007/s10762-014-0057-0.

[13]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[14]

P. C. Hansen and J. Jorgensen, AIR tools Ⅱ: Algebraic iterative reconstruction methods, improved implementation, Numerical Algorithms, 79 (2018), 107-137.  doi: 10.1007/s11075-017-0430-x.

[15]

S. Hubmer and R. Ramlau, Convergence analysis of a two-point gradient method for nonlinear ill-posed problems, Inverse Problems, 33 (2017), 095004, http://stacks.iop.org/0266-5611/33/i=9/a=095004. doi: 10.1088/1361-6420/aa7ac7.

[16]

Y. Jin, G. Kim and S. Jeon, Terahertz Dielectric Properties of Polymers, Journal of the Korean Physical Society, 49 (2006), 513-517. 

[17]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, in Radon Series on Computational and Applied Mathematics, Vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[18]

B. Littau, J. Tepe, G. Schober, S. Kremling, T. Hochrein and P. Heidemeyer, Entwicklung und Evaluierung der Potenziale von Terahertz-Tomografie-Systemen, SKZ - Das Kunststoffzentrum (Eds.), Shaker Verlag, Aachen, 2016.

[19]

A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, 1989. doi: 10.1007/978-3-322-84808-6.

[20]

A. K. Louis and P. Maass, Contour Reconstruction in 3-D X-Ray CT, IEEE Transactions on Medical Imaging, 12 (1993), 764-769.  doi: 10.1109/42.251129.

[21]

S. Mukherjee and J. Federici, Study of structural defects inside natural cork by pulsed terahertz tomography, in 2011 International Conference on Infrared, Millimeter, and Terahertz Waves, (2011). doi: 10.1109/irmmw-THz.2011.6104965.

[22]

F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284.

[23]

J. Neu and C. A. Schmuttenmaer, Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS), Journal of Applied Physics, 124 (2018), 231101. doi: 10.1063/1.5047659.

[24]

D. M. Oberlin and E. M. Stein, Mapping properties of the radon transform, Indiana University Mathematics Journal, 31 (1982), 641-650.  doi: 10.1512/iumj.1982.31.31046.

[25]

B. Recur, J. P. Guillet, L. Bassel, C. Fragnol, I. Manek-Hönninger, J. Delagnes, W. Benharbone, P. Desbarats, J. Domenger and P. Mounaix, Terahertz radiation for tomographic inspection, Optical Engineering, 51 (2012), 1-8.  doi: 10.1117/1.OE.51.9.091609.

[26]

O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numerical Functional Analysis and Optimization, 17 (1996), 197-214.  doi: 10.1080/01630569608816691.

[27]

S. Sommer, M. Koch and A. A. Buda, Terahertz time-domain spectroscopy of plasticized poly(vinyl chloride), Analytical Chemistry, 90 (2018), 2409-2413.  doi: 10.1021/acs.analchem.7b04548.

[28]

J. Tepe, T. Schuster and B. Littau, A modified algebraic reconstruction technique taking refraction into account with an application in terahertz tomography, Inverse Problems in Science and Engineering, 25 (2017), 1448-1473.  doi: 10.1080/17415977.2016.1267168.

[29]

G. Trichopoulos and K. Sertel, Broadband terahertz computed tomography using a 5k-pixel real-time THz camera, Journal of Infrared, Millimeter, and Terahertz Waves, 36 (2015), 675-686.  doi: 10.1007/s10762-015-0144-x.

[30]

A. Wald and T. Schuster, Tomographic terahertz imaging using sequential subspace optimization, in New Trends in Parameter Identification for Mathematical Models, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-70824-9_14.

[31]

S. Wang and X.-C. Zhang, Pulsed terahertz tomography, Journal of Physics D: Applied Physics, 37 (2004), R1–R36. doi: 10.1088/0022-3727/37/4/R01.

[32]

M. Yahyapour, A. Jahn, K. Dutzi, T. Puppe, P. Leisching, B. Schmauss, N. Vieweg and A. Deninger, Fastest thickness measurements with a terahertz time-domain system based on electronically controlled optical sampling, Applied Sciences, 9 (2019), 1283. doi: 10.3390/app9071283.

Figure 1.  Schematic drawing and image of the measurement setup. The THz radiation is generated by a transmitting antenna (Tx). Two Off-Axis Parabolic Mirrors (OPM) are used to create a focussed THz beam. After interacting with an object, the THz beam is guided to the detecting antenna (Rx) by two OPMs again. Exemplarily, a measured reference signal through air and one measured signal through an object is shown
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Figure 2.  THz beam profile in the focal plane along the parallel scanning direction. The energy density distribution was measured by shifting a rectangular aperture through the focal spot in z-direction
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Figure 3.  Triangular plastic sample (left), the measured electric field $ E_ {i,j} $ corresponding to $ ( {s_i,\theta_j}) = (35,0) $ (right, blue), and the reference field $ E_{\rm{ref}} $ (right, orange)
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Figure 4.  An example for the presence of multiple peaks in a THz signal. (left) The THz beam partially travels through air and the object. (right) This gives rise to two dominant peaks in the THz signal, one arising from the pulse that travelled through air and a second one that travelled through the object respectively. For our reconstructions only the largest peak (main peak) is used
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Figure 5.  Pre-processed data $ -2\log(\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert) $ (left) and $ -\log(I_ {i,j}/ I_{\rm{ref}}) $ (right)
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Figure 6.  Simulated measurement data $ \left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert $ for the triangular plastic sample depicted in Figure 3 (left), and the resulting reconstruction obtained via the nonlinear Landweber approach introduced in Section 4.1 (right)
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Figure 7.  Pre-processed data $ \left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert $ obtained from THz measurements of the triangular plastic sample depicted in Figure 3 (left), and the resulting reconstruction obtained via the nonlinear Landweber approach introduced in Section 4.1 (right)
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Figure 8.  Reconstructions for Problem 2 obtained from the data $ -2\log(\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert) $ depicted in Figure 5 (right) via the application of the following reconstruction methods introduced in Section 4.2: filtered back-projection (top left), contour tomography (top right), Landweber iteration (bottom left), Tikhonov regularization (bottom right)
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Figure 9.  Reconstructions for Problem 3 obtained from the data $ -\log(I_ {i,j}/ I_{\rm{ref}}) $ depicted in Figure 5 (left) via the application of the following reconstruction methods introduced in Section 4.2: filtered back-projection (top left), contour tomography (top right), Landweber iteration (bottom left), Tikhonov regularization (bottom right)
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