Abstract
Terahertz tomography aims for reconstructing the complex refractive index of a specimen, which is illuminated by electromagnetic radiation in the terahertz regime, from measurements of the resulting (total) electric field outside the object. The illuminating radiation is reflected, refracted, and absorbed by the object. In this work, we reconstruct the complex refractive index from tomographic measurements by means of regularization techniques in order to detect defects such as holes, cracks, and other inclusions, or to identify different materials and the moisture content. Mathematically, we are dealing with a nonlinear parameter identification problem for the two-dimensional Helmholtz equation, and solve it with the Landweber method and sequential subspace optimization. The article concludes with some numerical experiments.
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References
W. Arendt, K. Urban, Partielle Differenzialgleichungen (Spektrum Akademischer Verlag, Berlin, 2010)
K. Atkinson, W. Han, Theoretical Numerical Analysis (Springer, New York, 2001)
G. Bao, P. Li, Inverse medium scattering for the Helmholtz equation at fixed frequency. Inverse Probl. 21(5), 16–21 (2005)
G. Bao, P. Li, Inverse medium scattering problems in near-field optics. J. Comput. Math. 25(3), 252–265 (2007)
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Science+Business Media, New York, 2011)
W.L. Chan, J. Deibel, D.M. Mittleman, Imaging with terahertz radiation. Rep. Prog. Phys. 70(8), 1325–1379 (2007)
D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, New York, 2013)
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics (American Mathematical Society, Providence, 1998)
B. Ferguson, X.-C. Zhang, Materials for terahertz science and technology. Nat. Mater. 1(1), 26–33 (2002)
D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2001)
J.P. Guillet, B. Recur, L. Frederique, B. Bousquet, L. Canioni, I. Manek-Hönninger, P. Desbarats, P. Mounaix, Review of terahertz tomography techniques. J. Infrared Millim. Terahertz Waves 35(4), 382–411 (2014)
P.R. Halmos, Measure Theory (Springer, Berlin, 2013)
M. Hanke, A. Neubauer, O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72(1), 21–37 (1995)
G. Narkiss, M. Zibulevsky, Sequential subspace optimization method for large-scale unconstrained optimization. Technical report, Technion - The Israel Institute of Technology, Department of Electrical Engineering, 2005
F. Natterer, The Mathematics of Computerized Tomography (Vieweg+Teubner Verlag, Berlin, 1986)
S. Sauter, C. Schwab, Boundary Element Methods (Springer, Berlin, 2011)
F. Scheck. Theoretische Physik 3 (Springer, Berlin, 2010)
F. Schöpfer, T. Schuster, Fast regularizing sequential subspace optimization in Banach spaces. Inverse Probl. 25(1), 015013 (2009)
F. Schöpfer, A.K. Louis, T. Schuster, Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods. J. Inverse Ill-Posed Probl. 16(5), 479–206 (2008)
J. Tepe, T. Schuster, B. Littau, A modified algebraic reconstruction technique taking refraction into account with an application in terahertz tomography. Inverse Probl. Sci. Eng. 25, 1448–1473 (2016)
A. Wald, T. Schuster, Sequential subspace optimization for nonlinear inverse problems. J. Inverse Ill-posed Probl. 25(4), 99–117 (2016)
D. Werner, Funktionalanalysis (Springer, New York, 2011)
Acknowledgements
The authors are indebted to Heiko Hoffmann for valuable discussions about the proof of Theorem 2.
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Wald, A., Schuster, T. (2018). Tomographic Terahertz Imaging Using Sequential Subspace Optimization. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_14
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DOI: https://doi.org/10.1007/978-3-319-70824-9_14
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