Top

Published in:

2020 | OriginalPaper | Chapter

The Singular Value Decomposition of the Operators of the Dynamic Ray Transforms Acting on 2D Vector Fields

Authors : Anna P. Polyakova, Ivan E. Svetov, Bernadette N. Hahn

Published in: Numerical Computations: Theory and Algorithms

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter An Online Learning Approach to a Multi-player N-armed Functional Bandit

next chapter Numerical Investigation into a Combined Ecologically Pure Plasma-Vortex Generator of Hydrogen and Thermal Energy

Sharafutdinov, V.A.: Integral geometry of tensor fields. VSP, Utrecht (1994). https://doi.org/10.1515/9783110900095

Svetov, I.E.: Reconstruction of the solenoidal part of a three-dimensional vector field by its ray transforms along straight lines parallel to coordinate planes. Numer. Anal. Appl. 5(3), 271–283 (2012). https://doi.org/10.1134/S1995423912030093CrossRefMATH

Derevtsov, E.Yu., Svetov, I.E.: Tomography of tensor fields in the plain. Eur. J. Math. Comput. Appl. 3(2), 24–68 (2015)

Derevtsov, E.Yu., Kashina, I.G.: Numerical solution of a vector tomography problem with the help of polynomial bases. Siberian J. Numer. Math. 5(3), 233–254 (2002) (in Russian)

Derevtsov, E.Yu., Svetov, I.E., Volkov, Yu.S., Schuster, T.: Numerical \(B\)-spline solution of emission and vector 2\(D\)-tomography problems for media with absorbtion and refraction. In: Remy, J.-G. (eds.) Proceedings of the 2008 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering SIBIRCON 2008, pp. 212–217 (2008). https://doi.org/10.1109/SIBIRCON.2008.4602618

Svetov, I.E., Derevtsov, E.Yu., Volkov, Yu.S., Schuster, T.: A numerical solver based on B-splines for 2D vector field tomography in a refracting medium. Math. Comput. Simul. 97, 207–223 (2014). https://doi.org/10.1016/j.matcom.2013.10.002MathSciNetCrossRef

Derevtsov, E.Yu., Kazantsev, S.G., Schuster, T.: Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions. J. Inverse Ill-Posed Probl. 15(1), 1–38 (2007). https://doi.org/10.1515/JIIP.2007.002MathSciNetCrossRef

Derevtsov, E.Yu., Efimov, A.V., Louis, A.K., Schuster, T.: Singular value decomposi-tion and its application to numerical inversion for ray transforms in 2D vector tomography. J. Inverse Ill-Posed Probl. 19(4–5), 689–715 (2011). https://doi.org/10.1515/jiip.2011.047

Svetov, I.E., Polyakova, A.P.: Comparison of two algorithms for the numerical solution of the two-dimensional vector tomography. Siberian Electron. Math. Rep. 10, 90–108 (2013) (in Russian). https://doi.org/10.17377/semi.2013.10.009

10.

Polyakova, A.P.: Reconstruction of a vector field in a ball from its normal radon transform. J. Math. Sci. 205(3), 418–439 (2015). https://doi.org/10.1007/s10958-015-2256-1MathSciNetCrossRef

11.

Polyakova, A.P., Svetov, I.E.: Numerical solution of the problem of reconstructing a potential vector field in the unit ball from its normal Radon transform. J. Appl. Ind. Math. 9(4), 547–558 (2015). https://doi.org/10.1134/S1990478915040110MathSciNetCrossRefMATH

12.

Rieder, A., Schuster, T.: The approximate inverse in action III: 3D-Doppler tomography. Numer. Math. 97(2), 353–378 (2004). https://doi.org/10.1007/s00211-003-0512-7MathSciNetCrossRefMATH

13.

Schuster, T.: Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method. Inverse Prob. 21(1), 75–91 (2005). https://doi.org/10.1088/0266-5611/21/1/006MathSciNetCrossRefMATH

14.

Svetov, I.E., Maltseva, S.V., Polyakova, A.P.: Numerical solution of 2D-vector tomography problem using the method of approximate inverse. In: Ashyralyev, A., Lukashov, A. (eds.) ICAAM 2016, AIP Conference Proceedings, vol. 1759, p. 020132. AIP Publishing LLC, Melville (2016). https://doi.org/10.1063/1.4959746

15.

Derevtsov, E.Yu., Louis, A.K., Maltseva, S.V., Polyakova, A.P., Svetov, I.E.: Numerical solvers based on the method of approximate inverse for 2D vector and 2-tensor tomography problems. Inverse Prob. 33(12), 124001 (2017). https://doi.org/10.1088/1361-6420/aa8f5aMathSciNetCrossRef

16.

Hahn, B.: Efficient algorithms for linear dynamic inverse problems with known motion. Inverse Prob. 30(3), 035008 (2014). https://doi.org/10.1088/0266-5611/30/3/035008MathSciNetCrossRefMATH

17.

Hahn, B.: Dynamic linear inverse problems with moderate movements of the object: ill-posedness and regularization. Inverse Prob. Imaging 9(2), 395–413 (2015). https://doi.org/10.3934/ipi.2015.9.395MathSciNetCrossRefMATH

18.

Hahn, B.: Null space and resolution in dynamic computerized tomography. Inverse Prob. 32(2), 025006 (2016). https://doi.org/10.1088/0266-5611/32/2/025006MathSciNetCrossRefMATH

Title: The Singular Value Decomposition of the Operators of the Dynamic Ray Transforms Acting on 2D Vector Fields
Authors: Anna P. Polyakova
Ivan E. Svetov
Bernadette N. Hahn
Publisher: Springer International Publishing
Book: Numerical Computations: Theory and Algorithms
Print ISBN: 978-3-030-40615-8

Electronic ISBN: 978-3-030-40616-5

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-40616-5_42

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner