Skip to main content
Springer Nature Link
Log in

Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields

  • Conference paper
  • First Online:
Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

  • 784 Accesses

Abstract

Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components \(\xi ^j\) of direction vector \(\xi \) participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.

The reported study was funded by Russian Foundation for Basic Research (RFBR) and German Science Foundation (DFG) according to the joint German-Russian research project 19-51-12008 and by German Science Foundation (DFG) under project Lo 310/17-1.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Budinger, T., Gullberg, G., Huesman, R.: Emission computed tomography. In: Herman, G. (ed.) Image Reconstruction from Projections: Implementation and Applications, pp. 147–246. Springer, Heidelberg (1979). https://doi.org/10.1007/3-540-09417-2_5

    Chapter  Google Scholar 

  2. Natterer, F.: The Mathematics of Computerized Tomography. Wiley, Chichester (1986)

    MATH  Google Scholar 

  3. Natterer, F.: Inverting the attenuated vectorial Radon transform. J. Inverse Ill Posed Probl. 13(1), 93–101 (2005). https://doi.org/10.1515/1569394053583720

    Article  MathSciNet  MATH  Google Scholar 

  4. Kazantsev, S., Bukhgeim, A.: Inversion of the scalar and vector attenuated X-ray transforms in a unit disc. J. Inverse Ill Posed Probl. 15(7), 735–765 (2007). https://doi.org/10.1515/jiip.2007.040

    Article  MathSciNet  MATH  Google Scholar 

  5. Tamasan, A.: Tomographic reconstruction of vector fields in variable background media. Inverse Probl. 23(5), 2197–2205 (2007). https://doi.org/10.1088/0266-5611/23/5/022

    Article  MathSciNet  MATH  Google Scholar 

  6. Ainsworth, G.: The attenuated magnetic ray transform on surfaces. Inverse Probl. Imaging 7(1), 27–46 (2013). https://doi.org/10.3934/ipi.2013.7.27

    Article  MathSciNet  MATH  Google Scholar 

  7. Sadiq, K., Tamasan, A.: On the range characterization of the two-dimensional attenuated doppler transform. SIAM J. Math. Anal. 47(3), 2001–2021 (2015). https://doi.org/10.1137/140984282

    Article  MathSciNet  MATH  Google Scholar 

  8. Monard, F.: Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces. SIAM J. Math. Anal. 48(2), 1155–1177 (2016). https://doi.org/10.1137/15M1016412

    Article  MathSciNet  MATH  Google Scholar 

  9. Aben, H., Puro, A.: Photoelastic tomography for three-dimensional flow birefringence studies. Inverse Probl. 13(2), 215–221 (1997). https://doi.org/10.1088/0266-5611/13/2/002

    Article  MathSciNet  MATH  Google Scholar 

  10. Ainola, L., Aben, H.: Principal formulas of integrated photoelasticity of characteristic parameters. J. Opt. Soc. Am. A 22(6), 1181–1186 (2005). https://doi.org/10.1364/JOSAA.22.001181

    Article  MathSciNet  Google Scholar 

  11. Lionheart, W.R.B., Withers, P.J.: Diffraction tomography of strain. Inverse Probl. 31(4), 045005 (2015). https://doi.org/10.1088/0266-5611/31/4/045005

    Article  MathSciNet  MATH  Google Scholar 

  12. Karassiov, V.P.: Polarization tomography of quantum radiation: theoretical aspects and operator approach. Theor. Math. Phys. 145(3), 1666–1677 (2005). https://doi.org/10.1007/s11232-005-0189-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Panin, V.Y., Zeng, G.L., Defrise, M., Gullberg, G.T.: Diffusion tensor MR imaging of principal directions: a tensor tomography approach. Phys. Med. Biol. 47(15), 2737–2757 (2002). https://doi.org/10.1088/0031-9155/47/15/314

    Article  Google Scholar 

  14. Schmitt, J.M., Xiang, S.H.: Cross-polarized backscatter in optical coherence tomography of biological tissue. Opt. Lett. 23(13), 1060–1062 (1998). https://doi.org/10.1364/OL.23.001060

    Article  Google Scholar 

  15. Kuranov, R.V., Sapozhnikova, V.V., et al.: Complementary use of cross-polarization and standard OCT for differential diagnosis of pathological tissues. Opt. Express 10(15), 707–713 (2002). https://doi.org/10.1364/OE.10.000707

    Article  Google Scholar 

  16. Gelikonov, V.M., Gelikonov, G.V.: New approach to cross-polarized optical coherence tomography based on orthogonal arbitrarily polarized modes. Laser Phys. Lett. 3(9), 445–451 (2006). https://doi.org/10.1002/lapl.200610030

    Article  Google Scholar 

  17. Sharafutdinov, V.: A problem of integral geometry for generalized tensor fields on \(R^n\). Sov. Math. Dokl. 33(1), 100–102 (1986)

    MATH  Google Scholar 

  18. Sharafutdinov, V.: Integral Geometry of Tensor Fields. VSP, Utrecht (1994)

    Book  Google Scholar 

  19. Derevtsov, E.Yu., Polyakova, A.P.: Solution of the integral geometry problem for 2-tensor fields by the singular value decomposition method. J. Math. Sci. 202(1), 50–71 (2014). https://doi.org/10.1007/s10958-014-2033-6

    Article  MathSciNet  Google Scholar 

  20. 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.2013.10.002

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  22. Derevtsov, E.Yu., Maltseva, S.V.: Reconstruction of the singular support of a tensor field given in a refracting medium by its ray transform. J. Appl. Ind. Math. 9(4), 447–460 (2015). https://doi.org/10.1134/S1990478915040018

    Article  MathSciNet  Google Scholar 

  23. Monard, F.: Efficient tensor tomography in fan-beam coordinates. Inverse Probl. Imaging 10(2), 433–459 (2016). https://doi.org/10.3934/ipi.2016007

    Article  MathSciNet  MATH  Google Scholar 

  24. Monard, F.: Efficient tensor tomography in fan-beam coordinates. II: attenuated transforms. Inverse Probl. Imaging 12(2), 433–460 (2018). https://doi.org/10.3934/ipi.2018019

    Article  MathSciNet  MATH  Google Scholar 

  25. Mueller, R.K., Kaveh, M., Wade, G.: Reconstructive tomography and applications to ultrasonic. Proc. IEEE 67(4), 567–587 (1979). https://doi.org/10.1109/PROC.1979.11284

    Article  Google Scholar 

  26. Ball, J., Johnson, S.A., Stenger, F.: Explicit inversion of the Helmholtz equation for ultrasound insonification and spherical detection. In: Wang, K. (ed.) Acoustical Imaging, vol. 9. Springer, Boston (1980). https://doi.org/10.1007/978-1-4684-3755-3_26

    Chapter  Google Scholar 

  27. Schmitt, U., Louis, A.K.: Efficient algorithms for the regularization of dynamic inverse problems: I. Theory. Inverse Probl. 18(3), 645–658 (2002). https://doi.org/10.1088/0266-5611/18/3/308

    Article  MathSciNet  MATH  Google Scholar 

  28. Schmitt, U., Louis, A.K., Wolters, C., Vauhkonen, M.: Efficient algorithms for the regularization of dynamic inverse problems: II. Applications. Inverse Probl. 18(3), 659–676 (2002). https://doi.org/10.1088/0266-5611/18/3/309

    Article  MathSciNet  MATH  Google Scholar 

  29. Hahn, B., Louis, A.K.: Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotron-based X-ray tomography. Inverse Probl. 28(4), 045013 (2012). https://doi.org/10.1088/0266-5611/28/4/045013

    Article  MathSciNet  MATH  Google Scholar 

  30. Kireitov, V.R.: On the problem of determining an optical surface by its reflections. Funct. Anal. Appl. 10(3), 201–209 (1976). https://doi.org/10.1007/BF01075526

    Article  Google Scholar 

  31. Born, M., Wolf, E.: Principles of Optics. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  32. Goodman, J.: Introduction to Fourier optics. McGraw-Hill Book Company, New York (1968)

    Google Scholar 

  33. Kireitov, V.R.: Inverse Problems of the Photometry. Computing Center of the USSR Acad. Sci., Novosibirsk (1983). (in Russian)

    Google Scholar 

  34. Case, K., Zweifel, P.: Linear Transport Theory. Addison-Wesley Publishing Company, Boston (1967)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 630090, Novosibirsk, Russia

    Evgeny Yu. Derevtsov & Yuriy S. Volkov

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    Evgeny Yu. Derevtsov & Yuriy S. Volkov

  3. Saarland University, 66041, Saarbrücken, Germany

    Thomas Schuster

Authors
  1. Evgeny Yu. Derevtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuriy S. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Schuster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuriy S. Volkov .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Derevtsov, E.Y., Volkov, Y.S., Schuster, T. (2020). Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-40616-5_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-40615-8

  • Online ISBN: 978-3-030-40616-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics