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Radioelectronics and Communications Systems

, Volume 61, Issue 9, pp 406–418 | Cite as

Estimation of Optimal Parameter of Regularization of Signal Recovery

  • Evgeni D. Prilepsky
  • Jaroslaw E. Prilepsky
Article
  • 3 Downloads

Abstract

In this paper there are researched regularizing properties of discretization in a space of output signals for some linear operator equation with noisy data. The essence of proposed method is selection of discretization level which is a parameter of the regularization in this context by the principle of equality of random and deterministic components of the input signal recovering error. It is shown the method, i.e. the solution which is discrete by input signal is stable to small inaccuracies in input signal. At that in case of definite level of output signal measurements inaccuracy the recovering error of input signal is unambiguously defined by input signal sampling increment that allows to select reasonably the regularization parameter for specific criterion, for example, for definite measurements inaccuracy. Specific calculations and examples are represented in explicit form for single-dimension case but this does not restricts generality of proposed method.

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References

  1. 1.
    A. N. Tikhonov, V. Ya. Arsenin. The Methods of Ill-Conditioned Problems Solution [in Russian] (Nauka, Moscow, 1979).Google Scholar
  2. 2.
    V. A. Morozov, Methods of Regularization of Unstable Problems [in Russian] (Izd-vo Moskovskogo Un-ta, Moscow, 1987).Google Scholar
  3. 3.
    A. B. Bakushinskiy, A. V. Goncharovskiy, Ill-Conditioned Problems. Numerical Methods and Applications [in Russian] (Izd-vo Moskovskogo Un-ta, Moscow, 1989).Google Scholar
  4. 4.
    M. Benning, M. Burger, “Modern regularization methods for inverse problems,” Acta Numerica 27, 1 (2018). DOI: 10.1017/S0962492918000016.MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. P. Tanana, A. I. Sidikova, Optimal Methods for Ill-Posed Problems. With Applications to Heat Conduction (De Gruyter, Berlin-Boston, 2018). ISBN: 978-3-11-057721-1.CrossRefzbMATHGoogle Scholar
  6. 6.
    Ugayraj, K. Mulani, P. Talukdar, A. Das, R. Alagirusamy, “Performance analysis and feasibility study of ant colony optimization, particle swarm optimization and cuckoo search algorithms for inverse heat transfer problems,” Int. J. Heat Mass Transfer 89, 359 (2015). DOI: 10.1016/j.ijheatmasstransfer.2015.05.015.CrossRefGoogle Scholar
  7. 7.
    M. Stille, M. Kleine, J. Hägele, J. Barkhausen, T. M. Buzug, “Augmented likelihood image reconstruction,” IEEE Trans. Medical Imaging 35, No. 1, 158 (2016). DOI: 10.1109/TMI.2015.2459764.CrossRefGoogle Scholar
  8. 8.
    T. Gass, G. Székely, O. Goksel, “Consistency-based rectification of nonrigid registrations,” J. Medical Imaging 2, 014005 (2015). DOI: 10.1117/1.JMI.2.1.014005.CrossRefGoogle Scholar
  9. 9.
    S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, No. 3, 307 (2017). DOI: 10.1364/OPTICA.4.000307.CrossRefGoogle Scholar
  10. 10.
    J. Adler, O. Öktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Problems 33, No. 12, 124007 (2017). DOI: 10.1088/1361-6420/aa9581.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    B. Kaltenbacher, “Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems,” Inverse Problems 16, No. 5, 1523 (2000). DOI: 10.1088/0266-5611/16/5/322.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. Kaltenbacher, J. Offtermatt, “A convergence analysis of regularization by discretization in preimage space,” Math. Comp. 81, 2049 (2012). DOI: 10.1090/S0025-5718-2012-02596-8.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Kaltenbacher (Blaschke), H. W. Engl, W. Grever, M. Klibanov, “An application of Tikhonov regularization to phase retrieval,” Nonlinear World 3, 771 (1996).MathSciNetzbMATHGoogle Scholar
  14. 14.
    B. Kaltenbacher, “Boundary observability and stabilization for Westervelt type wave equations without interior damping,” Appl. Math. Optim. 62, No. 3, 381 (2010). DOI: 10.1007/s00245-010-9108-7.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D. V. Dovnar, K. G. Predko, “Method of eliminating rectilinear uniform blurring of an image,” Optoelectron. Instrument. Data Process., No. 6, 100 (1984).Google Scholar
  16. 16.
    D. V. Dovnar, K. G. Predko, “Use of orthogonalization of the mappings of basis functions for regularized restoration of a signal,” USSR Computational Mathematics and Mathematical Physics 26, 13 (1986). DOI: 10.1016/0041-5553(86)90070-4.CrossRefzbMATHGoogle Scholar
  17. 17.
    Yu. E. Voskoboynikov, “Estimation of the optimal regularization parameter of an iterative wavelet algorithm for signal recovery,” Optoelectron. Instrument. Data Process. 49, No. 2, 115 (2013). DOI: 10.3103/S8756699013020027.CrossRefGoogle Scholar
  18. 18.
    Yu. E. Voskoboynikov, V. A. Litasov, “Stable algorithm for recover of image in case of ill-conditioned instrument function,” Avtometriya 42, No. 6, 3 (2006). URI: https://www.iae.nsk.su/images/stories/5_Autometria/5_Archives/2006/6/3-15.pdf.Google Scholar
  19. 19.
    S. Pereverzev, E. Schock, “On the adaptive selection of the parameter in regularization of ill-posed problems,” SIAM J. Numerical Analysis 43, No. 5, 2060 (2006). URI: https://www.jstor.org/stable/4101307.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Y. Mints, E. D. Prilepskii, “Image discretization method applied for extended object restoration,” Optika i Spectroskopiya 75, 696 (1993).Google Scholar
  21. 21.
    S. P. Luttrell, “A new method of sample optimization,” Optica Acta 32, No. 3, 255 (1985). DOI: 10.1080/713821739.MathSciNetCrossRefGoogle Scholar
  22. 22.
    B. R. Frieden, “Image-restoration using a norm of maximum information,” Optical Engineering 19, No. 3, 290 (1980). DOI: 10.1117/12.7972512.CrossRefGoogle Scholar
  23. 23.
    K. Kido, Discrete Fourier Transform, in Digital Fourier Analysis: Fundamentals. Undergraduate Lecture Notes in Physics (Springer, New York, 2015). DOI: 10.1007/978-1-4614-9260-3_4.Google Scholar
  24. 24.
    M. Born, E. Volf, Basic Principles of Optic [in Russian] (Nauka, Moscow, 1973).Google Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Aston UniversityBirminghamUK

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