Estimation of optimal parameter of regularization of signal recovery
Keywords:signal recovery, image recovery, linear regularizing algorithm for estimation of the optimal signal parameter, regularization problem
AbstractIn 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.
TIKHONOV, A.N.; ARSENIN, V.Y. The Methods of Ill-Conditioned Problems Solution [in Russian]. Moscow: Nauka, 1979.
MOROZOV, V.A. Methods of Regularization of Unstable Problems [in Russian]. Moscow: Izd-vo Moskovskogo Un-ta, 1987.
BAKUSHINSKIY, A.B.; GONCHAROVSKIY, A.V. Ill-Conditioned Problems. Numerical Methods and Applications [in Russian]. Moscow: Izd-vo Moskovskogo Un-ta, 1989.
BENNING, M.; BURGER, M. “Modern regularization methods for inverse problems,” Acta Numerica, v.27, p.1-111, 2018. DOI: https://doi.org/10.1017/S0962492918000016.
TANANA, V.P.; SIDIKOVA, A.I. Optimal Methods for Ill-Posed Problems. With Applications to Heat Conduction. Berlin-Boston: De Gruyter, 2018. ISBN: 978-3-11-057721-1.
UGAYRAJ; MULANI, K.; TALUKDAR, P.; DAS, A.; ALAGIRUSAMY, R. “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, v.89, p.359-378, 2015. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.015.
STILLE, M.; KLEINE, M.; HÄGELE, J.; BARKHAUSEN, J.; BUZUG, T.M. “Augmented likelihood image reconstruction,” IEEE Trans. Medical Imaging, v.35, n.1, p.158-173, 2016. DOI: https://doi.org/10.1109/TMI.2015.2459764.
GASS, T.; SZÉKELY, G.; GOKSEL, O. “Consistency-based rectification of nonrigid registrations,” J. Medical Imaging, v.2, p.014005, 2015. DOI: https://doi.org/10.1117/1.JMI.2.1.014005.
TURITSYN, S.K.; PRILEPSKY, J.E.; LE, S.T.; WAHLS, S.; FRUMIN, L.L.; KAMALIAN, M.; DEREVYANKO, S.A. “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica, v.4, n.3, p.307-322, 2017. DOI: https://doi.org/10.1364/OPTICA.4.000307.
ADLER, J.; ÖKTEM, O. “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Problems, v.33, n.12, p.124007, 2017. DOI: https://doi.org/10.1088/1361-6420/aa9581.
KALTENBACHER, B. “Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems,” Inverse Problems, v.16, n.5, p.1523-1539, 2000. DOI: https://doi.org/10.1088/0266-5611/16/5/322.
KALTENBACHER, B.; OFFTERMATT, J. “A convergence analysis of regularization by discretization in preimage space,” Math. Comp., v.81, p.2049-2069, 2012. DOI: https://doi.org/10.1090/S0025-5718-2012-02596-8.
KALTENBACHER (BLASCHKE), B.; ENGL, H.W.; GREVER, W.; KLIBANOV, M. “An application of Tikhonov regularization to phase retrieval,” Nonlinear World, v.3, p.771-786, 1996.
KALTENBACHER, B. “Boundary observability and stabilization for Westervelt type wave equations without interior damping,” Appl. Math. Optim., v.62, n.3, p.381-410, 2010. DOI: https://doi.org/10.1007/s00245-010-9108-7.
DOVNAR, D.V.; PREDKO, K.G. “Method of eliminating rectilinear uniform blurring of an image,” Optoelectron. Instrument. Data Process., n.6, p.100, 1984.
DOVNAR, D.V.; PREDKO, K.G. “Use of orthogonalization of the mappings of basis functions for regularized restoration of a signal,” USSR Computational Mathematics and Mathematical Physics, v.26, p.13, 1986. DOI: http://doi.org/10.1016/0041-5553(86)90070-4.
VOSKOBOYNIKOV, Y.E. “Estimation of the optimal regularization parameter of an iterative wavelet algorithm for signal recovery,” Optoelectron. Instrument. Data Process., v.49, n.2, p.115, 2013. DOI: https://doi.org/10.3103/S8756699013020027.
VOSKOBOYNIKOV, Y.E.; LITASOV, V.A. “Stable algorithm for recover of image in case of ill-conditioned instrument function,” Avtometriya, v.42, n.6, p.3, 2006. URI: https://www.iae.nsk.su/images/stories/5_Autometria/5_Archives/2006/6/3-15.pdf.
PEREVERZEV, S.; SCHOCK, E. “On the adaptive selection of the parameter in regularization of ill-posed problems,” SIAM J. Numerical Analysis, v.43, n.5, p.2060-2076, 2006. URI: https://www.jstor.org/stable/4101307.
MINTS, M.Y.; PRILEPSKII, E.D. “Image discretization method applied for extended object restoration,” Optika i Spectroskopiya, v.75, p.696, 1993.
LUTTRELL, S.P. “A new method of sample optimization,” Optica Acta, v.32, n.3, p.255-257, 1985. DOI: https://doi.org/10.1080/713821739.
FRIEDEN, B.R. “Image-restoration using a norm of maximum information,” Optical Engineering, v.19, n.3, p.290-296, 1980. DOI: https://doi.org/10.1117/12.7972512.
KIDO, K. Discrete Fourier Transform, in Digital Fourier Analysis: Fundamentals. Undergraduate Lecture Notes in Physics. New York: Springer, 2015. DOI: https://doi.org/10.1007/978-1-4614-9260-3_4.
BORN, M.; VOLF, E. Basic Principles of Optic [in Russian]. Moscow: Nauka, 1973.