Systems and Means of Informatics
2020, Volume 30, Issue 4, pp 1424
AVERAGE PROBABILITY OF ERROR IN CALCULATING WAVELETVAGUELETTE COEFFICIENTS WHILE INVERTING THE RADON TRANSFORM
 A. A. Kudryavtsev
 O. V. Shestakov
Abstract
Image reconstruction methods based on decomposition of the image function in a special wavelet basis and subsequent thresholding of the decomposition coefficients are used to solve computational tomography problems. Their attractiveness lies in adaptation to spatial inhomogeneities of images and the possibility of reconstructing local areas of the image from incomplete projection data that is of key importance, for example, for medical applications where it is undesirable to expose a patient to an unnecessary dose of radiation. The analysis of errors of these methods is an important practical task, since it allows one to assess the quality of both the methods themselves and the equipment used. The paper considers the waveletvaguelette decomposition method for reconstructing tomographic images in a model with an additive Gaussian noise. The order of the loss function based on the average probability of error in calculating wavelet coefficients is estimated.
[+] References (9)
 Donoho, D. 1995. Nonlinear solution of linear inverse problems by waveletvaguelette decomposition. Appl. Comput. Harmon. A. 2:101126.
 Lee, N. 1997. Waveletvaguelette decompositions and homogenous equations. West Lafayette, IN: Purdue University. PhD Diss. 103 p.
 Abramovich, F., and B.W. Silverman. 1998. Wavelet decomposition approaches to statistical inverse problems. Biometrika 85(1): 115129.
 Markin, A. V., and O.V. Shestakov. 2010. Asimptotiki otsenki riska pri porogovoy obrabotke veyvletveyglet koeffitsientov v zadache tomografii [The asymptotics of risk estimate for waveletvaguelette coefficients' thresholding in the problems of tomography]. Informatika i ee Primeneniya  Inform. Appl. 4(2):3645.
 Mallat, S. 1999. A wavelet tour of signal processing. New York, NY: Academic Press. 857 p.
 Zakharova, T. V., and O.V. Shestakov. 2016. Teoriya veyvletov i ee primenenie v obrabotke signalov [Wavelet theory and its application in signal processing]. Moscow: MasterPrint. 180 p.
 Sadasivan, J., S. Mukherjee, and C. S. Seelamantula. 2014. An optimum shrinkage estimator based on minimumprobabilityoferror criterion and application to signal denoising. 39th IEEE Conference (International) on Acoustics, Speech and Signal Processing Proceedings. Piscataway, NJ: IEEE. 42494253.
 Kudryavtsev, A. A., and O. V. Shestakov. 2016. Asymptotic behavior of the threshold minimizing the average probability of error in calculation of wavelet coefficients. Dokl. Math. 93(3):295299.
 Kudryavtsev, A. A., and O. V. Shestakov. 2018. Minimizatsiya oshibok vychisleniya veyvletkoeffitsientov pri reshenii obratnykh zadach [Minimization of errors of calculating wavelet coefficients while solving inverse problems]. Informatika i ee Primeneniya  Inform. Appl. 12(2): 1723.
[+] About this article
Title
AVERAGE PROBABILITY OF ERROR IN CALCULATING WAVELETVAGUELETTE COEFFICIENTS WHILE INVERTING THE RADON TRANSFORM
Journal
Systems and Means of Informatics
Volume 30, Issue 4, pp 1424
Cover Date
20201210
DOI
10.14357/08696527200402
Print ISSN
08696527
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
Radon transform; waveletvaguelette decomposition; thresholding; loss function
Authors
A. A. Kudryavtsev and O. V. Shestakov ,
Author Affiliations
Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University, 152 Leninskie Gory, GSP1, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center "Computer Science
and Control", Russian Academy of Sciences, 442 Vavilov Str., Moscow 119333, Russian Federation
