APPLICATION OF TWO-DIMENSIONAL PADÉ-TYPE APPROXIMATIONS FOR IMAGE PROCESSING
DOI:
https://doi.org/10.15588/1607-3274-2023-1-10Keywords:
Padé-type approximants, Gibbs phenomenon, size of the image fileAbstract
Context. The Gibbs phenomenon introduces significant distortions for most popular 2D graphics standards because they use a finite sum of harmonics when image processing by expansion of the signal into a two-dimensional Fourier series is used in order to reduce the size of the graphical file. Thus, the reduction of this phenomenon is a very important problem.
Objective. The aim of the current work is the application of two-dimensional Padé-type approximations with the aim of elimination of the Gibbs phenomenon in image processing and reduction of the size of the resulting image file.
Method. We use the two-dimensional Padé-type approximants method which we have developed earlier to reduce the Gibbs phenomenon for the harmonic two-dimensional Fourier series. A definition of a Padé-type functional is proposed. For this purpose, we use the generalized two-dimensional Padé approximation proposed by Chisholm when the range of the frequency values on the integer grid is selected according to the Vavilov method. The proposed scheme makes it possible to determine a set of series coefficients necessary and sufficient for construction of a Padé-type approximation with a given structure of the numerator and denominator. We consider some examples of Padé approximants application to simple discontinuous template functions for both formulaic and discrete representation.
Results. The study gives us an opportunity to make some conclusions about practical usage of the Padé-type approximation and about its advantages. They demonstrate effective elimination of distortions inherent to Gibbs phenomena for the Padé-type approximant. It is well seen that Padé-type approximant is significantly more visually appropriate than Fourier one. Application of the Padétype approximation also leads to sufficient decrease of approximants’ parameter number without the loss of precision.
Conclusions. The applicability of the technique and the possibility of its application to improve the accuracy of calculations are demonstrated. The study gives us an opportunity to make conclusions about the advantages of the Padé-type approximation practical usage.
References
Timan A. F. Theory of approximation of functions of a real variable. New York, MacMillan, 1963, 631 p. https://doi.org/10.1016/c2013-0-05307-8
Mitra S. K. Digital Signal Processing: A Computer-Based Approach. New York, McGraw-Hill, 2001, 866 p. https://doi.org/10.1016/s0026-2692(98)00072-x
Helmberg G. Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions, Journal of Fourier Analysis and Applications, 2002, Vol. 8(1), pp. 29– 42. DOI: 10.1007/s00041-002-0002-9
Archibald R. and Gelb A. A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity, IEE Transactions of Medical Imaging, 2002, Vol. 21(4), pp. 305–319. DOI: 10.1109/TMI.2002.1000255
Veraart J., Fieremans E., Jelescu I. O., Knoll F., and Novikov D. S. Gibbs ringing in diffusion MRI, Magn. Reson. Med., 2016, Vol. 76, pp. 301–314. DOI: 10.1002/mrm.25866
Serov V. Fourier Series, Fourier Transform and Their Applications to Mathematical Physics. New York, Springer International Publishing, 2017, 534 p. DOI: 10.1007/978-3319-65262-7.
Maggioli F., Melzi S., Ovsjanikov M., Bronstein M. M., Rodolà E. Orthogonalized fourier polynomials for signal approximation and transfer, Computer Graphics Forum, 2021, Vol. 40(2), pp. 435–447. https://doi.org/10.1111/cgf.142645
Andrianov I., Awrejcewicz J., Danishevskyy V., Ivankov A. Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions. New York, John Wiley & Sons, 2014, 288 p. DOI: 201410.1002/9781118725184.
Olevska Yu. B., Olevskyi V. I., Olevskyi O. V. Using of fuzzy mathematical models in automated systems for recognition of high molecular substances, Application of Mathematics in Technical and Natural Sciences: 10th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences – AMiTaNS’18, Albena, 20–25 June: proceedings. New York, American Institute of Physics, Melville, NY, 2018, pp. 060003-1– 060003-9. (AIP Conference Proceedings, Vol. 2025(1)). https://doi.org/10.1063/1.5064911
Prots’ko I. O., Kuzminskij R. D., Teslyuk V. M. Efficient computation of the integer DCT-II for compressing images, Radio Electronics, Computer Science, Control, 2019, No. 2, pp. 151–157. https://doi.org/10.15588/1607-3274-2019-2-16
Baker J. A., Jr. and Graves-Morris P. Padé approximants. New York, Cambridge University Press, 1996, 746 p. https://doi.org/10.1017/cbo9780511530074
Andrianov I. V., Olevskyi V. I., Shapka I. V., Naumenko T. S. Technique of Padé-type multidimensional approximations application for solving some problems in mathematical physics, Application of Mathematics in Technical and Natural Sciences: 10th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences – AMiTaNS’18, Albena, 20–25 June, 2018: proceedings. New York, American Institute of Physics, Melville, NY, 2018, pp. 040002-1–040002-9. (AIP Conference Proceedings, Vol. 2025 (1)). DOI: 10.1063/1.5064886
Bosuwan N., López Lagomasino G. Inverse Theorem on Row Sequences of Linear Padé-orthogonal Approximation, Comput. Methods Funct. Theory, 2015, Vol. 15, pp. 529– 554. https://doi.org/10.1007/s40315-015-0121-3
Labych Yu. A., Starovoitov A. P. Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients, Sb. Math, 2009, Vol. 200(7), pp. 1051– 1074. DOI: 10.1070/SM2009v200n07ABEH004027
Buslaev V. I., Suetin S. P. On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions, J. Approx. Theory, 2016, Vol. 206, pp. 48–67. DOI: 10.1016/j.jat.2015.08.002
Sablonniere P. Padé-Type Approximants for Multivariate Series of Functions, Lecture Notes in Mathematics, 1984, Vol. 1071, pp. 238–251. https://doi.org/10.1007/bfb0099622
Kida S. Padé-type and Padé approximants in several variables, Appl. Numer. Math, 1989/90, Vol. 6, pp. 371–391. https://doi.org/10.1016/0168-9274(90)90027-D
Olevska Yu. B., Olevskyi V. I., Shapka I. V., and Naumenko T. S. Application of two-dimensional Padé-type approximants for reducing the Gibbs phenomenon, Application of Mathematics in Technical and Natural Sciences: 11th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences – AMiTaNS’19, Albena, 20–25 June, 2019: proceedings. New York, American Institute of Physics, Melville, NY, 2018, pp. 060014-1–060014-8. (AIP Conference Proceedings, Vol. 2164). https://doi.org/10.1063/1.5130816
Daras N. J. The convergence of Padé-type approximants to holomorphic functions of several complex variables, Appl. Numer. Math, 1989/90, Vol. 6, pp. 341–360. https://doi.org/10.1016/0168-9274(90)90025-B
TESTIMAGES free collection of digital images for testing [Electronic resource]. Access mode: https://testimages.org/Received 00.00.2023.
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Copyright (c) 2023 V. I. Olevskyi, V. V. Hnatushenko, G. M. Korotenko, Yu. B. Olevska, Ye. O. Obydennyi
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