Application of Two-Dimensional Padé-Type Approximations for Image Processing

Abstract

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.

Raster image processing using 2D Padé-type approximations

“…To accomplish this, we propose utilizing the two-dimensional Padé-type approximants method that we have previously developed. This method has shown promise in achieving these objectives in our previous work [12].…”

“…In our approach, we aim to construct Padé-type approximants for the harmonic two-dimensional Fourier series as a subset of power series [12]. In order to achieve this, we will present some basic principles of construction.…”

“…Traditionally, the focus has been on approximation techniques that utilize polynomials [8,9] and trigonometric functions [6,7,10]. One particularly successful type of approximation technique is the use of fractional rational functions [8,11,12]. The interest in this type of approximation has been steadily growing due to its wide range of applications in various fields, including theoretical physics, applied mechanics, geophysics, and more [8,11,12].…”

“…One particularly successful type of approximation technique is the use of fractional rational functions [8,11,12]. The interest in this type of approximation has been steadily growing due to its wide range of applications in various fields, including theoretical physics, applied mechanics, geophysics, and more [8,11,12]. These techniques enable the generalized summation of series and the extension of the function being approximated into the meromorphic domain.…”

“…Lately, there has been a significant focus on extending the classical theory of approximation using fractional-rational functions. This includes exploring different types of basis functions and various methods of constructing approximants, such as Padé-type approximations [12,[14][15][16][17][18][19]. By carefully selecting the construction method, significant enhancements in the desirable properties of the approximants can be achieved for specific classes of functions [12][13][14].…”