Abstract

Research focuses on inventing a methodology for creating linear splines with a customized grid of glue nodes. The results aim to improve the approximation properties of the spline function for processing various digital content. In particular, the processing of satellite signals, as well as the performance of image filtering or compression using the proposed results, play a key role in the design of the compression process of digital graphics to make it cheaper to store and send. Hermit spline, multiple-scale analysis, image compression, digital image According to Cisco's forecasts [1], global digital transformation from 2016 to 2021 will continue to CMiGIN 2022: 2nd International Conference on Conflict Management in Global Information Networks, November 30, 2022, Kyiv, Ukraine ORCID: 0000-0002-1301-9611 (O. Kolganova); 0000-0001-8183-9016 (L. Tereshchenko); 0000-0003-0399-7013 (V. Kravchenko); 0000(V. Volkogon); 0000-0002-9761-5583 (V. Shutko); 0000-0002-3531-7724 (M. Shutko); 0000-0003-3810-6460 (N. Mykhalchyshyn)

1. Introduction

significantly affect the demands and needs of IP networks.

This is primarily indicated by: the projected growth in the number of Internet users (from 3.3 to 4.6 billion, i.e. 58% of people worldwide), a quickening growth in the use of personal devices and machineto-machine connections, an increase in the average capacity of broadband access and an increase in video traffic.

Regarding its proportion of all IP traffic and the expansion of Internet traffic in general, the video will continue to rule. Compared to 2016, when this indicator was 67% of Internet traffic, at the end of 2021 it was already 80%. By the end of 2021, there will be almost 2 billion active consumers of video content on Internet. It eliminates those who only communicate through mobile devices. It eliminats those who only communicate through mobile devices (excluding those who use exclusively mobile communications). For comparison, this indicator was almost 1.5 billion active consumers in 2016. At the end of 2021, via the global Internet per month 3 trillion minutes of video were transmitted (equivalent to 5 million video years per month or 1 million video minutes per second).

We must also realize that regardless of our desires our daily lives are filled with instances of computer vision [31]. First, photo filtering and compression are ubiquitous in our news feeds, social networks, online shops, books. An image should be thought of as functions that represent it in the form of a matrix of pixel values. In this case, filters serve as systems for creating an improved version of the image based on a combination of pixel values of the original image. For compression purposes, images are also considered in a two-dimensional array of pixels (Fig. 1) [18-21].

2022 Copyright for this paper by its authors.

In order to reduce the volume of data arrays to facilitate their processing, storage and transmission, coding methods have become widely used [9-11, 17, 25-27]. These methods are based on approximation or partial data extraction. These include lossy compression methods (irreversible compression). More components are deleted when the compression ratio is higher, resulting in coarser images. It is the opposite of reversible data compression (lossless compression), which does not degrade data. When compared to lossless approaches, the lossy compression technique offers a significantly higher level of data reduction [8].

Lossy compression significantly reduces file size before the end user notices a deterioration in quality. Even strong compression may be desirable to further reduce the amount of data (to cut back on storage space or transmission times, for example). Most lossy compression algorithms make use of the discrete cosine transform (DCT), which was first presented by N. Ahmed, T. Natarajan, and K. R. Rao in 1974 [6].

The most typical application of this technology is to compress multimedia data, more so in streaming multimedia and Internet telephony. In comparison, compression of text files and data files (text elements and financial records) is usually done using a lossless method. For this purpose, the original lossless master file will be the basis for creating additional copies in the future. It may be useful to create a lossless master file that will then serve as a backbone for additional copying. This way, there is no need to make other lossy compressed copies of the original file, accumulating additional artifacts and data loss [12, 13, 15, 16].

We can depict a picture made up of individual pixels as a function of to better comprehend the characteristics of images [28-30] and the technical process utilized to process them. Every pixel has a distinct meaning. Regarding grayscale images, pixel's intensity ranges from 0 to 255, where black is 0 and white is 255 (Fig. 2).

Assume that the function ( , ), is defined on a rectangular matrix with a finite range of intensity at the pixel position ( , )

: [ , ] [ , ] → [0,255]. ( 1 )

This method is simply continued in a color image. In its place, the function ( , ) is now a vector of three values: ( 2 ) .

Mixing red, green, and blue colors form a palette of color images according to the RGB palette. Accordingly, three channels, which we consider in the form of 1x3 vectors, represent each pixel. Since the integer values from 0 to 255 correspond to each of the specified colors, the number of color combinations is calculated as 256 × 256 × 256 = 16777216. Thus, the union of the three functions will correspond to the function ( , ) (Fig. 3).

Image warping and image filtering are common technologies of image processing. Realization of the range of the image by changing the value of the pixels is possible by using image filtering. In this case, the image colors change without changing the pixel positions. Changing the image area by changing the position of the pixels is possible using image warping. In this case, the points in the exposure area are matched without changing colors [11].

In the process of compressing information, the data is usually presented as a sequence of instantaneous values. The resulting data sequence is used to construct a function of some class that approximates the input signal in the sense of the selected criterion. In the future, when performing various transformations, a constructed function is used instead of a signal, which approximates it. This numerical-analytical approach is increasingly used in the modern signal processing theory, which is explained by computational considerations [23, 39, 40]. Quite important is the degree of adequacy of the numerical-analytical model constructed in this way to the real signal under study, the error of approximation of its individual characteristics [24, 41, 42]. Naturally, the approximation error depends on the selected class of functions [2-7, 22].

2. Related works and problem statement

Mathematicians have long encountered an intuitive method for using piecewise functions in approximation issues [34]. But, as N. P. Korniychuk notes, the intrusion of splines into the theory of approximation occurred due to interpolation problems, due to their good computational and approximate properties [7].

The beginning of the development of spline interpolation theory and the term spline itself is deduced from 1946 in the article by Isaac Jacob Schoenberg [3, 4]. Its intensive development took place in the 50s and 70s, and the traditional field of application of interpolation splines became computer-aided design systems at this time. However, the potential of splines is much broader than just describing some curves [32]. In the real world, many physical processes are by their very nature splines. In mechanics, it is the deformation of a flexible plate or rod fixed at individual points, or the trajectory of the object, if the force acting on it, changes stepwise (the trajectory of an artificial space object with active and inertial segments of motion, the trajectory of the aircraft with a step change of thrust and change the profile of the wing, etc.). In thermodynamics, this is the heat exchange in a rod composed of fragments with different heat transfers. In chemistry - diffusion through layers of different substances. In electricity - the propagation of electromagnetic fields through heterogeneous media. That is, the spline is not a contrived mathematical abstraction, and in many cases, it is a solution of differential equations that describe very real physical processes [38].

There are a large number of structures called splines. So let's try to make some detail in this variety.

Type of spline fragments. The fact that the spline consists of fragments of the same type is one of the key features that distinguishes it from other piece functions. The most famous splines, consist of fragments - algebraic polynomials not higher than a given degree. As a rule, these are cubic polynomials, or polynomials of non-even degrees. Linear, cubic, fifth degree. Higher degrees are rarely used, due to the complexity of the calculations and the complexities described in the previous section. Their main advantage is the simplicity of calculations and analysis. The disadvantage is that relatively few real physical processes correspond to this dependence [33].

Setting objectives. Of particular interest is the class of polynomial splines, which successfully combines the advantages of static polynomials and the ability to control the smoothness of approximations [2, 5-7].

The choice of polynomial splines as reduction functions in interpolation and approximation necessitates the resolution of a number of issues, the most important of which are the issues with selecting the degree of spline and the quantity of nodes in the interpolation grid [36, 40-42].

The spline degree used to recover the signal is based on a priori information about the differential properties of the signal, namely, the degree is selected so that the differential properties of the spline coincide with the a priori known differential properties of the signal. Due to the fact that in practice there is information only about the initial derivatives of the signal, for approximation it is advisable to use Hermitian splines with different tasks of the values of the derivatives in the interpolation nodes.

In some cases, consider functions that are close to the boundary between splines and ordinary functions and splines and piecewise functions. It: • Splines consisting of two fragments. They have a simplified version of the construction, but special attention should be paid to boundary conditions • A piecewise constant spline function has no continuity even values. A trivial option that does not have the main advantage of splines - smoothness. As well as the laman, has rather methodical value for developing technology of work with splines.

The grid of interpolation points can significantly affect the efficiency of calculations. Important are the cases of uniform grid and uniform grid, with the distance between the points of a multiple of the distance between the nodes of the spline. But the grid can be uneven [14].

Consider the process of creating a linear spline using this spline's modified grid of glue nodes. When using multiscale analysis to compress and filter graphical data, it is interesting to take splines into consideration [35, 37].

3. Iterative method of constructing linear splines

In practical problems of smoothing experimental data, it is often advisable to dwell on piecewise linear approximation. So, for example, according to the given observations ( xi , yi ) , i = 1, N , where a  xi  b , it is necessary to construct a spline P( x) - r -frames broken line so that  = [P] = iN=1i (P( xi ) − yi )2  min where i  0 – set weights.

We describe the spline P( x) by a set of 2r numbers, namely: ~x1  ~x2    ~xr −1 - nodes in the interval [a, b] (for commonality we also put, ~x0 = a , ~xr = b ) and the value of the function in the nodes P(~xk ) = ak , 0  k  r (Fig. 4).  k = [ ~xk −1, ~xk ) when 1  k  r −1 ,  r = [ ~xr −1, ~xr ]

 k =  k [P] = k i (P(xi ) − yi )2 

− where k the amount by i, for which xi   k .

The following iterative algorithm for solving the problem is proposed. ( 3 ) (4) Step 0. The initial values ~xk = ~xk(0) are set and the corresponding values ak = ak(0) are determined, ~ which minimize the functional  when xk are fixed.

Step 1. a) Fix the odd nodes, that is the nodes x1, ~x3 ,  and values of the function in them: ~ a1(0) , a3(0) ,  and under this condition we calculate the optimal values of ~x0 = a = ~x0( 1 ) , ~x2 = ~x ( 1 ) 2 , ~x4 = ~x ( 1 ) 4 , … and the corresponding values of a2 = a2( 1 ) , a4 = a( 1 ) 4 ,….

~x2( 1 ) , , ~x6( 1 ) ,  b) For fixed even nodes, we mean are given, we optimize the odd nodes, so ~x1 = ~x1( 1 ) , ~x3 = ~x( 1 )

3 ,… and determine the value of the function in the nodes a1( 1 ) , a3( 1 ) , .

The result is nodes ~x0( 1 ) = a , ~x1( 1 ) , ~x2( 1 ) ,…, ~xr(−1)1 , ~xr( 1 ) = b and values in nodes: a0( 1 ) , a1( 1 ) , a2( 1 ) , .

Suppose the n − 1 steps are done and we have nodes Pn−1(~xk(n−1) ) = ak(n−1) , 0  k  r .

The spline built on this information will be initialized P(x) and defined [P1 ] . ~x (n−1) k and a spline

Pn−1 (x) ;

Let’s calculate [Pn−1] . Based on the specified accuracy of calculations  , check the condition [Pn−2 ] − [Pn−1 ]   . If it is executed, we stop the iterative process and consider P(x) = Pn−1 (x) . Otherwise, we proceed to the n -th step, which differs from the above-described 1st step in that the superscript (0) is replaced by (n −1) , and ( 1 ) – by (n) .

~ The movement of xk is illustrated in Figure 5.

~ Figure 5: Movement of xk

Because of the dependence on the value of ak only values of k and k+1 , then the minimum of the  conditions takes the form: 1 = 0 k + k+1 = 0 r = 0 (9) a0 ; ak ak , 1 k  r −1; ar .

Let’s initialize the a1 and so on to the ar . To do this, denote

ak = Aka0 + Bk , 0  k  r .

Let's put, A0 =1, B0 = 0. Then with (14) we have

A1 = −1i pi2 1i piqi ,

B1 = 1iyi pi 1i piqi .

Substituting (17) into (15), we obtain: (6) (7) (8) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) Obviously, pi + qi =1.

Expressions (6), (7), (8) take a simple form k = ki(ak−1pi + akqi − yi)2, qi = ~xxk −−~x~xkk−−11 , pi = ~x~xkk−−~xxki−1 .

i 1 k = k i(ak−1pi + akqi − yi)pi 2 ak−1 , 12 akk = k i(ak−1pi + akqi − yi)qi .

System (9) is disclosed as follows:

a01i pi2 + a11i piqi = 1iyi pi, ak−1ki piqi + ak (kiqi2 + k+1i pi2)+ ak+1k+1i piqi = = kiyiqi + k+1iyiqi , 1 k  r −1, ar−1ri piqi + arriqi2 = riyiqi .

We will consider a0 as indefinite size and we will express the a1 through the a0 , the a2 through (21) (22)

,  Bk = i yi qi +  y p − Bk −2 i pi qi −

i i i k−1 k k−1   − Bk −1 iqi2 +  p2 

i i   k−1 k  i piqi k , 2  k  r .

A

B r .

Now let's subst20ute

Thus, we consistently calculate k , up to r , ar −1 = Ar −1a0 + Br −1 and ar = Ar a0 + Br in the equation (16):

A

Bk

a From this equation we define and substitute 0 in (17), where

a values of k defined completely.

4. Conclusions

A k and

B k we already know. Thus,

This approach makes it possible to vary the grid of interpolation points, choosing its characteristics in a specific situation to achieve the optimum outcome. The creation of a method for creating a linear spline that uses a customized grid for its glue nodes allows for a reduction in the spline's standard deviations from the approximated function. In other words, it enhances the spline function's approximation qualities, which can be utilized to process a range of digital data. We implement them specifically while processing satellite signals, filtering and compressing graphic data. This paper did not set out the task of developing a complete method for compression of digital images. The research will be continued for the creation of a mathematical approach to develop spline functions of the third degree with adaptive calculation of the gluing nodes grid of this spline. Such splines are of interest regarding their practical use in signal and image processing problems. We can conclude that this strategy offers room for advancement.

5. References

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