Mathematical method of graphic data spline-processing

Abstract

When you need to process experimental data, there is a problem of adequate reflection of physical processes arises. In engineering practice, situations are quite common when, based on a limited amount of numerical data (experimental or calculated), it is necessary to determine the nature of the functional dependence that they represent and calculate the values of this dependence for an arbitrary argument. In this case, it becomes necessary to replace a complex dependence with a simpler one, which, however, would convey the character of a complex one with an accuracy acceptable for practical purposes. Also, in the process of filtering or compressing any information, data is usually represented as a sequence of values. The data sequence obtained in this way is used further to construct a function of a certain class that approximates the input signal in the sense of the selected criterion. Further, when carrying out various transformations, a constructed function is used instead of a signal, which approximates it. To process such data, it is proposed to use spline transformations, which are one of the most progressive methods of data processing. The theory of interpolation and smoothing computational schemes is currently widely developed. This approach, called the numerical-analytical approach, is being applied increasingly in modern signal processing theory, which is explained by computational considerations. In this case, the degree of adequacy of the numerical-analytical model constructed in this way to the real investigated signal, the error in the approximation of its individual characteristics is also important. The article is devoted to the development of a method for constructing a linear spline with an adapted mesh of gluing nodes of this spline to improve the approximation properties of the spline function. For this, an iterative method of constructing splines is used. The development of a method for constructing a linear spline with an adapted mesh of the gluing nodes of this spline makes it possible to reduce the standard deviations of the spline from the function being approximated. That is, it improves the approximation properties of the spline function, which can be used to process various digital data. In particular, we use them in the tasks of filtering and compressing graphic information, processing satellite signals.