Modification of the Mathematical Apparatus of almost Periodic Analysis for Processing Spatio-Temporal Data with an Ordered Argument

Abstract

This paper discusses the mathematical apparatus of almost periodic analysis based on shift functions to solve the problem of identifying characteristic almost periods using vector computation. A process for converting an image from a Cartesian coordinate system to a polar coordinate system is presented, which allows structural almost periodic analysis using vector representation of the computation of the generalized shift function underlying almost periodic analysis. The process consists of two steps: the first step involves the determination of four rectangular grid nodes, for each polar grid node; and the second step interpolates the brightness values for the polar grid nodes from the rectangular grid. Thus, the image in Cartesian coordinates is converted to polar coordinates for further analysis. The mechanism for estimating a set of nearly periods in empirical data with ordered argument, represented by nonlinear oscillations with trend, is to investigate the results of matching the shift parameters on the argument Δt and the parameter τ - the nearly period in the generalized shift function. The inner summands of the generalized shift function are the result of trend exclusion. The paper considers the case of trend exclusion based on geometric progression. As a result, a modified method of calculating the results of the generalized shift function based on vector calculations is proposed in order to calculate the resulting near-periods from image slices. Such calculation method allows obtaining the results of almost periodic analysis of centric structures in images at a qualitative level and with an acceptable speed of data processing.