Analysis of the correlation properties of the wavelet transform coefficients of typical images

The increasing flow of photo and video information transmitted through the channels of infocommunication systems and complexes stimulates the search for effective compression algorithms that can significantly reduce the volume of transmitted traffic, while maintaining its quality. In the general case, the compression algorithms are based on the operations of converting the correlated brightness values of the pixels of the image matrix into their uncorrelated parameters, followed by encoding the obtained conversion coefficients. Since the main known decorrelating transformations are quasi-optimal, the task of finding transformations that take into account changes in the statistical characteristics of compressed video data is still relevant. These circumstances determined the direction of the study, related to the analysis of the decorrelating properties of the generated wavelet coefficients obtained as a result of multi-scale image transformation. The main result of the study was to establish the fact that the wavelet coefficients of the multi-scale transformation have the structure of nested matrices defined as submatrices. Therefore, it is advisable to carry out the correlation analysis of the wavelet transformation coefficients separately for the elements of each submatrix at each level of decomposition (decomposition). The main theoretical result is the proof that the core of each subsequent level of the multi-scale transformation is a matrix consisting of the wavelet coefficients of the previous level of decomposition. It is this fact that makes it possible to draw a conclusion about the dependence of the corresponding elements of neighboring levels. In addition, it has been found that there is a linear relationship between the wavelet coefficients within the local area of the image with a size of $8\times8$ pixels. In this case, the maximum correlation of submatrix elements is directly determined by the form of their representation, and is observed between neighboring elements located, respectively, in a row, column or diagonally, which is confirmed by the nature of the scattering. The obtained results were confirmed by the analysis of samples from more than two hundred typical images. At the same time, it is substantiated that between the low-frequency wavelet coefficients of the multi-scale transformation of the upper level of the expansion, approximately the same dependences are preserved uniformly in all directions. The practical significance of the study is determined by the fact that all the results obtained in the course of its implementation confirm the presence of characteristic dependencies between the wavelet transform coefficients at different levels of image decomposition. This fact indicates the possibility of achieving higher compression ratios of video data in the course of their encoding. The authors associate further research with the development of a mathematical model for adaptive arithmetic coding of video data and images, which takes into account the correlation properties of wavelet coefficients of a multi-scale transformation.