Wavelets and digital filters designed and synthesized in the time and frequency domains

Abstract

<abstract> <p>The relevance of the problem under study is due to the fact that the comparison is made for wavelets constructed in the time and frequency domains. The wavelets constructed in the time domain include all discrete wavelets, as well as continuous wavelets based on derivatives of the Gaussian function. This article discusses the possibility of implementing algorithms for multiscale analysis of one-dimensional and two-dimensional signals with the above-mentioned wavelets and wavelets constructed in the frequency domain. In contrast to the discrete wavelet transform (Mallat algorithm), the authors propose a multiscale analysis of images with a multiplicity of less than two in the frequency domain, that is, the scale change factor is less than 2. Despite the fact that the multiplicity of the analysis is less than 2, the signal can be represented as successive approximations, as with the use of discrete wavelet transform. Reducing the multiplicity allows you to increase the depth of decomposition, thereby increasing the accuracy of signal analysis and synthesis. At the same time, the number of decomposition levels is an order of magnitude higher compared to traditional multi-scale analysis, which is achieved by progressive scanning of the image, that is, the image is processed not by rows and columns, but by progressive scanning as a whole. The use of the fast Fourier transform reduces the conversion time by four orders of magnitude compared to direct numerical integration, and due to this, the decomposition and reconstruction time does not increase compared to the time of multiscale analysis using discrete wavelets.</p> </abstract>

Application of Discrete and Fast Fourier Transforms to Increase the Speed of Multiscale Image Analysis

“…The Pearson correlation coefficient of the original signal with the reconstructed signal for sampling 1024 samples is at least 0.999. The profiling of the program shows that the time of the wavelet transformation using FFT is 15,000 times less than with direct numerical integration for sampling a signal of 32,768 samples [11], [15].…”

Implementation of Machine Learning and Deep Learning Techniques for the Detection of Epileptic Seizures Using Intracranial Electroencephalography

“…The fundamental step of wavelet transformation is decomposing the signal into a series of approximations and details representing different time and frequency scales [65]. This process can be recursively repeated on successive approximations, creating a wavelet tree.…”

Image Recognition of Group Point Objects under Interference Conditions

“…Snakes provide a unified account of a number of visual problems, including the detection of edges, lines, and subjective contours, motion tracking, and stereo matching. The use of the wavelet transform [11] in the processing of images of nanoparticles obtained by the dynamic scattering method increases the resolution of the method and creates better images [12]. The methods of merging images in the frequency domain have been considered [13,14].…”