At present, discrete wavelet transform (Mallat algorithm) is used for signal decomposition and reconstruction. Discrete wavelets are asymmetrical, not smooth functions and do not allow decomposition of signals with a multiplicity of less than two, which limits the number of decomposition levels. Continuous wavelet transform has a number of positive properties (symmetry, smoothness of the basis function) which are necessary for signal analysis and synthesis. The paper proposes algorithms for calculating direct and inverse continuous wavelet transforms in the frequency domain, which allows decomposing, reconstructing and filtering the image with high speed and accuracy. It is established that application of fast Fourier transform reduces the conversion time by four orders of magnitude in compared to direct numerical integration. The results of applying algorithms to the images obtained with an optical microscope are presented. Orthogonal symmetric and anti-symmetric wavelets with rectangular amplitude frequency response are also presented. It is shown that these firstly designed wavelets allow one to reconstruct the signal even faster than the algorithms created using fast Fourier transform. Continuous wavelet transform has been found to allow multiscale analysis of signals with a multiplicity of less than two. In addition, the construction of orthogonal wavelets in the frequency domain with the maximum possible number of zero moments allows one to analyze the finer (high-frequency) structure of the signal, as well as to suppress its slowly changing components, which makes it possible to concentrate energy in a few significant coefficients, which is a prerequisite for compression. INDEX TERMS algorithms, image filtering, signal analysis, wavelet transforms

In the Mallat algorithm, calculations are performed in the time domain. To speed up the signal conversion at each level, the wavelet coefficients are sequentially halved. This paper presents an algorithm for increasing the speed of multiscale signal analysis using fast Fourier transform. In this algorithm, calculations are performed in the frequency domain, which is why the authors call this algorithm multiscale analysis in the frequency domain. For each level of decomposition, the wavelet coefficients are determined from the signal and can be calculated in parallel, which reduces the conversion time. In addition, the zoom factor can be less than two. The Mallat algorithm uses non-symmetric wavelets, and to increase the accuracy of the reconstruction, large-order wavelets are obtained, which increases the transformation time. On the contrary, in our algorithm, depending on the sample length, the wavelets are symmetric and the time of the inverse wavelet transform can be faster by 6–7 orders of magnitude compared to the direct numerical calculation of the convolution. At the same time, the quality of analysis and the accuracy of signal reconstruction increase because the wavelet transform is strictly orthogonal.

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<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>