The Orthogonal Wavelets in the Frequency Domain Used for the Images Filtering

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 article 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.