Abstract
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.
Highlights
In various scientific problems, the phenomenon of Raman scattering (Raman) employed in the most modern spectrometers is widely used for the study of the structure and properties of various materials
Wavelets based on derivatives of the Gauss function allow us to reconstruct a signal, perform a multiresolution analysis, filtering one-dimensional and two-dimensional signals
The profiling of the program shows that the wavelet transform time using the fast Fourier transform (FFT) is 15,000 times lesser than with direct numerical integration for 32768 samples of the signal
Summary
In various scientific problems, the phenomenon of Raman scattering (Raman) employed in the most modern spectrometers is widely used for the study of the structure and properties of various materials. A lot of time is required to calculate the WT by direct numerical integration, the wavelet scalogram is calculated in the frequency domain using the fast Fourier transform (FFT). In order to move from the frequency domain to the time domain, it is necessary to calculate the inverse Fourier transform of the obtained complex conjugate spectrum. In order to calculate the WT of the signal in the frequency domain, it is necessary to obtain the Fourier spectra of the signal and wavelet for different scale values a, find the complex conjugate spectrum and the inverse Fourier transform of complex conjugate spectra to obtain the wavelet scalogram of the signal. The algorithm for the numerical calculation of the direct continuous fast WT of a signal S(t) according to formula (1) in the frequency domain includes the following steps: 1. 2. The coefficients of the trigonometric series a (n),b (n) of the wavelet ψ(k) are calculated using the FFT:
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