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

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 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

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

“…The algorithm for numerical calculation of the direct fast WT signal 𝑆(𝑡) in the frequency domain using wavelets based on derivatives of the Gaussian function is described in [10]. In this algorithm for even wavelets, in this algorithm steps 4 and 7 are not performed.…”

“…The algorithm for numerical calculation of the inverse fast WT signal 𝑆(𝑡) in the frequency domain using wavelets based on derivatives of the Gaussian function is described in [10].…”

“…The authors have developed ways to increase the speed of MSA in the frequency domain based on wavelets constructed in the frequency domain [9]. The authors proposed an MSA algorithm using wavelets based on derivatives of the Gauss function [10], [11]. The proposed algorithm enables reconstructing the image.…”

John von Neumann’s Space-Frequency Orthogonal Transforms

“…From Equation (24), it results that the analysis of 2D signal A is equivalently performed by using a couple of 1D dictionaries (according to Definition ( 12)) or with the help of sfas taken from a 2D dictionary (according to Definition (23)).…”

“…Since their inception by Meyer, Mallat and Daubechies (more than 35 years ago), the literature reporting how these wavelets work in conjunction with images has become so vast that any attempt to encompass all the sound (and sometimes amazing) results is very likely doomed to fail. One can only cite a few interesting recent works, such as [23] (with application to image filtering), [24] (in which principal component analysis is developed by means of wavelets, aiming to achieve image denoising), [25] (where image fusion is performed) and [26] (which deals with the efficient implementation of 2D wavelet transforms).…”