Wavelet System Research Articles

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups

The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

Norm inequalities with fractional integrals

Fractional spline wavelet systems are considered. Together with the Battle–Lemarié scaling and wavelet functions of natural orders, they are used to find conditions that ensure certain inequalities between the norms of images and pre-images of fractional integration operators in Besov spaces with Muckenhoupt weights on R \mathbb {R} .

Some Conditions and Perturbation Theorem of Irregular Wavelet/Gabor Frames in Sobolev Space

Due to its potential applications in image restoration and deep convolutional neural networks, the study of irregular frames has interested some researchers. This paper addresses irregular wavelet systems (IWSs) and irregular Gabor systems (IGSs) in Sobolev space HsR. We obtain the sufficient and necessary conditions for IWS and IGS to be frames. By applying these conditions, we also derive the characterizations of IWS and IGS to be frames. Finally, we discuss the perturbation theorem of irregular wavelet frames (IWFs) and irregular Gabor frames (IGFs). We also provided some examples to support our results.

Tight wavelet frames on the space of M-positive vectors

Wavelets on the sets of [Formula: see text]-positive vectors in the Euclidean space are studied. These sets are multidimensional analogs of the half-line in the Walsh analysis. Following the ideas of the Walsh analysis, the space of [Formula: see text]-positive vectors is equipped with a coordinate-wise addition. Harmonic analysis on this space is also similar to the Walsh harmonic analysis, and the Fourier transform is such that there exists a class of so-called test functions (with a compact support of the function itself and of its Fourier transform). Tight wavelet frames consisting of the test functions are studied. A complete description of the masks generating such frames is given, and an algorithmic method for constructing them is developed. These frames may be very useful for applications to signal processing because some examples of such systems on the half-line were already investigated in this aspect, and it appeared that they have an advantage over classical wavelet systems when used for processing fractal signals and images.

Crypto Trend Prediction Based on Wavelet Transform and Deep Learning Algorithm

Crypto currency price prediction is one of the biggest challenges for any prediction system owing to its volatility and dependence on many factors. Artificial intelligence powered by the development of sophisticated computing technology and developments in research has opened up a wide variety of applications and new possibilities to solve complicated problems. Extraction of appropriate features and input with features affecting the output is equally important in time series prediction problems. Wavelet transform is considered an effective method to split a non-stationary signal into uncorrelated components. This work attempts to extract the advantage of wavelets to extract features from an available data set and to predict the trend of the crypto price for next 16 days using the Bi-LSTM network. Historic data of crypto currencies available in Yahoo finance through the yFinance API with the python code is used as the primary data for this work. Due to the gaining importance of SHIB crypto due to its multiplication of value from its start, we have taken SHIB trend analysis as a case study with the use of a data set provided by Yahoo Finance API. The novelty of the proposed system lies largely on the combination of wavelet and deep learning system to improve the performance measures. The proposed systems evaluates the wavelet transform coefficients of the historic data available from the Yahoo Finance API and feed the wavelet transform values to the input of Bi-LSTM network to predict the wavelet transform coefficients for the next 16 days. Finally an inverse wavelet transform is used to get the price of the crypto currency for next 16 days. The results obtained shows that the model was able to estimate the trend in crypto price. The trend analysis shows whether there is an increase or decrease of prices in upcoming days and this could be useful to invest or to withdraw the funds to improve the returns. The proposed scheme resulted in lower Mean Square Error (MSE), Mean Average Error (MAE) and Root Mean Square Error (RMSE) compared with the currently available methods. The method could be useful for trend analysis of other crypto currencies as well. Though it is practically impossible to get the trend analysis from the price and volume history alone, the proposed scheme gives a reasonable estimate of the trend of the crypto price. The work may be extended in future to applications like prediction of agricultural product price, variation in climatic conditions, and availability of certain agricultural products like Onion in the market.

Discrete Periodic Multiresolution Analysis

Periodic multiresolution analyses in the space of periodic complex-valued functions of an integer argument are studied. A characterization of multiresolution analyses in terms of the Fourier coefficients of functions forming a scaling sequence is obtained. An example of a multiresolution analysis with a scaling sequence that consists of trigonometric polynomials with a minimal possible spectrum is presented. A wavelet system associated with such multiresolution analysis is presented.

Matrix-valued nonstationary frames associated with the Weyl–Heisenberg group and the extended affine group

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space [Formula: see text]. First, we show that a diagonal matrix-valued window function constitutes a frame for [Formula: see text] whenever each diagonal entry constitutes a frame for the space [Formula: see text]. This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on [Formula: see text]. It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on [Formula: see text] may not be a frame for [Formula: see text]. In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group.

Anisotropic Triebel–Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

This paper provides maximal function characterizations of anisotropic Triebel–Lizorkin spaces associated to general expansive matrices for the full range of parameters p in (0,infty ), q in (0,infty ] and alpha in {mathbb {R}}. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of mild decay, moment, and smoothness conditions.

Multivariate tile $\mathrm{B}$-splines

Tile $\mathrm{B}$-splines in $\mathbb R^d$ are defined as autoconvolutions of indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $\mathbb R^d$. These functions are not piecewise-polynomial, however, being direct generalizations of the classical $\mathrm{B}$-splines, they enjoy many of their properties and have some advantages. In particular, exact values of the Hölder exponents of tile $\mathrm{B}$-splines are evaluated and are shown, in some cases, to exceed those of the classical $\mathrm{B}$-splines. Orthonormal systems of wavelets based on tile B-splines are constructed, and estimates of their exponential decay are obtained. Efficiency in applications of tile $\mathrm{B}$-splines is demonstrated on an example of subdivision schemes of surfaces. This efficiency is achieved due to high regularity, fast convergence, and small number of coefficients in the corresponding refinement equation.

Generalized inequalities for nonuniform wavelet frames in linear canonical transform domain

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in L2(R) was considered by Gabardo and Nashed. In this setting, the associated translation set is a spectrum ? which is not necessarily a group nor a uniform discrete set, given ? = {0, r/N} + 2Z, where N ? 1 (an integer) and r is an odd integer with 1 ? r ? 2N?1 such that r and N are relatively prime and Z is the set of all integers. In this article, we continue this study based on non-standard setting and obtain some inequalities for the nonuniform wavelet system {f?j,?(x) = (2N)j/2f((2N)jx??)e???A/B (t2??2), j ? Z, ? ? ?}to be a frame associated with linear canonical transform in L2(R). We use the concept of linear canonical transform so that our results generalise and sharpen some well-known wavelet inequalities.

Characterization of tight wavelet frames with composite dilations in L2(Rn)

Tight wavelet frames are different from the orthonormal wavelets because of redundancy. By sacrificing orthonormality and allowing redundancy, the tight wavelet frames become much easier to construct than the orthonormal wavelets. Guo, Labate, Lim, Weiss, and Wilson [Electron. Res. Announc. Am. Math. Soc. 10 (2004), 78-87] introduced the theory of wavelets with composite dilations in order to provide a framework for the construction of waveforms defined not only at various scales and locations but also at various orientations. In this paper, we provide the characterization of composite wavelet system to be tight frame for L2(Rn).

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

AbstractWe exhibit the necessary range for which functions in the Sobolev spaces can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.

Step wavelets on Vilenkin groups

The construction of wavelet bases and frames on the Vilenkin group Gp is studied. Wavelet systems consisting of functions that are compactly supported and band-limited at the same time are of our interest. A complete description of all refinable functions providing such systems is given.

Reconstruction of multidimensional digital signals

We study the reconstruction of multidimensional digital signals in multirate digital signal processing. We show that the composition of sampling operator and interpolation operator forms the frame operator of multivariate discrete time wavelet (MDTW) system of scale matrix [Formula: see text]. By taking a collection of [Formula: see text] filters, we construct dual MDTW tight frames. Also, we discuss the existence and construction of MDTW Parseval frames in multidimensional sequence space. The necessary and sufficient condition for the existence of orthonormal basis of scale matrix [Formula: see text] is discussed. Using dual MDTW tight frame, we characterize biorthogonal pair of Riesz bases. Further, we define multivariate Parseval frame of weighted exponentials, dual multivariate tight frame of weighted exponentials of scale matrix [Formula: see text] in periodic space [Formula: see text]. Finally, we give the applications of MDTW frames in the theory of frames of weighted exponentials of scale matrix [Formula: see text] in periodic space [Formula: see text].

Comparative analysis of wavelet transform filtering systems for noise reduction in ultrasound images.

Wavelet transform (WT) is a commonly used method for noise suppression and feature extraction from biomedical images. The selection of WT system settings significantly affects the efficiency of denoising procedure. This comparative study analyzed the efficacy of the proposed WT system on real 292 ultrasound images from several areas of interest. The study investigates the performance of the system for different scaling functions of two basic wavelet bases, Daubechies and Symlets, and their efficiency on images artificially corrupted by three kinds of noise. To evaluate our extensive analysis, we used objective metrics, namely structural similarity index (SSIM), correlation coefficient, mean squared error (MSE), peak signal-to-noise ratio (PSNR) and universal image quality index (Q-index). Moreover, this study includes clinical insights on selected filtration outcomes provided by clinical experts. The results show that the efficiency of the filtration strongly depends on the specific wavelet system setting, type of ultrasound data, and the noise present. The findings presented may provide a useful guideline for researchers, software developers, and clinical professionals to obtain high quality images.

Wavelet approximation in Orlicz spaces

Approximation properties of wavelet decompositions in the Orlicz spaces are investigated. Both dual wavelet frames and frame-like wavelet systems are considered. The order of approximation (in the sense of modular convergence) of wavelet frame decompositions satisfying a number of natural conditions is found for an arbitrary Orlicz space. For the Orlicz spaces satisfying Δ2-condition, an error estimate providing approximation order of such wavelet expansions is given in terms of the Luxemburg norm. Similar results are obtained for appropriate frame-like systems under the assumption that an Orlicz space satisfies Δ′-condition.

A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space – the so called window–signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window–signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window–signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window–signal space. As one example application, we show how continuous wavelet multipliers (also called Calderón–Toeplitz operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

Speaker Verification using CWT DWT Transform and Neural Network

In this Present study, the technique of wavelet transform and neural network were developed for speech based text-dependent and text0independent speaker identification. 390 feature were fed to feed-forward back propagation neural network for classification The function of feature extraction and classification are performed using wavelet and neural network system. The declared result shows that the proposed method can make an effective analysis with average identification rate reaching 98%. The best recognition rate selection obtained was for FFBPNN (Feed Forward Back Propagation Neural Network).

A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.

Multidimensional periodic discrete wavelets

In this paper, the definition of a periodic discrete multiresolution analysis is provided. The characterization of such systems is obtained in terms of properties of scaling sequences. Wavelet systems generated by such multiresolution analyses are defined and described. Decomposition and reconstruction formulas for the associated discrete wavelet transform are provided.