Dual Wavelet Research Articles

On density order of quasi-projection operators and dual wavelet frames

From two functions in the Hilbert space L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} whose Fourier transform has certain decay, one defines a quasi-projection operator. In this paper, we prove necessary and sufficient conditions on those functions in order to the associated quasi-projection operator provides a desired approximation order and density order. To give our conditions we will use the classical notion of approximate continuity. As a consequence, we obtain approximation properties of dual wavelet frame constructed by Mixed Oblique Extension Principle. We show our results in the context of reducing subspaces of L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} with a dilation given by an expansive linear map preserving the integer lattice.

Some Compactly Supported Dual Wavelets Associated to The Quincunx Matrix in ℝ3

Some Compactly Supported Dual Wavelets Associated to The Quincunx Matrix in ℝ³

Dual wavelet frame packets with non-square iterative matrices

Traditional wavelet frame packets are based on square iterative matrices and frame scaling functions. Since the Fourier transform of frame scaling functions are discontinuous, the resulting wavelet frame packets do not possess time-frequency localization. In this study, we will develop the new construction of wavelet frame packets by using non-square iterative matrices: one is to construct wavelet frame packets directly from known non-GMRA wavelet frames to obtain finer time-frequency structure, the other is based on GMRA wavelet frames constructed from smooth generalization scaling functions, the corresponding quasi-projection formula and fast algorithm are also given.

Dual Wavelet Attention Networks for Image Classification

Global average pooling (GAP) plays an important role in traditional channel attention. However, there is the disadvantage of insufficient information to use the result of GAP as the channel scalar. At the same time, the existing spatial attention models focus on the areas of interest using average pooling or convolutional networks, but there is a loss of feature information and neglect of the structural feature. In this paper, dual wavelet attention is proposed, which can effectively alleviate the aforementioned problems and enhance the representation ability of CNNs. Firstly, the equivalence between the sum of the low-frequency subband coefficients of 2D DWT (Haar) and GAP is proved. On this basis, the statistical characteristics of low-frequency and high-frequency subbands are effectively combined to obtain the channel scalars, which can better measure the importance of each channel. In addition, 2D DWT can effectively capture the approximate and detailed structural features. Thus, wavelet spatial attention is proposed, which can effectively focus on the key spatial structural features. Different from traditional spatial attention, it can better curve the structural and spatial attention for different channels. The experiments are verified on four natural image data sets and three remote sensing scene classification data sets, which shows the effectiveness and versatility of the proposed methods. The code of this paper will be available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/yutinyang/DWAN</uri> .

Frame Wavelets of Local Fields with No Dual Wavelet Frames

Frame Wavelets of Local Fields with No Dual Wavelet Frames

Lung Sound Recognition Method Based on Wavelet Feature Enhancement and Time-Frequency Synchronous Modeling.

Lung diseases are serious threats to human health and life, therefore, an accurate diagnosis of lung diseases is significant. The use of artificial intelligence to analyze lung sounds can aid in diagnosing lung diseases. Most of the existing lung sound recognition methods ignore the correlation between the time-domain and frequency-domain information of the lung sounds. Additionally, the spectrograms used in these models do not adequately capture the detailed features of the lung sounds. This paper proposes a model based on wavelet feature enhancement and time-frequency synchronous modeling, comprising a dual wavelet analysis module (DWAM), a cubic network, and an attention module. DWAM in the model performed a dual wavelet transformation on the spectrograms to extract the detailed features of the lung sounds. The cubic network comprised multiple cubic gated recursive units to capture the correlation of the time-frequency of the lung sounds using the time-frequency synchronous modeling. The attention module, which includes temporal and channel attention, was used to enhance the time-domain and channel dimension features. In the combined dataset and the International Conference on Biomedical and Health Informatics 2017 dataset, the suggested framework outperforms existing models by more than 1.36% and 4.28%, respectively.

Non homogeneous dual wavelet frames and oblique extension principles in Hs(K)

In this paper, we introduce the notion of nonhomogeneous dual wavelet frames in Sobolev spaces over local fields. We provide the complete characterization of nonhomogeneous dual wavelet frames on local fields. Furthermore, we obtain a mixed oblique extension principle for such frames.

On the nonhomogeneous wavelet bi-frames for reducing subspaces of Hs(K)

Ahmad and Shiekh in Filomat 34: 6(2020), 2091-2099 have constructed dual wavelet frames in Sobolev spaces on local fields of positive characteristic. We continued the study and provided the characterization of nonhomogeneous wavelet bi-frames. First of all we introduce the reducing subspaces of Sobolev spaces over local fields of prime characteristics and then provide the way to characterize the nonhomogeneous wavelet bi-frames over such fields. Our results are better than those established by Ahmad and Shiekh.

A characterization of nonhomogeneous dual and weak dual wavelet superframes for Walsh‐reducing subspace of L2(ℝ+,ℂL)$$ {L}&amp;amp;amp;amp;amp;#x0005E;2\left({\mathbb{R}}_{&amp;amp;amp;amp;amp;#x0002B;},{\mathbb{C}}&amp;amp;amp;amp;amp;#x0005E;L\right) $$

The concept of superframe was first introduced by Balan in the context of signal multiplexing and by Han and Larson from the perspective of pure mathematics. Since then, the theory of super wavelet (Gabor) frames has been well developed. In recent years, due to enjoying the same fast algorithm as affine wavelet frames and having more freedom to design wavelet masks, nonhomogeneous wavelet frames have attracted much attention of many mathematicians. This paper addresses (weak) nonhomogeneous dual wavelet superframes in the setting of Walsh‐reducing subspaces of . We present a characterization of nonhomogeneous (weak) dual wavelet superframe pairs in a Walsh‐reducing subspace of . Some examples are also provided.

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.

Dual graph wavelet neural network for graph-based semi-supervised classification

Vertex classification is an important graph mining technique and has important applications in fields such as social recommendation and e-Commerce recommendation. Existing classification methods fail to make full use of the graph topology to improve the classification performance. To alleviate it, we propose a Dual Graph Wavelet neural Network composed of two identical graph wavelet neural networks sharing network parameters. These two networks are integrated with a semi-supervised loss function and carry out supervised learning and unsupervised learning on two matrixes representing the graph topology extracted from the same graph dataset, respectively. One matrix embeds the local consistency information and the other the global consistency information. To reduce the computational complexity of the convolution operation of the graph wavelet neural network, we design an approximate scheme based on the first type Chebyshev polynomial. Experimental results show that the proposed network significantly outperforms the state-of-the-art approaches for vertex classification on all three benchmark datasets and the proposed approximation scheme is validated for datasets with low vertex average degree when the approximation order is small.

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.

Characterization of nonhomogeneous dual wavelet frames in Sobolev spaces

Characterization of nonhomogeneous dual wavelet frames in Sobolev spaces

Approximately dual pairs of wavelet frames

This paper deals with structural issues concerning wavelet frames and their dual frames. It is known that there exist wavelet frames {aj/2ψ(aj⋅−kb)}j,k∈Z in L2(R) for which no dual frame has wavelet structure. We first generalize this result by proving that there exist wavelet frames for which no approximately dual frame has wavelet structure. Motivated by this we show that by imposing a very mild decay condition on the Fourier transform of the generator ψ∈L2(R), a certain oversampling {aj/2ψ(aj⋅−kb/N)}j,k∈Z indeed has an approximately dual wavelet frame; most importantly, by choosing the parameter N∈N sufficiently large we can get as close to perfect reconstruction as desired, which makes the approximate dual frame pairs perform equally well as the classical dual frame pairs in applications.

Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

Adaptive Dual Wavelet Threshold Denoising Function Combined with Allan Variance for Tuning FOG-SINS Filter

Allan variance (AV) stochastic process identification method for inertial sensors has successfully combined the wavelet transform denoising scheme. However, the latter usually employs a traditional hard threshold or soft threshold that presents some mathematical problems. An adaptive dual threshold for discrete wavelet transform (DWT) denoising function overcomes the disadvantages of traditional approaches. Assume that two thresholds for noise and signal and special fuzzy evaluation function for the signal with range between the two thresholds assure continuity and overcome previous difficulties. On the basis of AV, an application for strap-down inertial navigation system (SINS) stochastic model extraction assures more efficient tuning of the augmented 21-state improved exact modeling Kalman filter (IEMKF) states. The experimental results show that the proposed algorithm is superior in denoising performance. Furthermore, the improved filter estimation of navigation solution is better than that of conventional Kalman filter (CKF).

A higher order Faber spline basis for sampling discretization of functions

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.

Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in $$L^2({\mathbb {K}})$$

In this paper, we introduce the structure of nonuniform nonhomogeneous dual wavelet frames over non-Archimedean fields. A characterization of nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces over non-Archimedean fields is obtained.

Wavelet frames in $L^2(\mathbb{R}^d)$

We obtain a necessary and sufficient condition for the existence of wavelet frames. We define and study the synthesis and analysis operators associated with wavelet frames. We discuss some applications of operator value (OPV) frames in the theory of wavelet frames. Also, we discuss the minimal property of wavelet frame coefficients and study the property of over completeness of wavelet frames. Various characterizations of wavelet frame, Riesz wavelet basis and orthonormal wavelet basis are given. Further, dual wavelet frames are discussed and a characterization of dual wavelet frames is given. Finally, we give a characterization of a pair of biorthogonal Riesz bases.

Combining dual-tree complex wavelets and multiresolution in iterative CT reconstruction with application to metal artifact reduction

BackgroundThis paper investigates the benefits of data filtering via complex dual wavelet transform for metal artifact reduction (MAR). The advantage of using complex dual wavelet basis for MAR was studied on simulated dental computed tomography (CT) data for its efficiency in terms of noise suppression and removal of secondary artifacts. Dual-tree complex wavelet transform (DT-CWT) was selected due to its enhanced directional analysis of image details compared to the ordinary wavelet transform. DT-CWT was used for multiresolution decomposition within a modified total variation (TV) regularized inversion algorithm.MethodsIn this study, we have tested the multiresolution TV (MRTV) approach with DT-CWT on a 2D polychromatic jaw phantom model with Gaussian and Poisson noise. High noise and sparse measurement settings were used to assess the performance of DT-CWT. The results were compared to the outcome of the single-resolution reconstruction and filtered back-projection (FBP) techniques as well as reconstructions with Haar wavelet basis.ResultsThe results indicate that filtering of wavelet coefficients with DT-CWT effectively removes the noise without introducing new artifacts after inpainting. Furthermore, adoption of multiple resolution levels yield to a more robust algorithm compared to varying the regularization strength.ConclusionsThe multiresolution reconstruction with DT-CWT is also more robust when reconstructing the data with sparse projections compared to the single-resolution approach and Haar wavelets.