Orthogonal Wavelet Bases Research Articles

Spherical-Orthogonal-Symmetric Haar Wavelet for Driver's Visual Detection

This work proposes a spherical-orthogonal-symmetric Haar wavelet to decompose and reconstruct spherical iris signals to obtain stronger geometric features of iris surface. It compares its feature extraction abilities of spherical harmonics, semi-orthogonal and nearly orthogonal spherical Haar wavelet. The developed spherical-orthogonal-symmetric Haar wavelet with a convolutional neural network is also proposed for drivers’ iris recognition. It can effectively capture the local fine features of iris spherical surface, and has stronger ability of iris recognition than semi-orthogonal or nearly orthogonal spherical Haar wavelet bases.

A supervised functional Bayesian inference model with transfer-learning for performance enhancement of monitoring target batches with limited data

To increase the monitoring performance of the batch process with serious nonlinearity, uneven-length, and limited-data issues, a supervised transfer-learning based functional Bayesian inference method is developed in this study. The raw uneven-length batch data are firstly transformed into functional data by choosing appropriate orthogonal wavelet basis functions (WBFs). Using the approximation coefficients representing the latent features that are inherent to the raw data, a Gaussian process model and a Bayesian inference method are applied based on the coefficients in each source batch process and the established models are transferred to enhance modeling performance for the target process with limited batches. In the proposed functional method, the unfolding operation is not needed for preprocessing, avoiding distortion of the raw data structure. With the compact support property of WBFs, within-batch detection can be implemented effectively to recognize faults earlier. The advantages of the proposed model are verified using a numerical case and an industrial polytetrafluoroethylene process.

Nonorthogonal wavelet transformation for reconstructing gravitational wave signals

Detections of gravitational-wave signals from compact binary coalescences have enabled us to study extreme astrophysical phenomena and explore fundamental physics. A crucial requisite for these studies is to have accurate signal models with characteristic morphologies, which have been challenging for many decades, and researchers are still endeavoring to incorporate important physics. Therefore, morphology-independent methods have been developed for identifying a signal and its reconstruction. The reconstructed signal allows us to test the agreement between the observed signal and the waveform posterior samples from parameter estimation. These methods model observed signals using a nearly orthogonal wavelet basis in the frame of continuous wavelet transformation. Here, we propose log-uniform scales to construct the wavelets, which are are highly redundant (nonorthogonal) compared to the conventional octave scales but more efficient for reconstructing the signals at high frequencies. And we introduce a semi-model-dependent reconstruction method using the posterior samples of the events, where we model the signal using Gabor-Morlet wavelets with log-uniform scales. We demonstrate the ability to detect deviation using a numerical simulation of an eccentric binary black hole merger, where the signal in the data does not belong to the search template waveform manifold. Finally, we apply this method to each binary black hole merger event in GWTC-1. We have found that the signal produced by the GW150914 event has 96% agreement with the waveform posterior samples. As the detector sensitivity improves and the detected population of black hole mergers grows, we expect the proposed method will provide even stronger tests.

Comparative study on wavelet functional partial least squares soft sensor for complex batch processes

Conventional data-driven models for batch processes conduct unfolding operations and neglect the continuous property. A novel soft sensor method is proposed based on the wavelet functional partial least squares (WFPLS). The three-dimensional (3D) batch data is transferred into an even two-dimensional function matrix no matter whether it has even-length or uneven-length. The orthogonal wavelet basis functions are selected by the proposed active algorithm to approximate the variables' trajectories. Two versions of WFPLS are developed to predict quality based on different assumptions of the loading function. By comparing one of the proposed methods and the classical multi-way partial least squares (MPLS), the root means square errors are relatively descended by 20.6566% and 63.9925%, R2 are improved by 2.5813% and 83.2518%, and the corresponding computation time of the proposed method is only 5.7% and 41% of the time used by MPLS in the numerical case and in the industrial case, respectively.

Highly Efficient Spatial-Temporal Correlation Basis for 5G IoT Networks.

One of the major concerns in 5G IoT networks is that most of the sensor nodes are powered through limited lifetime, which seriously affects the performance of the networks. In this article, Compressive sensing (CS) technique is used to decrease transmission cost in 5G IoT networks. Sparse basis is one of the important steps in the CS. However, most of the existing sparse basis-based method such as DCT (Discrete cosine transform) and DFT (Discrete Fourier Transform) basis do not capture data structure characteristics in the networks. They also do not take into consideration multi-resolution representations. In addition, some of sparse basis-driven methods exploit either spatial or temporal features, resulting in performance degradation of CS-based strategies. To address these challenging problems, we propose a novel spatial–temporal correlation basis algorithm (SCBA). Subsequently, an optimal basis algorithm (OBA) is provided considering greedy scoring criteria. To evaluate the efficiency of OBA, orthogonal wavelet basis algorithm (OWBA) by employing NS (Numerical Sparsity) and GI (Gini Index) sparse metrics is also presented. In addition, we discuss the complexity of the above three algorithms, and prove that OBA has low numerical rank. After experimental evaluation, we found that OBA is capable of the sparsest representing original signal compared to spatial, DCT, haar-1, haar-2, and rbio5.5. Furthermore, OBA has the low recovery error and the highest efficiency.

Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Image deblurring is a fundamental problem in imaging, usually solved with computationally intensive optimization procedures. The goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: (i) a sparse approximation of the blur operator in wavelet bases, (ii) a diagonal preconditioner and (iii) an implementation on massively parallel architectures. Combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation. For instance, a 1024 × 1024 image can be deblurred in 0.15 s.

Bounds and Conditions for Compressive Digital Holography Using Wavelet Sparsifying Bases

Numerous experiments have been conducted with success in the field of compressive digital holography, but the theory to determine optimal measurement conditions is lagging behind. In contrast to a prior study that expects object wavefields to be sparse in the spatial domain, we investigate how the configuration of the interferometer influences the reconstruction of wavefields that are sparse in a multiresolution orthogonal wavelet basis. In particular, we derive expressions for the coherence between the free-space wave propagation operator and the basis functions of a Shannon multiresolution representation as a function of the wavelength, the propagation distance, the image sensor's pixel pitch, and the scale of the basis functions. These expressions reveal that the coherence as a function of the Fresnel number is subject to specific scaling and translating rules as the scale of the basis functions changes. For a multiresolution orthogonal wavelet representation and digital holograms that are recorded in the near field, we deduce subsequently the optimal configuration of the interferometer and we show by means of hypothesis testing that the associated phase transition bound coincides with the weak threshold for block-sparse compressive sensing with a block length of 2, which is an optimal bound for the class of complex-valued compressive sensing problems. By means of experiments with a USAF 1951 resolution target and an angle grid, we validate our findings and demonstrate that the reconstructed object wavefields are resilient to sparsity defects and additive noise.

Realization of the Spline-Wavelet Decomposition of the First Order

The aim of the paper is to present an orthogonal basis for the discrete wavelets in the general structure of the spline-wavelet decomposition. Decomposition of an original numerical flow without embedding in the standard functional spaces is discussed. It makes it possible to concentrate on simplification of the realization formulas: here, the simple formulas of decomposition and reconstruction are presented, an orthogonal wavelet basis is constructed, and an illustrative example is given. Finally, some estimates of the complexity of the method discussed for different software environments are provided. Bibliography: 6 titles.

On non-compactly supported p-adic wavelets

For any prime p, we prove the existence of non-compactly supported orthogonal p-adic wavelet bases in the Hilbert space L2(Qp), and construct the first explicit example of such a basis. The reasons are based on a special parametrization of the set of eigen standard Haar vector-functions. It should be noted that all previously known orthogonal p-adic wavelet bases were modifications of the p-adic Haar basis.

A New Class of Hilbert Pairs of Almost Symmetric Orthogonal Wavelet Bases

A New Class of Hilbert Pairs of Almost Symmetric Orthogonal Wavelet Bases

Modal Analysis of Ambient PMU Measurements Using Orthogonal Wavelet Bases

This paper proposes a new method, called wavelet scale decomposition, for modal analysis of ambient synchrophasor data using orthogonal wavelet bases. Wavelet formulation of the problem enables reliable estimation results even with short data analysis windows that are a few minutes long. Also, in addition to finding the mode frequency and damping ratio of oscillatory modes, the proposed method is able to estimate their mode shapes as well. The method is tested on some archived real phasor measurement unit data sets and on simulations from Kundur two area system. The results show that the proposed method is able to track damping ratio variations in power systems effectively with short analysis windows.

Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena

An orthogonal wavelet basis is characterized by its approximation order, which relates to the ability of the basis to represent general smooth functions on a given scale. It is known, though perhaps not widely known, that there are ways of exceeding the approximation order, i.e., achieving higher-order error in the discretized wavelet transform and its inverse. The focus here is on the development of a practical formulation to accomplish this first for 1D smooth functions, then for 1D functions with discontinuities and then for multidimensional (here 2D) functions with discontinuities. It is shown how to transcend both the wavelet approximation order and the 2D Gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet-basis grid.

Wavelets Centered on a Knot Sequence: Theory, Construction, and Applications

We develop a general notion of orthogonal wavelets ‘centered’ on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the $$\tau $$ -integers where $$\tau $$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $$L^2(\mathbf {R})$$ with scaling factor $$\tau $$ .

Construction of Compactly Supported Nonseparable Orthogonal Wavelet with Dilation 4

In this paper, we give a method for the construction of high dimension nonseparable and compactly supported orthogonal wavelet bases and the method to construct orthogonal wavelet bases are nonseparable . The orthogonal wavelets are associated with dilation 4.

Regular orthogonal basis on Heisenberg group and application to function spaces

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.

On orthogonal p-adic wavelet bases

A variety of different orthogonal wavelet bases has been found for L2(R) for the last three decades. Such bases are not only interesting by themselves, a number of theoretical and applied problems were solved using them. It appeared that similar non-trivial wavelet constructions also exist for functions defined on some other algebraic structures, such as local fields of positive characteristic including the Vilenkin/Cantor groups. We note that most of the bases involved consist of Bruhat–Schwartz functions, i.e. compactly supported and band-limited ones. There were several attempts to develop wavelet theory in a more general setting, e.g., for zero-dimensional groups. Analyzing these papers, one can see that actually nothing except for the Haar basis and its trivial modifications was constructed. In the present paper we give an explanation for these failures. Namely, we describe the situation for the field Qp of p-adic numbers and prove that any orthogonal wavelet basis for L2(Qp) that consists of band-limited (periodic) functions, is a modification of the Haar basis.

On the vaguelet and Riesz properties of [formula omitted]-unbounded transformations of orthogonal wavelet bases

In this work, we prove that certain L2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.

Localization of the Sound Source Based on Directional Orthogonal Periodic Wavelet Basis

This paper introduces a smooth and adjustable periodic interpolatory wavelet which is sensitive to the direction of the signal.It is produced by orthogonalization ,in this way it constitutes directional orthogonal wavelet basis in the two-dimensional space ; the position of the sound signal is detected by fast transform of this wavelet and the result is presented.

Design of Compactly Supported Nonseparable Orthogonal Wavelet with Arbitrary Dilation in <i>L<sup>2</sup>(R<sup>d</sup>)</i>

In this paper, we present a method for the construction of nonseparable and compactly supported orthogonal wavelet bases in ，and orthogonal wavelet bases with this method are nonseparable . The orthogonal wavelets are associated with arbitrary dilation matrix, where is the identity matrix of order and is the arbitrary integer.

Wavelet methods to eliminate resonances in the Galerkin-truncated Burgers and Euler equations

It is well known that solutions to the Fourier-Galerkin truncation of the inviscid Burgers equation (and other hyperbolic conservation laws) do not converge to the physically relevant entropy solution after the formation of the first shock. This loss of convergence was recently studied in detail in [S. S. Ray et al., Phys. Rev. E 84, 016301 (2011)], and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution. In this work, we propose a way to remove this resonance by filtering a wavelet representation of the Galerkin-truncated equations. A method previously developed with a complex-valued wavelet frame is applied and expanded to embrace the use of real-valued orthogonal wavelet basis, which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space. We also apply the complex-valued wavelet based method to the 2D Euler equation problem, showing that it is able to filter the resonances in this case as well.