Multiresolution Analysis Theory Research Articles

Power allocation strategy for fuel cell distributed drive electric tractor based on adaptive multi-resolution analysis theory

Fuel cell distributed drive electric tractor (FCDET) provided a novel pathway for the development of green agricultural machinery technology. However, FCDET faced problems such as low traction efficiency, short endurance, and high hydrogen consumption. Relevant studies have shown that reasonable and effective power allocation strategy can improve the efficiency and reduce the energy consumption of fuel cell systems. In this paper, a drive power allocation strategy based on adaptive multi-resolution analysis theory (AMRA) was proposed. The strategy could realize effective decoupling between various energy sources, reduce the problems of frequent start-stop and large power fluctuation for fuel cells. We started with an equivalent circuit model of the fuel cell system, a total efficiency solution model, an energy dissipation model, and a drive motor response model. The first-time reconstruction strategy was based on Tunable Q-factor Wavelet Transform (TQWT) to obtain a subsequence that responds to the oscillatory characteristics of the power signal. The second-time reconstruction strategy took the low-frequency subsequence through Variational Mode Decomposition (VMD) into a number of discrete sub-signals with special sparse properties. Meanwhile, the Sparrow Search Algorithm (SSA) was used to obtain the optimal combined values of the modal decomposition layers and the quadratic penalty factor in the VMD on real time. Finally, the second-time decomposed subsequences and sub-signals were reconstructed according to the frequency characteristics, and the reconstructed power signals were redistributed among the individual energy sources. In order to validate the proposed power allocation strategy, we used the ET504-H prototype as a research object. The power information of plowing and transportation operating conditions was collected in Mengjin test base of China YTO Group. The MATLAB/Simulink-PXI joint simulation platform was set up and the driving motor bench test was carried out in the New Energy Key Laboratory of Henan Province. The results show that the AMRA-based FCDET drive power allocation strategy can effectively improve the energy utilization of the fuel cell system and the economy of the whole tractor operating unit. Compared to the power-following strategy under plowing and transportation conditions, the average efficiency of the fuel cell system was improved by 8.3 % and 3.3 %, the equivalent hydrogen consumption was reduced by 35.6 % and 43.89 %, respectively. The average efficiency of the drive motor was improved by 2.06 % and 2.08 %, the total energy consumption was reduced by 3.73 % and 2.6 %, respectively. This study can provide a theoretical foundation and a novel technical approach for the development of FCDET control system.

Fuzzy wavelet network based on extended Kalman filter training algorithm combined with least square weight estimation: Efficient and improved chromatographic QSRR/QSPR models

Inspired by the theory of multiresolution analysis (MRA) of wavelet transforms and fuzzy concepts, a fuzzy wavelet network (FWN) is proposed for predicting the gas chromatographic retention time of phenol derivatives on fused-silica, wide-bore, and open-tubular columns with different polarities in terms of quantitative structure-retention relationship (QSRR) studies. The FWN consists of a set of fuzzy rules. Each rule corresponding to a sub-wavelet neural network (WNN) consists of single-scaling wavelets. The proposed FWN follows a two-stage efficient learning algorithm scheme, which undertakes the extended Kalman filter (EKF) training algorithm for adjusting all translation parameters, dilation parameters, and weight coefficients of the WNNs and then least square estimation for updating all the weights. 4-3-1 FWNs were developed using the descriptors selected by the multiple linear regression (MLR) models as inputs. By learning the translation parameters of the wavelets and adjusting the shape of membership functions, the model accuracy was remarkably improved. The accuracy and robustness of the generated models were illustrated using leave-one-out and leave-multiple-out cross-validations and also by Y-randomization. Sensitivity analysis by sequential zeroing of weights revealed that the molecular geometry and electronic interactions play a major role in the retention behavior of these compounds on non-polar and semi-polar columns, respectively.

Image compression using fractal multiwavelet transform

Multiwavelets occur in wavelet theory for more general multiresolution analysis (MRA) using several scaling functions. The muliwavelets feature orthogonality, short support, symmetry, approximation order and regularity at a time which is not possible by using only a single wavelet system. In the present work, theory of MRA and wavelet using several scaling functions is reviewed to define Discrete Multiwavelet Transform and Fractal Multiwavelet Transform in matrix form, using a multifilter. As an application the defined transforms are applied to an image for its compression and the resulting entropy and PSNR values are compared with those occurring with earlier used transforms like Haar and D4 transforms.

Scaling functions on the spectrum

Abstract A generalization of Mallat’s classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed [4] for which the translation set Λ = {0, r/N}+2 ℤ is no longer a discrete subgroup of ℝ but a spectrum associated with a certain one-dimensional spectral pair. In this short communication, we characterize the scaling functions associated with such a nonuniform multiresolution analysis by means of some fundamental equations in the Fourier domain.

Sparsity Enhanced Topological Fractal Decomposition for Smart Machinery Fault Diagnosis

Automatic fault feature extraction-based smart fault diagnosis is becoming more and more popular, as it does not require excessive expertise of on-site staff. Advanced signal processing techniques are of significant importance in order to ensure efficient and effective fault feature analysis. Multi-resolution analysis is an effective tool utilized to decouple multiple signal modes within the measured vibration signal. However, current multi-resolution analyzing methods still cannot enable continuous spectral refinements around fixed analyzing frequencies. To address this problem, a novel theory of topological fractal multi-resolution analysis (TFMRA) is proposed. With the concept of nested centralized wavelet packet cluster (NCWPC), TFMRA is equipped with the ability to extract multiple fault features simultaneously. Mathematically, we prove that: 1) each NCWPC is a topology subset of spectral domain of the investigated signal and 2) all sets of NCWPC share a common self-similar fractal property in geometry. This paper reveals an important intrinsic relation between classical dyadic multi-resolution analysis and TFMRA. That is, each dyadic wavelet packet can be uniquely associated with an NCWPC according to the definitions of TFMRA, and classical wavelet packet spaces are regarded as proper subsets of the proposed NCWPCs. Combining signal decomposition using TFMRA and damage information of a mechanical system, we propose an improved sparsity promoted vibration signature analyzing methodology to investigate repetitive transient fault features. This method was applied to extract abnormal vibration signatures from an experimental rotor test rig with rub-impact faults. Processing results demonstrate that nanocomponents of transient vibrations, which are produced by rub-impact faults, were successfully identified. These results are compared with those of some other comparison techniques based on sparse representation. It is verified that the proposed fault diagnosis method possesses more robust noise resisting capability.

3D multi-resolution wavelet convolutional neural networks for hyperspectral image classification

Hyperspectral images contain abundant spectral information, and three-dimensional (3D) feature extraction methods have been shown to be effective for classification. In this paper, we propose a hyperspectral image classification method that uses 3D multi-resolution wavelet convolutional network (3D MWCNNs) in which wavelets are first characterized by their time-frequency and multi-resolution. Then, the 3D-MWCNNs extract features from coarse to fine scales. In addition, 3D-MWCNNs work stably and effectively for approximation. In the conventional implementation of wavelets, empirical parameters must be determined in advance and the feature extraction process is not adaptive. Convolutional neural networks (CNNs) have strong adaptive learning capabilities and can extract features from low to high levels; however, they lack the theoretical underpinnings to perform multi-resolution approximation for filter learning. Therefore, by combining the CNNs framework with multi-resolution analysis theory, a model called 3D MWCNNs is proposed to extract the 3D features from different scales and different depths adaptively. 3D MWCNNs model is better at feature representation and approximation from 3D cube data; therefore, they capture the spatial and spectral features more discriminatively to improve the classification accuracy. Experimental results on three well-known hyperspectral images demonstrate that the proposed framework achieves considerably higher classification accuracy than do several state-of-the-art algorithms.

Wavelets as basis functions to represent the coarse-graining potential in multiscale coarse graining approach

In this paper, we apply Multiresolution Analysis (MRA) to develop sparse but accurate representations for the Multiscale Coarse-Graining (MSCG) approximation to the many-body potential of mean force. We rigorously framed the MSCG method into MRA so that all the instruments of this theory become available together with a multitude of new basis functions, namely the wavelets. The coarse-grained (CG) force field is hierarchically decomposed at different resolution levels enabling to choose the most appropriate wavelet family for each physical interaction without requiring an a priori knowledge of the details localization. The representation of the CG potential in this new efficient orthonormal basis leads to a compression of the signal information in few large expansion coefficients. The multiresolution property of the wavelet transform allows to isolate and remove the noise from the CG force-field reconstruction by thresholding the basis function coefficients from each frequency band independently. We discuss the implementation of our wavelet-based MSCG approach and demonstrate its accuracy using two different condensed-phase systems, i.e. liquid water and methanol. Simulations of liquid argon have also been performed using a one-to-one mapping between atomistic and CG sites. The latter model allows to verify the accuracy of the method and to test different choices of wavelet families. Furthermore, the results of the computer simulations show that the efficiency and sparsity of the representation of the CG force field can be traced back to the mathematical properties of the chosen family of wavelets. This result is in agreement with what is known from the theory of multiresolution analysis of signals.

Scaling functions generating fractional Hilbert transforms of a wavelet function

It is well-known that an orthonormal scaling function generates an orthonormal wavelet function in the theory of multiresolution analysis. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (resp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. We also show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.

Wavelet packet transform for detection of single events in acoustic emission signals

Acoustic emission signals in tribology can be used for monitoring the state of bodies in contact and relative motion. The recorded signal includes information which can be associated with different events, such as the formation and propagation of cracks, appearance of scratches and so on. One of the major challenges in analyzing these acoustic emission signals is to identify parts of the signal which belong to such an event and discern it from noise. In this contribution, a wavelet packet decomposition within the framework of multiresolution analysis theory is considered to analyze acoustic emission signals to investigate the failure of tribological systems. By applying the wavelet packet transform a method for the extraction of single events in rail contact fatigue test is proposed. The extraction of such events at several stages of the test permits a classification and the analysis of the evolution of cracks in the rail.

Semi-orthogonal wavelet frames on local fields

AbstractWe investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. The theory of frame multiresolution analysis recently proposed by Shah [J. Operators (2015), Article ID 216060] on local fields is used to establish equivalent conditions for a finite number of functions

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field \(K\) of positive characteristic. The associated subspace \(V_0\) of \(L^2(K)\) has an orthonormal basis, a collection of translates of the scaling function \(\varphi \) of the form \(\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }\) where \(\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1\) is an integer and \(r\) is an odd integer such that \(r\) and \(N\) are relatively prime and \(\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}\). We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter \(m_{0}\) with appropriate conditions.

Estimating the Statistical Characteristics of Remote Sensing Big Data in the Wavelet Transform Domain

Since it is difficult to deal with big data using traditional models and algorithms, predicting and estimating the characteristics of big data is very important. Remote sensing big data consist of many large-scale images that are extremely complex in terms of their structural, spectral, and textual features. Based on multiresolution analysis theory, most of the natural images are sparse and have obvious clustering and persistence characters when they are transformed into another domain by a group of basic special functions. In this paper, we use a wavelet transform to represent remote sensing big data that are large scale in the space domain, correlated in the spectral domain, and continuous in the time domain. We decompose the big data set into approximate multiscale detail coefficients based on a wavelet transform. In order to determine whether the density function of wavelet coefficients in a big data set are peaky at zero and have a heavy tailed shape, a two-component Gaussian mixture model (GMM) is employed. For the first time, we use the expectation-maximization likelihood method to estimate the model parameters for the remote sensing big data set in the wavelet domain. The variance of the GMM with changing of bands, time, and scale are comprehensively analyzed. The statistical characteristics of different textures are also compared. We find that the cluster characteristics of the wavelet coefficients are still obvious in the remote sensing big data set for different bands and different scales. However, it is not always precise when we model the long-term sequence data set using the GMM. We also found that the scale features of different textures for the big data set are obviously reflected in the probability density function and GMM parameters of the wavelet coefficients.

A flexible family of density functions

The aim of this paper is to define a new family of probability density functions (MR pdf) based on the multiresolution analysis theory. Each function of this family can be seen as a particular type of density mixture.The MR pdf has advantages with regards to estimation over conventional mixtures and it is suitable to model a large variety of square integrable probability density functions.

Mixtures of Mixtures Based on Multiresolution Analysis Theory

In this article, the approaches for exploiting mixtures of mixtures are expanded by using the Multiresolution family of probability density functions (MR pdf). The flexibility and the properties of local analysis of the MR pdf facilitate the location of subpopulations into a given population. In order to do this, two algorithms are provided. The MR model is more flexible in adapting to the different subpopulations than the traditional mixtures. In addition, the problems of identification of mixtures distributions and the label-switching do not appear in the MR pdf context.

The Stability and Perturbation of Bivariate Gabor Frames and Applications in Material Engineering

Material science broadly encompasses the fundamental study of solid matter with the goal of engineering new materials with superior properties, and ultimately enabling altogether new types of devices The window functions and bivariate Gabor frames are introduced. The existence of bivariate Gabor frames with compact support is discussed. Sufficient conditions for irregular bivariate Gabor system to be frames are presented by means of frame multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal bivariate vector-valued wavelets with compact support is proposed, and their properties are investigated. The pyramid decomposition scheme is derived based on a generalized multiresolution structure.

Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform

The fractional wavelet transform (FRWT), which generalizes the classical wavelet transform, has been shown to be potentially useful for signal processing. Many fundamental results of this transform are already known, but the theory of multiresolution analysis and orthogonal wavelets is still missing. In this paper, we first develop multiresolution analysis associated with the FRWT and then derive a construction of orthogonal wavelets for the FRWT. Several fractional wavelets are also presented. Moreover, some applications of the derived results are discussed.

Why are Haar bases in various structures the same?

The Haar system [1] is a model example of a wavelet basis in L2([0, 1]). The active study of wavelets began at the end of the twentieth century in connection with the development of multiresolution analysis. Soon interest in wavelet bases defined on various structures arose. In [2] and [3], a theory of multiresolution analysis on the Cantor group was developed, and various wavelet bases substantially different from Haar bases were found. The group of 2-adic numbers is largely similar to the Cantor group (in that they have the same canonical representation of elements and the same metric); the difference is only in the group operation “+,” but this results in dramatically different wavelet theories and multiresolution analyses. In particular, it is shown in [4] that, in contrast to the case of the Cantor group, there does not exist a p-adic multiresolution analysis with orthogonal scaling function other than Haar multiresolution analysis. At the same time, the p-adic Haar basis (first written out in [5]) has exactly the same form as the Haar basis on the Cantor group for p = 2 or on the Vilenkin group for p > 2. Haar bases on the interval [0, 1] with dilation factor pj depending on the level number j were studied in [6]. Haar bases on the rings of integer p-adic numbers and on the projective limits of finite cyclic groups in the oneand many-dimensional cases were constructed in [7]–[10]. The resemblance between the Haar basis on the zero-dimensional group and the Haar basis [6] on [0, 1] was indicated in [8, Remark 2.4]. In the present paper, we give a simple explanation of this resemblance by showing that all the above-mentioned Haar bases can be defined by the same formulas. We describe a sufficiently general scheme for constructing Haar bases which does not even rely on the presence of the group operation “+.” (Although the use of the group operation makes the notation of the Haar functions convenient, it can be completely avoided.) One only needs a “not too bad” topology on a measure space and a suitable sequence of partitions of the space. The construction of the Haar system and the proof of its basis property are exactly the same as for the classical Haar basis. Let μ be a measure on Ω, let Σ be the σ-algebra of measurable sets, and let p = {pj}j=−∞ be a sequence of integers pj > 1. Assume that there exist set collections {Ωjn}, n ∈ Z+, pairwise disjoint for each j ∈ Z and such that μΩ0n = 1 for each n ∈ Z+, Ω = ⋃ nΩjn for each j ∈ Z, and every Ωj−1,n is partitioned into pj subsets Ωj,nk all of which have the same measure, nk = nk(n, j) ∈ Z+. The family of pairs (j, n), numbering the sets Ωjn will be denoted by I, the set {Ωjn}(j,n)∈I will be called an (H,p)-partition, and its elements will be called cells. Let φjn be the characteristic function of the set Ωjn multiplied by a factor normalizing in L2(Ω). Set

Designment and Traits of Quarternary Multiwavelets and Applications in Solid-State Physics and Materials Science

Information science focuses on understanding problems from the perspective of the stakeholders involved and then applying information and other technologies as needed. New physics emerge because of the diverse new material properties which need to be explained. In this work, the notion of vector-valued multiresolution analysis and biorthogonal binary multi- tiwavelets is introduced. The existence of compactly supported biorthogonal vector-valued wavelets associated with a pair of biorthogonal compactly supported vector-valued scaling functions is investigated. A new method for constructing a class of biorthogonal compactly supported vector-valued wavelet functions is presented by using multiresolution analysis and matrix theory.

Designment Algorithm on Orthogonal Two-Directional Vector-Valued Wavelets with Three-Scale Dilation Factor

In this work, the notion of orthogonal two-directional shortly supported wavelets with poly-scale is developed. A new method for designing two-directional orthogonal wavelet wraps is proposed and their properties is investigated by means of time-frequency analysis methodand, operator theory. The existence of shortly supported orthogonal two-directional vector-valued wavelets associated with a pair of orthogonal compactly supported vector-valued scaling functions is investigated. A new method for designing a sort of orthogonal shortly supported two-directional vector-valued wavelet functions is presented by using multiresolution analysis and matrix theory.

Higher Rank Wavelets

Abstract A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in L2(ℝd). While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.