Construction Of Multiwavelets Research Articles

Standard pairs and construction of multiwavelets using refinement masks satisfying sum rules of order one

A multiwavelet is typically constructed starting from a vector-valued function satisfying a matrix refinement equation. The approximation order of such a refinable function vector is related to the sum rules of order [Formula: see text] satisfied by the corresponding refinement mask. A refinable function vector can be obtained using cascade algorithm by constructing a refinement mask which satisfies the sum rules of order [Formula: see text] A standard pair associated with a refinement mask gives information about its spectral properties. In this paper, we present a procedure for constructing refinement masks satisfying the sum rules of order [Formula: see text], starting from standard pairs. How this helps in the construction of asymmetric multiwavelets using standard pairs is illustrated through examples. A sufficient condition on a standard pair and a necessary and sufficient condition on a left standard pair are established so that the corresponding refinement mask satisfies the sum rules of order [Formula: see text].

A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

In recent years the concept of multiresolution-based adaptive discontinuous Galerkin (DG) schemes for hyperbolic conservation laws has been developed. The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defined on a hierarchy of nested grids. Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element. For non-uniform grid hierarchies multiwavelets have to be constructed for each element and, thus, becomes extremely expensive. To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

Shortest multi-wavelet matrix filters for refinable vector splines

In multi-wavelet decomposition techniques for the analysis of a given vector-valued signal, it is desirable with respect to the computational efficiency of the associated algorithms that the low-pass and high-pass matrix filters sequences be as short as possible. By applying a multi-wavelet construction method based on the solving of a system of four matrix Laurent polynomial identities, and using as main building block a class of arbitrarily smooth refinable vector splines, we establish an explicitly formulated class of shortest possible associated matrix filter sequences for decomposition, as well as minimally supported spline multi-wavelets for these optimal filter sequences.

Standard pairs and existence of symmetric multiscaling functions

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

Biorthogonal bases of multiwavelets

A method for the construction of biorthogonal bases of multiwavelets from known bases of multiscaling functions is given. It is similar to the method presented in the author’s 2014 paper joint with N.I. Chernykh and is based on the same principle: the construction of multiwavelets based on k multiscaling functions employs an analog of the vector product of vectors in a 2k-dimensional space.

Multifractal entropy based adaptive multiwavelet construction and its application for mechanical compound-fault diagnosis

Compound-fault diagnosis of mechanical equipment is still challenging at present because of its complexity, multiplicity and non-stationarity. In this work, an adaptive redundant multiwavelet packet (ARMP) method is proposed for the compound-fault diagnosis. Multiwavelet transform has two or more base functions and many excellent properties, making it suitable for detecting all the features of compound-fault simultaneously. However, on the other hand, the fixed basis function used in multiwavelet transform may decrease the accuracy of fault extraction; what’s more, the multi-resolution analysis of multiwavelet transform in low frequency band may also leave out the useful features. Thus, the minimum sum of normalized multifractal entropy is adopted as the optimization criteria for the proposed ARMP method, while the relative energy ratio of the characteristic frequency is utilized as an effective way in automatically selecting the sensitive frequency bands. Then, The ARMP technique combined with Hilbert transform demodulation analysis is then applied to detect the compound-fault of bevel gearbox and planetary gearbox. The results verify that the proposed method can effectively identify and detect the compound-fault of mechanical equipment.

A Comparative Study on Multiwavelet Construction Methods and Customized Multiwavelet Library for Mechanical Fault Detection

Inner product transform principle reveals that the basis functions most relevant or similar to the fault features are pivotal to the meaningful fault detection. Customized multiwavelet methods and practices have continued to improve over the recent years, focused on two-scale similarity transform (TST), lifting transform (LT), and lifting scheme (LS). Due to the respective advantages and disadvantages, a comparative study on the multiwavelet construction methods by TST, symmetric and dissymmetric LT, and LS is discussed in the paper, covering the differences of construction theories, the synthetic analyses of construction strategies, and the comparison of waveform characteristics along with their applicable occasions. Comprehensively utilizing the capabilities of the construction methods, a novel customized multiwavelet library is established for the accurate fault detection. The proposed method is applied to incipient fault detection of rolling bearing for electric locomotive to verify the effectiveness and feasibility.

Construction of Multiwavelets on an Interval

Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions.

Separation and Extraction of Electromechanical Equipment Compound Faults Using Lifting Multiwavelets

Compound faults of electromechanical equipment always occur simultaneously and in a cascading way. Based on signal matching,multiwavelets with good properties have several scaling functions and wavelet functions. Therefore,they could better match various features of compound faults,and separate and extract compound features at one time. The construction of multiwavelets based on lift frame does not depend on the Fourier transform,and the algorithm is simple and fast. Hence,multiwavelet transforms via lift frame like the second wavelet transforms are studied. Based on Hermite spline interpolation,the new multiwavelets with short support,biorthogonality,symmetry and fourth approximate order are designed,and the corresponding preprocessing algorithm is given. Aimed at the problem of compound faults,posttreatment procedure is modified and decoupled features are clearly presented in different channels of analysis results of multiwavelets. The proposed method is applied to compound faults diagnosis for bearing test bench and traveling unit of electric locomotive. The results show that this method can effectively separate and extract the features of bearing inner-race fault and shaft anomalous bending,and successfully decouple the bearing outer-race defect and roller defect of the locomotive.

The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets

Multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing, due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks. In order to process and represent vector-valued discrete data efficiently and sparsely by a multiwavelet transform, a multiwavelet has to be prefiltered or balanced. Balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature. Dual multiframelets include (bi)orthogonal multiwavelets as special cases, but their fundamental prefiltering and balancing property has not yet been investigated in the literature. In this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform. This approach, to our best knowledge, has not been considered so far in the literature even for multiwavelets, but it reveals the essential structure of prefiltering and the balancing property of multiwavelets and multiframelets. We prove that every biorthogonal multiwavelet can be prefiltered with the balancing order matching the order of its vanishing moments; that is, from every given compactly supported multivariate biorthogonal multiwavelet, one can always build another (essentially equivalent) compactly supported biorthogonal multiwavelets with its balancing order matching the order of the vanishing moments of the original one. More generally, we show that if a dual multiframelet can be prefiltered, then it can be equivalently transformed into a balanced dual multiframelet with the same balancing order. However, we notice that most available dual multiframelets in the literature cannot be simply prefiltered with its balancing order matching its order of vanishing moments and they must be designed to possess high balancing orders. The key ingredient of our approach is based on investigating some properties of the subdivision and transition operators acting on discrete vector polynomial sequences, which play a central role in a discrete multiframelet transform and are of interest in their own right. We also establish a new canonical form of a matrix mask, which greatly facilitates the investigation and construction of multiwavelets and multiframelets. In this paper, we obtain a complete criterion and the essential structure for balanced or prefiltered dual multiframelets in the most general setting. Our investigation of the balancing property of a multiframelet deepens our understanding of the multiframelet transform in signal processing and scientific computation.

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x):= (φ1(x), ..., φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x):= (φ1new(x), ..., φrnew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally, we reveal the relation between Φ(x) and Φnew(x). To embody our results, we construct a symmetric refinable function vector with approximation order 6 from Hermite cubics which provides approximation order 4.

Explicit construction of high-pass filter sequence for orthogonal multiwavelets

For the construction of multiwavelets, there are no unified, explicit formulas as that in the scalar case available so far. In this paper, by studying the relationship between length 3 and length 4 filter sequences of orthogonal multiwavelets based on the result of Chui for the construction of length 3 orthogonal multiwavelets, a set of explicit formulas is given for the construction of length 4 high-pass filter sequence. Examples demonstrate that our proposed approach not only provides explicit formulas for the construction of length 4 high-pass filter sequence with multiplicity r, but also yields a set of new low-pass and high-pass filter sequences with length 3 and multiplicity 2 r .

Multiwavelet construction via an adaptive symmetric lifting scheme and its applications for rotating machinery fault diagnosis

Multiwavelets and the lifting scheme are two important developments of wavelet theory. Multiwavelets outperform scalar wavelets in many applications due to their better properties. The lifting scheme is a method to construct a new wavelet with prescribed properties. In this paper, multiwavelets are integrated with the lifting scheme, synthesizing their advantages. Due to multiple wavelet bases, the lifting scheme of multiwavelets is more flexible than that of scalar wavelets. With supplement of a symmetric condition, a novel adaptive symmetric lifting scheme of multiwavelets is presented. Kurtosis is chosen to be the performance measurement of lifting coefficients, and the genetic algorithm is used to optimize the free parameters in the lifting scheme. The proposed method, constructing a new multiwavelet via an adaptive lifting scheme, is applied to analyze the simulation of a rolling bearing and gearbox vibration signals. The results demonstrate that the adaptive symmetric lifting of multiwavelets is more effective in extracting fault features of rotating machinery than conventional diagnosis techniques with scalar wavelets and non-adaptive multiwavelets.

Some properties and construction of multiwavelets related to different symmetric centers

In this paper we are interested in discuss the symmetry property and construction of an m-band compactly supported orthonormal multiwavelets related to the filters with different symmetric centers. With the development of the several equivalent conditions on this type of symmetry in terms of filter sequences and polyphase matrices, we derive several necessary constraints on the number of symmetric filters of the system, which is crucial for the construction of multiwavelets associated with given multiscaling functions with different symmetry centers. Then, we show how to construct multiwavelets with desired symmetric property by matrix extensions. Finally, to illustrate our proposed general scheme, we give two examples in this paper.

A fast algorithm for constructing orthogonal multiwavelets

AbstractMultiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, a ∈ Z). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

Construction of multi-wavelets from compactly supported refinable super functions

In this work, a new method for constructing multi-wavelets with arbitrary approximation order is presented. The approach uses refinable super functions which enable the formulation of approximation order condition in terms of a generalized eigenvalue equation. The generalized left eigenvectors of the resulting finite down-sampled convolution matrix are recognized as the coefficients that enter the finite linear combination of multi-scaling functions that produce the desired super function. The method is demonstrated by constructing a specific class of multi-wavelets with approximation order 2, which includes Geronimo–Hardin–Massopust (GHM) multi-wavelet as one of its parameterized solutions. A new multi-wavelet, possessing approximation order three and multiplicity two with support [0,2] and [0,3] is also constructed.

Multivariate Refinement Equations and Convergence of Cascade Algorithms in L p (0 < p < 1) Spaces

We consider the solutions of refinement equations written in the form $$ \begin{array}{*{20}c} {{\varphi {\left( x \right)} = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a{\left( \alpha \right)}\varphi {\left( {Mx - \alpha } \right)} + g{\left( x \right)}} },}} & {{x \in \mathbb{R}^{s} ,}} \\ \end{array} $$ where the vector of functions ϕ = (ϕ 1, ..., ϕ r ) T is unknown, g is a given vector of compactly supported functions on ℝ s , a is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence ϕ n , n = 1, 2, ..., by the iterative process $$ \begin{array}{*{20}c} {{\varphi _{n} {\left( x \right)} = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a{\left( \alpha \right)}\varphi _{{n - 1}} {\left( {Mx - \alpha } \right)} + g{\left( x \right)}} },}} & {{x \in \mathbb{R}^{s} ,}} \\ \end{array} $$ from a starting vector of function ϕ 0. We characterize the L p -convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.

On vector subdivision

In this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity of the associated limit function. These results extend and complete our earlier work on vector subdivision and its use in the construction of multiwavelets.

Matrix extension and biorthogonal multiwavelet construction

Suppose that P( z) and P ̃ (z) are two r × n matrices over the Laurent polynomial ring R[ z], where r < n, which satisfy the identity P(z) P ̃ (z)∗ = I r on the unit circle T . We develop an algorithm that produces two n × n matrices Q( z) and Q ̃ (z) over R[ z], satisfying the identity Q(z) Q ̃ (z)∗ = I n on T such that the submatrices formed by the first r rows of Q( z) and Q ̃ (z) are P( z) and P ̃ (z) respectively. Our algorithm is used to construct compactly supported biorthogonal multiwavelets from multiresolutions generated by univariate compactly supported biorthogonal scaling functions with an arbitrary dilation parameter m ∈ Z, where m >1.

Oblique and Hierarchical Multiwavelet Bases

We develop the theory of oblique multiwavelet bases, which encompasses the orthogonal, semiorthogonal, and biorthogonal cases, and we circumvent the noncommutativity problems that arise in the construction of multiwavelets. Oblique multiwavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of the theory to accommodate a wider variety of wavelet bases. For example, for a given multiresolution, we can construct supercompact wavelets for which the support is half the size of the shortest orthogonal, semiorthogonal, or biorthogonal wavelet. The theory also produces the h-type, piecewise linear hierarchical bases used in finite element methods, and it allows us to construct new h-type, smooth hierarchical bases, as well as h-type hierarchical bases that use several template functions. For the hierarchical bases, and for all other types of oblique wavelets, the expansion of a function can still be implemented with a perfect reconstruction filter bank. We illustrate the results using the Haar scaling function and the Cohen–Daubechies–Plonka multiscaling function. We also construct a supercompact spline uniwavelet of order 3 and a hierarchical basis that is based on the Hermit cubic spline, and we explicitly give the coefficients of the corresponding filter bank.