Biorthogonal Multiwavelets Research Articles

B-spline quarklets and biorthogonal multiwavelets

In this paper, we show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces [Formula: see text] of [Formula: see text] which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters, they even satisfy all those properties. Moreover, we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore, we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from [Formula: see text]. Finally, we also verify that quarks and quarklets can be used to define sequences of subspaces [Formula: see text] of [Formula: see text] that yield non-orthogonal decompositions of [Formula: see text].

FPGA Implementation of Balanced Biorthogonal Multiwavelet Using Direct Pipelined Mapping Method for Image Compression Applications

FPGA Implementation of Balanced Biorthogonal Multiwavelet Using Direct Pipelined Mapping Method for Image Compression Applications

Wavelets on intervals derived from arbitrary compactly supported biorthogonal multiwavelets

(Bi)orthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. A lot of problems in applications are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. From any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on $[0,\infty)$ or $[0,1]$ with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. (Multi)wavelets on intervals satisfying homogeneous boundary conditions will also be addressed. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. Several examples of orthogonal and biorthogonal multiwavelets on the interval $[0,1]$ will be provided to illustrate our construction approaches and proposed algorithms.

Chebyshev Biorthogonal Multiwavelets and Approximation

In this paper, we construct Chebyshev biorthogonal multiwavelets, and use this multiwavelets to approximate signals (functions). The convergence rate for signal approximation is derived. The fast signal decomposition and reconstruction algorithms are presented. The numerical examples validate the theoretical analysis.

Biorthogonal multiwavelets on the interval for solving multidimensional fractional optimal control problems with inequality constraint

SummaryThis article proposes a new numerical approach for solving fractional optimal control problems including state and control inequality constraints using new biorthogonal multiwavelets. The properties of biorthogonal multiwavelets are first given. The Riemann‐Liouville fractional integral operator for biorthogonal multiwavelets is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well‐developed algorithms may be applied. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. The method is computationally very attractive and gives very accurate results.

High performance angiogram sequence compression using 2D bi-orthogonal multi wavelet and hybrid speck-deflate algorithm

Fast communication of medical information is a big challenge for today’s researchers. Either bandwidth or information needs to be sacrificed in order to communicate the entire information within a short time with minimal bandwidth. Compression of medical information before transmission helps us to communicate the entire information within short time with minimal cost. This paper concentrates on reducing the transmission bandwidth by compressing the entire medical information using a Hybrid algorithm which uses 2D Bi-orthogonal multi-wavelet transform and SPECK – Deflate encoder. The algorithm is studied with the standard test dataset GRUSELAMBIX. The real time data was analyzed to ensure the perseverance of the diagnostic information. 2D Bi-orthogonal multi-wavelet transform was applied on each frame on the DICOM file and the co-efficients are encoded using SPECK encoder followed by the deflate compression algorithm. To preserve the original diagnostic information, motion estimation and compensation was not applied on the medical videos. The qualitative performance of this algorithm is evaluated using Compression Ratio, PSNR, MSE, UIQIand SSIM. The Mean opinion Score from experts and non-experts was obtained on the real time data to evaluate the quantitative performance. The proposed algorithm revealed a better compression ratio than the traditional compression algorithm.

Optimal control of linear time‐delay systems by a hybrid of block‐pulse functions and biorthogonal cubic Hermite spline multiwavelets

SummaryThis paper proposes a new numerical approach for finding the solution of linear time‐delay control systems with a quadratic performance index using new hybrid functions. This method is based on a hybrid of block‐pulse functions and biorthogonal multiwavelets that consist of cubic Hermite splines on the primal side. The excellent properties of the hybrid functions together with the operational matrices of integration, product, and delay are presented. Using these matrices, the solution of the optimal control of delay systems is reduced to the solution of algebraic equations. Because of the sparsity nature of these matrices, this method is computationally very attractive and reduces CPU time and computer memory. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. The effectiveness of the method and the accuracy of the solution are shown in comparison with some other methods by illustrative examples.

Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation

To numerically solve the Burgers’ equation, in this paper we propose a general method for constructing wavelet bases on the interval [0,1] derived from symmetric biorthogonal multiwavelets on the real line. In particular, we obtain wavelet bases with simple structures on the interval [0,1] from the Hermite cubic splines. In comparison with all other known constructed wavelets on the interval [0,1], our constructed wavelet bases on the interval [0,1] from the Hermite cubic splines not only have good approximation and symmetry properties with extremely short supports, but also employ a minimum number of boundary wavelets with a very simple structure. These desirable properties make them to be of particular interest in numerical algorithms. Our constructed wavelet bases on the interval [0,1] are then used to solve the nonlinear Burgers’ equation. Our method is based on the finite difference formula combined with the collocation method. Therefore, our proposed numerical scheme in this paper is abbreviated as MFDCM (Mixed Finite Difference and Collocation Method). Some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy.

Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity $$r=2$$r=2, a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.

Biorthogonal balanced multiwavelets with high armlets order and their application in image denoising

Multiple approaches can be used to meet the nth order wavelet analysis consistency requirement between balanced multiwavelets and armlets. Due to the complexity of the n-balancing conditions, it is difficult to construct a family of n-balanced orthogonal multiwavelets for a relatively large n. In this paper, we introduce a convenient way to construct symmetric/antisymmetric biorthogonal multiwavelets provided that both the balanced order n≥2 and the high armlets order conditions are met. First, we introduce the concept of biorthogonal armlets and illustrate a relationship among the four filter banks of a biorthogonal system. Then we present an algorithm for constructing biorthogonal balanced multiwavelets with armlets order (BBMA) and compute a novel class of such multiwavelets. Finally, we investigate the performances of BBMA and existing wavelets on the application of image denoising. Simulation results show that BBMA typically result in improved denoising.

Balanced Biorthogonal Multiwavelet with Symmetric/Antisymmetric Filter Banks

A class of the balanced biorthogonal multiwavelets was constructed by defining a specific matrix filter structure, in which the multifilter banks of multiwavelets have had the desired symmetry. Here, the multifilter banks have possess symmetric/antisymmetric, which resembled filters of scalar wavelet and have in favor of application, notwithstanding the multiwavelets constructed in this paper have lost the linear phase, so they have formed a new type of multiwavelets undoubtedly.

BIORTHOGONAL MULTIWAVELETS RELATED BY DIFFERENTIATION

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.

High Balanced Biorthogonal Multiwavelets with Symmetry

Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. Yang et al. (2006) constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.

Divergence-Free Multiwavelets on the Half Plane

We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition.

Explicit construction of compactly supported biorthogonal multiwavelets via matrix extension

Polyphase matrix extension of the scaling vector functions plays an important role in the construction of compactly supported biorthogonal multiwavelets. However, the involved computations are very complicated, and there is no unified, direct formula available so far. In this paper, abstract algebraic methods are used to investigate the canonical forms of polyphase matrices of the scaling vector functions. According to the related properties of canonical forms of polyphase matrices, it is proved that the matrix extension equations are always solvable so that two explicit formula groups of the solution set corresponding to different canonical forms can be derived. All the explicit formulas are represented via the submatrices of polyphase matrices directly. Furthermore, for a given matrix extension problem, any solution can be obtained from these explicit formulas via product-preserving transformations, which means that, the proposed algorithm provides a complete solution set. Computational examples demonstrated that by using the explicit formulas, our matrix extension algorithm is direct and effective. Finally, a simple application by using multiwavelets for denoising is presented and the experimental results showed that the multiwavelets outperformed the scalar wavelets under different test signals.

De-Noising Signal of The Quartz Flexural Accelerometer by Multiwavelet Shrinkage

Abstract Methods of de-noising the output signal of the JSD-I/A quartz flexural accelerometer based on five types of multiwavelets are comparatively investigated in this paper. Firstly, the theory of multiwavelet transform and the generalized cross validation criterion are analyzed. Secondly, because the JSD-I/A quartz flexural accelerometer which is fixed in SCT-1 two-axis rotation platform by the appropriative clamp has a start-up procedure of 3 minutes, the output signal of the quartz flexural accelerometer are sampled after applying the voltage for 5 minutes. Thirdly, based on the soft threshold function and the universal threshold, GHM orthogonal multiwavelet, SA4 orthogonal multiwavelet, CL orthogonal multiwavelet, Cardbal2 balanced multiwavelet and BIGHM biorthogonal multiwavelet are applied to de-noise the sampled signal of the JSD-I/A quartz flexural accelerometer with 4 decomposition level, respectively. Lastly, the generalized cross validation criterion is used to evaluate the de-noising effects of the above five multiwavelets. The experimental results shows that the generalized cross validation value of BIGHM biorthogonal multiwavelet is effective in de-nosing the the output signal of the JSD-I/A quartz flexural accelerometer, and offer the best performance than GHM orthogonal multiwavelet, SA4 orthogonal multiwavelet, CL orthogonal multiwavelet and Cardbal2 balanced multiwavelet.

Explicit construction of symmetric compactly supported biorthogonal multiwavelets via group transformations

Polyphase matrix extension of multiscaling vectors is a fundamental approach for the construction of compactly supported biorthogonal multiwavelets. In this paper, canonical form of polyphase matrices of multiscaling vectors and some properties of solutions satisfying the matrix equation of PR condition are studied through unimodular groups over the Laurent polynomial ring. Under the canonical form of polyphase matrices of multiscaling vectors with the same symmetric center and different symmetric centers, explicit symmetric extension formulas expressed by the polyphase matrices of multiscaling vectors can be obtained. As a result, a novel approach for the construction of symmetric compactly supported biorthogonal multiwavelets with multiplicity 2 is proposed. Finally, several examples are given for verification of our proposed approach.

Totally interpolating biorthogonal multiwavelet systems with approximation order

In this article, compactly supported totally interpolating biorthogonal multiwavelet systems are studied. Necessary and sufficient conditions for such systems to have given approximation orders are stated in simple equations. It is shown that the shorter nontrivial filter component that has the minimum possible length for a given approximation order is uniquely determined up to a discrete parameter. Among systems with such property, we provide all totally interpolating biorthogonal stable multiwavelet systems of approximation orders 2 and 3 with minimal total length whose scaling vectors have minimal lengths as well.

APPROXIMATION AND BALANCING ORDERS FOR TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS

We consider totally interpolating biorthogonal multiwavelet systems with finite impulse response two-band multifilter banks, a study balancing order conditions of such systems. Based on FIR and interpolating properties, we show that approximation order condition is completely equivalent to balancing order condition. Consequently, a prefiltering can be avoided if a totally interpolating biorthogonal multiwavelet system satisfies suitable approximation order conditions. An example with approximation order 4 is provided to illustrate the result.

Matrix extension with symmetry and construction of biorthogonal multiwavelets with any integer dilation

In this paper, we investigate the biorthogonal matrix extension problem with symmetry and its application to construction of biorthogonal multiwavelets. Given a pair of biorthogonal matrices (P,P˜), the biorthogonal matrix extension problem is to find a pair of extension matrices (Pe,P˜e) of Laurent polynomials with symmetry such that the submatrix of the first r rows of Pe,P˜e is the given matrix P,P˜, respectively; Pe and P˜e are biorthogonal satisfying PeP˜e⋆=Is; and Pe and P˜e have the same compatible symmetry. We satisfactorily solve the biorthogonal matrix extension problem with symmetry and provide a step-by-step algorithm for constructing the desired pair of extension matrices (Pe,P˜e) from the given pair of matrices (P,P˜). Moreover, our results cover the case for paraunitary matrix extension with symmetry (i.e., the given pair satisfies P=P˜). Matrix extension plays an important role in many areas such as wavelet analysis, electronic engineering, system sciences, and so on. As an application of our general results on biorthogonal matrix extension with symmetry, we obtain a satisfactory algorithm for constructing univariate biorthogonal multiwavelets with symmetry for any dilation factor d from a given pair of biorthogonal d-refinable function vectors with symmetry. Correspondingly, pairs of d-dual filter banks with the perfect reconstruction property and with symmetry can be derived by applying our algorithm to a given pair of d-dual low-pass filters with symmetry. Several examples of symmetric biorthogonal multiwavelets are provided to illustrate our results in this paper.