Multiscaling Functions Research Articles

Bauer’s spectral factorization method for low order multiwavelet filter design

Para-Hermitian polynomial matrices obtained by matrix spectral factorization lead to functions useful in control theory systems, basis functions in numerical methods or multiscaling functions used in signal processing. We introduce a fast algorithm for matrix spectral factorization based on Bauer’s method. We convert Bauer’s method into a nonlinear matrix equation (NME). The NME is solved by two different numerical algorithms (Fixed Point Iteration and Newton’s Method) which produce approximate scalar or matrix factors, as well as a symbolic algorithm which produces exact factors in closed form for some low-order scalar or matrix polynomial matrices, respectively. Convergence rates of the two numerical algorithms are investigated for a number of singular and nonsingular scalar and matrix polynomials taken from different areas. In particular, one of the singular examples leads to new orthogonal multiscaling and multiwavelet filters. Since the NME can also be solved as a Generalized Discrete Time Algebraic Riccati Equation (GDARE), numerical results using built-in routines in Maple 17.0 and six Matlab versions are presented.

Multiwavelet-based ECG compressed sensing

This paper presents a novel Multiwavelet-based hybrid method for compressed sensing (CS) and reconstruction of ECG signals. The proposed method is patient-agnostic and does not require any prior patient-specific information. The proposed method removes low-frequency components by using multi-scaling wavelet functions, which sparsify the ECG signals in the multi-wavelet domain. A novel way of CS acquisition is used to acquire the high-frequency component of the ECG signal. The compressed sensed signal acquisition is performed based on sample difference thresholding. This approach ensures that a significant portion of the ECG signal will be acquired, without a need for a QRS complex or an R peak detector. The main compression is achieved by CS and linear and nonlinear quantization. Arithmetic encoding is used to improve compression further. The results show good reconstruction quality while maintaining a high compression ratio. The proposed method was evaluated for different multi-wavelet functions.

FACE IDENTIFICATION USING BACK-PROPAGATION ADAPTIVE MULTIWAVENET

Face Identification is an important research topic in the field of computer vision and pattern recognition and has become a very active research area in recent decades. Recently multiwavelet-based neural networks (multiwavenets) have been used for function approximation and recognition, but to our best knowledge it has not been used for face Identification. This paper presents a novel approach for the Identification of human faces using Back-Propagation Adaptive Multiwavenet. The proposed multiwavenet has a structure similar to a multilayer perceptron (MLP) neural network with three layers, but the activation function of hidden layer is replaced with multiscaling functions. In experiments performed on the ORL face database it achieved a recognition rate of 97.75% in the presence of facial expression, lighting and pose variations. Results are compared with its wavelet-based counterpart where it obtained a recognition rate of 10.4%. The proposed multiwavenet demonstrated very good recognition rate in the presence of variations in facial expression, lighting and pose and outperformed its wavelet-based counterpart.

An algorithm for constructing the two-direction Armlet multiwavelet

In this paper, an algorithm is discussed for constructing the two-direction Armlet multiwavelet. First, the definition of two-direction Armlet multiwavelet is presented in this paper. A two-direction multiwavelet can be changed to a special multi-wavelet. By Two-scale Similar Transform (TST), a transform can be taken on the two-scale matrix symbols of a two-direction multi-wavelets. This transform keeps the orthogonality of the two-direction multi-wavelets. However, the condition is discussed which the two-direction multi-wavelet corresponding to a two-direction multi-scaling function is an Armlet with order n. An approach is given for constructing the transform matrix. Finally, an example is given for discussing the two-direction Armlet multi-wavelet with order 2.

Construction of symmetric multiwavelets using standard pairs

A multiscaling equation in the Fourier domain accommodates a trigonometric matrix polynomial. This trigonometric matrix polynomial is known as the symbol function. The existence and properties of a multiscaling function, which is the solution of a multiscaling equation, depend on the symbol function. It is possible to construct symbol functions corresponding to compactly supported and symmetric multiscaling functions from standard pairs. A standard pair carries the spectral information about the symbol function. In this paper, we briefly explain the construction of compactly supported and symmetric multiscaling functions and the corresponding mulitwavelets using standard pairs. We derive the necessary as well as sufficient condition, on the eigenspace of the square matrix in the standard pair, for the existence of a symbol function corresponding to a multiscaling equation with a compactly supported solution. We create a pseudo bi-orthogonal pair of symmetric and compactly supported multiscaling functions and the corresponding multiwavelets using standard pairs.

Toward new multi-wavelets: associated filters and algorithms. Part I: theoretical framework and investigation of biomedical signals, ECG, and coronavirus cases.

Biosignals are nowadays important subjects for scientific researches from both theory, and applications, especially, with the appearance of new pandemics threatening the humanity such as the new coronavirus. One aim in the present work is to prove that wavelets may be a successful machinery to understand such phenomena by applying a step forward extension of wavelets to multi-wavelets. We proposed in a first step to improve multi-wavelet notion by constructing more general families using independent components for multi-scaling and multi-wavelet mother functions. A special multi-wavelet is then introduced, continuous, and discrete multi-wavelet transforms are associated, as well as new filters, and algorithms of decomposition, and reconstruction. Applied breakthroughs of the paper may be summarized in three aims. In a first direction, an approximation (reconstruction) of a classical (stationary, periodic) example dealing with Fourier modes has been conducted in order to confirm the efficiency of the HSch multi-wavelets in approximating such signals and in providing fast algorithms. The second experimentation is concerned with the decomposition and reconstruction application of the HSch multi-wavelet on an ECG signal. The last experimentation is concerned with a de-noising application on a strain of coronavirus signal permitting to localize approximately the transmembrane segments of such a series as neighborhoods of the local maxima of an numerized version of the strain. Accuracy of the method has been evaluated by means of error estimates and statistical tests.

Approximation order of two-direction multiscaling functions

We investigate Condition E and the basic regularity conditions for the two-direction multiscaling functions $\boldsymbol {\phi }$, and then investigate approximation order of $\boldsymbol {\phi }$. By investigating a structure of approximation vectors, we are able to find simple and efficient criteria for Condition E, the basic regularity conditions, and the approximation order of $\boldsymbol {\phi }$. Examples for illustrating the general theory are given.

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.

Combination of epsilon and Ritz methods with multiscaling basis for solving a class of fractional optimal control problems

This paper presents an approximate direct method for solving a class of fractional optimal control problems by combining the epsilon and Ritz methods and utilizing multiscaling basis functions. By applying the epsilon method, we transform the given optimal control problem into a variational problem. Then, we solve the variational problem approximately by the Ritz direct method with multiscaling polynomial basis functions. The choice of multiscaling basis functions provides a high rate of convergence for the Ritz method. The convergence of the derived approximate minimal to the exact one is extensively discussed. At the end some illustrative test examples are included to demonstrate the efficiency of the technique.

Hermite interpolant multiscaling functions for numerical solution of the convection diffusion equations

A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diusion equation to the solution of algebraic equations. Because of sparsity of these matrices, this method is computationally very attractive and reduces the CPU time and computer memory. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Matrix spectral factorization for SA4 multiwavelet

In this paper, we investigate Bauer's method for the matrix spectral factorization of an r-channel matrix product filter which is a halfband autocorrelation matrix. We regularize the resulting matrix spectral factors by an averaging approach and by multiplication by a unitary matrix. This leads to the approximate and exact orthogonal SA4 multiscaling functions. We also find the corresponding orthogonal multiwavelet functions, based on the QR decomposition.

Nonlinear Constrained Optimal Control Problems and Cardinal Hermite Interpolant Multiscaling Functions

AbstractIn this paper, a numerical method for solving nonlinear quadratic optimal control problems with inequality constraints is presented. The method is based upon cardinal Hermite interpolant multiscaling function approximation. The properties of these multiscaling functions are presented first. These properties are then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one, to which existing algorithms may be applied. Illustrative examples are included to demonstrate the efficiency and applicability of the technique.

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.

Cardinal Hermite interpolant multiscaling functions for solving a parabolic inverse problem

An effective method based upon cardinal Hermite interpolant multiscaling functions is proposed for the solution of the one-dimensional parabolic partial differential equation with given initial condition and known boundary conditions and subject to overspecification at a point in the spatial domain. The properties of multiscaling functions are first presented. These properties together with a collocation method are then utilized to reduce the parabolic inverse problem to the solution of algebraic equations. The scheme described is efficient. The numerical results obtained using the present algorithms for test problems show that this method can solve the model effectively.

Construction of multiscaling functions using the inverse representation theorem of matrix polynomials

Wavelet analysis deals with finding a suitable basis for the class of $$L^2$$ functions. Symmetric basis functions are very useful in various applications. In the case of all wavelets other than the famous Haar wavelet, the simultaneous inclusion of compact supportedness, orthogonality and symmetricity is not possible. Theory of multiwavelets assumes significance since it offers orthogonal, compact frames without losing symmetry. We can also construct symmetric, compactly supported and pseudo-biorthogonal bases which are also possible only in the case of multiwavelets. The properties of a multiwavelet directly depends on the corresponding multiscaling function. A multiscaling function is characterized by a unique symbol function, which is a matrix polynomial in complex exponential. A matrix polynomial can be constructed from its spectral data. Our aim is to find the necessary as well as sufficient conditions a spectral data must satisfy so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function $$\varPhi (x)$$ . We will construct such a multiscaling function $$\varPhi (x)$$ and its dual $$\tilde{\varPhi }(x)$$ such that the functions $$\varPhi (x)$$ and $$\tilde{\varPhi }(x)$$ form a pair of pseudo-biorthogonal multiscaling functions.

Numerical Solution Of Black-scholes Equation Using Bernstein Multi-scaling Functions

A numerical method for solving Black-Scholes equation is presented. The method is based upon Bernstein multi-scaling basis approximations. The properties of Bernstein multi-scaling functions are first presented. These properties together with the forward Euler and Ritz-Galerkin method are then utilized to reduce the Black-Scholes equation to the solution of algebraic equations. Illustrative example is included to demonstrate the validity and applicability of the new technique.

Point Values and Normalization of Two-Direction Multi-wavelets and their Derivatives

Abstract. A two-direction multiscaling function ˚ satis es a recursion relation that usesscaled and translated versions of both itself and its reverse. This o ers a more general andexible setting than standard one-direction wavelet theory. In this paper, we investigatehow to nd and normalize point values and derivative values of two-direction multiscalingand multiwavelet functions. Determination of point values is based on the eigenvalue ap-proach. Normalization is based on normalizing conditions for the continuous moments of˚. Examples for illustrating the general theory are given. 1. IntroductionA standard (one-direction) multiscaling function satis es a recursion relationwhich uses scaled and translated versions of itself (see (2.1) below). A two-direction multiscaling function ˚satis es a recursion relation that uses scaledand translated versions of both itself and its reverse (see (2.2) below). A two-direction multiwavelet is formed by ˚and its associated multiwavelet functions (s)

Analysis of time-varying delay systems by hybrid of block-pulse functions and biorthogonal multiscaling functions

This paper proposes new effective methods for analysis of time-varying delay systems using new hybrid functions. The excellent properties of the hybrid functions which consist of block pulse functions and biorthogonal multiscaling functions are presented. We utilise operational matrices of product, delay and integration to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

Construction of orthogonal multiwavelet bases

We propose a method for constructing orthogonal multiwavelet bases of the space L 2(ℝ) for any known multiscaling functions that generate a multiresolution analysis of dimension greater than 1.

Minimum-Energy Multiwavelet Frame on the Interval[0,1]

Drawing inspiration from the idea of combining multiwavelets on the interval with frame theory organically, we study minimum-energy multiwavelet frame on the interval[0,1](MEMWFI). Firstly, left boundary multiscaling functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward, and the equivalent characterizations of MEMWFI are given. Then, two algorithms of constructing MEMWFI are proposed. Finally, the decomposition formula, reconstruction formulas, and numerical examples are given.