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When a signal is embedded in an additive Gaussian noise, its estimation is often done by finding a wavelet basis that concentrates the signal energy over few coefficients and by thresholding the noisy coefficients. However, in many practical problems such as medical X-ray image, astronomical and low-light images, the recorded data is not modeled by Gaussian noise but as the realization of a Poisson process. Multiwavelet is a new development to the body of wavelet theory. Multiwavelet simultaneously offers orthogonality, symmetry and short support which are not possible in scalar 2-channel wavelet systems. After reviewing this recently developed theory, a new theory and algorithm for denoising medical X-ray images using multiwavelet multiple resolution analysis (MRA) are presented and investigated in this paper. The proposed covariance shrink (CS) method is used to threshold wavelet coefficients. The form of thresholds is carefully formulated which is the key to more excellent results obtained in the extensive numerical simulations of medical image denoising compared to conventional methods.

The aim of any fingerprint image compression technique is to achieve a maximum amount of compression with an adequate quality compressed image which is suitable for fingerprint recognition. Currently available techniques in the literature provide 100% recognition only up to a compression ratio of 180:1. The performance of any identification technique inherently depends on the techniques with which images are compressed. To improve the identification accuracy while the images are highly compressed, a multiwavelet-based identification approach is proposed in this paper. Both decimated and undecimated coefficients of SA4 (Symmetric Antisymmetric) multiwavelet are used as features for identification. A study is conducted on the identification performance of the multiwavelet transform with various sizes of images compressed using both wavelets and multiwavelets for fair comparison. It was noted that for images with size power of 2, the decimated multiwavelet-based compression and identification give a better performance compared to other combinations of compression/identification techniques whereas for images with size not a power of 2, the undecimated multiwavelet transform gives a better performance compared to other techniques. A 100% identification accuracy was achieved for the images from NIST-4, NITGEN, FVC2002DB3_B, FVC2004DB2_B and FVC2004DB1_B databases for compression ratios up to 520:1, 210:1, 445:1, 545:1 and 1995:1, respectively.

A pre-processing design using neural networks is proposed for multiwavelet filters. Various numerical experiments are presented and a comparison is given between neural network pre-processing and a pre-processing for solving linear systems. Neural network pre-processing produces a good approximation for a large number of terms and converges rapidly.

In this paper we construct a one-parameter family of interpolating multigenerators which generalize the scalar generators constructed in Ref. 4 in a natural way. Furthermore we extend this approach to design a family of interpolating orthonormal multigenerators. The construction is based on the factorization techniques introduced in Ref. 27.

A new algorithm for modeling of chaotic systems is presented in this paper. First, more information is acquired utilizing the reconstructed embedding phase space, and the multiwavelets transform provides a sensible decomposition of the data so that the underlying temporal structures of the original time series become more tractable. Second, based on the Recurrent Least Squares Support Vector Machines (RLS-SVM), modeling of the chaotic system is realized. To demonstrate the effectiveness of our algorithm, we use the power spectrum and dynamic invariants involving the Lyapunov exponents and the correlation dimension as criterions, and then apply our method to Chua's circuit time series. The similarity of dynamic invariants between the original and generated time series shows that the proposed method can capture the dynamics of the chaotic time series more effectively.

The parametrization for two kinds of multifilter banks generating balanced multiwavelets is presented in this paper. In case (I), both lowpass and highpass filters are flipping filters. Filters in case (II) have different lengths, and both lowpass and highpass filters are symmetric (antisymmetric). Based on these parametric expressions, some balanced multiwavelets and analysis-ready multiwavelets (armlets) are constructed. Moreover, the application of these multiwavelets constructed in image processing is also studied.

The multiscaling function of Alpert multiwavelet system and Chui–Lian multiwavelet system consist of one symmetric and one antisymmetric scaling functions. In its single level decomposition, only one of 16 subband blocks may be considered as a core part because the antisymmetric scaling function may play the high pass filter. We complete the Alpert multiwavelet system by determining the high pass filter using the algorithm of paraunitary extension principle. Taking the advantage of the symmetric/antisymmetric (SA) feature of Alpert and Chui–Lian multiscaling functions, our iteration (SA iteration) of multiwavelet decomposition continues to the core part (1/16 of subbands) for the image denoising problem while the traditional iteration applies to the one fourth of 16 subbands as core parts. The numerical results demonstrate that our method of symmetric/antisymmetric iteration exhibits performance superior to the traditional ways.

The solutions of convection and diffusion equations are analyzed using Multiresolution Multiwavelet (MRMW) basis functions of multidimension including time. The distribution representing the solution reconstructed in non-standard form of the higher order MRMW reproduces the original solution in the multidimensional space much more efficiently than the lower order one. Smooth movement of the distribution is observed except for the initial area even using the MRMW of zeroth order but the shape of the distribution is not well reproduced for the case. The initial area is better reproduced by the MRMW of higher order and the wavelet component decays much quicker both in low and high levels of resolution.

Vector-valued refinable interpolatory functions with multiplicity r are discussed in this paper. This kind of refinable functions have a sampling property like Shannon's sampling theorem, and corresponding matrix-valued refinable masks possess special structure. In the context of multiwavelets, some properties of multifilter banks related will be present. Based on these properties, it will be shown that there are no symmetric (or antisymmetric) vector-valued refinable functions with interpolatory property. In the practical application, multiwavelets are always required to possess a certain degree of smoothness, which is related to three different concepts: balancing order, approximation order and analysis-ready order. In the general case, three notions are different. But if the scaling function is interpolatory, three concepts will be verified to equal to each other. Finally, a complete characterization of multifilter banks {H, G} will also be given and it will be used to construct some new balanced multiwavelets with interpolatory property for case r = 2, corresponding to which, multifilter banks have rational coefficients.

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.

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 $p$ 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 $1.$ 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 $1$, 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 $1$.

In this paper, using standard pairs, we present a method to construct symbols satisfying the sum rules of order $p$ for any given $p$. It is shown that using matrix polynomial theory symbols, symmetric or non-symmetric, satisfying the sum rules of order $p$ can be constructed efficiently. The construction is illustrated using various examples.