In this paper, using orthogonally of TCHEBYCHEV polynomials, we present an orthonormal wavelet basis for L2[0, 1]. We use this basis for solving Neumann problems with Galerkin method. The property of this basis is that a variety of integral operators is represented in this basis as sparse matrices, to high precision. Some examples are solved to illustrate the efficiency and accuracy of this method.

This paper presents a new fingerprint coding technique using non-linear approximation of multiwavelet coefficients followed by multi-stage vector quantization. Fingerprints contain oscillatory patterns in the vertical, horizontal and diagonal direction. Wavelet transform often fails to accurately capture the oscillatory patterns in the fingerprints, especially at low bit rates. Due to implementation constraints scalar wavelets do not posses all the properties such as orthogonality, short support, linear phase symmetry, and a high order of approximation through vanishing moments simultaneously, which are very much essential for efficient image compression. New class of wavelets called ‘Multiwavelets’ which posses more than one scaling function overcomes this problem. The effective multiwavelet coefficients are quantized by multistage vector quantization which can achieve very low encoding and storage complexity in comparison to unstructured vector quantization. Entropy coding of quantized coefficients is done using Huffman coding. The performance of the proposed scheme is compared with the results obtained from scalar wavelets. The performance of the proposed scheme is better than the existing wavelet based coding at low bit rate.

Motivated by [B. Han and Q. Mo, Adv. Comp. Math. 18(2003) 211-245] and [B. Han and Z. Shen, Constr. Approx. 29 (2009) 369-406], we propose dual two-direction frames in dual Sobolev spaces (Hs (R), H-s (R)), with s>0. Based on the dual two-direction frames from a pair of two-direction refinable functions, dual multiwavelet frames with symmetry{ Y l(x): = (y1l(x), y2l(x)) T}l=1 d and {y~ l (x): = (y~l1(x), y~l2(x)) T} l=1d can be constructed very easily. The vanishing moment of the constructed multiwavelet frames is discussed. An example is given to illustrate our results.

There are some methods for solving integro-differential equations. In this work, we solve the general-order Feredholm integro-differential equations. The Petrov-Galerkin method by considering Chebyshev multiwavelet basis is used. By using the orthonormality property of basis elements in discretizing the equation, we can reduce an equation to a linear system with small dimension. For numerical examples, the solutions may be produced with good accuracy, by choosing suitable trial and test spaces in Petrov-Galerkin method.

This work was intended as an attempt to motivate readers for a comparison study of constructions of Legendre multiwavelet and Chebyshev multiwavelet. It is also shown how to use them in Petrov-Galerkin approach for solving Fredholm integro-differential equation of high orders of the second kind. In fact, a numerical technique for the discretization method of Fredholm integro-differential equations is presented that yields linear system. The important point to note here is the convergence of presented methods. For the first time, two conditions are proved for convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method. The proof of these conditions with using linear algebra and matrix theory ensures that Petrov-Galerkin methods has a unique approximation. Finally, some relevent numerical examples, for which the exact solution is known, will indicate accuracy and applicability of the proposed method.

In this paper vibration analysis for local faults and transient phenomena detection, using multiwavelet systems is developed. Unlike the scalar wavelet systems in which their coefficients are scalar parameters, the transformation parameters of multiwavelet systems are vector valued, and their calculation requires some special techniques. In this investigation, having considered the technique used to obtain the scalar wavelet system coefficients as well as the orthogonality of the scale and wavelet functions of multiwavelet systems, the transformation coefficients of the multiwavelet system are calculated, and then some artificial vibration signals are analyzed using the multiwavelet system. The results are compared with those obtained through the scalar wavelet systems and frequency analysis techniques. The combination of the contributions of the analyzed signal at different levels of the multiscale and multiwavelet function spaces results in original signal which shows the validity of the results. One of the main advantages of the multiwavelet systems over the scalar wavelet systems is their ability to analyze the signal in more frequency intervals. Using this property, the detection of the local and transient phenomena from the vibration signals, which may be caused by a small and local defects in a mechanical system, may be performed more efficiently.

In this paper, we introduce Legendre multiwavelet functions and use them as a set of base functions to approximate the solution of a parabolic differential equation with an unknown time-dependent coefficient in an inverse problem. Using the expansion formula of a known function in terms of the Legendre multiwavelet base, we define integral and product operational matrices from a general point of view. With the help of these matrices, we transform the problem into a system of algebraic equations. By solving the obtained system of algebraic equations using the existing optimization algorithms, we provide an approximation for the solution of the problem in the form of its expansion in terms of the Legendre multiwavelet base. In addition to expressing the algorithm of the proposed numerical method, we perform the proposed method on two examples and report its numerical results. We also compare the results of the proposed method with the results reported from other methods.

Backgorund: Nowadays, everybody’s life is dominated by COVID-19, which might have been the source of severe acute respiratory syndrome coronavirus 2. This virus disrupts the lungs first of all. Recently, it has been found that coronavirus may affect the brain. Because all body actions rely on the brain, hence investigating its healthy is an essential item in coronavirus effects. Method: Brain image segmentation can be helpful in the detection of the regions damaged by the effects of coronavirus. Since every image given by photography devices may have noises, therefore, first of all, the brain magnetic resonance angiography (MRA) images must be denoised for best investigation. In the present paper, we have presented the construction of multishearlets based on multiwavelets for the first time and have used them for the purpose of denoising. Multiwavelets have some advantages to wavelets. Therefore, we have used them in the shearlet system to expand the properties of multiwavelets in all directions. After denoising, we have proposed a scheme for the automatic characterization of the initial curve in the active contour model for segmentation. Detecting the initial curve is a challenging task in active contour? based segmentation because detecting an initial curve far from the desired region can lead to unfavorable results. Results: The results show the performance of using multishearlets in detecting affected regions by COVID-19. Using multishearlets has led to the high value of peak signal? to? noise ratio and Structural similarity index measure in comparison with original shearlets. Original shearlets are constructed from wavelets whereas we have constructed multishearlets from multiwavelets. Conclusion: The results show that multishearlets can neutralize the effect of noise in MRA images in a good way rather than shearlets. Moreover, the proposed scheme for segmentation can lead to 0. 99 accuracy.