This paper investigates the mathematical framework of multiresolution analysis based on irregularly spaced knots sequence. Our presentation is based on the construction of nested nonuniform spline multiresolution spaces. From these spaces, we present the construction of orthonormal scaling and wavelet basis functions on bounded intervals. For any arbitrary degree of the spline function, we prov...

This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar functi...

This paper proposes to adopt hierarchical tree construction of directional lapped orthogonal transforms (DirLOTs) to image denoising. The DirLOTs are 2-D non-separable lapped orthogonal transforms with directional characteristics. The bases are allowed to be anisotropic with the fixed-criticallysubsampling, overlapping, orthogonal, symmetric, real-valued and compact-support property. As well, i...

Abstract. We investigate the use of compactly supported divergence-free wavelets for the representation of solutions of the Navier-Stokes equations. After reviewing the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for two and three-dimensional incompressible flows. We also propose a new method to practically comput...

This paper considers the classical Shannon sampling theorem in multiresolution spaceswith scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which...

Suppose we have observations yi = si+zi, i = 1; :::; n, where (si) is signal and (zi) is i.i.d. Gaussian white noise. Suppose we have available a library L of orthogonal bases, such as the Wavelet Packet bases or the Cosine Packet bases of Coifman and Meyer. We wish to select, adaptively based on the noisy data (yi), a basis in which best to recover the signal (\de-noising"). Let Mn be the tota...

The method of signal denoising via wavelet thresholding was popularised by Donoho and Johnstone (1994, 1995) and is now widely applied in science and engineering. It is based on thresholding of wavelet coefficients arising from the standard scalar orthogonal discrete wavelet transform (DWT). Recently this approach has been extended to incorporate thresholding coefficients arising from the discr...

The aim of this paper is the detailed investigation of trigono-metric polynomial spaces as a tool for approximation and signal analysis. Sample spaces are generated by equidistant translates of certain de la Vall ee Poussin means. The diierent de la Vall ee Poussin means enable us to choose between better time-or frequency-localization. For nested sample spaces and corresponding wavelet spaces,...

In this paper we study the algebraic and geometric structure of the space of compactly supported biorthogonal wavelets. We prove that any biorthogonal wavelet matrix pair (which consists of the scaling lters and wavelet lters) can be factored as the product of primitive parau-nitary matrices, a pseudo identity matrix pair, an invertible matrix, and the canonical Haar matrix. Compared with the f...

In this paper, diffusion wavelet-based multiscale linear minimum mean square-error estimation (LMMSE) scheme for image denoising in conjunction to neural network is proposed, and the determination of the optimal wavelet basis with respect to the proposed scheme is also discussed. Genrally ,the over complete wavelet expansion (OWE) is more effective than the orthogonal wavelet transform (OWT) sp...