Multiwavelets--theory and applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (leaves 93-99).by Vasily Strela.Ph.D

Signal and Image Denoising via Wavelet Thresholding: Orthogonal and Biorthogonal, Scalar and Multiple Wavelet Transforms

In this paper we discuss wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into account the covariance structure of the transform. The form of the universal threshold is carefully formulated. The results of numerical simulations in signal and image denoising are presented. Multiwavelets outperform scalar wavelets for three out of four noisy 1D test signals, and the Chui-Lian scaling functions and wavelets combined with repeated row preprocessing appears to be a good general method. Vector thresholding does not always outperform scalar thresholding. Multiwavelets generally outperform scalar wavelets for image denoising for all four noisy 2D test images, and the results are visually very impressive. Only for `Lenna' and `fingerprints' with signal to noise ratios of 2 do scalar wavelets perform best. As f..

Compactly supported refinable functions with infinite masks, preprint

A compactly supported scaling function can come from a refinement equation with infinitely many nonzero coefficients (an infinite mask). In this case we prove that the symbol of the mask must have the special rational form ã(Z) = ˜ b(Z 2)˜c(Z) / ˜ b(Z). Any finite combination of the shifts of a refinable function will have such a mask, and will be refinable. We also study compactly supported solutions of vector refinement equations with infinite masks. Our characterization is based on the two-scale similarity transform which plays an essential role in the investigation of multiple wavelets. This concept is used to characterize refinable subspaces of refinable shift-invariant spaces. One advantage of our approach is to provide the refinement masks for generators of refinable subspaces

From Wavelets to Multiwavelets

. This paper gives an overview of recent achievements of the multiwavelet theory. The construction of multiwavelets is based on a multiresolution analysis with higher multiplicity generated by a scaling vector. The basic properties of scaling vectors such as L 2 -stability, approximation order and regularity are studied. Most of the proofs are sketched. 1. Introduction Wavelet theory is based on the idea of multiresolution analysis (MRA). Usually it is assumed that an MRA is generated by one scaling function, and dilates and translates of only one wavelet # # L 2 (IR) form a stable basis of L 2 (IR). This paper considers a recent generalization allowing several wavelet functions # 1 , . . . , # r . The vector ## # # # # # ## = (# 1 , . . . , # r ) T is then called a multiwavelet. Multiwavelets have more freedom in their construction and thus can combine more useful properties than the scalar wavelets. Symmetric scaling functions constructed by Geronimo, Hardin, and Massop..

Image denoising using scale mixtures of Gaussians in the wavelet domain

Abstract — We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimate over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error