In this paper we report on the performance of Lemarie uniwavelets and biwavelets for solving the ill-posed inverse problem of recovering the derivative of a noisy signal. The noise under consideration can be white gaussian or other noises like Tukey noise. The denoising procedures utilized are wavelet and biwavelet Efromovich-Pinsker (EP) estimators, which have been shown to be universal for both estimating the function and its derivative.
KEYWORDS: Wavelets, Signal processing, Statistical analysis, Error analysis, Error control coding, Space operations, Chemical elements, Signal analysis, Mathematics, Lanthanum
We investigate several issues surrounding the general question of when a function in a finitely generated shift invariant subspace of L2(R) can be determined by certain of its sample values just as a function band limited to (-1/2, ½) can be expressed in terms of its integer samples. The main theme here is how answers to this question depend on general properties of the generators of the shift invariant space, such as orthogonality properties, scaling relations, smoothness and so forth. One of the main issues that we address is the question of how to control aliasing error.
We use wavelets based ona modification of the Geronimo- Hardin-Massopust construction to define localized extension/restriction operators form half-spaces to their full spaces/boundaries respectively. These operations are continuous in Sobolev and Morrey space norms. We also prove estimates for multiresolution projections of pointwise products of functions in these spaces. These are two of the key steps in extending results of Federbush and of Cannone and Meyer concerning solutions of Navier-Stokes with initial data in Sobolev and Morrey spaces to the case of half spaces and, ultimately, to more general domains with boundary.
This manuscript gives a construction of divergence-free multiwavelets which combines the Hardin-Marasovich (HM) construction with a recipe of Strela for increasing or decreasing regularity of biorthogonal wavelets. Strela's process preserves symmetry of the HM wavelets. This enables the divergence-free wavelets to be suitably adapted to the analysis of divergence-free vector fields whose boundary traces are tangent vectors.
A theory of frames that extend Gabor analysis by including chirping is discussed. The chirping parameter in these 'time-frequency localization frames' depends on time and/or frequency shift parameters that can be adapted to analyze and detect chirps in noisy signals. Radar/sonar applications are outlined. The frame theory is motivated by a generalized notion of square-integrable group representation developed by Ali, Antoine, and Gazeau, together with ideas in Baraniuk's thesis on a metaplectic extension of Cohen's class.
Conference Committee Involvement (1)
Wavelets XI
31 July 2005 | San Diego, California, United States
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