Smooth Orthonormal Wavelet Libraries: Design And Application

For signal-based design of orthonormal (ON) wavelets, an optimization of a cost function over an N-dimensional angle space is required. However: (1) the N-dim space includes both smooth and non-smooth wavelets; (2) many of the smooth wavelets are similar in shape. A more practical approach for some applications may be to construct a library of smooth ON wavelets in advance---a library that consists of representative wavelet shapes for a given filter length. Existing ON wavelet libraries (Daubechies, nearlysymmetric, Coiflets) provide only one wavelet for each filter length. We construct ON wavelet libraries using local variation to determine wavelet smoothness and the discrete inner product to discriminate between wavelet shapes. The relationship between library size and the similarity threshold is investigated for various filter lengths. We apply an entropy-based wavelet selection algorithm to an example signal set, and investigate compactness in the wavelet domain as a function of li..

A Library-Based Approach To Design Of Smooth Orthonormal Wavelets

Recently we addressed the question of using comprehensive libraries of smooth orthonormal wavelets for signalbased wavelet design. Here we present the details of the algorithm for assembling a wavelet library from the continuum of compactly-supported orthonormal wavelets. The algorithm is based on a multi-level minimization, with local variation serving as the smoothness measure and the discrete inner product as the measure used to discriminate between wavelet shapes. An example library is provided. 1. INTRODUCTION An orthonormal (ON) wavelet library is a set of wavelets selected from the continuum of compactly-supported ON wavelets, such that they satisfy one or more criteria. Examples are the libraries formed by the Daubechies wavelets (for which the objective is the maximum number of vanishing wavelet moments), nearly-symmetric wavelets (filters closest to linear phase), and Coiflets (scaling function and wavelet have the same number of vanishing moments). Each of these libraries p..

A General Approach To The Generation Of Biorthogonal Bases Of Compactly-Supported Wavelets

Biorthogonal bases of compactly-supported wavelets are characterized by the FIR perfect-reconstruction filterbanks to which they correspond. In this paper we develop explicit representations of all such filterbanks, allowing us to generate every possible biorthogonal compactly-supported wavelet basis. For these filterbanks, the product H(z) = H(z) e H(z) of the two lowpass filters must have N 2 zeros at z = \Gamma1 . There are N + 1 minimal-length filterbanks for each N . The filterbanks associated with standard orthogonal and symmetric biorthogonal wavelet bases are found as a special case by using appropriate factorizations of symmetric H(z) with even N ; other filterbanks lead to novel biorthogonal bases. 1. INTRODUCTION The close relationship between orthonormal wavelet bases and quadrature -mirror filter (QMF) filterbanks is well-known [6, 3, 4]. Daubechies ' success in exploiting this relationship led her, along with Cohen and Feauveau, to construct biorthogonal wavelet bases..

Computing Derivatives Of Scaling Functions And Wavelets

This paper provides a general approach to the computation, for sufficiently regular multiresolution analyses, of scaling functions and wavelets and their derivatives. Two distinct iterative schemes are used to determine the multiresolution functions, the so-called `cascade' algorithm and an eigenvector-based method. We present a novel development of these procedures which not only encompasses both algorithms simultaneously but also applies to the computation of derivatives of the functions. With this we demonstrate that the differences between the two algorithms are due solely to their respective initializations. We prove that the cascade initialization can be used only to compute the functions themselves, while the eigenvector one works for their derivatives as well. Finally, as an alternative to a result of Daubechies and Lagarias, we derive a new, simpler normalization formula for the eigenvector method. 1. INTRODUCTION Wavelets, in particular those associated with multiresolution..