Generalized L-Spline Wavelet Bases

Thierry Blu, Ildar Khalidov, Michaël Unser
SPIE, 2005
Conference paper

Abstract

We build wavelet-like functions based on a parametrized family of pseudo-differential operators $L _{ v }$ that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of $L _{ v }$ , generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression a = $2 ^{ i }$ . The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm.

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https://infoscience.epfl.ch/record/211365?ln=en

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Thierry Blu, Ildar Khalidov, Michaël Unser
SPIE, 2005
Conference paper

Abstract

Official source

https://infoscience.epfl.ch/record/211365?ln=en

About this result

Related concepts (11)

Related concepts

Related publications (5)

Related publications

From Differential Equations to the Construction of New Wavelet-Like Bases

Ildar Khalidov, Michaël Unser

In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in

L _{ 2 }

. This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of

L _{ 2 }

-either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N − M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filterbank algorithm, where the filters depend on the decomposition level.

IEEE2006

In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the

L _{ p }

-sense and a sharper theorem stating that smoothness implies order.

IEEE2003

Wavelets and radial basis functions (RBF) are two rather distinct ways of representing signals in terms of shifted basis functions. An essential aspect of RBF, which makes the method applicable to non-uniform grids, is that the basis functions, unlike wavelets, are non-local—in addition, they do not involve any scaling at all. Despite these fundamental differences, we show that the two types of representation are closely connected. We use the linear splines as motivating example. These can be constructed by using translates of the one-side ramp function (which is not localized), or, more conventionally, by using the shifts of a linear B-spline. This latter function, which is the prototypical example of a scaling function, can be obtained by localizing the one-side ramp function using finite differences. We then generalize the concept and identify the whole class of self-similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that, for any compactly supported scaling function φ(x), there exists a one-sided central basis function

ρ _{ + }

(x) that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation of

ρ _{ + }

, without any dilation.

SPIE2000