Operator-Like Wavelet Bases of $L_{2}(\mathbb{R}^{d})$
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Operator-Like Wavelet Bases of L2 (Rd ) Ildar Khalidov · Michael Unser · John Paul Ward
Received: 5 October 2012 / Revised: 21 August 2013 / Published online: 14 November 2013 © Springer Science+Business Media New York 2013
Abstract The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator. Keywords Fourier multiplier operators · Wavelets · Multiresolution · Stochastic differential equations Mathematics Subject Classification 42C40 · 42B15 · 60H15 1 Introduction In the past few decades, a variety of wavelets that provide a complete and stable multiscale representation of L2 (Rd ) have been developed. The wavelet decomposition is Communicated by Karlheinz Gröchenig. This research was funded in part by ERC Grant ERC-2010-AdG 267439-FUN-SP and by the Swiss National Science Foundation under Grant 200020-121763.
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I. Khalidov · M. Unser · J.P. Ward ( ) Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne, Switzerland e-mail: [email protected] I. Khalidov e-mail: [email protected] M. Unser e-mail: [email protected]
J Fourier Anal Appl (2013) 19:1294–1322
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very efficient from a computational point of view, due to the fast filtering algorithm. A fundamental property of traditional wavelet basis functions is that they behave like multiscale derivatives [17, 18]. Our purpose in this paper is to extend this concept by constructing wavelets that behave like a given Fourier multiplier operator L, which can be more general than a pure derivative. In our approach, the multiresolution spaces are characterized by generalized B-splines associated with the operator, and we show that, in a certain sense, the wavelet inherits properties of the operator. Importantly, the operator-like wavelet can be constructed directly from the operator, bypassing the scaling function space. What makes the approach even more attractive is that, at each scale, the wavelet space is generated by the shifts of a single function. Our work provides a generalization of some known constructions including: cardinal spline wavelets [5], elliptic wavelets [19], polyharmonic spline wavelets [25, 26], Wirtinger-Laplace operator-like wavelets [27], and exponential-spline wavelets [15]. In applications, it has been observed that many signals are well represented by a relatively small number of wavelet coefficients. Interestingly, the model that motivates our wavelet construction explains the origi
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