Construction of Orthonormal Piecewise Polynomial Scaling and Wavelet Bases on Non-Equally Spaced Knots
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Research Article Construction of Orthonormal Piecewise Polynomial Scaling and Wavelet Bases on Non-Equally Spaced Knots Anissa Zerga¨ınoh,1, 2 Najat Chihab,1 and Jean Pierre Astruc1 1 Laboratoire
de Traitement et Transport de l’Information (L2TI), Institut Galil´ee, Universit´e Paris 13, Avenue Jean Baptiste Cl´ement, 93 430 Villetaneuse, France 2 LSS/CNRS, Sup´ elec, Plateau de Moulon, 91 192 Gif sur Yvette, France Received 6 July 2006; Revised 29 November 2006; Accepted 25 January 2007 Recommended by Moon Gi Kang This paper investigates the mathematical framework of multiresolution analysis based on irregularly spaced knots sequence. Our presentation is based on the construction of nested nonuniform spline multiresolution spaces. From these spaces, we present the construction of orthonormal scaling and wavelet basis functions on bounded intervals. For any arbitrary degree of the spline function, we provide an explicit generalization allowing the construction of the scaling and wavelet bases on the nontraditional sequences. We show that the orthogonal decomposition is implemented using filter banks where the coefficients depend on the location of the knots on the sequence. Examples of orthonormal spline scaling and wavelet bases are provided. This approach can be used to interpolate irregularly sampled signals in an efficient way, by keeping the multiresolution approach. Copyright © 2007 Anissa Zerga¨ınoh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Since the last decade, the development of the multiresolution theory has been extensively studied (see, e.g., [1–4]). Many science and engineering fields exploit the multiresolution approach to solve their application problems. Multiresolution analysis is known as a decomposition of a function space into mutually orthogonal subspaces. The specific structure of the multiresolution provides a simple hierarchical framework for interpreting the signal information. The scaling and wavelet bases construction is closely related to the multiresolution analysis. The standard scaling or wavelet basis is defined as a set of translations and dilations of one prototype function. The derived functions are thus self-similar at different scales. Initially, the multiresolution theory has been mainly developed within the framework of a uniform sample distribution (i.e., constant sampling time). The proposed scaling and wavelet bases, in the literature, are built under the assumptions that the knots of the infinite sequence to be processed are regularly spaced. However, the nonuniform sampling situation arises naturally in many scientific fields such as geophysics, astronomy, meteorology, medical imaging, computer vision. The data is often generated or measured at sparse and irregular positions. The majority of
the theoretical tools developed in digital signal processing field are based on a unifor
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