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A Family of Polynomial Spline Wavelet Transforms

M. Unser, A. Aldroubi, M. Eden

Signal Processing, vol. 30, no. 2, pp. 141-162, January 1993.


This paper presents an extension of the family of orthogonal Battle/Lemarié spline wavelet transforms with emphasis on filter bank implementation. Spline wavelets that are not necessarily orthogonal within the same resolution level, are constructed by linear combination of polynomial spline wavelets of compact support, the natural counterpart of classical B-spline functions. Mallat's fast wavelet transform algorithm is extended to deal with these non-orthogonal basis functions. The impulse and frequency responses of the corresponding analysis and synthesis filters are derived explicitly for polynomial splines of any order n (n odd). The link with the general framework of biorthogonal wavelet transforms is also made explicit. The special cases of orthogonal, B-spline, cardinal and dual wavelets are considered in greater detail. The B-spline (resp. dual) representation is associated with simple FIR binomial synthesis (resp. analysis) filters and recursive analysis (resp. synthesis) filters. The cardinal representation provides a sampled representation of the underlying continuous functions (interpolation property). The distinction between cardinal and orthogonal representation vanishes as the order of the spline is increased; both wavelets tend asymptotically to the bandlimited sinc-wavelet. The distinctive features of these various representations are discussed and illustrated with a texture analysis example.

@ARTICLE(http://bigwww.epfl.ch/publications/unser9303.html,
AUTHOR="Unser, M. and Aldroubi, A. and Eden, M.",
TITLE="A Family of Polynomial Spline Wavelet Transforms",
JOURNAL="Signal Processing",
YEAR="1993",
volume="30",
number="2",
pages="141--162",
month="January",
note="")

© 1993 Elsevier. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Elsevier. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
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