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Design of regular nonseparable bidimensional wavelets using Grobner basis techniques

Jean-Charles Faugère 1, 2 Francois Moreau de Saint Martin Fabrice Rouillier 2, 1
1 CALFOR - Calcul formel
LIP6 - Laboratoire d'Informatique de Paris 6
2 POLKA - Polynomials, Combinatorics, Arithmetic
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The design of two-dimensional (2-D) filter banks yielding orthogonality and linear-phase filters and generating regular wavelet bases is a difficult task involving algebraic properties of multivariate polynomials. Using cascade forms implies dealing with nonlinear optimization. We turn the issue of optimizing the orthogonal linear-phase cascade from Kovacevic and Vetterli into a polynomial problem and solve it using Gr¨obner basis techniques and computer algebra. This leads to a complete description of maximally flat wavelets among the orthogonal linear-phase family proposed by Kovacevic and Vetterli. We obtain up to five degrees of flatness for a 16 x 16 filter bank, whose Sobolev exponent is 2.11, making this wavelet the most regular orthogonal linearphase nonseparable wavelet to the authors' knowledge.
Keywords : Calcul Formel wavelets computer algebra Bases de Groebner Ondelettes Groebner Basis
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https://hal.inria.fr/inria-00098579
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Submitted on : Monday, September 25, 2006 - 5:03:41 PM
Last modification on : Friday, February 26, 2021 - 3:28:02 PM

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Jean-Charles Faugère, Francois Moreau de Saint Martin, Fabrice Rouillier. Design of regular nonseparable bidimensional wavelets using Grobner basis techniques. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 1998, 46 (4), pp.845-856. ⟨10.1109/78.668541⟩. ⟨inria-00098579⟩

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