ADAPTIVE DIRECTIONAL SUBDIVISION SCHEMES AND SHEARLET MULTIRESOLUTION ANALYSIS

Autor(en): Kutyniok, Gitta
Sauer, Tomas
Stichwörter: APPROXIMATION ORDER; directional transforms; H-BASES; IDEALS; joint spectral radius; Mathematics; Mathematics, Applied; multiresolution analysis; MULTIVARIATE REFINABLE FUNCTIONS; POLYNOMIAL INTERPOLATION; refinement equation; REPRESENTATION; shearlets; subdivision schemes; TRANSFORM
Erscheinungsdatum: 2009
Herausgeber: SIAM PUBLICATIONS
Journal: SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volumen: 41
Ausgabe: 4
Startseite: 1436
Seitenende: 1471
Zusammenfassung: 
In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of nonstationary bivariate subdivision scheme, which allows us to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two-dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.
ISSN: 00361410
DOI: 10.1137/08072276X

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