Irregular Shearlet Frames: Geometry and Approximation Properties

Autor(en): Kittipoom, Pisamai
Kutyniok, Gitta
Lim, Wang-Q
Stichwörter: Amalgam spaces; DENSITY; Frame; Geometry; Homogeneous approximation property; Mathematics; Mathematics, Applied; Shearlet; Shearlet group; TRANSFORM; WAVELET FRAMES
Erscheinungsdatum: 2011
Herausgeber: BIRKHAUSER BOSTON INC
Journal: JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
Volumen: 17
Ausgabe: 4
Startseite: 604
Seitenende: 639
Zusammenfassung: 
Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far. In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L(2)(R(2)), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting in-sights into homogeneous approximation abilities of shearlet frames. We also present examples, such as oversampled shearlet systems and co-shearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.
ISSN: 10695869
DOI: 10.1007/s00041-010-9163-0

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