Irregular Shearlet Frames: Geometry and Approximation Properties

Kittipoom, Pisamai; Kutyniok, Gitta und Lim, Wang-Q (2011): Irregular Shearlet Frames: Geometry and Approximation Properties. In: Journal of Fourier Analysis and Applications, Bd. 17, Nr. 4: S. 604-639

Abstract

Recently, shearlet systems were introduced as a means to derive efficient encod- ing 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 L2(R2), 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 insights 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.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik > Professur für Mathematische Grundlagen des Verständnisses der künstlichen Intelligenz
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 1069-5869
• Sprache: Englisch