Compactly supported shearlets are optimally sparse

Kutyniok, Gitta und Lim, Wang-Q (2011): Compactly supported shearlets are optimally sparse. In: Journal of Approximation Theory, Vol. 163, No. 11: pp. 1564-1589

Abstract

Cartoon-like images, i.e., C2 functions which are smooth apart from a C2 dis- continuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do ex- hibit (almost) optimally sparse approximations within this model. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost impor- tance for applications.

In this paper, we now present the first complete proof of (almost) op- timally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of com- pactly supported elements. This class will be chosen as a subset of shearlet frames – not necessarily required to be tight – with shearlet generators having compact support and satisfying some weak moment conditions.

• Item Type: Journal article
• Keywords: Curvilinear discontinuities; edges; nonlinear approximation; optimal sparsity; shearlets; thresholding; wavelets
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Bavarian Chair for Mathematical Foundations of Artificial Intelligence
• Subjects: 500 Science > 510 Mathematics
• ISSN: 00219045
• Language: English