Compactly supported shearlets are optimally sparse

Gitta Kutyniok, Wang-Q Lim

Abstract: Cartoon-like images, i.e., C 2 functions which are smooth apart from a C 2 discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions. Highlights ► Approximation rates for bivariate functions exhibiting curvilinear singularities. ► Cartoon-like images are C 2 functions, which are smooth apart from a C 2 discontinuity curve. ► Compactly supported shearlet frames achieve the optimal approximation rate for cartoon-like images. ► First complete proof of optimally sparse approximations by a compactly supported system.