122 research outputs found
Compactly Supported Shearlets are Optimally Sparse
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
(non-linear) approximation properties of directional representation systems. It
was already shown that curvelets, contourlets, as well as shearlets do exhibit
(almost) optimally sparse approximation within this model. 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 now present the first complete proof of (almost) 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 shearlet frames --
not necessarily required to be tight -- with shearlet generators having compact
support and satisfying some weak moment conditions
Landau's necessary density conditions for LCA groups
H. Landau's necessary density conditions for sampling and interpolation may
be viewed as a general principle resting on a basic fact of Fourier analysis:
The complex exponentials ( in ) constitute an
orthogonal basis for . The present paper extends Landau's
conditions to the setting of locally compact abelian (LCA) groups, relying in
an analogous way on the basics of Fourier analysis. The technicalities--in
either case of an operator theoretic nature--are however quite different. We
will base our proofs on the comparison principle of J. Ramanathan and T.
Steger
Gabor Shearlets
In this paper, we introduce Gabor shearlets, a variant of shearlet systems,
which are based on a different group representation than previous shearlet
constructions: they combine elements from Gabor and wavelet frames in their
construction. As a consequence, they can be implemented with standard filters
from wavelet theory in combination with standard Gabor windows. Unlike the
usual shearlets, the new construction can achieve a redundancy as close to one
as desired. Our construction follows the general strategy for shearlets. First
we define group-based Gabor shearlets and then modify them to a cone-adapted
version. In combination with Meyer filters, the cone-adapted Gabor shearlets
constitute a tight frame and provide low-redundancy sparse approximations of
the common model class of anisotropic features which are cartoon-like
functions.Comment: 24 pages, AMS LaTeX, 4 figure
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