Preprint: Two papers on shearlets : Sparsity and Construction

[Wang-Q Lim (wlim@math.uso.de)]

Title : Compactly Supported Shearlets are Optimally Sparse

Authors : Gitta Kutyniok and Wang-Q Lim


Absatract :

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.



Title: Construction of Compactly Supported Shearlet Frames

Authors: Pisamai Kittipoom, Gitta Kutyniok and Wang-Q Lim

Abstract:

Shearlet tight frames have been extensively studied during the last years due to their optimal
approximation properties of cartoon-like images and their unified treatment of the continuum
and digital setting. However, these studies only concerned shearlet tight frames generated by
a band-limited shearlet, whereas for practical purposes compact support in spatial domain is
crucial.
In this paper, we focus on cone-adapted shearlet systems which
-- accounting for stability questions -- are associated with a general irregular set of parameters.
We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a
frame and provide explicit estimates for their frame bounds. Secondly, exploring these results
and using specifically designed wavelet scaling functions and filters,
we construct a family of cone-adapted shearlet frames consisting of compactly supported
shearlets. For this family, we derive
estimates for the ratio of their frame bounds and prove that they provide optimally
sparse approximations of cartoon-like images.


Preprints available at:

www.shearlet.org/papers/SparseShearSIAMfinal.pdf
www.shearlet.org/papers/ShearletFramesFinal.pdf