494 research outputs found
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
Mixed Needlets
The construction of needlet-type wavelets on sections of the spin line
bundles over the sphere has been recently addressed in Geller and Marinucci
(2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal
for needlets on this spin line bundle, in which needlet coefficients arise from
the usual, rather than the spin, spherical harmonics, as in the previous
constructions. We label this system mixed needlets and investigate in full
their properties, including localization, the exact tight frame
characterization, reconstruction formula, decomposition of functional spaces,
and asymptotic uncorrelation in the stochastic case. We outline astrophysical
applications.Comment: 26 page
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
Shearlets and Optimally Sparse Approximations
Multivariate functions are typically governed by anisotropic features such as
edges in images or shock fronts in solutions of transport-dominated equations.
One major goal both for the purpose of compression as well as for an efficient
analysis is the provision of optimally sparse approximations of such functions.
Recently, cartoon-like images were introduced in 2D and 3D as a suitable model
class, and approximation properties were measured by considering the decay rate
of the error of the best -term approximation. Shearlet systems are to
date the only representation system, which provide optimally sparse
approximations of this model class in 2D as well as 3D. Even more, in contrast
to all other directional representation systems, a theory for compactly
supported shearlet frames was derived which moreover also satisfy this
optimality benchmark. This chapter shall serve as an introduction to and a
survey about sparse approximations of cartoon-like images by band-limited and
also compactly supported shearlet frames as well as a reference for the
state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data",
Birkh\"auser-Springe
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