389 research outputs found
ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm
Multivariate problems are typically governed by anisotropic features such as
edges in images. A common bracket of most of the various directional
representation systems which have been proposed to deliver sparse
approximations of such features is the utilization of parabolic scaling. One
prominent example is the shearlet system. Our objective in this paper is
three-fold: We firstly develop a digital shearlet theory which is rationally
designed in the sense that it is the digitization of the existing shearlet
theory for continuous data. This implicates that shearlet theory provides a
unified treatment of both the continuum and digital realm. Secondly, we analyze
the utilization of pseudo-polar grids and the pseudo-polar Fourier transform
for digital implementations of parabolic scaling algorithms. We derive an
isometric pseudo-polar Fourier transform by careful weighting of the
pseudo-polar grid, allowing exploitation of its adjoint for the inverse
transform. This leads to a digital implementation of the shearlet transform; an
accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we
introduce various quantitative measures for digital parabolic scaling
algorithms in general, allowing one to tune parameters and objectively improve
the implementation as well as compare different directional transform
implementations. The usefulness of such measures is exemplarily demonstrated
for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu
Parabolic Molecules
Anisotropic decompositions using representation systems based on parabolic
scaling such as curvelets or shearlets have recently attracted significantly
increased attention due to the fact that they were shown to provide optimally
sparse approximations of functions exhibiting singularities on lower
dimensional embedded manifolds. The literature now contains various direct
proofs of this fact and of related sparse approximation results. However, it
seems quite cumbersome to prove such a canon of results for each system
separately, while many of the systems exhibit certain similarities.
In this paper, with the introduction of the notion of {\em parabolic
molecules}, we aim to provide a comprehensive framework which includes
customarily employed representation systems based on parabolic scaling such as
curvelets and shearlets. It is shown that pairs of parabolic molecules have the
fundamental property to be almost orthogonal in a particular sense. This result
is then applied to analyze parabolic molecules with respect to their ability to
sparsely approximate data governed by anisotropic features. For this, the
concept of {\em sparsity equivalence} is introduced which is shown to allow the
identification of a large class of parabolic molecules providing the same
sparse approximation results as curvelets and shearlets. Finally, as another
application, smoothness spaces associated with parabolic molecules are
introduced providing a general theoretical approach which even leads to novel
results for, for instance, compactly supported shearlets
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