186 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
Asymptotic Analysis of Inpainting via Universal Shearlet Systems
Recently introduced inpainting algorithms using a combination of applied
harmonic analysis and compressed sensing have turned out to be very successful.
One key ingredient is a carefully chosen representation system which provides
(optimally) sparse approximations of the original image. Due to the common
assumption that images are typically governed by anisotropic features,
directional representation systems have often been utilized. One prominent
example of this class are shearlets, which have the additional benefitallowing
faithful implementations. Numerical results show that shearlets significantly
outperform wavelets in inpainting tasks. One of those software packages,
www.shearlab.org, even offers the flexibility of usingdifferent parameter for
each scale, which is not yet covered by shearlet theory.
In this paper, we first introduce universal shearlet systems which are
associated with an arbitrary scaling sequence, thereby modeling the previously
mentioned flexibility. In addition, this novel construction allows for a smooth
transition between wavelets and shearlets and therefore enables us to analyze
them in a uniform fashion. For a large class of such scaling sequences, we
first prove that the associated universal shearlet systems form band-limited
Parseval frames for consisting of Schwartz functions.
Secondly, we analyze the performance for inpainting of this class of universal
shearlet systems within a distributional model situation using an
-analysis minimization algorithm for reconstruction. Our main result in
this part states that, provided the scaling sequence is comparable to the size
of the (scale-dependent) gap, nearly-perfect inpainting is achieved at
sufficiently fine scales
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