133 research outputs found
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
Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis
Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.Comment: 49 pages, 9 Figure
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