1,264 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
Insense: Incoherent Sensor Selection for Sparse Signals
Sensor selection refers to the problem of intelligently selecting a small
subset of a collection of available sensors to reduce the sensing cost while
preserving signal acquisition performance. The majority of sensor selection
algorithms find the subset of sensors that best recovers an arbitrary signal
from a number of linear measurements that is larger than the dimension of the
signal. In this paper, we develop a new sensor selection algorithm for sparse
(or near sparse) signals that finds a subset of sensors that best recovers such
signals from a number of measurements that is much smaller than the dimension
of the signal. Existing sensor selection algorithms cannot be applied in such
situations. Our proposed Incoherent Sensor Selection (Insense) algorithm
minimizes a coherence-based cost function that is adapted from recent results
in sparse recovery theory. Using six datasets, including two real-world
datasets on microbial diagnostics and structural health monitoring, we
demonstrate the superior performance of Insense for sparse-signal sensor
selection
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