Journal ArticleDOI
New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
TLDR
This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C2 edges.Abstract:
This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j/2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior.read more
Citations
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Probability Modeled Optimal K-Frame for Erasures
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Proceedings ArticleDOI
Anisotropic representations for superresolution of hyperspectral data
TL;DR: A method for superresolution based on anisotropic harmonic analysis based on the harmonic analytic technique of shearlets is developed in order to efficiently capture the directional information present in the image, which is then used to provide smooth, accurate images at higher resolutions.
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Anisotropic Multiscale Systems on Bounded Domains
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Proceedings ArticleDOI
Imaging with multiples accelerated by message passing
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TL;DR: This work proposes a method to image directly from the total up-going wavefield, including surface-related multiples, by sparse inversion by having the wave-equation solver carry out the multi-dimensional convolutions implicitly and cheaply by randomized subsampling and improves the overall performance by selecting new independent copies of the randomized modeling operator.
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The Use of Two Transform Methods in Fingerprints Recognition
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