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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Journal ArticleDOI
Infrared image enhancement with learned features
Zunlin Fan,Duyan Bi,Wenshan Ding +2 more
TL;DR: The significance of first layer in Stacked Sparse Denoising Auto-encoder is analyzed and a novel feature extraction is proposed for the proposed image enhancement scheme that achieves the best performance in infrared image enhancement.
Journal ArticleDOI
Uncertainty principles and optimally sparse wavelet transforms
Ron Levie,Nir Sochen +1 more
TL;DR: In this article, the authors introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform, which aims to design nonadaptive window functions that promote sparsity in some sense.
DissertationDOI
Sparse Recovery via Convex Optimization
TL;DR: The method of l_1 analysis is introduced and it is shown that it is guaranteed to give good recovery of a signal from a few measurements, when the signal can be well represented in a dictionary.
Posted Content
A Second-Order Method for Compressed Sensing Problems with Coherent and Redundant Dictionaries
TL;DR: This paper proposes a primal-dual Newton Conjugate Gradients (pdNCG) method and proves global convergence and fast local rate of convergence for pdNCG in Compressed Sensing problems where the signals to be recovered are sparse in coherent and redundant dictionaries.
Journal ArticleDOI
A primal-dual method for the Meyer model of cartoon and texture decomposition
TL;DR: Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.
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