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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Proceedings ArticleDOI
Contourlets and sparse image expansions
TL;DR: This paper introduces the directional vanishing moment condition for contourlet expansion and shows that with anisotropic scaling and sufficient directional vanishing moments, contourlets essentially achieve the optimal approximation rate, O((log M)3 M-2) square error with a best M-term approximation, for 2-D piecewise smooth functions with C2 contours.
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Harmonic analysis on directed graphs and applications: from Fourier analysis to wavelets
TL;DR: A novel harmonic analysis for functions defined on the vertices of a strongly connected directed graph of which the random walk operator is the cornerstone, and finds a frequency interpretation by linking the variation of the eigenvectors of the randomWalk operator obtained from their Dirichlet energy to the real part of their associated eigenvalues.
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Scattering in flatland: Efficient representations via wave atoms
Laurent Demanet,Lexing Ying +1 more
TL;DR: It is proved that a well-chosen fixed expansion, the non-standard wave atom form, provides a compression of the acoustic single- and double-layer potentials with wave number k as O(k)-by-O(k) matrices with Cεδk1+δ non-negligible entries, with δ>0 arbitrarily small, and ε the desired accuracy.
Journal ArticleDOI
Sensitivity Analysis of Wave-equation Tomography: A Multi-scale Approach
TL;DR: In this article, a wave-equation tomography is derived from cross correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events.
Proceedings ArticleDOI
Learning structured dictionaries for image representation
TL;DR: An over-complete code for sparse representation of natural images has been learnt from a set of real-world scenes, and the functions found have been organized into a hierarchical structure, allowing great flexibility in its design and lower computational complexity.
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