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
Curvelet transform to study scale-dependent anisotropic soil spatial variation
TL;DR: In this paper, a curvelet transform is used to characterize scale-dependent anisotropic soil spatial variation. But, it is not suitable for the presentation of soil variability information containing abrupt values or displaying discontinuity in its spatial distribution.
Journal ArticleDOI
Gabor phase retrieval is severely ill-posed
Rima Alaifari,Philipp Grohs +1 more
TL;DR: In this paper, the stability of the Gabor transform was shown to scale at least quadratically exponentially in the dimension of the subspaces of the signal domain L 2 (R ).
Journal ArticleDOI
Using the fibre structure of paper to determine authenticity of the documents: analysis of transmitted light images of stamps and banknotes.
TL;DR: A novel method is presented for distinguishing postal stamp forgeries and counterfeit banknotes from genuine samples using a curvelet-based algorithm for measuring overall fibre orientation distribution and quantifying anisotropy.
References
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Painless nonorthogonal expansions
TL;DR: In a Hilbert space H, discrete families of vectors {hj} with the property that f = ∑j〈hj ǫ à à hj à f à for every f in H are considered.
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Shiftable multiscale transforms
TL;DR: Two examples of jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored and the usefulness of these image representations for scale-space analysis, stereo disparity measurement, and image enhancement is demonstrated.