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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Curvelet processing and imaging: adaptive ground roll removal
TL;DR: The results suggest that these directional-selective basis functions provide a usefull tool for the removal of coherent noise such as ground roll.
Journal ArticleDOI
Detection of boundary curves on the piecewise smooth boundary surface of three dimensional solids
Robert Houska,Demetrio Labate +1 more
TL;DR: In this paper, a three dimensional solid with boundary surface S = S1[ [Sq, where each Sr is a smooth surface with boundary curve r] is considered, and multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of by precisely identifying the boundary set.
Journal ArticleDOI
Nonparametric estimation of composite functions
TL;DR: In this paper, the authors study the problem of nonparametric estimation of a multivariate function that can be represented as a composition of two unknown smooth functions and propose rate-optimal estimators for the sup-norm loss.
Journal ArticleDOI
A saliency detection model using shearlet transform
TL;DR: A novel saliency detection model using shearlet transform is presented, which obtains the global saliency based on global probability density distribution, and measures the local saliency by calculating the entropy of local areas.
Proceedings ArticleDOI
Image fusion algorithm based on adaptive pulse coupled neural networks in curvelet domain
TL;DR: Using the fast discrete curvelet transform, an image fusion algorithm based on adaptive pulse coupled neural networks (PCNNs) is proposed and demonstrates superiority of the proposed algorithm in terms of visual effect and objective evaluations.
References
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