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
A comparative study on mammographic image denoising technique using wavelet, curvelet and contourlet transforms
TL;DR: Comparisons of the discriminating power of the various multi-resolution based thresholding techniques - wavelet, curvelet, and contourlet for image denoising for mammogram images show that the curvelet-based thresholding can obtain a better image estimate than the wavelet- based and contouring-based restoration methods.
Journal ArticleDOI
Geological disaster survey based on Curvelet transform with borehole Ground Penetrating Radar in Tonglushan old mine site
TL;DR: Structural abnormalities of rock-mass in deep underground were surveyed with borehole ground penetrating radar (GPR) to find out whether there were any mined galleries or mined-out areas below the ruins with both the multiresolution analysis and sub-band directional of Curvelet transform.
Journal ArticleDOI
Critically Sampled Wavelets With Composite Dilations
Glenn R. Easley,Demetrio Labate +1 more
TL;DR: This paper investigates the constructions derived from this approach to develop critically sampled wavelets with composite dilations for the purpose of image coding and introduces new critically sampled discrete transforms that achieve much better nonlinear approximation rates than traditional discrete wavelet transforms and outperform the other critically sampled multiscale transforms recently proposed.
Proceedings ArticleDOI
Classification of Image Distortions for Image Quality Assessment
TL;DR: This paper constructed a set of optimal features that classify image distortions with a high accuracy rate and finds that the best performing classifier is multiclass classifier (Exhaustive Correction Code) with logistic regression as base classifier.
Journal ArticleDOI
PAC-Bayesian risk bounds for group-analysis sparse regression by exponential weighting
TL;DR: A powerful estimator by exponential weighted aggregation with a group-analysis sparsity promoting prior on the weights of a high-dimensional non-parametric regression model with fixed design and i.i.d. random errors is proposed.
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