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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Image Deconvolution Using a General Ridgelet and Curvelet Domain
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Seismic Random Noise Attenuation Based on PCC Classification in Transform Domain
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An algorithm for automatic curve detection
Z. Martínez,Carenne Ludeña +1 more
TL;DR: This article develops a two step model selection procedure based on a contourlet expansion of the image and proves the method is consistent in probability and applies it to synthetic images.
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Directional wavelet packets originating from polynomial splines.
TL;DR: A versatile library of quasi-analytic complex-valued wavelet packets (WPs) which originate from polynomial splines of arbitrary orders, which have a strong potential to be used in various image processing applications such as restoration of degraded images and extraction of characteristic features from the images.
Journal ArticleDOI
Palmprint Recognition Using Multiscale Transform, Linear Discriminate Analysis, and Neural Network
TL;DR: This paper uses PolyU hyperspectral palmprint database, and applies back-propagation neural network for recognition, linear discriminate analysis for dimensionality reduction, and 2D discrete wavelet, ridgelet, curvelet, and contourlet for feature extraction.
References
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Journal ArticleDOI
De-noising by soft-thresholding
TL;DR: The authors prove two results about this type of estimator that are unprecedented in several ways: with high probability f/spl circ/*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures.
Journal ArticleDOI
The curvelet transform for image denoising
TL;DR: In this paper, the authors describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform, which offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity.
Journal ArticleDOI
High performance scalable image compression with EBCOT
TL;DR: A new image compression algorithm is proposed, based on independent embedded block coding with optimized truncation of the embedded bit-streams (EBCOT), capable of modeling the spatially varying visual masking phenomenon.
Journal ArticleDOI
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.
Journal ArticleDOI
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.