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
Characterization of textural surfaces using wave atoms
TL;DR: In this paper, a wave atom transform combined with total variation minimization is proposed to characterize surfaces with oriented textural scratches, where wave atoms not only capture the coherence of the pattern along the oscillations but also the pattern across the oscillation.
Journal ArticleDOI
Scale Invariant and Noise Robust Interest Points With Shearlets
TL;DR: In this article, the authors consider blob-like features in the shearlets framework and derive a measure, which is very effective for blob detection, and, based on this measure, they propose a blob detector and a keypoint description, whose combination outperforms the state-of-the-art algorithms with noisy and compressed images.
Journal ArticleDOI
A Sparsity Basis Selection Method for Compressed Sensing
TL;DR: Numerical experiments show that the proposed SBSCS method improves the quality of signal recovery over the existing best basis compressed sensing method (BBCS) proposed by Peyré in 2010.
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Clustered Sparsity and Separation of Cartoon and Texture
TL;DR: This paper provides a theoretical study of the separation of a combination of cartoon and texture structures in a continuum model situation using this class of algorithms.
Journal ArticleDOI
Efficient processing of fluorescence images using directional multiscale representations.
TL;DR: The shearlet representation is applied to problems of soma detection of neurons in culture and extraction of geometrical features of neuronal processes in brain tissue, and proposed as a new framework for large-scale fluorescent image analysis of biomedical data.
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