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
Full length article: Compactly supported shearlets are optimally sparse
Gitta Kutyniok,Wang-Q Lim +1 more
TL;DR: This paper presents the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements.
Journal ArticleDOI
Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints
Massimo Fornasier,Holger Rauhut +1 more
TL;DR: In this article, a thresholded Landweber algorithm is proposed to compute solutions of linear inverse problems with joint sparsity regularization constraints by fast thresholded LDA, which is interpreted as a double-minimization scheme for a suitable target functional.
Journal ArticleDOI
Construction of Compactly Supported Shearlet Frames
TL;DR: In this paper, a family of cone-adapted irregular shearlet systems with a general irregular set of parameters is presented, and it is shown that they provide an optimally sparse N-term approximation of an M×M cartoon-like image with asymptotic computational cost O(M 2(1+τ) for some positive constant τ depending on M and N.
Journal ArticleDOI
Multiscale Hybrid Linear Models for Lossy Image Representation
TL;DR: The careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal-to-noise ratios than many existing methods, including wavelets.
Posted Content
Recovery algorithms for vector valued data with joint sparsity constraints
Massimo Fornasier,Holger Rauhut +1 more
TL;DR: This work shows how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms by discussing the adaptive choice of suitable weights appearing in the definition of sparsity measures.
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