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
Curvelet threshold denoising joint with empirical mode decomposition
Posted Content
Bendlets: A Second-Order Shearlet Transform with Bent Elements
TL;DR: In this paper, a second-order shearlet system based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator is proposed. But this system is not suitable for the measurement of the curvature of discontinuities.
Book ChapterDOI
Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings
TL;DR: The generalized shearlet dilation groups have been studied on a case-by-case basis in this article, and their suitability for the characterization of wavefront sets and embedding embeddings into the symplectic group has been investigated.
Journal ArticleDOI
Image inpainting using directional wavelet packets originating from polynomial splines
TL;DR: A recently designed versatile library of quasi-analytic complex-valued wavelet packets (qWPs) which originate from polynomial splines of arbitrary orders make them efficient in image processing applications, in particular, in dealing with the inpainting problem addressed in the paper.
Posted Content
A Unitary Extension Principle for Shearlet Systems
TL;DR: In this paper, the concept of adaptive MRA (AMRA) is introduced and a theory for fast decomposition algorithms associated with shearlet systems which encompasses tight shearlets frames with spatially compactly supported generators within such an AMRA structure is studied.
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