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
A fast tree-based algorithm for Compressed Sensing with sparse-tree prior
H.Q. Bui,C.N.H. La,Minh N. Do +2 more
TL;DR: A novel tree-based recovery algorithm, the Tree-based Orthogonal Matching Pursuit (TOMP) algorithm, is proposed to recover signals with sparse-tree priors, which gives reconstruction quality comparable to more sophisticated algorithms at less computational cost.
Proceedings ArticleDOI
Content-adaptive non-parametric texture similarity measure
TL;DR: In this article, a nonparametric texture similarity measure based on the singular value decomposition of the curvelet coefficients followed by a content-based truncation of the singular values is proposed.
Posted Content
Image Inpainting Using Directional Tensor Product Complex Tight Framelets.
Yi Shen,Bin Han,Elena Braverman +2 more
TL;DR: The grouping effect property for frame-based convex minimization models using the balanced approach is established and partially explains the effectiveness of models using a balanced approach for several image restoration problems.
Patent
Systems and methods for computer vision using curvelets
Shai Dekel,Alexander Sherman +1 more
TL;DR: In this paper, the authors present a system for computer vision including a plurality of images, a signature processor adapted to generate a signature based at least in part on a curvelet transform, and a matching algorithm adapted to receive a query image.
Journal ArticleDOI
Shearlet transform in aliased ground roll attenuation and its comparison with f-k filtering and curvelet transform
Seyed Abolfazl Hosseini,Abdolrahim Javaherian,Abdolrahim Javaherian,Hossien Hassani,Siyavash Torabi,Maryam Sadri +5 more
TL;DR: In this paper, a shot record is divided into several segments, and the appropriate mute zone is defined for all segments, then the inverse shearlet transform is applied to the filtered segment.
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