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
Accelerating seismic interpolation with a gradient projection method based on tight frame property of curvelet
TL;DR: A fast gradient projection method for seismic interpolation based on the tight frame property of curvelet transform that can overcome shortcomings of the L1 norm and improve the computational efficiency.
Patent
Image denoising techniques
TL;DR: In this paper, image denoising techniques include determining wavelet-domain noise model and a non-parametric multivariate wavelet description from the image signal for raw image data.
Journal ArticleDOI
Ground roll attenuation using a curvelet-SVD filter: a case study from the west of Iran
Bahareh Boustani,Siyavash Torabi,Abdorahim Javaherian,Abdorahim Javaherian,Seyed Ahmad Mortazavi +4 more
TL;DR: In this article, a curvelet-SVD filter was proposed to attenuate ground roll in high frequency subbands. But the ground roll energy in each subband was not considered.
Journal ArticleDOI
Morphologically Decoupled Structured Sparsity for Rotation-Invariant Hyperspectral Image Analysis
TL;DR: It is asserted that hyperspectral image (HSI) processing can benefit very significantly by decoupling data into geometrically distinct components since the resulting decoupled components are much more suitable for sparse representation-based classifiers.
Journal ArticleDOI
Ridgelet kernel regression
TL;DR: A ridgelet kernel regression method to approximate multi-dimensional functions, especially those with certain kinds of spatial inhomogeneities, based on ridgelet theory, kernel and regularization techniques and using particle swarm optimization to optimize the directions of ridgelets.
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