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
Directional Wavelets and a Wavelet Variogram for Two-Dimensional Data
TL;DR: In this article, two 2D generalizations of the 1D variogram that use one-and two-dimensional filters to remove different types of trend present in the data and to provide information on the underlying variation simultaneously are presented.
Journal ArticleDOI
Shearlet-based compressed sensing for fast 3D cardiac MR imaging using iterative reweighting.
Jackie Ma,Maximilian März,Stephanie Funk,Jeanette Schulz-Menger,Gitta Kutyniok,Tobias Schaeffter,Christoph Kolbitsch +6 more
TL;DR: The proposed technique had lower relative errors, higher structural similarity and higher diagnostic scores compared to the other reconstruction techniques especially for high undersampling factors, i.e. short scan times.
Separation of primaries and multiples by non-linear estimation in the curvelet domain
Felix J. Herrmann,D.J. Verschuur +1 more
TL;DR: Herrmann et al. as discussed by the authors proposed a non-linear estimation method for predicting multiple suppression from seismic data and matching the predicted multiples with the true multiples in the data.
Journal ArticleDOI
A Fast and Efficient Approach for Image Compression Using Curvelet Transform
TL;DR: A novel image compression technique using features of wavelet and curvelet transforms is proposed to improve efficiency and compression performance and reveals significant improvement in compression ratio and decoded peak-signal-to-noise-ratio.
Proceedings ArticleDOI
Imaging in Compressed domain using Dreamlets
Ru-Shan Wu,Bangyu Wu,Yu Geng +2 more
TL;DR: In this paper, the authors further investigated the theory and algorithm of wave propagation and imaging in the dreamlet domain for direct application of compressed seismic data, and they showed that the wave coefficients of the seismic data is also decreasing with the depth of migration in the receiver side, meanwhile the imaging quality using the compressed data remains in a similar accuracy.
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