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
An efficient medical image watermarking scheme based on FDCuT–DCT
TL;DR: The robustness of the scheme is better than an existing scheme for a similar set of medical images in terms of Normalized Correlation (NC), and experimental results show that scheme is robust to geometric attacks, signal processing attacks and JPEG compression attacks.
Journal ArticleDOI
A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform
TL;DR: A new hybrid method for image approximation is proposed that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture.
Journal ArticleDOI
Depth Reconstruction From Sparse Samples: Representation, Algorithm, and Sampling
TL;DR: This paper provides empirical evidence that depth maps can be encoded much more sparsely than natural images using common dictionaries, such as wavelets and contourlets, and proposes an alternating direction method of multipliers (ADMM) for depth map reconstruction.
Book ChapterDOI
Curvelets and Ridgelets
Jalal M. Fadili,Jean-Luc Starck +1 more
TL;DR: Glossary WT1D The one-dimensional Wavelet Transform as defined in [1].
Journal ArticleDOI
Fast Computation of Fourier Integral Operators
TL;DR: In this paper, a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography is introduced.
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