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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Book ChapterDOI
Transforms and Operators for Directional Bioimage Analysis: A Survey
TL;DR: The intent is to provide image-processing methods that can be deployed in algorithms that analyze biomedical images with improved rotation invariance and high directional sensitivity, and address the problem of matching directional patterns by proposing steerable filters.
Journal ArticleDOI
Directionlets: anisotropic multidirectional representation with separable filtering
TL;DR: This work presents a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT, which provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N/sup -1.55/), which, while slower than the optimal rate O-2/, is much better than O-1/ achieved with wavelets, but at similar complexity.
Book
Continuous curvelet transform
TL;DR: The Continuous Curvelet Transform (CCT) as mentioned in this paper is a continuous transform based on the Fourier Integral Operators (FIFO) that is closely related to the wavelet transform.
Journal ArticleDOI
Super-Resolution With Sparse Mixing Estimators
Stéphane Mallat,Guoshen Yu +1 more
TL;DR: A class of inverse problem estimators computed by mixing adaptively a family of linear estimators corresponding to different priors corresponding toDifferent priors are introduced, providing state-of-the-art numerical results.
Journal ArticleDOI
Optimal approximation of piecewise smooth functions using deep ReLU neural networks.
TL;DR: It is proved that one cannot approximate a general function f∈Eβ(Rd) using neural networks that are less complex than those produced by the construction, which partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions.
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