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
Quantitative Segmentation of Fluorescence Microscopy Images of Heterogeneous Tissue: Application to the Detection of Residual Disease in Tumor Margins.
Jenna L. Mueller,Zachary T. Harmany,Jeffrey K. Mito,Stephanie A. Kennedy,Yongbaek Kim,Leslie G. Dodd,Joseph Geradts,David G. Kirsch,Rebecca Willett,J. Quincy Brown,Nimmi Ramanujam +10 more
TL;DR: The combination of a subcellular-resolution microendoscope, acriflavine staining, and SCA+CT can be used to accurately isolate nuclei and quantify their density in anatomical images of heterogeneous tissue.
Journal ArticleDOI
Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames
Hui Ji,Zuowei Shen,Yufei Zhao +2 more
TL;DR: This paper constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation, and showed the advantages of the proposed discreteGabor frames over the existing wavelet frames in several image recovery experiments.
Journal ArticleDOI
Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets
TL;DR: In this paper, the authors studied the general reconstruction of a compactly supported function from its Fourier coefficients using Compactly Supported Shearlet (CSH) systems, assuming that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space.
Journal ArticleDOI
Image inpainting from partial noisy data by directional complex tight framelets
Yi Shen,Bin Han,Elena Braverman +2 more
TL;DR: An iterative thresholding algorithm using tight frames derived from TP-C-TF for the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise does not require us to tune parameters manually for reasonably good performance.
Proceedings ArticleDOI
Multifractional splines: application to seismic imaging
TL;DR: In this article, a non-adaptive Curvelet/Contourlet technique was proposed to image and preserve the singularities and a data adaptive Matching Pursuit method was used to characterize these imaged singularities by multi-fractional Splines.
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