Optimally Sparse Multidimensional Representation Using Shearlets

By Kanghui Guo, Demetrio Labate. SIAM Journal on Mathematical Analysis, 2007

741 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geophysics

View Paper: https://www.semanticscholar.org/paper/0a813d13300e25c25aac93fa99f9b1822e6048cb

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In Applied Mathematics

In Computer Vision and Pattern Recognition

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Citations in the corpus

An introduction to continuous optimization for imaging (2016, methodology): “There is a vast literature on optimization techniques for such problems, which have been tackled in this form at least since Mine and Fukushima (1981) and Fukushima and Mine (1981).”
Modern regularization methods for inverse problems (2018, background): “ natural images in wavelet bases (cf. [274, 225, 273, 341]). Further improvements were made by replacing the orthonormal bases by frames (cf. [126]) such as curvelets (cf. [97, 98]) or shearlets (cf. [257, 197, 256]). Sparsity is naturally measured by the ‘0-norm, the number of nonzero entries. Sinc”
The Curvelet Transform (2010, background): “Steerable wavelets [28], [59], Gabor wavelets [40], wedgelets [23], beamlets [24], bandlets [51], [54], contourlets [21], shearlets [39], [31], wave atoms [20], platelets [67], and surfacelets [42] have been proposed independently to identify and restore geometric features.”
Optimal approximation of piecewise smooth functions using deep ReLU neural networks (2017, background): “For instance, shearlets and curvelets provide optimal N -term approximation rates for piecewise C 2 ( R 2 ) functions with C 2 jump curves, [9, 29, 22, 39].”
Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach (2010, background): “Other analogous constructions to curvelets include contourlets (Do and Vetterli, 2002), surfacelets (Lu and Do, 2007), and, more recently, shearlets (Guo et al., 2006; Guo and Labate, 2007).”
Resolution of the wavefront set using continuous shearlets (2006, background): “We refer to [13, 15, 16, 17, 18, 22, 21] for more detail about the research about shearlets.”
Seislet transform and seislet frame (2010, background): “…(Pennec and Mallat, 2005), contourlets (Do and Vetterli, 2005), curvelets (Starck et al., 2002), directionlets (Velisavljevic, 2005), shearlets (Guo and Labate, 2007), etc. Unlike isotropic wavelets, directional transforms attempt to design basis functions so that they appear elongated…”
Full length article: Compactly supported shearlets are optimally sparse (2011, positive): “The expert reader will have noticed that in contrast to the proofs in [1] and [11], which also split the domain into smaller scale boxes, we do not apply a weight function to obtain a smooth partition of unity.”
A Shearlet Approach to Edge Analysis and Detection (2009, neutral): “As a consequence, this transform provides an optimally efficient representation of images with edges [21].”