Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames

@article{Kutyniok2012OptimallySA,
  title={Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames},
  author={Gitta Kutyniok and Jakob Lemvig and Wang-Q Lim},
  journal={SIAM J. Math. Anal.},
  year={2012},
  volume={44},
  pages={2962-3017}
}
We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter \beta controlling classical smoothness and one parameter \alpha controlling anisotropic smoothness. The class then consists of piecewise C^\beta-smooth functions with… 

Figures from this paper

Optimally Sparse Representations of Cartoon-Like Cylindrical Data
Sparse representations of multidimensional data have received a significant attention in the literature due to their applications in problems of data restoration and feature extraction. In this
Dualizable Shearlet Frames and Sparse Approximation
Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while
Efficient representation of spatio-temporal data using cylindrical shearlets
TLDR
A new multiscales directional system of functions based on cylindrical shearlets is introduced and it is proved that this new approach achieves superior approximation properties with respect to conventional multiscale representations.
The Role of $\alpha$-Scaling for Cartoon Approximation
The class of cartoon-like functions, classicly defined as piecewise C functions consisting of smooth regions separated by C discontinuity curves, is a well-established model for image data. The quest
Efficient Resolution of Anisotropic Structures
TLDR
A new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems and a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable isotropic regularity assumptions.
Shearlets: Theory and Applications
Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in
ShearLab 3D
TLDR
This article presents extensive numerical experiments in 2D and 3D concerning denoising, inpainting, and feature extraction, comparing the performance of ShearLab 3D with similar transform-based algorithms such as curvelets, contourlets, or surfacelets.
Nonseparable Shearlet Transform
  • W. Lim
  • Computer Science
    IEEE Transactions on Image Processing
  • 2013
TLDR
This paper introduces a new shearlet transform associated with a nonseparable shearlett generator, which improves the directional selectivity of previous shearlets and shows numerical experiments demonstrating the potential in 2D and 3D image processing applications.
An asymptotic analysis of separating pointlike and $C^{\beta}$-curvelike singularities
TLDR
This paper proposes a reconstruction framework with theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames, and constructs a dual pair of bandlimited α-shearlets which possesses a good time and frequency localization.
...
...

References

SHOWING 1-10 OF 37 REFERENCES
Optimally Sparse Representations of 3D Data with C2 Surface Singularities Using Parseval Frames of Shearlets
TLDR
It is proved that this 3D shearlet construction provides essentially optimal sparse representations for functions on $\mathbb{R}^3$ which are $C-regular away from discontinuities along $C^2$ surfaces, and it is shown that this asymptotic behavior significantly outperforms wavelet and Fourier series approximations.
Compactly supported shearlets are optimally sparse
Optimally sparse 3D approximations using shearlet representations
TLDR
The result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3-D data.
Optimally Sparse Multidimensional Representation Using Shearlets
In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f
Construction of Compactly Supported Shearlet Frames
Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital
Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators
TLDR
This paper shows that a construction having exactly these properties is obtained by using the framework of affine systems, which provide a Multiresolution analysis similar to the one associated with classical wavelets, which is very useful for the development of fast algorithmic implementations.
Compactly Supported Shearlets
Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic
Representation and Compression of Multidimensional Piecewise Functions Using Surflets
TLDR
Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets, and a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales is proposed.
New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
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. Conceptually, the
...
...