Image Compression using an Efficient Edge Cartoon + Texture Model
DCC '02 Proceedings of the Data Compression Conference
The contourlet transform: an efficient directional multiresolution image representation
IEEE Transactions on Image Processing
Adaptive Multiresolution Analysis Structures and Shearlet Systems
SIAM Journal on Numerical Analysis
Image separation using wavelets and shearlets
Proceedings of the 7th international conference on Curves and Surfaces
Bandlimited shearlet-type frames with nice duals
Journal of Computational and Applied Mathematics
Directional Multiscale Processing of Images Using Wavelets with Composite Dilations
Journal of Mathematical Imaging and Vision
Journal of Visual Communication and Image Representation
Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis
Journal of Mathematical Imaging and Vision
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Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.