Optimally sparse multidimensional representation using shearlets. (English) Zbl 1197.42017

Summary: 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 \(f\) which are \(C^2\) except for discontinuities along \(C^2\) curves. More specifically, if \(f_N^S\) is the \(N\)-term reconstruction of \(f\) obtained by using the \(N\) largest coefficients in the shearlet representation, then the asymptotic approximation error decays as \[ \| f-f_N^S\|_2^2 \asymp N^{-2} (\log N)^3,\quad N \to \infty, \] which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate \(N^{-1}\) associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.


42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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