Optimally sparse multidimensional representation using shearlets
Optimally sparse multidimensional representation using shearlets
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DOI：
10.1137/060649781
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发表时间：
2007-01-01
影响因子：
2
通讯作者：
Labate, Demetrio
中科院分区：
文献类型：
--
作者：
Guo, Kanghui;Labate, Demetrio
在本文中，我们表明，剪切波，仿射系统的功能，最近推出的作者和他们的合作者，基本上是最佳的表示2维函数f是C-2除了不连续沿着C-2曲线。更具体地，如果f（N）（S）是通过使用剪切波表示中的N个最大系数获得的f的N项重构，则渐近逼近误差平行于f-f（N）（S）平行于2渐近地衰减到2N-2（logN）（3），N ->无穷大，这基本上是最优的，并且大大优于与小波近似相关联的相应的渐近逼近速率N-1。与具有类似稀疏性的曲波不同，剪切波形成仿射系统，并且具有更简单的数学结构。事实上，这个系统的元素形成了一个Parseval框架，并通过对单个局部化良好的窗口函数应用伸缩、剪切变换和平移来生成。
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 parallel to f - f(N)(S)parallel to 2 asymptotic to 2 N-2 (logN)(3), N -> infinity, 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.