System Planning Corporation, Arlington, VA 22201, USA.
IEEE Trans Image Process. 2012 Feb;21(2):550-61. doi: 10.1109/TIP.2011.2164415. Epub 2011 Aug 12.
Wavelets with composite dilations provide a general framework for the construction of waveforms defined not only at various scales and locations, as traditional wavelets, but also at various orientations and with different scaling factors in each coordinate. As a result, they are useful to analyze the geometric information that often dominate multidimensional data much more efficiently than traditional wavelets. The shearlet system, for example, is a particular well-known realization of this framework, which provides optimally sparse representations of images with edges. In this paper, we further investigate the constructions derived from this approach to develop critically sampled wavelets with composite dilations for the purpose of image coding. Not only do we show that many nonredundant directional constructions recently introduced in the literature can be derived within this setting, but we also introduce new critically sampled discrete transforms that achieve much better nonlinear approximation rates than traditional discrete wavelet transforms and outperform the other critically sampled multiscale transforms recently proposed.
复合伸缩小波为构造不仅在不同尺度和位置（如传统小波），而且在不同方向和每个坐标中具有不同比例因子的波形提供了一个通用框架。因此，它们非常适合分析通常在多维数据中占主导地位的几何信息，比传统小波更有效。例如，剪切波系统是该框架的一个特殊知名实现，它为边缘图像提供了最佳稀疏表示。在本文中，我们进一步研究了从这种方法得出的构造，以开发用于图像编码的具有复合伸缩的临界采样小波。我们不仅表明可以在这种情况下推导出文献中最近引入的许多非冗余方向构造，而且还引入了新的临界采样离散变换，这些变换在非线性逼近率方面比传统离散小波变换要好得多，并且优于最近提出的其他临界采样多尺度变换。
