Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
IEEE Trans Image Process. 2011 Oct;20(10):2705-21. doi: 10.1109/TIP.2011.2138147. Epub 2011 Apr 7.
We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2-D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a one-to-many mapping (Nth-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M×M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallat-type multiresolution analysis of L(2)(R(d)), and provide a fast Fourier transform-based decomposition algorithm. We also propose a principal-component-based method for signal-adapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized principal component analysis), which consistently performs best.
我们提出了一个功能框架，用于设计任意维数的紧可转向小波框架。该方法的 2-D 版本可以看作是 Simoncelli 的可转向金字塔的推广，通过适当的参数化，可以访问更大的可转向小波调色板。我们构造的核心是一个原始各向同性小波框架，它提供了信号的多分辨率分解。可转向小波通过将原始小波应用于一对一映射（N 阶广义 Riesz 变换）来获得。可转向小波的形状由一个 M×M 的酉矩阵（其中 M 是小波通道的数量）控制，该矩阵可以任意选择；这比传统的等角配置（可转向金字塔）提供了更广泛的解决方案。我们给出了这些广义小波变换的完整功能描述，并推导出它们的转向方程。我们描述了一些具体的变换示例，包括一些基于 L(2)(R(d))的 Mallat 型多分辨率分析构建的示例，并提供了基于快速傅里叶变换的分解算法。我们还提出了一种基于主成分的信号自适应小波设计方法。最后，我们给出了一些说明性示例，并比较了各种可转向变换的去噪性能。结果支持优化的小波设计（等化主成分分析），它始终表现最好。
