Steerable Pyramids and Tight Wavelet Frames in L-2(R-d)

作者：Unser Michael^*; Chenouard Nicolas; Van De Ville Dimitri

来源：IEEE Transactions on Image Processing, 2011, 20(10): 2705-2721.

DOI：10.1109/TIP.2011.2138147

• Directional derivatives
• multiresolution decomposition
• Riesz transform
• steerable filters
• steerable pyramid
• tight frames
• wavelet transform

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 (N th-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M x 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.

• 出版日期2011-10