Steerable Pyramids and Tight Wavelet Frames in L-2(R-d)
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.
steerable pyramid multiresolution decomposition riesz transform wavelet transform directional derivatives steerable filters tight frames