title: Steerable pyramids and tight wavelet frames in L2(R(d)).
abstract: 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.
authors.affiliation: Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
authors.forename: Michael
authors.initials: M
authors.lastname: Unser
authors.forename: Nicolas
authors.initials: N
authors.lastname: Chenouard
authors.forename: Dimitri
authors.initials: D
authors.lastname: Van de Ville
mesh_terms: D000465:Algorithms; D005583:Fourier Analysis; D007091:Image Processing, Computer-Assisted; D025341:Principal Component Analysis; D058067:Wavelet Analysis