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.
- URL: http://bigwww.epfl.ch/publications/unser1103.html
- URL: http://bigwww.epfl.ch/publications/unser1103.pdf
- URL: http://bigwww.epfl.ch/publications/unser1103.ps
Keywords: Directional derivatives ; multiresolution decomposition ; Riesz transform ; steerable filters ; steerable pyramid ; tight frames ; wavelet transform ; Linear Inverse Problems ; Multiresolution Analysis ; Statistics ; Domain ; Enhancement ; Retrieval ; Order ; CIBM-SP
Record created on 2011-12-16, modified on 2016-08-09