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Steerable Pyramids and Tight Wavelet Frames in L2(ℝd)

M. Unser, N. Chenouard, D. Van De Ville

IEEE Transactions on Image Processing, vol. 20, no. 10, pp. 2705-2721, October 2011.



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 L2(ℝd), 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.


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AUTHOR="Unser, M. and Chenouard, N. and Van De Ville, D.",
TITLE="Steerable Pyramids and Tight Wavelet Frames in
        $L_{2}({\mathbb{R}}^{d})$",
JOURNAL="{IEEE} Transactions on Image Processing",
YEAR="2011",
volume="20",
number="10",
pages="2705--2721",
month="October",
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