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Scientific article
English

Steerable pyramids and tight wavelet frames in L2(R(d))

Published inIEEE transactions on image processing, vol. 20, no. 10, p. 2705-2721
Publication date2011
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.

Keywords
  • Algorithms
  • Fourier Analysis
  • Principal Component Analysis
  • Wavelet Analysis
  • Image Processing
  • Computer-Assisted
Citation (ISO format)
UNSER, Michael, CHENOUARD, Nicolas, VAN DE VILLE, Dimitri. Steerable pyramids and tight wavelet frames in L2(R(d)). In: IEEE transactions on image processing, 2011, vol. 20, n° 10, p. 2705–2721. doi: 10.1109/TIP.2011.2138147
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ISSN of the journal1057-7149
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