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Steerable Wavelet Frames in L2(ℝD)

M. Unser

Proceedings of the International Conference on Wavelets and Applications (ICWA'09), Saint-Petersburg, Russia, June 14-20, 2009, pp. 61-62.



We introduce an Nth-order extension of the Riesz transform in d dimensions. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner- product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of L2(ℝd) into another "steerable" wavelet frame, while preserving the frame bounds. Concretely, this means we can design reversible multi-scale decompositions in which the analysis wavelets (feature detectors) can be spatially rotated in any direction via a suitable linear combination of wavelet coefficients. The concept provides a rigorous functional counterpart to Simoncelli's steerable pyramid whose construction was entirely based on digital filter design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method using a Laplacian-like (or Mexican hat) polyharmonic spline wavelet transform as our primary frame. We display new wavelets that replicate the behavior of the Nth-order partial derivatives of an isotropic Gaussian kernel.


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