Wavelet Steerability and the Higher-Order Riesz Transform

Our main goal in this paper is to set the foundations of a general continuous-domain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions (d ¿ 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. 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
L

₂
( \BBR
d
) into another ¿steerable¿ wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli's steerable pyramid whose construction was primarily based on filterbank 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 with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially behave like the N th-order partial derivatives of an isotropic Gaussian kernel.