Abstract

We present two operator-based methods for the construction of wavelets with improved shift-invariance and/or rotation-invariance properties. The first is inspired by Kingsbury’s dual-tree complex wavelet transform and relies on the shifting action of the group of fractional Hilbert transform operators. It leads to a precise characterization of the shiftability property of complex wavelets whose real and imaginary parts are functionally related through the Hilbert transform. The second method is devoted to the construction of steerable wavelets. It relies on an Nth-order extension of the Riesz transform that has the remarkable property of mapping any primary wavelet frame (or basis) of L_2(R^d) 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. It allows for the specification of wavelets with any order of steerability in any number of dimensions. We illustrate the approach with the design of new steerable polyharmonic-spline wavelets that replicate the behavior of the Nth-order partial derivatives of an isotropic Gaussian kernel and demonstrate thei r suitability for image processing.