A Unifying Parametric Framework for 2D Steerable Wavelet Transforms

We introduce a complete parameterization of the family of two-dimensional steerable wavelets that are polar-separable in the Fourier domain under the constraint of self-reversibility. These wavelets are constructed by multiorder generalized Riesz transformation of a primary isotropic bandpass pyramid. The backbone of the transform (pyramid) is characterized by a radial frequency profile function h(.), while the directional wavelet components at each scale are encoded by an M x (2N + 1) shaping matrix U, where M is the number of wavelet channels and N the order of the Riesz transform. We provide general conditions on h(omega) and U for the underlying wavelet system to form a tight frame of L2(R-2) (with a redundancy factor 4/3M). The proposed framework ensures that the wavelets are steerable and provides new degrees of freedom (shaping matrix U) that can be exploited for designing specific wavelet systems. It encompasses many known transforms as particular cases: Simoncelli's steerable pyramid, Marr gradient and Hessian wavelets, monogenic wavelets, and Nth-order Riesz and circular harmonic wavelets. We take advantage of the framework to construct new generalized spheroidal prolate wavelets, whose angular selectivity is maximized, as well as signal-adapted detectors based on principal component analysis. We also introduce a curvelet-like steerable wavelet system. Finally, we illustrate the advantages of some of the designs for signal denoising, feature extraction, pattern analysis, and source separation.