A Unifying Parametric Framework for 2D Steerable Wavelet Transforms
Michael A. Unser, Nicolas Chenouard · SIAM Journal on Imaging Sciences · 2013
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 $h(\omega)$, while the directional wavelet components at each scale are encoded by an $M \times (2N+1)$ shaping matrix ${\bf U}$, where $M$ is the number of wavelet channels and $N$ the order of the Riesz transform. We provide general conditions on $h(\omega)$ and ${\bf U}$ for the underlying wavelet system to form a tight frame of $L_2(\mathbb{R}^2)$ (with a redundancy factor $4/3M$). The proposed framework ensures that the wavelets are steerable and provides new degrees of freedom (shaping matrix ${\bf U}$) 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 $N$th-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.