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 $h(\omega)$, 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 part...