Abstract. 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(ω), while the directional wavelet components at each scale are encoded by an M × (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(ω) 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 N thorder 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 signaladapted 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. Key words. wavelets, steerability, Riesz transform, frame DOI. 10.1137/120866014 1. Introduction. Scale and directionality are essential ingredients for visual perception and processing. This has prompted researchers in image processing and applied mathematics to develop representation schemes and function dictionaries that are capable of extracting and quantifying this type of information explicitly. The fundamental operation underlying the notion of scale is dilation, which calls for a wavelet-type dictionary in which dilated sets of basis functions that "live" at different scales coexist. The elegant aspect here is that it is possible to specify wavelet bases of L 2 (R 2 ) that result in an orthogonal decomposition of an image in terms of its multiresolution components [1] . The multiscale analysis achieved by these "classical" wavelets is well suited for extracting isotropic image features and isolated singularities, but is not quite as efficient for sparsely encoding edges and curvilinear structures. To capture directionality, the basis functions need to be angularly selective and the original dilation scheme complemented with spatial rotation. This can be achieved at the expense of
Further, steerability has recently been related to tight frames, presenting Simoncelli's Steerable Pyramid and multiple other Wavelets arising as a special case of the non-orthogonal Riesz transform REF .
A Unifying Parametric Framework for 2 D Steerable Wavelet Transforms
SIAM J. Imaging Sciences
SIAM J. Imaging Sciences
Mathematics Computer Science