An unified theory for steerable and quadrature filters

In this paper, a complete theory of steerable filters is presented which shows that quadrature filters are only a special case of steerable filters. Although there has been a large number of approaches dealing with the theory of steerable filters, none of these gives a complete theory with respect to the transformation groups which deform the filter kernel. Michaelis and Sommer (Michaelis and Sommer, 1995) and Hel-Or and Teo (Teo and Hel-Or, 1996; Teo and Hel-Or, 1998) were the first ones who gave a theoretical justification for steerability based on Lie group theory. But the approach of Michaelis and Sommer considers only Abelian Lie groups. Although the approach of Hel-Or and Teo considers all Lie groups, their method for generating the basis functions may fail as shown in this paper. We extend these steerable approaches to arbitrary Lie groups, like the important case of the rotation group SO(3) in three dimensions. Quadrature filters serve for computing the local energy and local phase of a signal. Whereas for the one dimensional case quadrature filters are theoretically well founded, this is not the case for higher dimensional signal spaces. The monogenic signal (Felsberg and Sommer, 2001) based on the Riesz transformation has been shown to be a rotational invariant generalization of the analytic signal. A further generalization of the monogenic signal, the 2D rotational invariant quadrature filter (Köthe, 2003), has been shown to capture richer structures in images as the monogenic signal. We present a generalization of the rotational invariant quadrature filter based on our steerable theory. Our approach includes the important case of 3D rotational invariant quadrature filters but it is not limited to any signal dimension and includes all transformation groups that own a unitary group representation.