Steerable Near-Quadrature Filter Pairs in Three Dimensions

Steerable filter pairs that are near quadrature have many image processing applications. This paper proposes a new methodology for designing such filters. The key idea is to design steerable filters by minimizing a departure-from-quadrature function. These minimizing filter pairs are almost exactly in quadrature. The polar part of the filters is nonnegative, monotonic, and highly focused around an axis, and asymptotically the filters achieve exact quadrature. These results are established by exploiting a relation between the filters and generalized Hilbert matrices. These near-quadrature filters closely approximate three dimensional Gabor filters. We experimentally verify the asymptotic mathematical results and further demonstrate the use of these filter pairs by efficient calculation of local Fourier shell correlation of cryogenic electron microscopy.