Complex Derivative Filters

Steerable filters are a valuable tool for various low-level vision tasks. In this paper, we argue for the use of complex analysis in the context of 2-D steerable filters. In particular, we recommend the use of complex partial derivatives as a computational basis. Complex derivatives have a major advantage in comparison to real derivatives: they show a canonical rotation behavior, namely a rotation affects the derivative just by a multiplication with a complex unit number. So, the complex derivatives can be steered in a more elegant way and above that they are less expensive to compute. We present several analytical formulas for common and new filter kernels in terms of complex derivatives. Further we relate the complex derivatives of a Gaussian with the Gauss-Laguerre transform and show that the Gauss-Laguerre functions provide an optimal signal representation for local and smooth images. We discuss various finite difference schemes for the realization of the derivatives and use them in practice. In a first experiment, we use a newly introduced filter kernel for anisotropic blurring. The complex formalism offers an elegant way to locally adapt the shape and orientation of the kernel. Second, we use the proposed filters as matched filters to detect vessels in retinal images.