The multidimensional isotropic generalization of quadrature filters in geometric algebra

In signal processing, the approach of the analytic signal is a capable and often used method. For signals of finite length, quadrature filters yield a bandpass filtered approximation of the analytic signal. In the case of multidimensional signals, the quadrature filters can only be applied with respect to a preference direction. Therefore, the orientation has to be sampled, steered or orientation adaptive filters have to be used. Up to now, there has been no linear approach to obtain an isotropic analytic signal which means that the amplitude is independent of the local orientation. In this paper, we present such an approach using the framework of geometric algebra. Our result is closely related to the Riesz transform and the structure tensor. It is seamless embedded in the framework of Clifford analysis. In a suitable coordinate system, the filter response contains information about local amplitude, local phase and local orientation of intrinsically one-dimensional signals. We have tested our filters on two- and three-dimensional signals