Closed-Form Design of 2D Filters with Elliptical and Circular Frequency Response

An analytical design procedure is described in this paper for a type of 2D recursive filter, with a frequency response elliptical or circular in shape. The design is based on a specially determined frequency mapping, applied to an digital prototype filter. This analytical design approach yields the transfer function of the desired 2D filter in a factored, closed form. Some efficient, accurate approximations are used, like Chebyshev-Padé method, without resorting to global optimization algorithms. The filter matrices are a convolution of smaller size matrices, an obvious advantage in implementation. Moreover, the obtained filter is tunable, since its coefficients depend on the specified orientation and bandwidth. As proven by given design examples, these 2D filters have a precise shape, with negligible distortions even near the margins of frequency plane, good selectivity and low order.