A Class of Directional Zero-Phase 2D Filters Designed Using Analytical Approach

This paper proposes an analytical design procedure for a class of 2D filters with a directional characteristic in the frequency plane. The method starts from two types of low-pass prototypes, namely a Gaussian filter and a wide-band filter. The two prototypes and the resulted 2D filters are zero-phase, a very useful property in image processing tasks. The prototypes are parametric and can be scaled along the frequency axis, thus resulting narrower or wider. A 1D to 2D frequency mapping, based on accurate approximations, is derived and applied to each prototype, obtaining the desired directional 2D frequency response with specified selectivity and orientation. The designed filters are adjustable, with a frequency response in factored matrix form depending on specifications. The Gaussian prototype yields very selective directional filters, used in detecting oriented straight lines in images, as show simulation results on several test images. Using oriented wide-band filters, other useful 2D filters result, namely square and parallelogram filters. The designed filters have accurate frequency responses, good directional selectivity and high linearity at relatively low order. An advantage of this analytical approach is that the filter transfer function results in parametric form and allows for easy tunability.