Optimal infinite impulse response zero crossing based edge detectors

We present formal optimality criteria and a complete design methodology for a family of zero crossing based, infinite impulse response (recursive) edge detection filters. In particular, we adapt the optimality criteria proposed by Canny (IEEE Trans. Pattern Anal. Mach. IntelligencePAMI-8, 1986, 679–714) to filters designed to respond with a zero crossing in the output at an edge location and additionally to impulse responses which are (allowed to be) infinite in extent. The spurious response criterion is captured directly by an appropriate measure of filter spatial extent for infinite responses. Infinite duration impulse responses may be implemented efficiently with recursive filtering techniques and so require constant computation time with respect to scale. As we will show, we can achieve both superior performance and increased speed by designing directly for an infinite impulse response than by any of the proposed finite duration approaches. We also show that the optimal filter which responds with a zero crossing in its output may not be implemented by designing the optimal peak responding filter (similar to Canny) and taking an additional derivative. It is necessary to formulate the criteria and design for a zero crossing response from the outset, else optimality is sacrificed. Filter parameters and performance criteria are presented for several designs, and experimental results are presented on a variety of images which demonstrate the behavior in the presence of very adverse noise, with respect to scale, and as compared to other “optimal” IIR filters which have been reported.