A discrete expression of Canny´s criteria for step edge detector performances evaluation

Optimal filters for edge detection are usually developed in the continuous domain and then transposed by sampling to the discrete domain. Simpler filters are directly defined in the discrete domain. We define criteria to compare filter performances in the discrete domain. Canny has defined (1983, 1986) three criteria to derive the equation of an optimal filter for step edge detection: good detection, good localization, and low-responses multiplicity. These criteria seem to be good candidates for filter comparison. Unfortunately, they have been developed in the continuous domain, and their analytical expressions cannot be used in the discrete domain. We establish three criteria with the same meaning as Canny´s. Some comparisons with experimental results confirm the validity of our approach. This study highlights the existence of two classes of derivative operators that are distinguished by whether or not the impulse response of the filter in continuous space domain is continuous on its center. These classes exhibit very different properties for the second and third criteria. We extend the use of the first and third criteria to the smoothing filters. We also define an optimal continuous filter according to the continuous third criterion and an optimal discrete filter according to the discrete third criterion. We compare the performances of the sampled version of the continuous filter to those of the optimal discrete filter