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<p><b>Abstract</b>—On one hand, optimal filters used for edge detection are usually developed in the continuous domain and then transposed by sampling to the discrete domain. On the other hand, the simpler filters like the Sobel filter are directly defined in the discrete domain. Most of the previous works on edge detection were made to elaborate optimal filters. But, few works present methods to compare them. In this paper, we define criteria to compare the performances of different filters in their application area: the discrete domain. Canny has defined three criteria to derive the equation of an optimal filter for step edge detection: (1) good detection, (2) good localization, and (3) low-responses multiplicity. These criteria seem to be good candidates for filters comparison. Unfortunately, they have been developed in the continuous domain, and their analytical expressions cannot be used in the discrete domain. Unlike previous works, our approach is based on a direct computation 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 highlighted 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. It appears that the sampled version of the continuous optimal filter is not optimal for the sampled data even in the case where the spectrum overlapping due to the sampling is reduced.</p>
Edge detection, Canny criteria, edge operators, localization criterion.

T. Kamlé and D. Demigny, "A Discrete Expression of Canny's Criteria for Step Edge Detector Performances Evaluation," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 19, no. , pp. 1199-1211, 1997.
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