Abstract: Edge detection is the most fundamental step in vision algorithms. A number of edge detectors have been discussed in the computer vision literature. Examples of classic edge detectors include the Marr-Hildreth edge operator, facet edge operator, and the Canny edge operator. Edge detection using morphological techniques are attractive because they can be efficiently implemented in near real time machine vision systems that have special hardware support. However, little performance characterization of edge detectors has been done. In general, performance characterization of edge detectors has been done mainly by plotting empirical curves of performance. Quantitative performance evaluation of edge detectors was first performed by Abdou and Pratt. It is the goal of this paper to perform a theoretical comparison of gradient based edge detectors and morphological edge detectors. By assuming that an ideal edge is corrupted with additive noise we derive theoretical expressions for the probability of misdetection (the probability of labeling of a true edge pixel as a nonedge pixel in the output). Further, we derive theoretical expressions for the probability of false alarm (the probability of labeling of a nonedge pixel as an output edge pixel) by assuming that the input to the operator is a region of flat graytone intensity corrupted with additive Gaussian noise of zero mean and variance $sigma$+2$/. Even though the blurring step in the morphological operator introduces correlation in the additive noise, we make an approximation that the output samples after blurring are i.i.d. Gaussian random variables with zero mean and variance $sigma$+2$//M where M is the window size of the blurring kernel. The false alarm probabilities obtained by using this approximation can be shown to be upperbounds of the false alarm probabilities computed without the approximation. The theory indicates that the blur-min operator is clearly superior when a 3 $MUL 3 window size is used. Since we only have an upperbound for the false alarm probability the theory is inadequate to confirm the superiority of the blur-min operator. Empirical evaluation of the performance indicates that the blur-min operator is superior to the gradient based operator. Evaluation of the edge detectors on real images also indicate superiority of the blur-min operator. Application of hysteresis linking, after edge detection, significantly reduces the misdetection rate, but increases the false alarm rate. !14
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