On optimal linear filtering for edge detection

In this paper, we will revisit the analytical expressions of the three Canny's criteria for edge detection quality: good detection, good localization, and low multiplicity of false detections. Our work differs from Canny's work on two essential points. Here, the criteria are given for discrete sampled signals, i.e., for the real, implemented filters. Instead of a single-step edge as input signal, we will use pulses of various width. The proximity of other edges affects the quality of the detection process. This is taken into account in the new expressions of these criteria. We will derive optimal filters for each of the criteria and for any combination of them. In particular, we will define an original filter which maximizes detection and localization and a simple approximation of the optimal filter for the simultaneous maximization of the three criteria. The upper bounds of the criteria are computed which will allow users to measure the absolute and relative performance of any filter (in this paper, exponential, Deriche, and first derivative of Gaussian filters will be evaluated). Our criteria can also be used to compute the optimal value of the scale parameter of a given filter when the resolution of the detection is fixed.