Towards the unification of band-limited derivative operators for edge detection

Abstract

Edge detection has been an important subject in image processing. Many linear operators for edge detection in noisy images, including some ‘optimal’ ones, have been proposed. Note that most of them were deduced from the mono-edge model, but in real images there exist always many edge points. So it will be interesting to analyse the problem of multi-edge detection in images. In the present paper, we present at first the optimal filter for mono- and multi-edge detection and show that an optimal filter for noise removing in edge detection is the symmetric exponential filter of an infinite size (ISEF). In more general multi-edge cases, the optimal filter is a linear combination of the output from different stages of a cascade of symmetric exponential filters. Based on the ISEF cascade for multi-edge detection, we analyse then some band-limited linear edge detectors, such as Gaussian filter, Gaussian cascade technique, box technique, fast filter transform, Canny's filter, DRF method, box difference and Gabor's filter used for edge detection. We show that these filters can be explained and unified by the cascade of exponential filters presented for multi-edge detection.