Editing Edge detection (section)

=== Kovalevsky ===
[[Vladimir Antonovich Kovalevsky|Vladimir A. Kovalevsky]]<ref>Kovalevsky, V., Image Processing with Cellular Topology, Springer 2021, ISBN 978-981-16-5771-9, pp. 113-138</ref> has suggested a quite different approach. He uses a preprocessing of the image with the Sigma filter<ref> Lee, J.-S., Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Information Processing. 1983, 24(2): 255-69 </ref> and with a special filter for the dilution of the ramps. This method uses no brightness of the image but only the intensities of the color channels which is important for detecting an edge between two adjacent pixels of equal brightness but different colors. The method scans the image two times: first along the horizontal lines and second along the vertical columns. In each horizontal line six consequent adjacent pixels are considered and five color difference between each two adjacent pixels are calculated. Each color difference is the sum of absolute differences of the intensities of the color channels Red, Green, and Blue of the corresponding adjacent pixels. If this sum is greater than a given threshold, then the sign of the color difference is set equal to the sign of the difference of the green intensities. If the green difference is zero, then the sign of the color difference is set equal to the sign of the difference of the red intensities. If, however, both the green and the red differences are zero, then the sign of the color difference is set equal to the sign of the blue difference which in this case cannot be zero since the sum is greater than the threshold. 
Certain conditions for the values and signs of the five color differences are specified in such way that if the conditions are fulfilled, then a short vertical stroke is put between the third and the fourth of the six pixels as the label of the edge.
Similar calculations are performed for the vertical columns. In this case a short horizontal stroke is put between the third and the fourth of the six subsequent pixels. The vertical and horizontal strokes (being the one-dimensional cells of an abstract cell complex corresponding to the image) mostly compose a connected sequence representing the edge.
This method is robust and very fast and, what is more important, it can detect edges between adjacent pixels of equal brightness’s if the color difference between these pixels is greater than the threshold.

The Canny–Deriche detector was derived from similar mathematical criteria as the Canny edge detector, although starting from a discrete viewpoint and then leading to a set of recursive filters for image smoothing instead of [[exponential filter]]s or Gaussian filters.<ref>R. Deriche (1987) ''Using Canny's criteria to derive an optimal edge detector recursively implemented'', Int. J. Computer Vision, vol 1, pages 167–187.</ref>

The [[Edge detection#Differential|differential edge detector]] described below can be seen as a reformulation of Canny's method from the viewpoint of differential invariants computed from a [[scale space representation]] leading to a number of advantages in terms of both theoretical analysis and sub-pixel implementation. In that aspect, [[Log Gabor filter]] have been shown to be a good choice to extract boundaries in natural scenes.<ref>Sylvain Fischer, Rafael Redondo, Laurent Perrinet, Gabriel Cristobal. [https://link.springer.com/content/pdf/10.1155/2007/90727.pdf Sparse approximation of images inspired from the functional architecture of the primary visual areas]. EURASIP Journal on Advances in Signal Processing, special issue on Image Perception, 2007</ref>