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September 18, 2007

The topological point of view on image segmentation

I was answering to an e-mail about where the main algorithm of ours has come from and how it is related to image segmentation especially the watershed algorithm. So I decided to quote it here. Books on image processing, like Gonzalez and Woods, spend very little time on topological issues and totally ignore well developed topological approaches and techniques. There is a good reason for that (and some bad reasons too). Traditionally topology can only be applied to geometric objects, i.e., binary images. This is very limiting because if you deal with gray scale images you have to start with thresholding every time and every time you loose information. When I discovered this, I decided to try to fix the problem by developing an algorithm for gray scale and color images based entirely on algebraic topological methods. The result is the method that we use here. It is similar to the watershed algorithm but only in the sense that both look for minima and maxima of the gray scale function. The difference comes from this simple topological idea that in the image there are objects and holes in them, and there is nothing else. In case of gray scale, this means that there are dark objects on light background with light holes in them, or vice versa. So, what the algorithm produces cannot even be called a “segmentation” which is supposed to split the image into non-overlapping areas.   

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