Dividing the Continent
Back home again, and plugged in to libraries as well as computers, I was not surprised to learn that others had gone before me in thinking about the nature of watersheds and divides. But I was surprised to learn just who my predecessors were.
Two of the best publications on the subject are short papers by Arthur Cayley (a founder of topology and graph theory) and James Clerk Maxwell (the author of the electromagnetic theory of light). Cayley and Maxwell do not focus on continental divides—perhaps the concept is not an obvious one for residents of an island nation—but their analysis of landforms in general clarifies aspects the divide problem. They emphasize peaks, pits and saddles as the keys to delineating the fundamental regions of a landscape.
Much as Euler gave a formula for the number of faces, edges and vertices in a polyhedron, Maxwell relates the number of topographic peaks, pits and saddles on a surface. In the case of a sphere, the formula is p+q–s=2, where p is the number of peaks, q the number of pits and s the number of saddles. Maxwell also outlines a procedure for dividing the landscape into watershed regions. Whereas all my own methods progressed either down from peaks or up from pits, Maxwell argues that the right way to do it is to start in the middle—at a saddle—and proceed toward both peaks and pits along lines of steepest ascent or descent.
The more recent literature on divides and watersheds held another surprise. I had expected to find work by geographers and cartographers, and indeed they have written extensively on the subject. But there is also a body of work by students of image analysis and artificial vision. The connection is clear once it’s pointed out. Just as a topographic map can be presented as an image in which elevation is encoded in brightness, so also a digitized image can be interpreted as a surface where altitude represents shades of gray or color. Finding watersheds in such a surface is a useful approach to identifying objects in the image.
The idea of using watersheds for image analysis was first proposed at the School of Mines in Paris in the 1970s. (The images that needed analyzing were micrographs of ore samples.) Workers there have continued to refine the method. The version of the algorithm I have found most helpful was devised by Luc Vincent and Pierre Soille when they were both working at the School of Mines.
The Vincent-Soille algorithm is related to what I have dubbed the global-warming method, but with a number of enhancements. One remarkably simple device greatly reduces the computational effort. In addition to storing the array of points that represents the landscape, they keep a list of the same points sorted in order of increasing elevation. With this list in hand, there is no need to search for points that will be submerged each time the water level rises; you can simply cross them off in order.
As it happens, one contemplated application of the watershed algorithm in image processing is the old dream of a car that drives itself. The "watersheds" detected in a video image might be the edges of a roadway. So perhaps the next time I cross the continental divide I’ll be able to pay more attention. I won’t have to keep my hands on the wheel.
© Brian Hayes