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# Map Algebra

Let us consider a set of raster maps, each containing a numeric attribute and having all exactly the same pixel size, number of rows and columns, and pixel coordinates (the ensemble of this information is sometimes called georeference, as mentioned in Chapter 3). In simple words, map algebra consists in applying algebraic expressions for each single pixel of the common georeference, using the values of the individual maps as input.

The results of the algebraic expression are stored in a raster map that has the same common georeference. For instance, if we have a map of the natural terrain elevation and a map of the height of buildings and infrastructure above ground, we can use map algebra to obtain a map of the actual ground surface elevation by summing for each pixel the corresponding values of the two maps.

Map algebra is in fact a very intuitive and practical way to overlay raster maps. Map algebra, as an elegant formal system for the combination of maps, was introduced in the 1980s by C. Dana Tomlin [1, 2, 12] and, in its general form, distinguishes among four classes of operators acting on raster maps: Local, Focal, Zonal, and Global.

Local operators are applied on a pixel-by-pixel basis, as in the example of buildings above. Focal operators analyze not only values at single pixels, but also in their neighborhood. A typical focal operator is the gradient of a continuous surface such as a DEM: the gradient is computed as a finite difference approximation ...

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