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Is there a way to find the compactness (ratio of perimeter to area) of various groups of contiguous cells using GRASS functions?

If you imagine I want to find the compactness of the three land parcels shown in the below picture enter image description here

Perhaps there are some reporting functions I can use to find the area and perimeter of these parcels and then I can determine the ratio myself?

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You can use v.to.db for this (see also keyword index, under "compactness"). Beforehand you need to vectorize your raster map with r.to.vect.

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