A Theoretical Analysis of Granulometry-Based Roughness Measures on Cartosat DEMs

The study of water bodies such as rivers is an important problem in the remote sensing community. A meaningful set of quantitative features reflecting the geophysical properties help us better understand the formation and evolution of rivers. Typically, river sub-basins are analyzed using Cartosat Digital Elevation Models (DEMs), obtained at regular time epochs. One of the useful geophysical features of a river sub-basin is that of a roughness measure on DEMs. However, to the best of our knowledge, there is not much literature available on theoretical analysis of roughness measures. In this article, we revisit the roughness measure on DEM data adapted from multiscale granulometries in mathematical morphology, namely
multiscale directional granulometric index (MDGI)
. This measure was classically used to obtain shape-size analysis in grayscale images. In earlier works, MDGIs were introduced to capture the characteristic surficial roughness of a river sub-basin along specific directions. Also, MDGIs can be efficiently computed and are known to be useful features for classification of river sub-basins. In this article, we ask the question
when does a MDGI fail to classify distinct DEMs?
We provide a theoretical analysis of a MDGI to answer this question. In particular, we identify nonrivial sufficient conditions on the structure of DEMs under which MDGIs cannot distinguish between distinct DEMs. These properties are illustrated with some fictitious DEMs. We also provide connections to a discrete derivative of volume of a DEM. Based on these connections, we provide intuition as to why a MDGI is considered a roughness measure. We empirically verify that MGDIs capture the topographical characteristics using the Lower-Indus, Wardha, and Barmer river sub-basins. We show that a simple decision tree based on MDGI features alone can successfully distinguish between sub-basins with a high accuracy. We obtain upto 94% accuracy while the baseline of random classifier is around 33.34%.