Topological generalization of continuous valued raster data

Abstract

We propose a novel method for generalizing continuous valued raster data with respect to topological constraints whereby smaller scale connected components and holes in the data sublevel sets are removed. The proposed method formulates the problem of generalization as an optimization problem with respect to persistent homology. We prove the objective function to be locally continuous with analytical gradients which can be used to perform optimization using gradient descent. Furthermore, we prove the convergence of gradient descent to a global optimal solution. The proposed method is general in nature and can be applied to raster data of any dimension. The utility of the method is demonstrated with respect to generalizing two-and three-dimensional raster data corresponding to digital elevation models (DEM) and subsurface mineral interpolation respectively.