Local digital algorithms for estimating the mean integrated curvature of r-regular sets

Consider the design based situation where an r-regular set is sampled on a random lattice. A fast algorithm for estimating the integrated mean curvature based on this observation is to use a weighted sum of 2×⋯×2 configuration counts. We show that for a randomly translated lattice, no asymptotically unbiased estimator of this type exists in dimension greater than or equal to three, while for stationary isotropic lattices, asymptotically unbiased estimators are plenty. Both results follow from a general formula that we state and prove, describing the asymptotic behavior of hit-or-miss transforms of r-regular sets.