  The surface area of a set which is only observed as a binary pixel image is
often estimated by a weighted sum of pixel configurations counts. In this paper
we examine these estimators in a design based setting -- we assume that the
observed set is shifted uniformly randomly. Bounds for the difference between
the essential supremum and the essential infimum of such an estimator are
derived, which imply that the variance is in $O(t^2)$ as the lattice distance
$t$ tends to zero. In particular, it is asymptotically neglectable compared to
the bias. A simulation study shows that the theoretically derived convergence
order is optimal in general, but further improvements are possible in special
cases.
