# Variances of surface area estimators based on pixel configuration counts

## TL;DR

This paper analyzes the variance of surface area estimators based on pixel configuration counts in binary images, showing that variance diminishes as lattice spacing decreases and is asymptotically negligible compared to bias.

## Contribution

It provides bounds on the variance of these estimators in a random shift setting and demonstrates the asymptotic behavior as lattice spacing tends to zero.

## Key findings

- Variance is in O(t^2) as lattice distance t tends to zero.
- Variance is asymptotically negligible compared to bias.
- Simulation confirms the theoretical convergence order is generally optimal.

## Abstract

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.