In this paper, we introduce a natural generalization of non-uniform systematic sampling to two dimensions. Under optimal auxiliary information about the function of interest, this design yields an estimator with zero variance. In a simulation study imitating sampling situations in microscopy, the 2D non-uniform systematic designs show similar efficiency as proportional-to-size sampling with replacement. An exception is area estimation where the 2D non-uniform systematic designs are superiour in a number of cases considered.

Keywords: Cavalieri estimator, efficiency, Horvitz-Thompson estimator, microscopy, systematic sampling, 2D sampling