Approximating boundaries using data recorded on a regular grid induces discrete rounding errors in both vertical and horizontal directions. In cases where grid points exhibit at least some degree of randomness, an extensive theory has been developed for local-polynomial boundary estimators. It is inapplicable to regular grids, however. In this paper we impose strict regularity of the grid and describe the performance of local linear estimators in this context. Unlike the case of classical curve estimation problems, pointwise convergence rates vary erratically along the boundary, depending on number-theoretic properties of the boundary’s slope. However, average convergence rates, expressed in the $L_1$ metric, are much less susceptible to fluctuation. We derive theoretical bounds to performance, coming within no more than a logarithmic factor of the optimal convergence rate.
"On global performance of approximations to smooth curves using gridded data." Ann. Statist. 26 (6) 2206 - 2217, December 1998. https://doi.org/10.1214/aos/1024691467