Monotonic Sampling of a Continuous Closed Curve with Respect to Its Gauss Digitization: Application to Length Estimation

In many applications of geometric processing, the border of a continuous shape and of its digitization (i.e., its pixelated representation) should be matched. Assuming that the continuous-shape boundary is locally turn bounded , we prove that there exists a mapping between the boundary of the digitization and the one of the continuous shape such that these boundaries are traveled together in a cyclic order manner. Then, we use this mapping to prove the multigrid convergence of perimeter estimators that are based on polygons inscribed in the digitization. Furthermore, convergence speed is given for this class of estimators. If, moreover, the continuous curves also have a Lipschitz turn, an explicit error bound is calculated.