Non-local estimators: A new class of multigrid convergent length estimators

An interesting property for curve length digital estimators is the convergence toward the continuous length and the associate convergence speed when the grid spacing tends to zero. On the one hand, DSS based estimators have been proved to converge but only under some convexity and smoothness or polygonal assumptions. On the other hand, we have introduced in a previous paper the sparse estimators and we proved their convergence for Lipschitz functions without convexity assumption. Here, we introduce a wider class of estimators, the non-local estimators, that intends to gather sparse estimators and DSS based estimators. We prove their convergence and give an error upper bound for a large class of functions.