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Many methods have been proposed to estimate differential geometric quantities like curvature on discrete data. A common characteristics is that they require (at least) one user-given scale parameter, that smooths data to take care of both the sampling rate and possible perturbations. Digital shapes are specific discrete approximation of Euclidean shapes, which come from their digitization at a given grid step. They are thus subsets of the digital plane Z^d. A digital geometric estimator is called multigrid convergent whenever the estimated quantity tends towards the expected geometric quantity as the grid step gets finer and finer. The problem is then: can we define curvature estimators that are multigrid convergent without such user-given parameter ? If so, what speed of convergence can we achieve? We present here three digital curvature estimators that aim at this objective: a first one based on maximal digital circular arc, a second one using a global optimisation procedure, a third one that is adigital counterpart to integral invariants and that works on 2D and 3D shapes.