In this paper we introduce a new curvature estimator based on global optimisation. This method called Global Min-Curvature ex- ploits the geometric properties of digital contours by using local bounds on tangent directions defined by the maximal digital straight segments. The estimator is adapted to noisy contours by replacing maximal seg- ments with maximal blurred digital straight segments. Experimentations on perfect and damaged digital contours are performed and in both cases, comparisons with other existing methods are presented.