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The automatic detection of noisy or damaged parts along digital contours is a difficult problem since it is hard to distinguish between information and perturbation without further a priori hypotheses. However, solving this issue has a great impact on numerous applications, including image segmentation, geometric estimators, contour reconstruction, shape matching, or image edition. We propose an original strategy to detect what the relevant scales are at which each point of the digital contours should be considered. It relies on theoretical results of asymptotic discrete geometry. A direct consequence is the automatic detection of the noisy or damaged parts of the contour, together with its quantitative evaluation (or noise level). Apart from a given maximal observation scale, the proposed approach does not require any parameter tuning and is easy to implement. We demonstrate its effectiveness on several datasets. We present different direct applications of this local measure to contour smoothing and geometric estimators whose algorithms initially required a noise/scale parameter to tune: They show the pertinence of the proposed measure for digital shape analysis and reconstruction.