Regularization of Discrete Contour by Willmore Energy

We propose a novel approach to reconstruct shapes from digital data. Contrarily to most methods, reconstructed shapes are smooth with a well-defined curvature field and have the same digitization as the input data: the range of application we have in mind is especially post-processing to image segmentation where labelled regions are digital objects. For this purpose, we introduce three new algorithms to regularize digital contours based on the minimization of Willmore energy: our first algorithm is based on tools coming from discrete geometry, the second is related to convex geometry while the third approach is a constrained phase field minimization. The three algorithms are described in details and the convergence of the phase field approach is investigated. We present a comparative evaluation of all three methods, in terms of the accuracy of curvature estimators and computation time.