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We present an algorithm for tracking moving objects using intrinsic minimal surfaces which handles particularly well the presence of severe and total occlusions even in the presence of weak object boundaries. We adopt an edge based approach and find the segmentation as a minimal surface in 3D space-time, the metric being dictated by the image gradient. Object boundaries are represented implicitly as the level set of a higher dimensional function, and no particular object model is assumed. We also avoid explicit estimation of a dynamic model since the problem is regarded as one of static energy minimization. A set of interior points provided by the user is used to constrain the optimization, which basically corresponds to selecting the object of interest within the video sequence. The constraints are such that they restrict the resulting surface to be star-shaped in the 3D spatio-temporal space. We present some challenging examples that show the robustness of the technique.