On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409-422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. 26-33, 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Almgren et al. (SIAM Journal on Control and Optimization 31(2):387-438, 1993) and Luckhaus and Sturzenhecker (Calculus of Variations and Partial Differential Equations 3(2):253-271, 1995), and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term.