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In this paper, we propose a robust and efficient Lagrangian approach, which we call Delaunay deformable models, for modeling moving surfaces undergoing large deformations and topology changes. Our work uses the concept of restricted Delaunay triangulation, borrowed from computational geometry. In our approach, the interface is represented by a triangular mesh embedded in the Delaunay tetrahedralization of interface points. The mesh is iteratively updated by computing the restricted Delaunay triangulation of the deformed objects. Our method has many advantages over popular Eulerian techniques such as the level set method and over hybrid Eulerian-Lagrangian techniques such as the particle level set method: localization accuracy, adaptive resolution, ability to track properties associated to the interface, seamless handling of triple junctions. Our work brings a rigorous and efficient alternative to existing topology-adaptive mesh techniques such as T-snakes.