Auxiliary Variables for Deformable Models

We present a new mathematical formulation for curve and surface reconstruction algorithms by introduction of auxiliary variables. For deformable models and templates, two-step iterative algorithms have been often used where, at each iteration, the model is first locally deformed according to the potential data attraction and then globally smoothed. We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a two-variables energy. This permits to transform an implicit data constraint defined by a non convex potential into an explicit convex reconstruction problem. We show some mathematical properties and results on this new auxiliary problem, in particular when the potential is a function of the distance to the closest feature point. We then illustrate our approach for some deformable models and templates and image restoration.