Auxiliary variables and two-step iterative algorithms in computer vision problems

We present a new mathematical formulation of some curve and surface reconstruction algorithms by the introduction of auxiliary variables. For deformable models and templates, the extraction of a shape is obtained through the minimization of an energy composed of an internal regularization term (not necessary in the case of parametric models) and an external attraction potential. 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 (or fitted in the parametric case). We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a two-variables energy. The first variable corresponds to the original model we are looking for, while the second variable represents an auxiliary shape close to the first one. This permits to transform an implicit data constraint defined by a non convex potential into an explicit convex reconstruction problem. This approach is much simpler since each iteration is composed of two simple to solve steps. Our formulation permits a more precise setting of parameters in the iterative scheme to ensure convergence to a minimum. 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.