Image Registration, Optical Flow and Local Rigidity

We address the theoretical problems of optical flow estimation and image registration in a multi-scale framework in any dimension. Much work has been done based on the minimization of a distance between a first image and a second image after applying deformation or motion field. Usually no justification is given about convergence of the algorithm used. We start by showing, in the translation case, that convergence to the global minimum is made easier by applying a low pass filter to the images hence making the energy convex enough. In order to keep convergence to the global minimum in the general case, we introduce a local rigidity hypothesis on the unknown deformation. We then deduce a new natural motion constraint equation (MCE) at each scale using the Dirichlet low pass operator. This transforms the problem to solving the energy minimization in a finite dimensional subspace of approximation obtained through Fourier Decomposition. This allows us to derive sufficient conditions for convergence of a new multi-scale and iterative motion estimation/registration scheme towards a global minimum of the usual nonlinear energy instead of a local minimum as did all previous methods. Although some of the sufficient conditions cannot always be fulfilled because of the absence of the necessary a priori knowledge on the motion, we use an implicit approach. We illustrate our method by showing results on synthetic and real examples in dimension 1 (signal matching, Stereo) and 2 (Motion, Registration, Morphing), including large deformation experiments.