Image Registration, Optical Flow, and Local Rigidity

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We address the theoretical problems of optical flow estimation and image registration in a multi-scale framework in any dimension. 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 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. We then use an implicit numerical approach. We illustrate our method on synthetic and real examples (Motion, Registration, Morphing).