Variational principles , Surface Evolution , PDE ' s , level set methods and the Stereo

1 Variational principles, Surface Evolution, PDE's, level set methods and the Stereo Problem Olivier Faugeras and Renaud Keriven Abstract We present a novel geometric approach for solving the stereo problem for an arbitrary number of images (greater than or equal to 2). It is based upon the de nition of a variational principle that must be satis ed by the surfaces of the objects in the scene and their images. The Euler-Lagrange equations which are deduced from the variational principle provide a set of PDE's which are used to deform an initial set of surfaces which then move towards the objects to be detected. The level set implementation of these PDE's potentially provides an e cient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformation, i.e. to deal with multiple objects. Results of an implementation of our theory also dealing with occlusion and vibility are presented on synthetic and real images.