Abstract : This thesis takes place in the context of image matching within the framework of large deformation diffeomorphisms. With important applications to medical imaging and computational anatomy, this approach uses the action of diffeomorphisms groups in order to classify images. One of the first issue to deal with is to compute the distance between objects on which can act the group of diffeomorphisms.
The case of discontinuous images was very partially understood. The first part of this thesis is devoted to fully tackle the case of discontinuous images in any dimension. Namely the images are assumed to be functions of bounded variations. We have provided technical tools to deal with discontinuous images within the diffeomorphism framework. The first application developed is a Hamiltonian formulation of the geodesic equations for a new model including a change of contrast in the images which is represented by an action of a diffeomorphism on the values of the level lines of the image. The second one is an extension of the metamorphosis framework developed by A.Trouvé and L.Younes to SBV functions, which points out that the geometry of such spaces is much more complicated than the one with smooth functions.
The second part of the thesis takes place in the probabilistic side of the field. Taking advantage of the Hamiltonian formulation, we aim to study stochastic perturbations of the geodesic equations. From a physical point of view, the perturbation we consider affects the forces on the particles and not the speed of the particles. In some sense, this model could be an interesting dynamical model for growth of shapes or at least random evolution of shapes.
We have proven that the solutions of the system in the case of landmarks are non exploding and that there exists a SDE in infinite dimension on a suitable Hilbert space (actually a sort of Besov or Sobolev space on a Haar basis) which extends the landmark case. In infinite dimension, the solutions are also defined for all times. Moreover an important convergence result of the landmark case to the infinite dimensional case is proven. Finally, let us precise that the structure of the noise is general enough to account for correlation between points of the curve in the noise.