A Variational Approach to Nonrigid Morphological Image Registration

A variational method to non rigid registration of multi-modal image data is presented. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. The aim is to correlate the morphologies of a template and a reference image under the deformation. Mathematically, the morphology of images can be described by the entity of level sets of the image and hence by its Gauss map. A class of morphological matching functionals is presented which measure the defect of the template Gauss map in the deformed state with respect to the deformed Gauss map of the reference image. The problem is regularized by considering a nonlinear elastic regularization energy. Existence of homeomorphic, minimizing deformation is proved under assumptions on the class of admissible deformations. With respect to actual medical applications suitable generalizations of the matching energies and the boundary conditions are presented. Concerning the robust implementation of the approach the problem is embedded in a multi-scale context. A discretization based on multi-linear finite elements is discussed and first numerical results are presented.