Anisotropic geometric diffusion in image and image-sequence processing
In this thesis nonlinear anisotropic geometric diffusion methods for the processing of static images and image-sequences are discussed. The models depend only on the morphology of the underlying data, and thus they are invariant under monotone transformations of the gray values. The evolution, which depends on the principal curvatures and the principal directions of curvature of level-sets, is capable of preserving important features of codimension 2, i.e. corners and edges of the level-sets. For the processing of image-sequences an anisotropic behavior in direction of the apparent motion of the level-sets is prescribed.
Important for the processing of noisy images is a suitable regularization of the data. Different approaches are discussed and the results of a local projection approach onto a polynomial space is compared with the convolution with kernels having compact support.
For the nonlinear problems the existence of viscosity solutions is shown by using a result of Giga et. al. for the linearized problems together with a fixed point argument.
The discretization of the models is done using a semi-implicit time-discretization together with finite elements on regular quadrilateral and hexahedral grids. Furthermore for the processing of image-sequences an operator splitting scheme is derived, which enables to solve this high dimensional problem with moderate effort.