ISBN 978-3-8439-3245-5

978-3-8439-3245-5, Reihe Mathematik

Jan Henrik Fitschen
Variational Models in Image Processing with Applications in the Materials Sciences

237 Seiten, Dissertation Technische Universität Kaiserslautern (2017), Softcover, A5

Zusammenfassung / Abstract

In this thesis, we propose various variational models to solve problems appearing in the materials sciences.

First, we deal with focused ion beam tomography, which is an imaging technique used to acquire three-dimensional images of specimens. The first problem we consider for such data is the removal of curtaining effects. These artifacts can be modeled as an additive composition of stripes and a laminar part. In order to remove them, we employ different directional regularization terms. Another problem we have to deal with are intensity inhomogeneities. Indeed, they make it often impossible to segment the images based on the gray values. To overcome this difficulty, we suggest a biconvex variational model that combines a total variation approach for segmentation with a multiplicative model for the intensity inhomogeneities.

Color image processing is another area where it is often desirable to correct wrong intensities in a preprocessing step. We consider the problem that we are given a badly exposed color image and want to compute an enhanced image possessing a given target intensity and incorporating the hue of the original image.

Next, we turn our attention to deformation experiments and present an optical flow model for strain analysis of a sequence of microstructural images. The crucial part of this model is the prior, which is chosen as the second order total generalized variation of the displacement field. This prior splits the strain into a smooth and a non-smooth part. The latter reflects the local damage, for example cracks, in which we are particularly interested.

Finally, we are interested in electron backscatter diffraction data as a special case of manifold-valued imaging. First, we suggest an iterative multiplicative filtering algorithm to partition manifold-valued images. Then, we focus on variational models and provide some results on the complexity of the minimization problems arising when transferring variational methods to manifolds. Further, we introduce intrinsic and extrinsic models that generalize the infimal convolution and total generalized variation to manifolds. This is the first attempt to generalize the infimal convolution of first and second order differences to manifold-valued images. For all problems, we propose models, analyze them with respect to existence and uniqueness of solutions, implement suitable algorithms, and discuss their convergence. The topics are completed by extensive experimental evaluations.