Diskretisierung und Approximation von Bildern und morphologischen Bildoperatoren

In the present thesis various aspects concerning discretisation and approximation of images and morphological image operators are systematically investigated. Binary images in the context of morphological image operators, which is considerably presented in the monograph Morphological Image Operators by Henk Heijmans, are modelled as subsets of a basic set. Thus the starting point is the discretisation of this basic set which will be assumed to be a locally compact Hausdorff space. A method to discretize such a space by locally finite ordered sets (that can also be viewed as T0-spaces) is presented for suitable spaces including paracompact thus metric spaces. There the notion of the considered space being approximated by its discretisations is given by the fact that it is a retract of the inverse limit of its system of discretisations. A way to discretize and approximate binary images which are modelled as closed subsets of the basic set is then derived from the mentioned method. That in turn leads to a way to discretize and approximate Scott-continuous image operators for binary images. Greyscale images in the context of morphological image operators are modelled as mappings from the basic set to a complete lattice of grey-values. In case of a strongly admissible grey-value lattice there are known results about lifting binary image operators to greyscale image operators by means of threshold-sets. However, little is yet known about strongly admissible lattices in general. They are therefore studied more precisely in the present work where in particular a characterization for the finite case is given. This leads to the assumption that these lattices are nearly chains. It is further shown that the terms “admissible lattice” and “completely distributive lattice” agree and that completely distributive lattices are more appropriate as they allow extensions of the known results. Next, threshold-sets are used in this work to derive a discretisation and approximation of greyscale images from the already mentioned discretisation of binary images. Again a discretisation and approximation of Scott-continuous operators can be achieved, this time for greyscale images. Moreover a discretisation and approximation of greyscale images is shown in the case that the grey-value lattice is discretized instead of the basic set. Finally, the possibility to model greyscale images by mass distributions is considered. At this, continuous valuations are used that are known from domain theory. It is shown that a discretisation and approximation of continuous valuations can