Medical imaging is still mainly focusing on visualization and data communication. However, applications of digital image analysis methods in biomedical research are also increasing, due to demands and developments in hardware, image analysis software, and methodology. Shape simplification procedures are already important modules of software packages. The literature offers a large number of papers about algorithms, which aim at deforming images into topologically equivalent images; the latter ones should represent the shape of complex objects in a simplified form. This thesis reviews and extends the diversity of approaches published in this area with respect to properties of algorithms and characterizations of simple points. It contributes in particular with new theoretical results. Topologic thinning methods deliver digital curves (skeletons), which are used to describe objects in digital images. The thesis shows that different applications require different constraints and adjustments of general thinning or curve analysis procedures. It also studies the effect of increased grid resolution (e.g., trying to utilize progress in hardware) for the potential accuracy of measurements based on skeletons. The thesis illustrates results by contributing to one particular application (i.e., analysis of astrocytes in human brain tissue), for example by verifying the efficiency of calculated features for classification.