The analysis of the geometry of objects is a fundamental property with important interpretations in most fields of natural science. As the fields of bioimaging, biology and pathology evolve, so should the computational and statistical methods which we use for the correction of imaging artifacts and for measuring the stochastic variations in the shape of geometric objects of interest.
In this Ph.D. thesis, we present novel methods either using or measuring the geometry of objects with several applications in biology and pathology.
Firstly, we present a novel correction method for restoring drifted FIB-SEM volumes of neuronal data by using the vesicle, a small spherical object, as a stochastic model for translational correction of an image stack.
Secondly, we present measures and statistics to assess if and how the shape of an object might be dependent on the shape and spatial distance to some reference object. We derive edge correction terms and assess the inverse problem by deriving the equivalence class under the measure in a simplified example.
Third and finally, we present a measure and statistic for point patterns which has migrated from some common source point, but where the knowledge of the point locations is sparse, limited to parallel planes. By assuming a uniform distribution of points across spherical shells from the source point, we estimate the dense point statistic using the available information.