# Research

My field of research is computational geometry and hence I am interested in both, geometry and algorithms/data structures. So far, my primary research focus was on skeleton structures, in particular Voronoi diagrams and straight skeletons. In the following you will find a reversed chronological list of my research activities:

• Inverse problems

Instead of computing the Voronoi diagram or the straight skeleton, we ask the reversed question: Given a tessellation of the plane into polygonal cells - is this a straight skeleton or a Voronoi diagram of some input? And if yes, can we (efficiently) reconstruct the input? And how can we (efficiently) determine the set of all possible inputs, in case that the reconstruction is not unique? These questions are partly motivated by biology but also have theoretical and practical implications in computational geometry.

• Straight skeletons and motorcycle graphs

The straight skeleton (blue) of a straight-line figure (black) is a geometric structure that was introduced to computational geometry in the mid 90s. Similar to Voronoi diagrams, it is used in a multitude of applications in science and industry, such as computing mitered offset curves (red), automatic generation of tool paths in NC machining, solving mathematical origami problems or reconstructing 3D bodies in medical imaging. A motorcycle graph is a geometric structure that is strongly related to the straight skeleton in different ways. More...

• Watermarking vector graphics

Watermarking refers to a process of embedding imperceptible statistical features into a digital asset (e.g., music, movies, vector graphics) that can be detected by those who possess a specific secret key. This enables the owner of a watermarked digial asset, such as geogaphic maps or a CAD drawing, to prove his ownership or retrace which collaborator illegitimately passed on the asset. More...

• Generalized Voronoi diagrams

Voronoi diagrams and their generalizations are key objects in computational geometry and have been thorougly investigated in the past four decades. Voronoi diagrams of straight-line segments are heavily used in science and industry and a few Voronoi-implementations are available that can cope with real-world datasets. For Voronoi diagrams of points, straight-line segments and circular arcs, however, ArcVroni is to our knowledge still the only implementation that is able to run on an industrial-strength level. More...