Prof. Dr. Konrad Polthier

Executive Board Member

FU Berlin Institut für Mathematik
Arnimallee 6
14195 Berlin
+49 (0) 30 +49(30)838-75871
Konrad.Polthier@fu-berlin.de

Scientist in Charge for Application Area Geometric Design and Visualization



Research focus

Mathematical geometry processing; discrete differential geometry and mathematical visualization

Projects as a project leader

  • CH18

    Boundary-Sensitive Hodge Decompositions

    Prof. Dr. Konrad Polthier

    Project heads: Prof. Dr. Konrad Polthier
    Project members: -
    Duration: 01.06.2017 - 31.12.2018
    Status: running
    Located at: Freie Universität Berlin

    Description

    Based on novel results for smooth and discrete Hodge-type decompositions on manifolds with boundary, this project aims to incorporate discrete boundary-sensitive Hodge decompositions as a central tool for the analysis of blood flow and parameterization of blood vessels. These decompositions provide the following two substantial improvements over existing methods: first, they are able to distinguish harmonic blood flow arising from boundary in- and out ow from harmonic circulations induced by the interior topology of the geometry. Second, they guarantee a theoretically-sound linkage of certain fields with controlled boundary behaviour to cohomological quantities of the geometry, which is the essential and still missing ingredient for the creation of periods to ensure global matching of parameter lines in modern parameterization techniques.

    http://www.mi.fu-berlin.de/en/math/groups/ag-geom/projects/ch18/index.html
  • GV-AP2

    Integrating discrete geometries and finite element spaces

    Prof. Dr. Konrad Polthier

    Project heads: Prof. Dr. Konrad Polthier
    Project members: -
    Duration: 01.07.2012 - 30.06.2016
    Status: completed
    Located at: Freie Universität Berlin

    Description

    Finite element methods are in every day use in engineering and modelling. The main idea with finite elements is to discretize objects such as machine parts or architectural elements in order to then simulate the movement and behaviour of these objects via discrete computations. Project A04 aims to link experiences from those applications of scientific computing with ideas from discrete geometry to improve the integration of technologies.

    http://www.discretization.de/en/projects/A04/
  • GV-AP16

    Computational and structural aspects of point set surfaces

    Prof. Dr. Konrad Polthier

    Project heads: Prof. Dr. Konrad Polthier
    Project members: Konstantin Poelke / M.Sc. Martin Skrodzki
    Duration: 01.07.2016 - 30.06.2020
    Status: running
    Located at: Freie Universität Berlin

    Description

    In the project “Computational and structural aspects of point set surfaces”, we will develop discrete differential geometric representations for point set surfaces and effective computational algorithms. Instead of first reconstructing a triangle based mesh, our operators act directly on the point set data. The concepts will have contact to meshless methods and ansatz spaces of radial basis functions. As proof of concept of our theoretical investigations we will transfer and implement key algorithms from surface processing, for example, for surface parametrization and for feature aware mesh filtering on point set surfaces. Point set surfaces have a more than 15 year long history in geometry processing and computer graphics as they naturally arise in 3D-data acquisition processes. A guiding principle of these algorithms is the direct processing of raw scanning data without prior meshing – a principle that has a long-established history in classical numerical computations. However, their usage mostly restricts to full dimensional domains embedded in R2 or R3 and a thorough investigation of a differential geometric representation of point set surfaces and their properties is not available. Inspired by the notion of manifolds, we will develop new concepts for meshless charts and atlases. These will be used to implement higher order differential operators including curvature descriptors. On this solid basis of meshless differential operators, we will develop novel algorithms for important geometry processing tasks, such as feature recognition, filtering operations, and surface parameterization.

    http://www.discretization.de/en/projects/C05/