Prof. Dr. Konrad Polthier

Projects as a project leader

• CH18

  Boundary-Sensitive Hodge Decompositions

  Prof. Dr. Konrad Polthier

  Project heads: Prof. Dr. Konrad Polthier
  Project members: Dr. Faniry Razafindrazaka
  Duration: 01.06.2017 - 31.12.2019
  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: Dr. 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/