ECMath CH18 - Boundary-Sensitive Hodge Decompositions

Principal Investigator:
Research Team:


Partner: 

  • Leo Goubergrits (Charité Berlin)
Funding:

Funded by Einstein Center for Mathematics Berlin - ECMath

Term:
Jun 01, 2017 — Dec 31, 2018
Boundary-Sensitive Hodge Decompositions

Boundary-Sensitive Hodge Decompositions

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 flow 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.

Publications

  1. Konstantin Poelke & Konrad Polthier (2018). Topology-Based Methods in Visualization. Springer (In review)
  2. F. H. Razafindrazaka, P. Yevtushenko, K. Poelke, K. Polthier & L. Goubergrits (2018). Hodge Decomposition of the wall shear stress vector fields characterizing biological flows. Journal of the Royal Society Interface. (In review)