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Dr. Behrend Heeren

Address: Institut für Numerische Simulation
Rheinische Friedrich-Wilhelms-Universität Bonn
Endenicher Allee 60
53115 Bonn
Germany
Room: Z2.065
Phone: +49 (0)228 - 73-3334
Fax: +49 (0)228 - 73-9015
eMail:

 

Research Interests

  • Shape Space Analysis

    • Discrete geodesic calculus in the space of thin shells
    • Time-dependent statistical analysis in the space of 3D shapes
  • Numerical Simulations with Subdivision Surfaces

Current Projects

BMBF competence network: CROP.SENSe.net: Networking sensor technology R&D for crop breeding and management
Project D4: 4D structural analysis of the sugar beet geometry

Our goal is developing techniques for model-based 3D reconstruction and classification of storage root geometry from tomographic data, i.e. data obtained from laser scans and MRI. We aim in comparing different cultivars subject to different soil, management and environmental conditions. Furthermore we investigate the temporal variation of sugar beet growth by applying and generalizing regression methods to the space of beets. Our tools are robust and non-supervised, that means they improve previous and-measured methods and corresponding empirical scales. Furthermore, recent results suggest that our methods are capable to statistically separate genotypic from environmental features. Hence the proposed techniques are relevant and effective tools to optimize plant breeding.

DFG SFB 1060: The Mathematics of Emergent Effects
Project C05: Discrete Riemannian calculus on shape space

The theory of shape spaces in vision is linked both to concepts from geometry and from physics. The flow of diffeomorphism approach and the optimal transportation approach are prominent examples for Riemannian metric structures on the space of shapes, intensively studied in the last decade. In general, the numerical realization of the underlying infinite dimensional Riemannian calculus poses enormous computational challenges. In this project we propose suitable time and space discrete approximations. They will be based on optimal matching deformations, which are significantly cheaper to compute but by construction non-Riemannian. We will also study abstract concepts of transportation between metric measure spaces with particular focus on spaces consisting only of a fixed number of points. In the time discrete calculus a discrete path energy is defined as the sum of pairwise deformation energies along a discrete path. Using a variational approach one can deduce from this step by step a discrete logarithm, a discrete exponential map, a discrete parallel transport, a discrete Levi-Civita connection, and finally a discrete Riemannian curvature tensor. This concept will be applied and analyzed in particular for the flow of diffeomorphism and the optimal transportation approach. Furthermore, with respect to a spatially discrete covering of shape space, the approach of deformation based shape dissimilarities will be combined with the diffusion map paradigm expedited by Coifman and coworkers to introduce physically sound and computationally efficient approximations of metric structures on shape spaces. Overall, we aim at combining methods from geometry, stochastic analysis and numerics to advance theory in computer vision and explore new applications with efficient computational tools.

FWF NFN S117: Geometry + Simulation
Project 5: Geodesic Paths in Shape Space

This project will provide robust and flexible tools for the quantitative analysis of shapes in the interplay between applied geometry and numerical simulation. Here, shapes S are curved surfaces which physically represent shell-type geometries, boundary geometries of volumetric physical objects, or material interfaces. In isogeometric analysis one faces a wide range of low- and moderate-dimensional descriptions of complicated and realistic geometries. Thus, the geometric description of shapes will be flexible, ranging from simple piecewise triangular to subdivision-generated spline type surface representations and from explicitly meshed volumes to descriptions via level set or characteristic functions. The fundamental tool for a quantitative shape analysis is the computation of a distance between shapes SA and SB as objects in a high- or even 1-dimensional Riemannian shape space. Hence, we aim at developing robust models and fast algorithms to compute geodesic paths in shape space. Both for boundary or interface contours and for shell-type surfaces the Riemannian metric will correspond to physical dissipation either of a viscous fluid filling the object volume or due to a visco-plastic behavior of the shells. The key tool of the proposed approach is a coarse time discretization combined with a variational scheme to minimize the underlying least action functional. For purposes of a geometric analysis not only the resulting value for the distance is of interest. In fact the geodesic paths are natural one parameter families of shapes on which physical simulations and PDE computations can be performed. In case of shapes being represented by shell type surfaces we will apply different approaches: subdivision surfaces generated from coarse surface triangulations and subdivision-based discrete function spaces as a modeling paradigm associated with spline models in CAD, application of methods from discrete exterior calculus on discrete surfaces to derive robust and geometrically consistent discrete shell models, the approximation of curvature-based functionals in shell models via variational principles on general classes of surface meshes. In the case of shapes being boundary contours or material interfaces of volumetric objects we aim at representing shapes via characteristic functions, working in the context of variational methods in BV. We will compare this approach with corresponding models based on level set or parametric descriptions. Here, recent results on global minimization of convex functionals on BV and coarse to fine multi-scale relaxation methodologies will be taken into account. For the numerical discretization we aim at using finite elements.

Short Curriculum Vitae

Jan 2017 —Post-Doctoral Researcher at the Institute for Numerical Simulation
Jan 2012 — Dec 2016PhD student at the Institute for Numerical Simulation
Thesis on "Numerical Methods in Shape Spaces and Optimal Branching Patterns", defended in Feb 2017
Supervisor: Martin Rumpf
Apr 2012 — May 2012 Visiting student at the California Institute of Technology, Pasadena, USA
Supervisor: Peter Schröder
Apr 2008 — Dec 2011 Student of Mathematics and Psychology at University of Bonn
Diplom in Mathematics at the Institute for Numerical Simulation
on "Geodätische im Raum von Schalenformen"
Supervisor: Martin Rumpf
Mar 2008 — Mar 2011 Scholarship of the Studienstiftung des deutschen Volkes
Jul 2009 — Dec 2009 Student of Mathematics and Oceanography at the University of New South Wales
and the Climate Change Research Centre, Sydney, Australia
Oct 2005 — Mar 2008 Student of Mathematics and Psychology at the University of Münster

Publications of Dr. Behrend Heeren:

[1] S. Markett, M. Reuter, B. Heeren, B. Lachmann, B. Weber, and C. Montag. Working memory capacity and the functional connectome - insights from resting-state fMRI and voxelwise eigenvector centrality mapping. Brain Imaging and Behavior, 2017. accepted.
bib ]
[2] P. W. Dondl, B. Heeren, and M. Rumpf. Optimization of the branching pattern in coherent phase transitions. C. R. Math. Acad. Sci. Paris, 354(6):639-644, 2016.
bib | arXiv | Abstract ]
[3] B. Heeren. Numerical Methods in Shape Spaces and Optimal Branching Patterns. PhD thesis, University of Bonn, 2016.
bib ]
[4] B. Heeren, M. Rumpf, P. Schröder, M. Wardetzky, and B. Wirth. Splines in the space of shells. Comput. Graph. Forum, 35(5):111-120, 2016.
bib | .pdf 1 ]
[5] S. Markett, C. Montag, B. Heeren, R. Sariyska, B. Lachmann, B. Weber, and M. Reuter. Voxelwise eigenvector centrality mapping of the human functional connectome reveals an influence of the catechol-o-methyltransferase val158met polymorphism on the default mode and somatomotor network. Brain Structure and Function, 221:2755-2765, 2016.
bib | DOI ]
[6] C. Zhang, B. Heeren, M. Rumpf, and W. Smith. Shell PCA: statistical shape modelling in shell space. In Proc. of IEEE International Conference on Computer Vision, 2015.
bib | .pdf 1 ]
[7] B. Heeren, M. Rumpf, P. Schröder, M. Wardetzky, and B. Wirth. Exploring the geometry of the space of shells. Comput. Graph. Forum, 33(5):247-256, 2014.
bib | .pdf 1 ]
[8] B. Berkels, P. T. Fletcher, B. Heeren, M. Rumpf, and B. Wirth. Discrete geodesic regression in shape space. In Proc. of International Conference on Energy Minimization Methods in Computer Vision and Pattern Recognition, volume 8081 of Lecture Notes in Computer Science, pages 108-122. Springer, 2013.
bib | DOI | .pdf 1 ]
[9] B. Heeren, M. Rumpf, M. Wardetzky, and B. Wirth. Time-discrete geodesics in the space of shells. Comput. Graph. Forum, 31(5):1755-1764, 2012.
bib | DOI | .pdf 1 ]
[10] B. Heeren. Geodätische im Raum von Schalenformen. diploma thesis, 2011.
bib | .pdf 1 ]

Talks

June 21st, 2016 Symposium on Geometry Processing in Berlin, Germany
Splines in the Space of Shells
May 12th, 2016 SIAM Conference on Mathematical Aspects of Material Sciences, Philadelphia, USA
Time-discrete geodesic calculus in the space of shells
January 28th, 2016 Oberwolfach Workshop on Mathematical Imaging and Surface Processing, Oberwolfach, Germany
Riemannian Splines in the Space of Shells
July 11th, 2014 Symposium on Geometry Processing in Cardiff, Wales
Exploring the Geometry in the Space of Shells
August 19th, 2013 EMMCVPR Conference in Lund, Sweden
Discrete Geodesic Regression in Shape Space
July 18th, 2013 European Conference on Computational Optimization 2013 in Chemnitz, Germny
Discrete Geodesic Calculus in the Space of Thin Shells
March 20th, 2013 GAMM Annual Meeting in Novi Sad, Serbia
Discrete Geodesic Calculus in the Space of Thin Shells
Sep 11th, 2012 Conference on Scientific Computing Algoritmy 2012 in Podbanske, Slovakia
Time-discrete Geodesics in the Space of Shells
July 18th, 2012 Symposium on Geometry Processing in Tallinn, Estonia
Time-discrete Geodesics in the Space of Shells
March 29th, 2012 Workshop Navigating the Space of Surfaces in Bonn, Germany
Time-discrete Geodesics in the Space of Shells
March 28th, 2012 GAMM Annual Meeting in Darmstadt, Germany
Time-discrete Geodesics in the Space of Shells

Teaching

Winter term 2017/18: Seminar on Shape spaces and shape representations
Summer term 2017: Lecture course on Geometry Processing and Discrete Shells
Assistant for Algorithmische Mathematik II given by Prof. Dr. A. Eberle and Prof. Dr. A. Umschmajew.
Winter term 2016/17: Assistant for Ingenieurmathematik III given by Dr. Martin Lenz
Summer term 2016: Assistant for Ingenieurmathematik II given by Dr. Martin Lenz
Winter term 2015/16: Computer lab for Einführung in die Grundlagen der Numerik given by Prof. Martin Rumpf
Summer term 2015: Tutorial and computer lab for Algorithmische Mathematik II given by Prof. Martin Rumpf
Winter term 2014/15: Assistant for Ingenieurmathematik I given by Dr. Antje Kiesel
Summer term 2014: Assistant for Ingenieurmathematik II given by Dr. Martin Lenz
Winter term 2013/14: Organizing Seminar on Modeling and Simulation with PDEs given by Prof. Martin Rumpf and Dr. Peter Hornung
Summer term 2013: Computer lab Numerical Methods for Thin Elastic Sheets with Ricardo Perl
Winter term 2012/13: Assistant for Numerical Algorithms with Benedict Geihe