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Geometric Reconstruction in Refraction- and Diffraction-based Tomography



Principal Investigators: Prof. Dr. Peter Gritzmann, Dr. Andreas Alpers
PhD Students: Stefan König, Carl Georg Heise
Funding: Deutsche Forschungsgemeinschaft , GR 993/10-1, AL 1431/1-1, GR 993/10-2
1st Funding Period: May 2011 - April 2013
2nd Funding Period: May 2013 - April 2016
Collaborators: Prof. Dr. Franz Pfeiffer (TU München, Germany), Prof. Henning Friis Poulsen (DTU, Denmark) , Prof. Gabor T. Herman (CUNY, USA) , Prof. Rafal Dunin-Borkowski (Research Centre Jülich, Germany) 



Computerized tomography (CT), the process of obtaining the density distribution within a specimen from multiple X-ray projections, has revolutionized diagnostic radiology over the past three decades. While standard CT is based on the principle of absorption, we focus on three novel experimental techniques that are based on refraction and (crystal) diffraction. The goal is to further enhance the resolution of previously largely invisible parts of the objects. From the technical side, these methods were developed by and are still at the research focus of our project partners. They are known as tomography by differential phase contrast, 3-dimensional X-ray diffraction (3DXRD) with synchrotron X-rays, and 3DXRD with X-rays generated by X-ray free electron lasers (XFELs). Our aim is to initiate and develop a general theory of geometric reconstruction in refraction- and diffractionbased tomography. Here, geometric means that the specimen - as in the applications discussed in this project - can be considered as geometric objects, such as polytopes, convex bodies, or certain lattice sets (or unions of those). This geometric point of view should allow us to devise and explore robust special-purpose reconstruction algorithms for the three experimental techniques mentioned above. Refraction


  • P. Gritzmann, B. Langfeld, and M. Wiegelmann: "Uniqueness in Discrete Tomography: Three Remarks and a Corollary," SIAM J. Discrete Math., 25, 1589--1599, 2011
  • R.S. Pennington, S. König, A. Alpers, C.B. Boothroyd, and R.E. Dunin-Borkowski: "Reconstruction of an InAs Nanowire using Geometric and Algebraic Tomography," Journal of Physics: Conference Series, 326, 012045 (4 pp.), 2011
  • A. Alpers, R.J. Gardner, S. König, R.S. Pennington, C.B. Boothroyd, L. Houben, R.E. Dunin-Borkowski, and K.J. Batenburg: "Geometric Reconstruction Methods for Electron Tomography," Ultramicroscopy, 128 (C), 42--54, 2013
  • C.G. Heise, K. Panagiotou, O. Pikhurko, and A. Taraz: "Coloring d-Embeddable k-Uniform Hypergraphs," submitted, 2012
  • A. Alpers, P. Gritzmann, D. Moseev, and M. Salewski: "3D Particle Tracking Velocimetry using Dynamic Discrete Tomography," submitted, 2013

Supervised Dissertations

Completed Theses and Projects

  • Marius Ritter - Particle Tracking Using Network-flow-based Discrete Tomography (Master's Thesis, 2014)
  • Katia Nedelec - Mathematische Grundlagen des Phase-Unwrappings (Master's Thesis, 2013)
  • Dominik Billing - Geometric Reconstruction of InAs-Nanowires (Diploma Thesis, 2011)
  • Anna-Marie Eisgruber - Tomographische Rekonstruktion von 3D Liniensegmenten (Diploma Thesis, 2011)
  • Ruth Bosse - Über Rysers Vermutung zu Matchings in Hypergraphen (Bachelor's Thesis, 2014)
  • Tobias Friedrich - Dreidimensionale Refraktionsvisualisierung (Project, 2014)
  • Stefano Turchetta - Discrete Optimization Methods for Particle Tracking Velocimetry (Interdisciplinary Project, 2014)

Student Research Assistants

  • Phi-Long Phan (2011)
  • Andrej Winokurow (2012)
  • Martin Schwenk (2013)

Research Unit M9

Department of Mathematics
Boltzmannstraße 3
85748 Garching b. München
phone:+49 89 289-16858
fax:+49 089 289-16859


Prof. Dr. Peter Gritzmann
Applied Geometry and Discrete Mathematics

Prof. Dr. Andreas S. Schulz
Mathematics of Operations Research
(affiliated member of M9)

Prof. Dr. Stefan Weltge
Discrete Mathematics


Jan 25th, 2019
Case Studies 2019: Preliminary Meeting on Wed, Feb 6th, at 16:00 in room MI 03.06.011.