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WP6 - Algorithms

This work package involves the development of computer-intensive algorithms, including a study of their mathematical and statistical properties.

WP6 - Subprojects

WP6.1: Digital grey-value stereology
WP6.2: Shape reconstruction from tensors
WP6.3: Reconstruction and segmentation of 3D images
WP6.4: Algorithms for single particle cryo-electron microscopy

Research questions

In the second funding period of CSGB, we study as new projects (a) estimation of Minkowski tensors from digital grey-value images and (b) shape reconstruction from finitely many tensors.

  • Preliminary calculations of the CSGB group and collaborators from the Karlsruhe stochastic geometry group show that it is possible to develop global algorithms for estimating Minkowski tensors from binary images. An extension of a stability result in Mérigot et al. (2011, IEEE Trans. Visual. Comp. Graphics) can be used to show multigrid convergence. In the long term, an adaptation of this new algorithm to digital grey-value images should be possible. It may be beneficial to use the idea of doubling the resolution (for instance by interpolating grey values linearly with the neighbouring values) and subsequently applying the new algorithm to the thresholded double resolution image.
  • The aim is to reconstruct (an approximation of) a set from finitely many tensors, possibly distorted by noise. The focus will be on consistency and convergence issues. One route of research involves translation invariant Minkowski tensors. It may be advantageous to work with trace-free Minkowski tensors, due to their close relation to spherical harmonics.

In the second funding period of CSGB, we also want to develop new algorithms for tomographic reconstruction and segmentation of 3D data with complex content. The focus will be on situations with non-linear image formation, missing data or very low signal-to-noise ratio. In these situations, the reconstruction may be improved by Bayesian formulations and implementation as variational methods.

The development of algorithms for single-particle cryo-electron microscopy, started in the first funding period, will be continued.

Selected references

  • Hug, D., Kiderlen, M. & Svane, A.M. (2017): Voronoi-based estimation of Minkowski tensors from finite point samples. Discrete Comput. Geom. 57, 545-570.
  • Kousholt, A. (2017): Reconstruction of n-dimensional convex bodies from surface tensors. Adv. Appl. Math. 83, 115-144.
  • Sander, B. & Golas, M.M. (2016): Molecular electron microscopy: an approach to study macromolecular assemblies. In Neuromethods 115, pp. 205-216. Springer.
  • Svane, A.M. (2017): Valuations in image analysis. In Tensor Valuations and their Applications in Stochastic Geometry and Imaging (eds. M. Kiderlen and E.B.V. Jensen), Lecture Notes in Mathematics 2177, Springer, pp. 435-454.
  • Svane, H. & Feragen, A. (2019): Reconstructing objects from noisy images. Proceedings of the GbR2019 Workshop.

Project leaders

Markus Kiderlen, AU-math
François Lauze, KU

Participating scientists

AU-math:
Rikke Krog Eriksen
Andrew du Plessis
Helene Svane

KU:
Sune Darkner
Aasa Feragen
Mads Nielsen

AU-bio:
Björn Sander

Focus points

  • new computer-intensive algorithms are developed and their mathematical and statistical properties scrutinized
  • an important theme is the development of estimators of Minkowski tensors from digital grey-value images
  • shape reconstruction from finitely many tensors is also studied

Applications

The developed algorithms are typically used in the other work packages. In some cases, the algorithms represent an independent contribution to the research performed at CSGB.

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Revised 20.09.2019