Stefan Sommer
Professor at Department of Computer Science, University of Copenhagen
DIGIFABA Kick-off
Stefan Sommer, University of Copenhagen
Faculty of Science, University of Copenhagen
Ringe, January, 2026
Mathematical models of shapes
Shape models should
- apply to landmarks, curves, surfaces and images
- be independent of discretization
- preserve shape structure
- equivariant to acting groups
- be recovered from discretizations
\(\Large\Rightarrow\) - model correlations between points
- nonlinear
Shapes, deformations and nonlinearity
E_{s_0,s_1}(\phi)=R(\phi)+\frac1\lambda S(\phi.s_0,s_1)
action: \(\phi.s=\phi\circ s\) (shapes)
\(\phi.s=s\circ\phi^{-1}\) (images)
\( \phi \)
\( \phi \) warp of domain \(\Omega\) (2D or 3D space)
landmarks: \(s=(x_1,\ldots,x_n)\)
curves: \(s: \mathbb S^1\to\mathbb R^2\)
surfaces: \(s: \mathbb S^2\to\mathbb R^3\)
s_0
s_1
Geometric + metric view
R(\phi_t)=\int_0^T\|\partial_t \phi_t\|_{\phi_t}^2dt
\( \phi_t:[0,T]\to\mathrm{Diff}(\Omega) \) path of diffeomorphisms (parameter t)
\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\phi_t
LDDMM: Grenander, Miller, Trouve, Younes, Christensen, Joshi, et al.
\partial_t \phi_t
\phi
Action on leafs
- define action on leaf shape and internal structure
- define fiber of internal structure change
- geometric and metric structure on fiber bundle
- statistics
People
- Thomas Besnier (postdoc from Jan 1 2026, 50%)
- Lili Bao (postdoc from Mar 1 2026, 50%)
- Gabriel D'hulst (PhD, from Jun 1 2026)
MSc students:
- spring 2026: Mark, Nynne
WP 2, 3
WP2 Computer vision for detection and segmentation
2.1) Identify relevant object detection and segmentation models. Set up computational pipelines for processing of the field images,
2.2) Develop a fine-tuning methodology to further optimise the base models,
2.3) Evaluate the results with a particular focus on the robustness of the models. For this, adequate fine-grained evaluation metrics need to be developed.
WP3 Shape analysis for plant morphology
3.1) Adapt infinite-dimensional shape models based on actions of diffeomorphisms to plant shape analysis, particularly defining appropriate actions for the overall shape and fine-grained internal structure,
3.2) Develop methodology for low-dimensional representation and visualisation of shape data using the diffeomorphic models,
3.3) Develop statistical methodology for regression analysis, hypothesis testing and analysis of time series of shape data.
Geometry, stochastics, geometric statistics
JaxGeometry: https://github.com/computationalevolutionarymorphometry/jaxgeometry CCEM: http://www.ccem.dk
Hyperiax: https://github.com/computationalevolutionarymorphometry/hyperiax slides: https://slides.com/stefansommer
References:
- Grong, Sommer: Most probable paths for developed processes, https://arxiv.org/abs/2211.15168
- Grong, Sommer: Most probable flows for Kunita SDEs, https://arxiv.org/abs/2209.03868
- Sommer, Schauer, v. d. Meulen: Stochastic flows and shape bridges, Oberwolfach, 2021
- Baker, Besnier, Sommer: A function space perspective on stochastic shape evolution, https://arxiv.org/abs/2302.05382
- Yang, Baker, Severinsen, Hipsley, Sommer: Simulating infinite-dimensional nonlinear diffusion bridges, https://arxiv.org/abs/2405.18353
- Baker, Yang, Severinsen, Hipsley, Sommer: Conditioning non-linear and infinite-dimensional diffusion processes, https://arxiv.org/abs/2402.01434
- Hansen, Eltzner, Huckemann, Sommer: Diffusion Means in Geometric Spaces, Bernoulli, 2023, arXiv:2105.12061
- Grong, Sommer: Most probable paths for anisotropic Brownian motions on manifolds, FoCM 2022, arXiv:2110.15634
- Philipp Harms, Peter W. Michor, Xavier Pennec, Stefan Sommer: Geometry of sample spaces, Diff. Geom. and its Appl., 2023, arXiv:2010.08039
- Arnaudon, v.d. Meulen, Schauer, Sommer: Diffusion bridges for stochastic Hamiltonian systems and shape evolutions,SIIMS,2022,arXiv:2002.00885
- Højgaard Jensen, Sommer: Simulation of Conditioned Diffusions on Riemannian Manifolds, 2021, arXiv:2105.13190.
- Arnaudon, Holm, Sommer: A Geometric Framework for Stochastic Shape Analysis, Foundations of Computational Mathematics, 2019, arXiv:1703.09971.
- Sommer, Svane: Modelling Anisotropic Covariance using Stochastic Development and Sub-Riemannian Frame Bundle Geometry, JoGM, 2017, arXiv:1512.08544.
- Arnaudon, Holm, Sommer: A Stochastic Large Deformation Model for Computational Anatomy, IPMI 2017, arXiv:1612.05323.
DIGIFABA Kick-off
By Stefan Sommer
