Supervised by Karsten Matthies, Dept of Mathematical Sciences, University of Bath
                      and David Tsang, Dept of Physics, University of Bath

Abstract

Related Work

Galley, Chad. Classical Mechanics of Nonconservative Systems. Physical Review Letters, 2013
Galley, Tsang and Stein. The principle of stationary nonconservative action for classical mechanics and field theories 2014.

Documents

Poster Euler-Poincaré equations for non-conservative systems (presented at BAMC 2021)

In collaboration with Barbara Gris, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris)
Supervised by Prof. Jan Lellmann, Intitute of Mathematics and Image Computing, Universität zu Lübeck (Lübeck)

Abstract

The deformation module approach introduced in [1] is a generic framework for integrating prior knowledge about desired deformation fields in shape and image registration. Deformation modules allow to build diffeomorphic deformations that satisfy a given structure, for example locally affine transformations, scalings and rotations, or a combination. The diffeomorphisms are built by integrating a flow of constrained velocity fields. However, in applications such as modeling of respiratory motion in abdominal images for operation planning, one often encounters situations where a single deformation model is not adequate. While it makes sense to consider diffeomorphic deformations of the seperate organs, breathing motion results in a non-smooth and possibly discontinuous displacement field at organ boundaries. To be able to use different constrained deformations for different organs, we combine the model with a framework to introduce region boundary constraints for multiple interacting objects in a shape or image, as introduced in [2]. Deformation fields are built for each object separately, while their interaction is incorporated by constraints on the deformation of the object boundaries. We derive theoretical results on the existence of optimal trajectories for the constrained registration problem and provide numerical results to demonstrate the potential of our approach.

Related Work

[1] Gris, Barbara. Modular approach on shape spaces, Sub-Riemannian geometry and computational anatomy. Diss. 2016.
[2] Arguillere, Sylvain. Géométrie sous-riemannienne en dimension infinie et applications à l'analyse mathématique des formes. Diss. 2014.

Documents

Thesis
Poster (presented at the European Women in Mathematics Conference 2019 in Leipzig)

Supervised by Dr. Carola B. Schönlieb, Department of Mathematics and Theoretical Physics, University of Cambridge
In collaboration with Ozan Öktem, KTH Royal Institute of Technology (Stockhholm)
                                         Barbara Gris, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris)

Abstract

In my project 'Indirect Image Registration: Geodesic Shooting for LDDMM and Deep Learning Approaches' the question was how to do registration of images where the template image is only given in the data space, e.g. as a sinogram of a CT. I familiarized with the LDDMM model, which is based on modeling the registration as an evolution of the template image over time to match the reference image, motivated by fluid mechanics. By geodesic shooting it is possible to describe this evolution over time by only a momentum at the staring point. The optimization process to find the optimal momentum needs complicated and time-consuming calculations. I worked out different formulations for using this approach together with a Machine Learning approach for faster, almost real-time computation.

Related Work

Chen, Chong, and Ozan Öktem. Indirect image registration with large diffeomorphic deformations. SIAM Journal on Imaging Sciences 11.1 (2018): 575-617.

Documents

Report