Arian Bërdëllima, Gabriele Steidl
01.09.2020 – 31.08.2022
The aim of this project is the modeling, analysis and application of dynamic imaging with a focus on functions defined on special non-Euclidean geometries as well as vector- and manifold-valued functions. A challenge consists in utilizing template based approaches including learning techniques. We will start with dynamic tomography problems related to Radon-type transforms on special geometries appearing in certain applications and consider vector-and manifold-valued images in a long term.
Depending on the concrete application, there are quite different (variational) approaches to model dynamics in imaging as i) optical flow, ii) dynamic optimal transport, iii) large deformation diffeomorphic mapping (LDDMM), and iv) metamorphosis. On the one hand, we intend to enlarge our research from iii) and vi) to a larger class of operators than the classical Radon transform which will require to deal with functions on special manifolds
and to investigate the question how to incorporate multiple prior information into the model. On the other hand, we will deal with inverse problems for vector- and manifold-valued problems, superesolution and compression, segmentation in time evolving frames of images.
The project can be seen as a long term one which merges dynamic inverse problems with manifold structures as well as prior knowledge on the data. We intend to establish new models for solving problems in this emerging field, to provide a mathematical analysis of the models and ecient minimization algorithms with convergence guarantees. The questions arising in this project come from certain applications related to novel imaging modalities and acquisition techniques as prescribed above. However, the mathematical methods will not be restricted to these applications, but will be of interest on their own.
A. Bërdëllima, R. Beinert, M. Gräf and G. Steidl, On the dynamical system of principal curves in R^d, Communications in Statistic: Simulation and Computation, 12:1–20, 2022.
A. Bërdëllima and G. Steidl. On α-firmly nonexpansive operators in r-uniformly convex spaces. Results in Mathematics, 76(4):1–27, 2021.
C. Kirisits, M. Quellmalz, M. Ritsch-Marte, O. Scherzer, E. Setterqvist, and G. Steidl. Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations. Inverse Problems, 37(11):115002, 2021.
S. Neumayer, J. Persch, and G. Steidl. Regularization of inverse problems via time discrete geodesics in image spaces. Inverse Problems, 35(5):055005, 28, 2019.
S. Neumayer, J. Persch and G. Steidl, Morphing of manifold-values images inspired by discrete geodesics in image spaces, SIAM Journal of Imaging Sciences, vol. 11, no. 3, pp. 1898–1930, 2018.
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