EF3 – Model-Based Imaging

Project

EF3-6

Deformation Based Regularization of Inverse Problems in Manifolds

Project Heads

Gabriele Steidl

Project Members

Arian Bërdëllima, Gabriele Steidl

Project Duration

01.09.2020 – 31.08.2022

Located at

TU Berlin

Description

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 fi eld, 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.

Project Webpages

Selected Publications

 
  1. R. Beinert and M. Quellmalz. Total variation-based reconstruction and phase retrieval for diffraction tomography. SIAM Journal on Imaging Sciences, 15(3):1373–1399, 2022.
  2. A. Bërdëllima and G. Steidl, Quasi α-firmly nonexpansive mappings in Wasserstein spaces, arXiv:2203.04851, (2022).

  3. 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.

  4. S. Dahlke, F. D. Mari, E. D. Vito, M. Hansen, M. Hasannsab, M. Quellmalz, G. Steidl, and G. Teschke. Continous wavelet frames on the sphere: The group-theoretic approach revisited. Applied and Computational Harmonic Analysis, 56:123–149, 2022.

  5. A. Bërdëllima and G. Steidl. On α-firmly nonexpansive operators in r-uniformly convex spaces. Results in Mathematics, 76(4):1–27, 2021.

  6. F. Faucher, C. Kirisits, M. Quellmalz, O. Scherzer, and E. Setterqvist. Diffraction tomography, fourier reconstruction, and full waveform inversion. arXiv preprint arXiv:2110.07921, 2021.

  7. 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.

  8. 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.

  9. 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.

  10. J. H. Fitschen, F. Laus, and G. Steidl. Transport between RGB images motivated by dynamic optimal transport. Journal of Mathematical Imaging and Vision, 56(3):409-429, 2016.
  11. J. Yuan, Ch. Schnörr, and G. Steidl. Simultaneous higher order optical flow estimation and decomposition. SIAM Journal of Scienti c Computing, 29(6):2283-2304, 2007.
 

Selected Pictures

Dynamical System of Principal Curves in R^d
Morphing path between a part of the YZ-slices 49 and 51 of the Camino dataset

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