Most tomographic modalities record the data sequentially, i.e. temporal changes of the object lead to inconsistent measurements. Consequently, suitable models and algorithms have to be developed in order to provide artefact free images. In this talk, we provide an overview of different strategies, including a data-driven approach to extract explicit motion maps which can then be incorporated within direct or iterative reconstruction methods for the underlying dynamic inverse problem. Our methods are illustrated by numerical results from real as well as simulated data of different imaging modalities.

In this talk we introduce a Frank-Wolfe-type algorithm for sparse optimization in Banach spaces. The functional we want to optimize consist of the sum of a smooth fidelity term and of a convex one-homogeneous regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. We apply this algorithm to the problem of particles tracking from heavily undersampled dynamic MRI data. This talk is based on the works cited below.

[1] K.Bredies, M.Carioni, S.Fanzon, D.Walter. Asymptotic linear convergence of Fully-Corrective Generalized Conditional Gradient methods. Mathematical Programming, 2023.

[2] K.Bredies, S.Fanzon. An optimal transport approach for solving dynamic inverse problems in spaces of measures. ESAIM:M2AN, 54(6): 2351-2382, 2020.

[3] K.Bredies, M.Carioni, S.Fanzon, F.Romero. A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Found Comput Math, 2022

[4] K.Bredies, M.Carioni, S.Fanzon. On the extremal points of the ball of the Benamou–Brenier energy. Bull. London Math. Soc., 53: 1436-1452, 2021.

[5] K.Bredies, M.Carioni, S.Fanzon. A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients. Communications in Partial Differential Equations, 47(10): 2023-2069, 2022.

Tomographic X-ray imaging on the nano-scale is an important tool to visualize the structure of materials such as alloys or biological tissue. Due to the small scale on which the data acquisition takes place, small perturbances caused by the environment become significant and cause a motion of the object relative to the scanner during the scan. Since this motion is hard to estimate and its incorporation into the reconstruction process strongly increases the numerical effort, we aim at a different approach for a stable reconstruction: We interpret the object motion as a modelling inexactness in comparison to the model in the static case. This inexactness is estimated and included in an iterative regularization scheme called sequential subspace optimization. Data-driven techniques are investigated to estimate the modelling error and to improve the obtained reconstructions.

Magnetic particle imaging (MPI) is a preclinical imaging modality exploiting the nonlinear magnetization response of magnetic nanoparticles to applied dynamic magnetic fields. We focus on MPI using a field-free line for spatial encoding because under ideal assumptions such as static objects, ideal magnetic fields and sequential line rotation, the MPI data are obtained by Radon transformed particle distributions. In practice, field imperfections and moving objects occur such that we have to adapt the Radon transform and jointly reconstruct time dependent particle distributions and adapted Radon data by means of total variation regularization.