Minisymposia Abstracts

Speaker: Kristian Bredies
University of Graz
Title: TGV2-regularization for inverse problems and applications in medical imaging
Abstract: The concept of total generalized variation (TGV) has recently been introduced as a penalty functional for images which measures smoothness adaptively on different scales. As a side effect, it is able to preserve edges in images while preventing undesired staircasing artifacts to appear in smooth regions. In the talk, we focus on the regularization properties of TGV of second order, in particular for ill-posed linear inverse problems in imaging. Applications of TGV2-regularization in medical imaging are presented, with attention on the reconstruction of magnetic resonance images from highly undersampled data. For this problem, regularization with second-order total generalized variation leads to more faithful representation of soft tissue while sharp transitions are still preserved.

Speaker: Matthias Gehre
University of Bremen
Title: Electrical Impedance Tomography with the complete electrode model and sparsity constraints
Abstract: We consider the complete electrode model for Electrical Impedance Tomography and show some of its analytical properties. Those results allow us to justify a reconstruction approach based on sparsity constraints. Finally some promising numerical results will be presented.

Speaker: Antoine Laurain
Humboldt-University of Berlin
Title: A shape and topology optimization method for the resolution of inverse problems
Abstract: We propose a general shape optimization approach for the resolution of different inverse problems in tomography. For instance, in the case of Electrical Impedance Tomography (EIT), we reconstruct the electrical conductivity while in the case of Fluorescence Diffuse Optical Tomography (FDOT), the unknown is a fluorophore concentration. In the two cases, the underlying partial differential equation are different but the reconstruction method essentially stays the same. These problems are in general severely ill-posed, and a standard cure is to make additional assumptions on the unknowns to regularize the problem. Our approach consists in assuming that the functions to be reconstructed are piecewise constants.
Thanks to this hypothesis, we are looking for the shape of one or several inclusions in the domain and the problem essentially boils down to a shape optimization problem. The sensitivity of a certain cost functional with respect to small perturbations of the shapes of these inclusions is analysed. The algorithm consists in initializing the inclusions using the notion of topological derivative, which measures the variation of the cost functional when a small inclusion is introduced in the domain, then to reconstruct the shape of the inclusions by modifying their boundaries with the help of the so-called shape derivative.

Speaker: Matthias Schlottbom RWTH Aachen
Title: Analysis of forward and inverse models in fluorescence optical tomography
Abstract: In this talk we will investigate forward and inverse models in fluorescence optical tomography. Fluorescence optical tomography is an imaging modality where a fluorophore concentration is reconstructed from boundary measurements of near- infrared light. The photon propagation can be modelled by an elliptic system of equations. We will show that this approach leads to a compact forward operator and in turn to an ill-posed inverse problem. For regularization Tikhonov-type least-squares functionals are considered. We will focus on smooth regularization functionals which approximate non-smooth regularization functionals, e.g. a smooth version of a total variation penalty. Numerical experiments show that reconstructions of the fluorophore concentration computed with these penalties are superior to reconstructions calculated with standard L2/H1 penalties.


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