Inverse Problems and Optimization II

Date & Time: Tuesday, September 29th, 2026 | 17 - 18:40 
Place: tba
Chairs: Marvin Knöller 

Speaker: Aada Hakula (Aalto University)
Title: Handling model-related uncertainties in diffuse optical tomography
Abstract: In this talk, we consider model-related uncertainties in diffuse optical tomography (DOT), a functional neuroimaging method that aims to reconstruct hemodynamic changes on the cerebral cortex using the boundary measurements of near-infrared light. More specifically, the measured changes in the detected amplitude and phase shift are linearly connected to the changes in the absorption within the brain. This image reconstruction problem is severely ill-posed and requires the segmented anatomical model of the target as well as the tissue-specific baseline optical parameters and the optode configuration, all of which are known only approximately in practice. We present uncertainties related to these factors and ways to handle them within the Bayesian inversion framework, with examples generated using Monte Carlo (MC) simulations in a neonatal voxel-based head model.

Speaker: Allti Jääskeläinen (Aalto University)
Title: Projection-based handling of uncertainties in head imaging with EIT
Abstract: This talk introduces a method for mitigating the effects of different types of uncertainties on head imaging reconstructions. We discuss the method in the context of electrical impedance tomography (EIT), where our aim is to reduce errors caused by unknown contact conductivities on the electrode surfaces, as well as conductivity changes in parts of the domain that we are not interested in. The effectiveness of our method is demonstrated both with simulated data and real-world measurements taken from a water tank.

The main principle of the method is to compute the Jacobian matrix of the forward map from the unknown auxiliary variables onto the measured values, and project the forward problem we wish to invert onto the orthogonal complement of the range of this Jacobian. While the Jacobian depends on the initial estimates for the unknown variables, we find that the range of the Jacobian is nearly independent of these values. For high-dimensional nuisance parameters, such as the conductivity in some part of the domain, we must limit the nullspace of the projection by choosing a suitable subspace of the range of the Jacobian. Additional prior information can be incorporated into this choice via a weighting matrix, for example the covariance matrix of a Gaussian prior for the auxiliary variables.

Speaker: Teresa Rauscher (University of Graz)
Title: A Bloch-Torrey Approach to Quantitative Susceptibility Mapping in Magnetic Resonance Imaging
Abstract: Quantitative susceptibility mapping (QSM) aims to recover the spatial distribution of magnetic susceptibility from magnetic resonance phase data. We formulate QSM within the Bloch–Torrey framework as a two-stage inverse problem. First, the equilibrium magnetization, transverse relaxation rate, and susceptibility-induced magnetic field are identified from time-resolved coil measurements. By combining spin-echo and gradient-echo measurements, we derive explicit relations in the diffusion-free case that show how these quantities can be distinguished. Building on coefficient-identification results for the Bloch–Torrey equation, we investigate local uniqueness and stability of this first identification step. In the second stage, the magnetic susceptibility is recovered from the reconstructed field through dipole inversion.

Speaker: Vigdis Toresen (Aalto University)
Title: Total variation regularization with reduced basis in electrical impedance tomography
Abstract: This talk presents the use of reduced basis techniques in combination with (smoothened) total variation regularization in electrical impedance tomography. We show that reduced basis methods can speed up a reconstruction algorithm while preserving both reconstruction quality and the edge-enhancing nature of total variation regularization. The approach is numerically tested in three dimensions on unstructured finite element meshes with both simulated and experimental data. The results demonstrate a substantial shortening of online reconstruction times.