Abstract. Multi-class labeling is one of the core problems in image analysis. We show how this combinatorial problem can be approximately solved using tools from convex optimization. We suggest a novel functional based on a multidimensional total variation formulation, allowing for a broad range of data terms. Optimization is carried out in the operator splitting framework using Douglas-Rachford Splitting. In this connection, we compare two methods to solve the Rudin-Osher-Fatemi type subproblems and demonstrate the performance of our approach on single-and multichannel images.

We introduce a linearly weighted variant of the total variation for vector fields in order to formulate regularizers for multi-class labeling problems with non-trivial interclass distances. We characterize the possible distances, show that Euclidean distances can be exactly represented, and review some methods to approximate non-Euclidean distances in order to define novel total variation based regularizers. We show that the convex relaxed problem can be efficiently optimized to a prescribed accuracy with optimality certificates using Nesterov's method, and evaluate and compare our approach on several synthetical and realworld examples.

Graph neural networks (GNNs) have emerged as a powerful machine learning approach for the prediction of molecular properties. In particular, recently proposed advanced GNN models promise quantum chemical accuracy at a fraction of the computational cost. While the capabilities of such advanced GNNs have been extensively demonstrated on benchmark datasets, there have been few applications in real atomistic simulations. Here, we therefore put the robustness of GNN interatomic potentials to the test, using the recently proposed GemNet architecture as a testbed. Models are trained on the QM7-x database of organic molecules and used to perform extensive MD simulations. We find that low test set errors are not sufficient for obtaining stable dynamics and that severe pathologies sometimes only become apparent after hundreds of ps of dynamics. Nonetheless, highly stable and transferable GemNet potentials can be obtained with sufficiently large training sets.

We develop a concept for the median filtering of tensor data. The main part of this concept is the definition of median for symmetric matrices. This definition is based on the minimisation of a geometrically motivated objective function which measures the sum of distances of a variable matrix to the given data matrices. This theoretically wellfounded concept fits into a context of similarly defined median filters for other multivariate data. Unlike some other approaches, we do not require by definition that the median has to be one of the given data values. Nevertheless, it happens so in many cases, equipping the matrix-valued median even with root signals similar to the scalarvalued situation.Like their scalar-valued counterparts, matrix-valued median filters show excellent capabilities for structure-preserving denoising. Experiments on diffusion tensor imaging, fluid dynamics and orientation estimation data are shown to demonstrate this. The orientation estimation examples give rise to a new variant of a robust adaptive structure tensor which can be compared to existing concepts.For the efficient computation of matrix medians, we present a convex programming framework.By generalising the idea of the matrix median filters, we design a variety of other local matrix filters. These include matrix-valued mid-range filters and, more generally, M-smoothers but also weighted medians and α-quantiles. Mid-range filters and quantiles allow also interesting cross-links to fundamental concepts of matrix morphology.

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