arXiv subjects

Christoph Brauer

Publications and source records attributed to Christoph Brauer.

8 recordsLinked to original sources

arXiv2026

Automatic ply-specific analyses of CFRP micrographs using shortest-path-based ply distinction

We present an automated approach to distinguish between ply instances in semantic segmentation masks of high-resolution carbon-fiber reinforced polymer micrographs. Interpreting the segmentation mask as a graph with pixels as vertices, enables us to use a shortest-path algorithm yielding the ply-separating paths. Thereby, we bridge the gap between semantic segmentation and ply instance segmentation using global information. We successfully apply our approach on high-resolution micrographs featuring a broad range of characteristics like artificially added gaps in single or multiple plies, different stacking sequences and ply traversing cracks. Assigning each fiber pixel to a ply based on the calculated paths, allows for a comprehensive, quantitative ply analysis with respect to its microstructural properties like the local fiber volume fraction as well as locally resolved ply and interleaf layer thickness. These insights help to reveal manufacturing-induced inhomogeneities, draw conclusions on manufacturing parameters and link mechanical properties to underlying microstructural imperfections.

cs.CV↗

arXiv2026

Voronoi-Based Vacuum Leakage Detection in Composite Manufacturing

In this article, we investigate vacuum leakage detection problems in composite manufacturing. Our approach uses Voronoi diagrams, a well-known structure in discrete geometry. The Voronoi diagram of the vacuum connection positions partitions the component surface. We use this partition to narrow down potential leak locations to a small area, making an efficient manual search feasible. To further reduce the search area, we propose refined Voronoi diagrams. We evaluate both variants using a novel dataset consisting of several hundred one- and two-leak positions along with their corresponding flow values. Our experimental results demonstrate that Voronoi-based predictive models are highly accurate and have the potential to resolve the leakage detection bottleneck in composite manufacturing.

math.MG↗

arXiv2025

Why the noise model matters: A performance gap in learned regularization

This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine reconstruction, i.e. the best possible affine linear map for minimizing the mean squared error. It is known that this optimal reconstruction can be achieved using Tikhonov regularization, but this requires precise knowledge of the noise covariance to properly weight the data fidelity term. However, in many practical applications, noise statistics are unknown. We therefore investigate the performance of regularization methods learned without access to this noise information, focusing on Tikhonov, Lavrentiev, and quadratic regularization. Our theoretical analysis and numerical experiments demonstrate that for non-white noise, a performance gap emerges between these methods and the optimal affine reconstruction. Furthermore, we show that these different types of regularization yield distinct results, highlighting that the choice of regularizer structure is critical when the noise model is not explicitly learned. Our findings underscore the significant value of accurately modeling or co-learning noise statistics in data-driven regularization.

math.NA↗

arXiv2023

Learning Variational Models with Unrolling and Bilevel Optimization

In this paper we consider the problem of learning variational models in the context of supervised learning via risk minimization. Our goal is to provide a deeper understanding of the two approaches of learning of variational models via bilevel optimization and via algorithm unrolling. The former considers the variational model as a lower level optimization problem below the risk minimization problem, while the latter replaces the lower level optimization problem by an algorithm that solves said problem approximately. Both approaches are used in practice, but unrolling is much simpler from a computational point of view. To analyze and compare the two approaches, we consider a simple toy model, and compute all risks and the respective estimators explicitly. We show that unrolling can be better than the bilevel optimization approach, but also that the performance of unrolling can depend significantly on further parameters, sometimes in unexpected ways: While the stepsize of the unrolled algorithm matters a lot (and learning the stepsize gives a significant improvement), the number of unrolled iterations plays a minor role.

stat.ML↗

arXiv2018

Primal-dual residual networks

In this work, we propose a deep neural network architecture motivated by primal-dual splitting methods from convex optimization. We show theoretically that there exists a close relation between the derived architecture and residual networks, and further investigate this connection in numerical experiments. Moreover, we demonstrate how our approach can be used to unroll optimization algorithms for certain problems with hard constraints. Using the example of speech dequantization, we show that our method can outperform classical splitting methods when both are applied to the same task.

stat.ML↗

arXiv2018

A Sinkhorn-Newton method for entropic optimal transport

We consider the entropic regularization of discretized optimal transport and propose to solve its optimality conditions via a logarithmic Newton iteration. We show a quadratic convergence rate and validate numerically that the method compares favorably with the more commonly used Sinkhorn--Knopp algorithm for small regularization strength. We further investigate numerically the robustness of the proposed method with respect to parameters such as the mesh size of the discretization.

math.OC↗

arXiv2016

A Primal-Dual Homotopy Algorithm for $\ell_{1}$-Minimization with $\ell_{\infty}$-Constraints

In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.

math.OC↗

arXiv2016

Rank-optimal weighting or "How to be best in the OECD Better Life Index?"

We present a method of rank-optimal weighting which can be used to explore the best possible position of a subject in a ranking based on a composite indicator by means of a mathematical optimization problem. As an example, we explore the dataset of the OECD Better Life Index and compute for each country a weight vector which brings it as far up in the ranking as possible with the greatest advance of the immediate rivals. The method is able to answer the question "What is the best possible rank a country can achieve with a given set of weighted indicators?" Typically, weights in composite indicators are justified normatively and not empirically. Our approach helps to give bounds on what is achievable by such normative judgments from a purely output-oriented and strongly competitive perspective. The method can serve as a basis for exact bounds in sensitivity analysis focused on ranking positions. In the OECD Better Life Index data we find that 19 out the 36 countries in the OECD Better Life Index 2014 can be brought to the top of the ranking by specific weights. We give a table of weights for each country which brings it to its highest possible position. Many countries achieve their best rank by focusing on their strong dimensions and setting the weights of many others to zero. Although setting dimensions to zero is possible in the OECD's online tool, this contradicts the idea of better life being multidimensional in essence. We discuss modifications of the optimization problem which could take this into account, e.g. by allowing only a minimal weight of one. Methods to find rank-optimal weights can be useful for various multidimensional datasets like the ones used to rank universities or employers.

math.OC↗