A curve is a continuously bending line with no angles that can be found anywhere in the real world, forming shapes and outlines. They are also the building blocks of historic watermarks, imprinted images on paper that may be used to identify its manufacturers. Their shapes consist of curves as bent wires are used in their production process. Often, the processing of scans of those curves may introduce gaps or a degraded quality which could be corrected by reconstructing the curves in those gaps. Curve reconstruction is a fundamental problem with many research applications, one of which is the reconstruction of curves for binarised scans of historic watermarks. In this paper, a data generation approach is proposed for the simulation of the watermark curves domain through singular automatically generated curves and human-drawn sketches which are then used along binarised watermark scans. I propose a hybrid method combining machine-learning and analytical approaches for curve reconstruction, aiming to leverage their advantages together. The method is compared to its components separately. Quantitative results against them demonstrate the superiority of the pure machine learning approach, as well as the need for more research into potentially better analytical components and a more realistic domain simulation.

Dimensionality reduction is essential for analyzing high-dimensional datasets across various fields. While t-SNE is a popular method for this purpose in Euclidean spaces, recent advancements suggest that hyperbolic spaces can better represent hierarchical structures. However, there is a notable lack of data structures and algorithms tailored for hyperbolic spaces. This research addresses this gap by implementing a hyperbolic quadtree structure in the upper half-plane model and integrating it into the hyperbolic t-SNE framework. Our goal is to accelerate the optimization of the hyperbolic t-SNE while maintaining reasonable precision and recall. We conduct rigorous benchmarking experiments to evaluate the performance of this approach, comparing it to existing methods. The findings provide insights into the practical utility of using the hyperbolic quadtree structure in the upper half-plane model in hyperbolic t-SNE embeddings.

In this work we aim to implement a variaton of the acceleration of hyperbolic t-SNE done by Skrodzki et. al. [19]. This variation aims to embed the points in the Klein Disk model of hyperbolic space instead of the Poincar ́e Disk model using an altared version of a polar quadtree to speed up the computation in a similar fashion as the Barnes-Hut scheme for the Euclidean versino of t-SNE. We analyze our results to prove our acceleration works for the Klein Disk model and compare the efficiency of our implemen- tation to the one for the Poincar ́e model in terms of quality of results and runtime.

Watermarks have an essential role in identifying the origins and age of specific documents. However, this is often a laborious process. One of the main issues in automatic watermark segmentation is the presence of text that obstructs it, making it difficult to properly reconstruct a watermark. Image processing and machine learning techniques face limitations, requiring time, training of data, or manual parameter selection. This research introduces a new method using wavelets transform to locate and remove text from a watermarked image, while preserving the underlying watermark. This method manages to outperform classic image processing techniques for the case when text is thicker than the watermark outline.

Feature extraction techniques for content-based image retrieval are explored, focusing on black-and-white images in the context of historical watermarks. Orthogonal moments and texture features are found to be most applicable. Seven methods are evaluated: four different orthogonal moments, Gabor features, and two novel combinations of orthogonal moments with Gabor features. Retrieval effectiveness is judged based on the precision-recall curve and mean average precision, and watermarks are considered when unchanged, rotated, sheared and both rotated and sheared. The results demonstrate that research into improving efficient grayscale image representation does not translate over to improvements with black-and-white images. As it stands, the basic Zernike moments and novel Gabor-Zernike features are most effective.

This paper describes the process and evaluation methods by which we adapted a Reaction-Diffusion model and an order parameter to monitor its segregation state in a 2D Cellular Automaton model. The model simulates Turing pattern formations, whose behavior will be studied by an order parameter formulated for this project. This order parameter acts as an indicator of the mixture state of the world. The implementation process of such a cellular automaton and an order parameter is described in detail, as well as the evaluation of this model and its order parameter. Different evaluation methods will be performed on this order parameter to show it as a valid indicator of the segregation state of the world in this model, as well as showcase different interesting properties and limitations of the parameters. Such methods will involve plots like Evolution, Phase-Transition, or Linear Regression plots, and the paper will conclude by summing up all the work taken into this project to confirm that this order parameter, with its formulation in this paper, is a valid and interesting one to use.

Recent years saw a rapid increase in conference formats that take place either fully online or in a hybrid fashion with some people on-site and others online. While these formats brought new challenges, they also opened up new opportunities. In the present article, we first outline advantages and disadvantages of different conference formats as discussed in the literature. We then share our own experiences based on two mathematics and art events that occurred during the respective annual meetings of the German Mathematical Society in 2020 and 2021. This is to illustrate the main benefits of online formats, in particular for the MathArt community. We conclude by highlighting two specific aspects–the facilitated presentation of large artworks and the availability of talk recordings–and give a brief outlook on hybrid events.

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Chip-firing was first introduced as a probabilistic game. Subsequently, it was generalized to arbitrary graph configurations and investigated mostly with regard to two-dimensional quad-grid layouts. In this paper, we lift chip-firing to the third dimension. Aside from the arising three-dimensional shapes, we are interested in the internal, two-dimensional structures. Furthermore, we explore the different shapes obtained by chip firing processes on various neighborhoods, such as the face-centered and the cube-centered grid as well as on a neighborhood inspired by knight moves.@en