Data visualisation is an important area of research: as the amount of data keeps increasing, we have to find ways of showcasing this data to provide an intuition for trends and patterns within it. This can be a particular challenge for high-dimensional data, since we cannot perceive it as is. A common approach is to use dimensionality-reduction techniques to bring the high-dimensional data into lower dimensions, which can then be visualised. One such technique is t-distributed Stochastic Neighbour Embedding (t-SNE), which produces good visualisations but struggles with long runtimes. This paper explores the effect of using sampled data instead of the full dataset to produce t-SNE embeddings, reducing the runtime of the algorithm and hence providing visualisations faster. We show that both visually and numerically, uniform random sampling and Poisson disk sampling can result in much faster runtimes while producing similar, or even more meaningful embeddings than the embedding of the entire dataset.

Modern data analysis often involves working with large multidimensional datasets. Visualizing this kind of data helps leverage human intuition and pattern recognition to reveal hidden relationships. t-SNE is a widely used tool for creating such visualizations. Despite its popularity, it suffers drawbacks in the form of hard-to-tune parameters with no heuristic for guaranteed best results. Due to the size of the data researchers have to work with, the algorithm can often exceed the available memory and lead to slowdowns and crashes. This paper investigates the behaviour of memory usage with respect to the tunable parameter perplexity and the size of the data. It provides a reliable way for researchers to predict the memory consumption before running the algorithm for the popular openTSNE implementation of t-SNE. In addition, a modification to reduce the peak memory usage of the implementation is presented. Together, these contributions improve the reliability and efficiency of t-SNE pipelines in memory-constrained environments.

Dimensionality reduction techniques, such as t-SNE, are widely used to visualize high-dimensional data and have a crucial role in practical tasks such as biological data exploration, anomaly detection, or clustering large datasets. However, they are highly dependent on hyperparameters or sampling strategies. This paper investigates whether the structural similarity between sampled and full embeddings can be measured using Procrustes analysis by comparing the structural similarity of the embeddings. This work provides a reproducible framework that quantifies the difference between visualizations produced by sampling t-SNE. These insights provide users a medium to create visualizations with t-SNE without exhaustive experimentation (for example, creating all visualizations), making t-SNE more accessible and reliable.

T-SNE is widely used for visualising high-dimensional data in lower dimensions.
To reduce the costs of parameter optimisation, t-SNE is performed on a sample of the original data. After sampling the points, the distances between them need to be calculated, which is expensive due to the high dimensionality of the data. We apply Dijkstra's algorithm to approximate the distances and similarities between the sampled points. This reduces the algorithm's execution time by up to 40% without degrading the quality of the projections.

Visualizing high-dimensional data is a key challenge in modern data analysis. T-distributed Stochastic Neighbor Embedding (t-SNE) is a popular nonlinear dimensionality reduction technique that maps such data into a low-dimensional embedding while preserving local relationships. A critical hyperparameter in t-SNE is perplexity. Choosing an appropriate value of perplexity for a particular use-case is non-trivial, especially for large datasets, where repeated t-SNE computations become computationally prohibitive. To mitigate this, the sample-based approach runs t-SNE twice: first on a downsampled subset of the data and then on the full dataset. This introduces two perplexity parameters: sample perplexity for the first run and full perplexity for the second run.

In this work, we systematically investigate the impact of varying combinations of sample perplexity and full perplexity on the quality of the final t-SNE embedding. Our findings show that sample perplexity predominantly determines the global layout of the embedding, while full perplexity influences local refinement. We also compare our approach with different strategies for choosing perplexity values, and find that while some offer better preservation of structural details, they provide less flexibility.

With Virtual Reality, we can create and explore an infinite number of environments. These environments can have multiple applications, such as in education, training, or entertainment. However, we need a way to move through these environments. The most natural way is to walk, but we are limited by our physical space. A solution would be hyperbolic space. In this thesis, we explore navigational performance in hyperbolic space and its relation to the properties of hyperbolic geometry. By using the existing application Holonomy VR, we can use hyperbolic space and virtual reality to create an infinite world in a confined physical space. We let participants navigate in this virtual experience to study if they can navigate and learn about its properties. To achieve this, the performance of Holonomy VR is improved and new features are added that allow for a textured top view of the hyperbolic plane in the form of a 2D experience. Also, a new mode is added that requires users to find multiple landmarks in the environment. The beacon cue is developed to let users see targets outside of their current reach. A user study is conducted in collaboration with Leiden University to test the navigational performance and the understanding of the hyperbolic properties. The results show that some people can adapt to navigating in a confined hyperbolic space and that the 2D experience results in the fastest training time. However, it is found that training by exploring the environment and learning about the landmarks before the evaluation does not improve the navigational performance. This contradicts the findings about navigation in a Euclidean scene. It is also found that some hyperbolic properties can be learned and that the 2D experience made it easier to learn about the diverging paths property. Participants who grasped the concept found the experience fun and interesting, and some even indicated that they wanted to learn more about hyperbolic space. Overall, this study shows that people can navigate a confined hyperbolic environment and that they can learn about its properties.

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.

Dimensionality reduction is an important task in high-dimensional data visualisation. Among the popular algorithms for achieving this is t-SNE, which aims to preserve local neighbourhoods in the lower-dimensional embeddings. While t-SNE traditionally works in Euclidean space, embedding in hyperbolic space offers several advantages, specifically for data of arbitrary size and exponential growth, such as tree-based structures. We propose a new solution to approximate and accelerate the calculation of t-SNE gradients using a uniform grid structure. This new method produces embeddings with better neighbourhood preservation than previous solutions, while also providing better runtime performance.

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.

This paper investigates a method for accelerating hyperbolic t-SNE — a popular high-dimensional data visualization technique. In particular, it focuses on building a hyperbolic t-SNE variant that uses a different model of hyperbolic space (called the Lorentz Hyperboloid model) for representing the low-dimensional embeddings. An acceleration algorithm based on a tree data-structure is then used to achieve a better asymptotic runtime complexity compared to the original version. The paper then compares this implementation with other alternatives — including accelerated variants — and shows that it computes embeddings of better quality at a similar rate.

With the rapid growth in data collection, efficient data processing is critical. Dimensionality reduction methods, like t-distributed stochastic neighbour embedding (t-SNE), compress high-dimensional data into embeddings that preserve the key features of the datasets making data less sparse and easier to process further. Recent improvements suggest that using hyperbolic space for data representation can benefit embeddings. As it is a new technique, it remains computationally expensive. Previous work suggests that a Barnes-Hut approximation with a polar quadtree can be applied to the Poincaré disk model to approximate the result of hyperbolic t-SNE and accelerate its calculation. However, the polar quadtree is proposed as a solution to accelerate the calculation without exploring alternative approaches. Aiming to close this gap, we propose an acceleration method using Barnes-Hut approximation with a Cartesian quadtree. We experimentally compare our acceleration method to a polar quadtree and showcase its lower execution time without the loss of quality of the embeddings.
Implementation and scripts for the experiments and plots are available at https://github.com/Sne4kers/hyperbolic-tsne

This paper tackles the problem of evaluating the task of finding similar scanned historical watermarks - small images embedded in historical paper that have been digitized to be processed on a computer - using pre-trained neural networks. This research aims to identify an efficient and accurate alternative to the traditional, time-consuming manual detection methods for finding similar watermarks. The primary issue addressed is the inefficiency of these manual methods. The evaluation focuses on finding similar watermarks for a specific query watermark, assessing the efficacy of neural networks in comparison to a prior art system that employs traditional image processing techniques. This comparison aims to determine how well these neural networks perform in the task of watermark similarity detection. The study involves a dataset of 500 labeled images tested in two distinct contexts: one using unprocessed images and another using images processed to keep only the watermark outline. The results show that pre-trained models achieve higher accuracy and time efficiency compared to the prior art system that uses image processing. These models demonstrate significant effectiveness in watermark recognition and comparison, with each network achieving over 80% accuracy for traced watermarks. EfficientNetB0 achieved 94.66%, VGG16 89.33%, ResNet50 86.67%, and InceptionV3 84%, while the prior art system gets 64,8%. These results conclude that these models are valuable tools in the field of watermark recognition and comparison.

A watermark image is a scan of a historical paper document that contains a watermark, which is a motif embedded in the paper that provides valuable information on the origins of a document. Developing tools to automatically identify watermarks can make this information more accessible to researchers. This paper focuses on one specific binarization technique, thresholding. Thresholding selects a threshold value, which is used to turn an image binary such that one color represents foreground and the other represents background. Ideally, binarization isolates the watermark’s shape by representing it as foreground, and removes unwanted information. This research compares the effectiveness of different thresholding techniques when applied to watermark images. Eight algorithms are selected from the literature, and a novel algorithm is proposed that seeks to improve on the other algorithms when applied to watermarks. The nine total algorithms are evaluated quantitatively on synthetic data, and qualitatively through a survey where participants select which algorithm appears best and rate it. The results show that there is no clear algorithm which works best for all images, however a logical adaptive approach may work marginally better than other approaches. Additionally, the presented algorithms do not adequately remove non-watermark information from the images. Further research should be conducted to analyze different binarization techniques in this context.

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.

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.

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.

Navigation is a core aspect of exploring virtual environments. To assist players, a mini-map is a commonly used navigational tool. Navigation in an unknown space can be difficult. This difficulty is only increased when a player finds themselves in a non-Euclidean space. This paper explores the effect the mini-map's positioning has on the player's navigational performance in Virtual Reality using hyperbolic space. For this two positions are used: the first mini-map is positioned on the Heads-Up Display and the second is positioned on the player's right hand. Two player groups are randomly created to take part in a player experiment. Players get to read an information sheet and complete a small tutorial before having to complete 3 levels. Players are measured on how much time and how many steps it takes them to complete each level, how much their time and steps improve over the levels, their total time, and their total amount of steps. Combined these results indicate that there is no clear effect on the player's performance navigating a hyperbolic space in Virtual Reality.

Simulating non-Euclidean geometry in virtual reality is of interest to a wide variety of fields of research. However it is still quite a challenge. Various methods are already known, but they vary greatly in performance and applicability. This paper compares some methods of rendering a non-Euclidean space. We focus on the order-5 square tiling that can be found in the hyperbolic plane, but the methods used are also relevant to other non-Euclidean spaces. We render this space using portals with two different implementations: one using render textures, an one using stencil polygons. Through an experiment where we measured and compared the frame rate of each method, we have found that, even with a small number of portals, the stencil polygon approach is more than two times as efficient. However, this method is limited in the number of portals, whereas render textures can be used for any number of portals.

Non-Euclidean spaces are spaces that do not satisfy all of Euclid’s postulates. An example of such a space is hyperbolic space. In this paper, a method is discussed to draw a tessellation of hyperbolic space in a manner that fits with the virtual reality game "Holonomy", a game which takes place in hyperbolic space. The main result shows that the new approach is better in terms of simplicity than the old approach and is able to draw the game world more faithfully than the old approach. The features of the new approach could still be significantly expanded upon.

Virtual Reality allows for the ultimate immersion in environments not naturally encountered. Still, hyperbolic environments are extremely difficult to get used to. This paper explores whether immersion in virtual hyperbolic environments can be enhanced by introducing a procedurally generated world. We propose a Wave Function Collapse (WFC) algorithm that can divide the hyperbolic plane into distinct areas and populate the space with dynamically generated objects. This can assist players in recognising areas visited prior, which can create an intuitive feel for how hyperbolic space operates. Although 5-order square tiling is used in this paper, the proposed algorithm can be extended to other tiling approaches. Evidence suggests players in a populated hyperbolic environment need on average fewer steps to reach their objective compared to a control group with an empty environment, although the time taken for both groups is roughly the same. From this and a qualitative analysis we infer that players feel more immersed in the environment when it is procedurally generated instead of mostly empty.