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Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100x on the benchmark datasets while maintaining an error rate below 0.1%. 

Comparing two road maps is a basic operation that arises in a variety of situations. A map comparison method that is commonly used, mainly in the context of comparing reconstructed maps to ground truth maps, is based ongraph sampling. The essential idea is to first compute a set of point samples on each map, and then to match pairs of samples—one from each map—in a one-to-one fashion. For deciding whether two samples can be matched, different criteria, e.g., based on distance or orientation, can be used. The total number of matched pairs gives a measure of how similar the maps are. Since the work of Biagioni and Eriksson [11, 12], graph sampling methods have become widely used. However, there are different ways to implement each of the steps, which can lead to significant differences in the results. This means that conclusions drawn from different studies that seemingly use the same comparison method, cannot necessarily be compared. In this work we present a unified approach to graph sampling for map comparison. We present the method in full generality, discussing the main decisions involved in its implementation. In particular, we point out the importance of the sampling method (GEO vs. TOPO) and that of the matching definition, discussing the main options used, and proposing alternatives for both key steps. We experimentally evaluate the different sampling and matching options considered on map datasets and reconstructed maps. Furthermore, we provide a code base and an interactive visualization tool to set a standard for future evaluations in the field of map construction and map comparison. 

Human mobility data science using trajectories or check-ins of individuals has many applications. Recently, we have seen a plethora of research efforts that tackle these applications. However, research progress in this field is limited by a lack of large and representative datasets. The largest and most commonly used dataset of individual human trajectories captures fewer than 200 individuals, while datasets of individual human check-ins capture fewer than 100 check-ins per city per day. Thus, it is not clear if findings from the human mobility data science community would generalize to large populations. Since obtaining massive, representative, and individual-level human mobility data is hard to come by due to privacy considerations, the vision of this work is to embrace the use of data generated by large-scale socially realistic microsimulations. Informed by both real data and leveraging social and behavioral theories, massive spatially explicit microsimulations may allow us to simulate entire megacities at the person level. The simulated worlds, which do not capture any identifiable personal information, allow us to perform “in silico” experiments using the simulated world as a sandbox in which we have perfect information and perfect control without jeopardizing the privacy of any actual individual. In silico experiments have become commonplace in other scientific domains such as chemistry and biology, permitting experiments that foster the understanding of concepts without any harm to individuals. This work describes challenges and opportunities for leveraging massive and realistic simulated alternate worlds for in silico human mobility data science. 

We present a series of nine Computational Geometry Concept Videos, available on Youtube. The videos are aimed at a general audience and introduce concepts ranging from closest and farthest pairs to data structures for range searching and for point location. The video series grew out of the development of an online graduate course on computational geometry, and the beginning portions of the videos are used in the course to motivate the concept and to tie it to a "real" problem in New Orleans. Thus our videos serve a dual purpose of outreach and education. 

This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral component of visualization and analysis pipelines. Although not all topological features present in data are of equal importance. To date, the default definition of feature importance is often assumed and fixed. This work shows how proven explainable deep learning approaches can be adapted for use in topological classification. In doing so, it provides the first technique that illuminates what topological structures are important in each dataset in regards to their class label. In particular, the approach uses a learned metric classifier with a density estimator of the points of a persistence diagram as input. This metric learns how to reweigh this density such that classification accuracy is high. By extracting this weight, an importance field on persistent point density can be created. This provides an intuitive representation of persistence point importance that can be used to drive new visualizations. This work provides two examples: Visualization on each diagram directly and, in the case of sublevel set filtrations on images, directly on the images themselves. This work highlights 

This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral component of visualization and analysis pipelines. Although not all topological features present in data are of equal importance. To date, the default definition of feature importance is often assumed and fixed. This work shows how proven explainable deep learning approaches can be adapted for use in topological classification. In doing so, it provides the first technique that illuminates what topological structures are important in each dataset in regards to their class label. In particular, the approach uses a learned metric classifier with a density estimator of the points of a persistence diagram as input. This metric learns how to reweigh this density such that classification accuracy is high. By extracting this weight, an importance field on persistent point density can be created. This provides an intuitive representation of persistence point importance that can be used to drive new visualizations. This work provides two examples: Visualization on each diagram directly and, in the case of sublevel set filtrations on images, directly on the images themselves. This work highlights real-world examples of this approach visualizing the important topological features in graph, 3D shape, and medical image data.