Merge trees are a powerful tool from topological data analysis that is frequently used to analyze scalar fields. The similarity between two merge trees can be captured by an interleaving: a pair of maps between the trees that jointly preserve ancestor relations in the trees. Interleavings can have a complex structure; visualizing them requires a sense of (drawing) order which is not inherent in this purely topological concept. However, in practice it is often desirable to introduce additional geometric constraints, which leads to variants such as labeled or monotone interleavings. Monotone interleavings respect a given order on the leaves of the merge trees and hence have the potential to be visualized in a clear and comprehensive manner.In this paper, we introduce ParkView: a schematic, scalable encoding for monotone interleavings. ParkView captures both maps of the interleaving using an optimal decomposition of both trees into paths and corresponding branches. We prove several structural properties of monotone interleavings, which support a sparse visual encoding using active paths and hedges that can be linked using a maximum of 6 colors for merge trees of arbitrary size. We show how to compute an optimal path-branch decomposition in linear time and illustrate ParkView on a number of real-world datasets.

Points of interest on a map such as restaurants, hotels, or subway stations, give rise to categorical point data: data that have a fixed location and one or more categorical attributes. Consequently, recent years have seen various set visualization approaches that visually connect points of the same category to support users in understanding the spatial distribution of categories. Existing methods use complex and often highly irregular shapes to connect points of the same category, leading to high cognitive load for the user. In this paper we introduce SimpleSets, which uses simple shapes to enclose categorical point patterns, thereby providing a clean overview of the data distribution. SimpleSets is designed to visualize sets of points with a single categorical attribute; as a result, the point patterns enclosed by SimpleSets form a partition of the data. We give formal definitions of point patterns that correspond to simple shapes and describe an algorithm that partitions categorical points into few such patterns. Our second contribution is a rendering algorithm that transforms a given partition into a clean set of shapes resulting in an aesthetically pleasing set visualization. Our algorithm pays particular attention to resolving intersections between nearby shapes in a consistent manner. We compare SimpleSets to the state-of-the-art set visualizations using standard datasets from the literature.

Isolines visually characterize scalar fields by connecting all points of the same value by a closed curve at repeated intervals. They work only as a set which gives the viewer an indication of the shape of the underlying field. Hence, when simplifying isolines it is important that the correspondence - the harmony - between adjacent isolines is preserved whenever it is present. The majority of state-of-the-art simplification methods treat isolines independently; at best they avoid collisions between adjacent simplified isolines. A notable exception is the work by Van Goethem et al. (2021) who were the first to introduce the concept of harmony between adjacent isolines explicitly as an algorithmic design principle. They presented a proof-of-concept algorithm that harmoniously simplifies a sequence of polylines. However, the sets of isolines of scalar fields, most notably terrain, consist of closed curves which are nested in arbitrarily complex ways and not of an ordered sequence of polylines. In this paper we significantly extend the work by Van Goethem et al. (2021) to capture harmony in general sets of isolines. Our new simplification algorithm can handle sets of isolines describing arbitrary scalar fields and is more efficient, allowing us to harmoniously simplify terrain with hundreds of thousands of vertices. We experimentally compare our method to the results of Van Goethem et al. (2021) on bundles of isolines and to general simplification methods on isolines extracted from DEMs of Antartica. Our results indicate that our method efficiently preserves the harmony in the simplified maps, which are thereby less noisy, cartographically more meaningful, and easier to read.