2D Embeddings of Multi-dimensional Partitionings

TL;DR

An algorithm for computing 2D embeddings of multi-dimensional partitioning and optimizing the area sizes and joint boundary lengths of the embedded segments to match the respective sizes and lengths in the multi-dimensional domain is proposed.

Abstract

Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models, where each segment of the partitioning represents a region of similar model behavior. Having computed a partitioning, one is commonly interested in understanding how large the segments are and which segments lie next to each other. While visual representations of 2D domain partitionings that reveal sizes and neighborhoods are straightforward, this is no longer the case when considering multi-dimensional domains of three or more dimensions. We propose an algorithm for computing 2D embeddings of multi-dimensional partitionings. The embedding shall have the following properties: It shall maintain the topology of the partitioning and optimize the area sizes and joint boundary lengths of the embedded segments to match the respective sizes and lengths in the multi-dimensional domain. We demonstrate the effectiveness of our approach by applying it to different use cases, including the visual exploration of 3D spatial domain segmentations and multi-dimensional parameter space partitionings of simulation ensembles. We numerically evaluate our algorithm with respect to how well sizes and lengths are preserved depending on the dimensionality of the domain and the number of segments.

Figures (17)

• Figure 1: (a) Topology is preserved, if the number of segment changes (grey crosses) when clockwise traversing the Moore neighborhood of the current cell (grey boundary) is $\leq3$. Otherwise, topology might change. (b) To prevent topology violations by simultaneously changing neighboring cells, only non-adjacent cells may change simultaneously.
• Figure 2: To improve the embedding, crossings between segments that become unnecessary can be removed. These crossings are either duplicates (grey circle) or crossings where one side is not required anymore to preserve the topology (black circles).
• Figure 3: Keeping the edge crossing position fixed might result in unnecessarily complex embeddings (a), while allowing them to move leads to more compact segments (b). Both examples have been created based on the same initialization.
• Figure 4: We create a height profile with plateaus in the center of the segments and valleys for separators between segments. Here, we set the height $h=1$ and the width for height changes $w=0.5r$, where $r$ is the number of pixels for each cell of the cellular automaton.
• Figure 5: Shading the segments and visually encoding special features like edge crossings and background areas allows for a visualization of segments independent of the color coding.