2D Embeddings of Multi-dimensional Partitionings

TL;DR

This paper introduces an algorithm to create 2D visualizations of multi-dimensional partitionings that preserve topology, segment sizes, and boundary lengths, aiding in the analysis of complex high-dimensional data.

Contribution

The paper presents a novel algorithm for embedding multi-dimensional partitionings into 2D while maintaining topological and size-related properties.

Findings

Effective preservation of segment sizes and boundary lengths in embeddings

Successful application to 3D spatial and parameter space partitionings

Numerical evaluation shows how embedding quality varies with dimensionality and segments

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.