Functional Decomposition for Bundled Simplification of Trail Sets

Abstract

Abstract-Bundling visually aggregates curves to reduce clutter and help finding important patterns in trail-sets or graph drawings. We propose a new approach to bundling based on functional decomposition of the underling dataset. We recover the functional nature of the curves by representing them as linear combinations of piecewise-polynomial basis functions with associated expansion coefficients. Next, we express all curves in a given cluster in terms of a centroid curve and a complementary term, via a set of so-called principal component functions. Based on the above, we propose a two-fold contribution: First, we use cluster centroids to design a new bundling method for 2D and 3D curve-sets. Secondly, we deform the cluster centroids and generate new curves along them, which enables us to modify the underlying data in a statistically-controlled way via its simplified (bundled) view. We demonstrate our method by applications on real-world 2D and 3D datasets for graph bundling, trajectory analysis, and vector field and tensor field visualization.

Route‐Aware Edge Bundling for Visualizing Origin‐Destination Trails in Urban Traffic

“…Geometry‐based methods cluster trails using a skeleton‐like control mesh that specifies how similar edges are routed. Such methods differ mainly in how they construct control meshes, e.g ., using hierarchical graph drawing [Hol06], triangulation [ZYC∗08,LBA10b], complex polygons [CZQ∗08], and functional decomposition [HPNT18]. Control meshes make geometry‐based methods flexible.…”

“…Constrained Bundling: Specialized methods have been proposed to bundle data with specific spatial constraints. These include minimizing ambiguities in visually following O‐to‐D connections [LLCM12, BRH∗17]; separating trails having different directions [SHH11,PHT15]; and bundling specific types of data such as paths constrained to a 3D curved surface [LBA10a] and connection paths in the human brain [BSL∗14, YSD∗17,HPNT18]. Closely related to our scope, KDEEB [HET12] presents an experiment where bundles are repelled by so‐called obstacles by a force field equal to the obstacles' distance transform gradient.…”

Geospatial Information Visualization and Extended Reality Displays

“…Despite some progress, many questions remain (see the "Visualizing Movement" section). Some potential solutions such as using abstraction and schematization when visualizing urban datasets in Digital Earth can be found in the fields of data and information visualization (Hurter et al 2018).…”

“…The reviewed approaches can be used with digital globes, or a future Digital Earth with virtual dashboards through which one can integrate analytical operations within an AR or VR system. Hurter et al (2018) show how interactions in a 3D immersive environment (see the "Immersive Technologies-From Augmented to Virtual Reality" section) can enable the exploration of large numbers of individual 3D trajectories. Next, we review the current state of the art in immersive technologies.…”

An Evolutionary Algorithm for an Optimization Model of Edge Bundling

“…Regarding the bundling methodologies, an edge bundling method can construct bundles in an explicit or in an implicit way [15]. In explicit approaches, the edges are first clustered and then a rendering module draws each group of edges [7,[16][17][18][19].…”