title: Interactive Visualization for Singular Fibers of Functions f : R³ → R².
abstract: Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from scientific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers-inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualizations. This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R³ → R². This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Validation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians.
authors.forename: Daisuke
authors.initials: D
authors.lastname: Sakurai
authors.forename: Osamu
authors.initials: O
authors.lastname: Saeki
authors.forename: Hamish
authors.initials: H
authors.lastname: Carr
authors.forename: Hsiang-Yun
authors.initials: HY
authors.lastname: Wu
authors.forename: Takahiro
authors.initials: T
authors.lastname: Yamamoto
authors.forename: David
authors.initials: D
authors.lastname: Duke
authors.forename: Shigeo
authors.initials: S
authors.lastname: Takahashi
mesh_terms: D000465:Algorithms; D003196:Computer Graphics; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D012107:Research Design; D013499:Surface Properties; D014584:User-Computer Interface