Reeb spaces of piecewise linear mappings

Edelsbrunner, Herbert; Harer, John; Patel, Amit

doi:10.1145/1377676.1377720

Cited by 93 publications

(93 citation statements)

References 11 publications

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“…Equally, Edelsbrunner et al [12] extracted the Jacobi set from discrete samples of two scalar field functions, but did not extract the entire Reeb space. Instead, the Reeb space was formulated later [14], and can now be seen to be identical to the Stein factorization [24]: in neither case was a practical computation given.…”

Section: Related Workmentioning

confidence: 99%

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Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Sakurai

Saeki

Carr

et al. 2016

IEEE Trans. Visual. Comput. Graphics

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Fig. 1. Singular fibers of the tangle cube function f (x, y, z) = (−x 4 − y 4 − z 4 + 5(x 2 + y 2 + z 2 ) − 10, z) after interactive perturbation. The domain view on the left visualizes isosurfaces of individual axis function in yellow and green with their intersection along the current fiber in red. In the center, the range view shows a cross mark at the function value defining the current fiber. Critical function values are shown in different colors according to the topological types of their fibers. Arranged around this view are thumbnails of the domain space that pop up when a value is selected. The Reeb space view shows the connectivity of connected components in the fiber, constructed by using a 3D layout of the Joint Contour Net.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 R R R 3 → R R R 2 . 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.Index Terms-Singular fibers, fiber topology, mathematical visualization, design study.• Daisuke Sakurai is with the

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Section: Related Workmentioning

confidence: 99%

“…In short, this analysis depends on understanding the Reeb space [24,14]. Although singular fiber theorists have considered the concept [19], sketching the topology has not been a common strategy when understanding a function.…”

Section: Drawing the Fibersmentioning

confidence: 99%

Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Sakurai

Saeki

Carr

et al. 2016

IEEE Trans. Visual. Comput. Graphics

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“…Work to date has included the concept of the Jacobi Set [13], and of the Reeb space [14] fields to a set of fields. However, there are few computationally effective tools for analysis of the exact Reeb space generated from a set of samples defined over a mesh.…”

Section: The Joint Contour Netmentioning

confidence: 99%

Skeletons for distributed topological computation

Duke

Hosseini

2015

Proceedings of the 4th ACM SIGPLAN Workshop on Functional High-Performance Computing

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Parallel implementation of topological algorithms is highly desirable, but the challenges, from reconstructing algorithms around independent threads through to runtime load balancing, have proven to be formidable. This problem, made all the more acute by the diversity of hardware platforms, has led to new kinds of implementation platform for computational science, with sophisticated runtime systems managing and coordinating large threadcounts to keep processing elements heavily utilized. While simpler and more portable than direct management of threads, these approaches still entangle program logic with resource management. Similar kinds of highly parallel runtime system have also been developed for functional languages. Here, however, language support for higher-order functions allows a cleaner separation between the algorithm and 'skeletons' that express generic patterns of parallel computation. We report results on using this technique to develop a distributed version of the Joint Contour Net, a generalization of the Contour Tree to multifields. We present performance comparisons against a recent Haskell implementation using shared-memory parallelism, and initial work on a skeleton for distributed memory implementation that utilizes an innovative strategy to reduce inter-process communication overheads.

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“…In practice, the analysis of multivariate data samples begins with extracting the Jacobi set, which has been tackled by Edelsbrunner et al [6,7]. They successfully developed an algorithm for extracting such Jacobi sets from functions to R 2 by identifying sample points where the gradients of the two corresponding scalar functions are parallel to each other.…”

Section: Analyzing Samples Of a Function R N → R Mmentioning

confidence: 99%

“…For a function f : R n → R m , by contracting each connected component of its inverse images to a point, we get a space R f , which is called the Reeb space [7] of f . This is expected to play an important role similar to that of a contour tree.…”

Section: Analyzing Samples Of a Function R N → R Mmentioning

confidence: 99%

Visual data mining based on differential topology: a survey

Saeki

Takahashi

2014

Pac. J. Math. Ind.

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In this article, we describe techniques for visual data mining based on differential topology. Data scientists have been working long on the analysis of data obtained from a wide variety of sources. The data is often represented as discrete sample points of a function R n → R m , while the dimensions of the data domain and range have rapidly increased due to recent advancement in computational power and measurement technology. Mathematical formulations of differential topology effectively help us to analyze such data in a hierarchical fashion and to visually extract significant features from it. We present new algorithms and application examples as well as existing ones, including the authors' recent results, so that we can fully elucidate the potential power of this approach especially in data visualization.

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Reeb spaces of piecewise linear mappings

Cited by 93 publications

References 11 publications

Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Skeletons for distributed topological computation

Visual data mining based on differential topology: a survey

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Reeb spaces of piecewise linear mappings

Cited by 93 publications

References 11 publications

Interactive Visualization for Singular Fibers of Functions f : R3 → R2

Interactive Visualization for Singular Fibers of Functions f : R3 → R2

Skeletons for distributed topological computation

Visual data mining based on differential topology: a survey

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Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Interactive Visualization for Singular Fibers of Functions f : R³ → R²