Distance between extremum graphs

Narayanan, Vidya; Thomas, Dilip Mathew; Natarajan, Vijay

doi:10.1109/pacificvis.2015.7156386

Cited by 35 publications

(27 citation statements)

References 28 publications

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“…Scale‐space based methods and topological techniques have also been used in this context (e.g. [WDC*07, NTN15]). Schneider et al .…”

Section: Related Workmentioning

confidence: 99%

Temporally Dense Exploration of Moving and Deforming Shapes

Frey

2020

Computer Graphics Forum

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We present our approach for the dense visualization and temporal exploration of moving and deforming shapes from scientific experiments and simulations. Our image space representation is created by convolving a noise texture along shape contours (akin to LIC). Beyond indicating spatial structure via luminosity, we additionally use colour to depict time or classes of shapes via automatically customized maps. This representation summarizes temporal evolution, and provides the basis for interactive user navigation in the spatial and temporal domain in combination with traditional renderings. Our efficient implementation supports the quick and progressive generation of our representation in parallel as well as adaptive temporal splits to reduce overlap. We discuss and demonstrate the utility of our approach using 2D and 3D scalar fields from experiments and simulations.

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“…Scale‐space based methods and topological techniques have also been used in this context (e.g. [WDC*07, NTN15]). Schneider et al .…”

Section: Related Workmentioning

confidence: 99%

Temporally Dense Exploration of Moving and Deforming Shapes

Frey

2020

Computer Graphics Forum

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“…Recently, the e-interleaving distance was introduced by Chazal et al 22 as yet another means to compare topological structures. Adaptions were proposed for specific structures, including merge trees 23 , the Reeb graph 24 , and extremum graphs and cluster trees 25,26 . However, no practically viable algorithms are known for approaches using this distance measure as its naive implementation leads to exponential complexity, which is the reason why we did not consider this distance measure in our effort.…”

Section: Scalar Functionsmentioning

confidence: 99%

Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data

Beketayev

Yeliussizov

Morozov

et al. 2018

Int. J. Comput. Geom. Appl.

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This paper studies the influence of the definition of neighborhoods and methods used for creating point connectivity on topological analysis of scalar functions. It is assumed that a scalar function is known only at a finite set of points with associated function values. In order to utilize topological approaches to analyze the scalar-valued point set, it is necessary to choose point neighborhoods and, usually, point connectivity to meaningfully determine critical-point behavior for the point set. Two distances are used to measure the difference in topology when different point neighborhoods and means to define connectivity are used: (i) the bottleneck distance for persistence diagrams and (ii) the distance between merge trees. Usually, these distances define how different scalar functions are with respect to their topology. These measures, when properly adapted to point sets coupled with a definition of neighborhood and connectivity, make it possible to understand how topological characteristics depend on connectivity. Noise is another aspect considered. Five types of neighborhoods and connectivity are discussed: (i) the Delaunay triangulation; (ii) the relative neighborhood graph; (iii) the Gabriel graph; (iv) the [Formula: see text]-nearest-neighbor (KNN) neighborhood; and (v) the Vietoris–Rips complex. It is discussed in detail how topological characterizations depend on the chosen connectivity.

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“…The robustness of this approach has been improved by Ji and Shen [12] with a global optimization correspondence algorithm based on the Earth Mover's Distance. Scale-space based methods and topological techniques have also been used here (e.g., [13,14]). Schneider et al [15] compare scalar fields on the basis of the largest contours.…”

Section: Visualization Of Spatio-temporal Datamentioning

confidence: 99%

Sampling and Estimation of Pairwise Similarity in Spatio-Temporal Data Based on Neural Networks

Frey

2017

Informatics

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Increasingly fast computing systems for simulations and high-accuracy measurement techniques drive the generation of time-dependent volumetric data sets with high resolution in both time and space. To gain insights from this spatio-temporal data, the computation and direct visualization of pairwise distances between time steps not only supports interactive user exploration, but also drives automatic analysis techniques like the generation of a meaningful static overview visualization, the identification of rare events, or the visual analysis of recurrent processes. However, the computation of pairwise differences between all time steps is prohibitively expensive for large-scale data not only due to the significant cost of computing expressive distance between high-resolution spatial data, but in particular owing to the large number of distance computations (O(|T| 2 )), with |T| being the number of time steps). Addressing this issue, we present and evaluate different strategies for the progressive computation of similarity information in a time series, as well as an approach for estimating distance information that has not been determined so far. In particular, we investigate and analyze the utility of using neural networks for estimating pairwise distances. On this basis, our approach automatically determines the sampling strategy yielding the best result in combination with trained networks for estimation. We evaluate our approach with a variety of time-dependent 2D and 3D data from simulations and measurements as well as artificially generated data, and compare it against an alternative technique. Finally, we discuss prospects and limitations, and discuss different directions for improvement in future work.

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Distance between extremum graphs

Cited by 35 publications

References 28 publications

Temporally Dense Exploration of Moving and Deforming Shapes

Temporally Dense Exploration of Moving and Deforming Shapes

Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data

Sampling and Estimation of Pairwise Similarity in Spatio-Temporal Data Based on Neural Networks

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