Local, smooth, and consistent Jacobi set simplification

Bhatia, Harsh; Wang, Bei; Norgard, Gregory; Pascucci, Valerio; Bremer, Peer-Timo

doi:10.1016/j.comgeo.2014.10.009

Title: Local, smooth, and consistent Jacobi set simplification

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Abstract

The relation between two Morse functions defined on a smooth, compact, and orientable 2-manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack fine-grained control over the process, and heavily restrict the type of simplifications possible. In this paper, we introduce a new framework that generalizes critical point cancellations in scalar functions to Jacobi set in two dimensions. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations,more »« less

Authors:: Bhatia, Harsh; Wang, Bei; Norgard, Gregory; Pascucci, Valerio; Bremer, Peer-Timo

Publication Date:: Fri May 01 00:00:00 EDT 2015

Research Org.:: Univ. of Utah, Salt Lake City, UT (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1556399

Alternate Identifier(s):: OSTI ID: 1221807; OSTI ID: 1360637

Report Number(s):: DOE-UTAH-PASCUCCI-0011
Journal ID: ISSN 0925-7721; S0925772114001163; PII: S0925772114001163

Grant/Contract Number:: AC52-07NA27344; NA0002375

Resource Type:: Published Article

Journal Name:: Computational Geometry

Additional Journal Information:: Journal Name: Computational Geometry Journal Volume: 48 Journal Issue: 4; Journal ID: ISSN 0925-7721

Publisher:: Elsevier

Country of Publication:: Netherlands

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Jacobi set; comparison of scalar functions; simplification; Morse functions; smoothness

Citation Formats


                    Bhatia, Harsh, Wang, Bei, Norgard, Gregory, Pascucci, Valerio, and Bremer, Peer-Timo. Local, smooth, and consistent Jacobi set simplification.  Netherlands: N. p., 2015. 
Web.  doi:10.1016/j.comgeo.2014.10.009.

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                    Bhatia, Harsh, Wang, Bei, Norgard, Gregory, Pascucci, Valerio, & Bremer, Peer-Timo. Local, smooth, and consistent Jacobi set simplification.  Netherlands.  https://doi.org/10.1016/j.comgeo.2014.10.009

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                    Bhatia, Harsh, Wang, Bei, Norgard, Gregory, Pascucci, Valerio, and Bremer, Peer-Timo. Fri .  
"Local, smooth, and consistent Jacobi set simplification".  Netherlands.  https://doi.org/10.1016/j.comgeo.2014.10.009.

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@article{osti_1556399,

  title        = {Local, smooth, and consistent Jacobi set simplification},

  author       = {Bhatia, Harsh and Wang, Bei and Norgard, Gregory and Pascucci, Valerio and Bremer, Peer-Timo},

  abstractNote = {The relation between two Morse functions defined on a smooth, compact, and orientable 2-manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack fine-grained control over the process, and heavily restrict the type of simplifications possible. In this paper, we introduce a new framework that generalizes critical point cancellations in scalar functions to Jacobi set in two dimensions. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some user-defined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal configuration, that is, one with no birth–death points (a birth–death point is a specific type of singularity within the Jacobi set where the level sets of the two functions and the Jacobi set have a common normal direction).},

  doi          = {10.1016/j.comgeo.2014.10.009},

  journal      = {Computational Geometry},

  number       = 4,

  volume       = 48,

  place        = {Netherlands},

  year         = {Fri May 01 00:00:00 EDT 2015},

  month        = {Fri May 01 00:00:00 EDT 2015}

}

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