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Optimal General Simplification of Scalar Fields on Surfaces

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Topological and Statistical Methods for Complex Data

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

We present a new combinatorial algorithm for the optimal general topological simplification of scalar fields on surfaces. Given a piecewise linear (PL) scalar field f, our algorithm generates a simplified PL field g that provably admits critical points only from a constrained subset of the singularities of f while minimizing the distance | | f − g | |  ∞ for data-fitting purpose. In contrast to previous algorithms, our approach is oblivious to the strategy used for selecting features of interest and allows critical points to be removed arbitrarily and additionally minimizes the distance | | f − g | |  ∞ in the PL setting. Experiments show the generality and efficiency of the algorithm and demonstrate in practice the minimization of | | f − g | |  ∞ .

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Notes

  1. 1.

    Note that the extremity s of the super-arc (m, s) admits a forward integral line ending in m.

  2. 2.

    Since f(s 0) < f(s 1) and t 0 > t 1, then \(\vert f(s_{0}) - g(s_{0})\vert = t_{0} - f(s_{0}) > t_{1} - f(s_{1}) = \vert f(s_{1}) - g(s_{1})\vert \). Thus, if T 0 and T 1 are the only sub-trees, \(\vert \vert f - g\vert \vert _{\infty } = t_{0} - f(s_{0})\).

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Acknowledgements

Data-sets are courtesy of AIM@SHAPE. This research is partially funded by the RTRA Digiteo through the unTopoVis project (2012-063D). The authors thank Hamish Carr for insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France

    Julien Tierny

  2. CNRS, UMR 7606, LIP6, F-75005, Paris, France

    Julien Tierny

  3. CNRS LTCI, Telecom ParisTech, 46 Rue Barrault, 75013, Paris, France

    Julien Tierny

  4. Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46, Rue Barrault, 75013, Paris, France

    David Günther

  5. School of Computing and SCI Institute, University of Utah, 72 S Central Campus Dr., Salt Lake City, UT, 84112, USA

    Valerio Pascucci

Authors
  1. Julien Tierny
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  2. David Günther
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  3. Valerio Pascucci
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Corresponding author

Correspondence to Julien Tierny .

Editor information

Editors and Affiliations

  1. Sandia National Laboratories, MS 9152, Livermore, California, USA

    Janine Bennett

  2. CEA CESTA, CS 60001, Le Barp CEDEX, France

    Fabien Vivodtzev

  3. School of Computing and SCI Institute, University of Utah, Salt Lake City, Utah, USA

    Valerio Pascucci

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Tierny, J., Günther, D., Pascucci, V. (2015). Optimal General Simplification of Scalar Fields on Surfaces. In: Bennett, J., Vivodtzev, F., Pascucci, V. (eds) Topological and Statistical Methods for Complex Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44900-4_4

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