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On Variational and PDE-Based Methods for Accurate Distance Function Estimation

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Abstract

A new variational problem for accurate approximation of the distance from the boundary of a domain is proposed and studied. It is shown that the problem can be efficiently solved by the alternating direction method of multipliers. Links between this problem and \(p\)-Laplacian diffusion are established and studied. Advantages of the proposed distance function estimation method are demonstrated by numerical experiments.

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REFERENCES

  1. F. Gibou, R. Fedkiw, and S. Osher, “A review of level-set methods and some recent applications,” J. Comput. Phys. 353, 82–109 (2017).

    Article  MathSciNet  Google Scholar 

  2. P. G. Tucker, “Hybrid Hamilton–Jacobi–Poisson wall distance function model,” Comput. Fluids 44 (1), 130–142 (2011).

    Article  Google Scholar 

  3. B. Roget and J. Sitaraman, “Wall distance search algorithm using voxelized marching spheres,” J. Comput. Phys. 241, 76–94 (2013).

    Article  MathSciNet  Google Scholar 

  4. A. Biswas, V. Shapiro, and I. Tsukanov, “Heterogeneous material modeling with distance fields,” Comput. Aided Geom. Des. 21, 215–242 (2004).

    Article  MathSciNet  Google Scholar 

  5. H. Xia and P. G. Tucker, “Fast equal and biased distance fields for medial axis transform with meshing in mind,” Appl. Math. Model. 35, 5804–5819 (2011).

    Article  MathSciNet  Google Scholar 

  6. I. Babuška, U. Banerjee, and J. E. Osborn, “Survey of meshless and generalized finite element methods: A unified approach,” Acta Numer. 12, 1–125 (2003).

    Article  MathSciNet  Google Scholar 

  7. M. Freytag, V. Shapiro, and I. Tsukanov, “Finite element analysis in situ,” Finite Elem. Anal. Des. 47 (9), 957–972 (2011).

    Article  Google Scholar 

  8. C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, I. Reid, and J. J. Leonard, “Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age,” IEEE Trans. Rob. 32, 1309–1332 (2016).

    Article  Google Scholar 

  9. F. Calakli and G. Taubin, “SSD: Smooth signed distance surface reconstruction,” Comput. Graphics Forum 30 (7), 1993–2002 (2011).

    Article  Google Scholar 

  10. M. Zollhöfer, A. Dai, M. Innmann, C. Wu, M. Stamminger, C. Theobalt, and M. Nießner, “Shading-based refinement on volumetric signed distance functions,” ACM Trans. Graphics 34 (4), 96:1–96:14 (2015).

  11. B. Zhu, M. Skouras, D. Chen, and W. Matusik, “Two-scale topology optimization with microstructures,” ACM Trans. Graphics 36 (5), 36:1–36:14 (2017).

  12. A. Belyaev, P.-A. Fayolle, and A. Pasko, “Signed Lp-distance fields,” Comput.-Aided Des. 45, 523–528 (2013).

    Article  MathSciNet  Google Scholar 

  13. K. Crane, C. Weischedel, and M. Wardetzky, “Geodesics in heat: A new approach to computing distance based on heat flow,” ACM Trans. Graphics 32, 152:1–152:11 (2013).

  14. A. Belyaev and P.-A. Fayolle, “On variational and PDE-based distance function approximations,” Comput. Graphics Forum 34 (8), 104–118 (2015).

    Article  Google Scholar 

  15. B. Lee, J. Darbon, S. Osher, and M. Kang, “Revisiting the redistancing problem using the Hopf–Lax formula,” J. Comput. Phys. 330, 268–281 (2017).

    Article  MathSciNet  Google Scholar 

  16. M. Royston, A. Pradhana, B. Lee, Y. T. Chow, W. Yin, J. Teran, and S. Osher, “Parallel redistancing using the Hopf–Lax formula,” J. Comput. Phys. 365, 7–17 (2018).

    Article  MathSciNet  Google Scholar 

  17. R. Glowinski, “On alternating direction methods of multipliers: A historical perspective,” in Modeling, Simulation, and Optimization for Science and Technology, Ed. by W. Fitzgibbon, Y. A. Kuznetsov, P. Neittaanmäki, and O. Pironneau (Springer, 2014), pp. 59–82.

    Google Scholar 

  18. A. Belyaev and P.-A. Fayolle, “A variational method for accurate distance function estimation,” in Lecture Notes in Computational Science and Engineering, Vol. 131: Numerical Geometry, Grid Generation, and Scientific Computing, NUMGRID 2018/Voronoi 150 (Springer International, Berlin, 2019).

  19. T. Bhattacharya, E. DiBenedetto, and J. Manfredi, “Limits as p → ∞ of Δpup = f and related extremal problems,” Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale Nonlinear PDEs (1989), pp. 15–68.

  20. B. Kawohl, “On a family of torsional creep problems,” J. Reine Angew. Math. 410 (1), 1–22 (1990).

    MathSciNet  MATH  Google Scholar 

  21. N. A. Wukie and P. D. Orkwis, “A p-Poisson wall distance approach for turbulence modeling,” in 23rd AIAA Computational Fluid Dynamics Conference (2017).

  22. G. Pólya and G. Szego, Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, 1951).

    Book  Google Scholar 

  23. P.-A. Fayolle and A. Belyaev, “p-Laplace diffusion for distance function estimation, optimal transport approximation, and image enhancement,” Comput. Aided Geom. Des. 67, 1–20 (2018).

    Article  MathSciNet  Google Scholar 

  24. P. Butzer and F. Jongmans, “P.L. Chebyshev (1821–1894): A guide to his life and work,” J. Approximation Theory 96, 111–138 (1999).

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

The Gargoyle mesh model is courtesy of AIM@SHAPE and the Lucy mesh model is courtesy of the Stanford Graphics Laboratory.

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Authors and Affiliations

  1. Computer Graphics Laboratory, University of Aizu, Aizu-Wakamatsu, Japan

    P.-A. Fayolle

  2. Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences Heriot-Watt University, Edinburgh, UK

    A. G. Belyaev

Authors
  1. P.-A. Fayolle
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  2. A. G. Belyaev
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Correspondence to P.-A. Fayolle or A. G. Belyaev.

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Fayolle, PA., Belyaev, A.G. On Variational and PDE-Based Methods for Accurate Distance Function Estimation. Comput. Math. and Math. Phys. 59, 2009–2016 (2019). https://doi.org/10.1134/S0965542519120066

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