On variational and PDE-based methods for accurate distance function estimation. (English) Zbl 1453.65042
Comput. Math. Math. Phys. 59, No. 12, 2009-2016 (2019) and Zh. Vychisl. Mat. Mat. Fiz. 59, No. 12, 2077-2085 (2019).
Summary: 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.
MSC:
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 65K10 | Numerical optimization and variational techniques |
| 35J60 | Nonlinear elliptic equations |
References:
| [1] | Gibou, F.; Fedkiw, R.; Osher, S., A review of level-set methods and some recent applications, J. Comput. Phys., 353, 82-109 (2017) · Zbl 1380.65196 |
| [2] | Tucker, P. G., Hybrid Hamilton-Jacobi-Poisson wall distance function model, Comput. Fluids, 44, 130-142 (2011) · Zbl 1271.76258 |
| [3] | Roget, B.; Sitaraman, J., Wall distance search algorithm using voxelized marching spheres, J. Comput. Phys., 241, 76-94 (2013) · Zbl 1349.76643 |
| [4] | Biswas, A.; Shapiro, V.; Tsukanov, I., Heterogeneous material modeling with distance fields, Comput. Aided Geom. Des., 21, 215-242 (2004) · Zbl 1069.65508 |
| [5] | Xia, H.; Tucker, P. G., Fast equal and biased distance fields for medial axis transform with meshing in mind, Appl. Math. Model., 35, 5804-5819 (2011) · Zbl 1228.76070 |
| [6] | Babuška, I.; Banerjee, U.; Osborn, J. E., Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12, 1-125 (2003) · Zbl 1048.65105 |
| [7] | Freytag, M.; Shapiro, V.; Tsukanov, I., Finite element analysis in situ, Finite Elem. Anal. Des., 47, 957-972 (2011) |
| [8] | Cadena, C.; Carlone, L.; Carrillo, H.; Latif, Y.; Scaramuzza, D.; Neira, J.; Reid, I.; Leonard, J. J., Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age, IEEE Trans. Rob., 32, 1309-1332 (2016) |
| [9] | Calakli, F.; Taubin, G., SSD: Smooth signed distance surface reconstruction, Comput. Graphics Forum, 30, 1993-2002 (2011) |
| [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). · Zbl 1334.68274 |
| [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] | Belyaev, A.; Fayolle, P.-A.; Pasko, A., Signed L_p-distance fields, Comput.-Aided Des., 45, 523-528 (2013) |
| [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] | Belyaev, A.; Fayolle, P.-A., On variational and PDE-based distance function approximations, Comput. Graphics Forum, 34, 104-118 (2015) |
| [15] | Lee, B.; Darbon, J.; Osher, S.; Kang, M., Revisiting the redistancing problem using the Hopf-Lax formula, J. Comput. Phys., 330, 268-281 (2017) · Zbl 1378.35292 |
| [16] | Royston, M.; Pradhana, A.; Lee, B.; Chow, Y. T.; Yin, W.; Teran, J.; Osher, S., Parallel redistancing using the Hopf-Lax formula, J. Comput. Phys., 365, 7-17 (2018) · Zbl 1395.65118 |
| [17] | Glowinski, R., Modeling, Simulation, and Optimization for Science and Technology (2014) |
| [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). · Zbl 1442.35117 |
| [19] | T. Bhattacharya, E. DiBenedetto, and J. Manfredi, “Limits as p → ∞ of Δ_pu_p = f and related extremal problems,” Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale Nonlinear PDEs (1989), pp. 15-68. |
| [20] | Kawohl, B., On a family of torsional creep problems, J. Reine Angew. Math., 410, 1-22 (1990) · Zbl 0701.35015 |
| [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] | Pólya, G.; Szego, G., Isoperimetric Inequalities in Mathematical Physics (1951), Princeton: Princeton University Press, Princeton · Zbl 0044.38301 |
| [23] | Fayolle, P.-A.; Belyaev, A., p-Laplace diffusion for distance function estimation, optimal transport approximation, and image enhancement, Comput. Aided Geom. Des., 67, 1-20 (2018) · Zbl 1505.65308 |
| [24] | Butzer, P.; Jongmans, F., P.L. Chebyshev (1821-1894): A guide to his life and work, J. Approximation Theory, 96, 111-138 (1999) · Zbl 0934.01017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
