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Article

Kósa, Balázs ; Haličková-Brehovská, Jana ; Mikula, Karol
New efficient numerical method for 3D point cloud surface reconstruction by using level set methods. (English). In: Mikula, Karol (ed.): Proceedings of Equadiff 14. Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017. Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, Bratislava, 2017. pp. 387-396
MSC: 53A05, 65D17, 65M06, 65Y20
Keywords:
Point cloud, level set methods, reconstruction
Summary:
In this article, we present a mathematical model and numerical method for surface reconstruction from 3D point cloud data, using the level-set method. The presented method solves surface reconstruction by the computation of the distance function to the shape, represented by the point cloud, using the so called Fast Sweeping Method, and the solution of advection equation with curvature term, which creates the evolution of an initial condition to the final state. A crucial point for efficiency is a construction of initial condition by a simple tagging algorithm which allows us also to highly speed up the numerical scheme when solving PDEs. For the numerical discretization of the model we suggested an unconditionally stable method, in which the semi-implicit co-volume scheme is used in curvature part and implicit upwind scheme in advective part. The method was tested on representative examples and applied to real data representing the historical and cultural objects scanned by 3D laser scanners.
References:
[1] Haličková, J., Mikula, K.: Level set method for surface reconstruction and its application in surveying. Journal of Surveying Engineering 143 (3). doi:10.1061/(ASCE)SU.19435428.0000159. DOI 10.1061/(ASCE)SU.19435428.0000159
[2] Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method. Computer Vision and Image Understanding 80 (2000) 295–319. doi:10.1006/cviu.2000.0875. DOI 10.1006/cviu.2000.0875
[3] Zhao, H. K.: A fast sweeping method for eikonal equations. Mathematics of Computation 74 (2004) 603–627. doi:10.1090/S0025-5718-04-01678-3. DOI 10.1090/S0025-5718-04-01678-3 | MR 2114640 | Zbl 1070.65113
[4] Corsaro, S., Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit covolume method in 3d image segmentation. SIAM Journal on Scientific Computing 28 (6) (2006) 2248–2265. doi:10.1137/060651203. DOI 10.1137/060651203 | MR 2272260
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