Abstract
In this paper we present a novel model for computing the oriented normal field on a point cloud. Differently from previous two-stage approaches, our method integrates the unoriented normal estimation and the consistent normal orientation into one variational framework. The normal field with consistent orientation is obtained by minimizing a combination of the Dirichlet energy and the coupled-orthogonality deviation, which controls the normals perpendicular to and continuously varying on the underlying shape. The variational model leads to solving an eigenvalue problem. If unoriented normal field is provided, the model can be modified for consistent normal orientation. We also present experiments which demonstrate that our estimates of oriented normal vectors are accurate for smooth point clouds, and robust in the presence of noise, and reliable for surfaces with sharp features, e.g., corners, ridges, close-by sheets and thin structures.
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References
Alexa, A., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.: Point set surfaces. In: Proceedings of IEEE Visualization, pp. 21–28 (2001)
Alliez, P., Cohen-Steiner, D., Tong, Y., Desbrun, M.: Voronoi-based variational reconstruction of unoriented point sets. In: Proceedings of Symposium on Geometry and Processing 07 (2007)
Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. In: SCG ’98: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, pp. 39–48 (1998)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15, 1373–1396 (2002)
Carr, J., Beatson, R., Cherrie, J., Mitchell, T., Fright, W., McCallum, B., Evans, T.: Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of SIGGRAPH 2001, pp. 67–76 (2001)
Dey, T., Li, G., Sun, J.: Normal estimation for point clouds: A comparison study for a Voronoi based method, pp. 39–46 (2005)
Gross, M., Pfister, H.: Point-Based Graphics, The Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann, San Mateo (2007)
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. In: Proceedings of SIGGRAPH’92, pp. 71–78 (1992)
Huang, H., Li, D., Zhang, H., Ascher, U., Cohen-Or, D.: Consolidation of unorganized point clouds for surface reconstruction. In: SIGGRAPH Asia ’09: ACM SIGGRAPH Asia 2009 Papers, pp. 1–7 (2009)
Kazhdan, M., Bolitho, M., Hoppe, H.: Poisson surface reconstruction. In: SGP’06: Proceedings of the Fourth Eurographics Symposium on Geometry Processing, pp. 61–70 (2006)
Levin, D.: Mesh-independent surface interpolation. In: Brunnett, G., Hamann, B., Mueller, H. (eds.) Geometric Modeling for Scientific Visualization, pp. 37–49. Springer, Berlin (2003)
Li, B., Schnabel, R., Klein, R., Cheng, Z., Dang, G., Jin, S.: Robust normal estimation for point clouds with sharp features. Comput. Graph. 34(2), 94–106 (2010)
Liu, S., Wang, C.C.L.: Technical section: Orienting unorganized points for surface reconstruction. Comput. Graph. 34(3), 209–218 (2010)
Mitra, N.J., Nguyen, A., Guibas, L.: Estimating surface normals in noisy point cloud data. Int. J. Comput. Geom. Appl. 14, 261–276 (2004)
Mount, D.M., Arya, S.: Ann: A library for approximate nearest neighbor searching. http://www.cs.umd.edu/~mount/ann/. (2006)
Ohtake, Y., Belyaev, A., Seidel, H.P.: 3D scattered data interpolation and approximation with multilevel compactly supported RBFs. Graph. Models 67(3), 150–165 (2005)
OuYang, D., Feng, H.Y.: On the normal vector estimation for point cloud data from smooth surfaces. Comput. Aided Des. 37(10), 1071–1079 (2005)
Pauly, M., Keiser, R., Kobbelt, L.P., Gross, M.: Shape modeling with point-sampled geometry. ACM Trans. Graph. 22(3), 641–650 (2003)
Stewart, G.W.: Matrix Algorithms, Volume II: Eigensystems SIAM: Society for Industrial and Applied Mathematics, Philadelphia (2001)
Sun, J., Smith, M., Smith, L., Farooq, A.: Examining the uncertainty of the recovered surface normal in three light photometric stereo. Image Vis. Comput. 25(7), 1073–1079 (2007)
Yang, Z., Kim, T.W.: Moving parabolic approximation of point clouds. Comput. Aided Des. 39(12), 1091–1112 (2007)
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Wang, J., Yang, Z. & Chen, F. A variational model for normal computation of point clouds. Vis Comput 28, 163–174 (2012). https://doi.org/10.1007/s00371-011-0607-6
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DOI: https://doi.org/10.1007/s00371-011-0607-6