Abstract
In this article, we present and discuss three statistical methods for surface reconstruction. A typical input to a surface reconstruction technique consists of a large set of points that has been sampled from a smooth surface and contains uncertain data in the form of noise and outliers. We first present a method that filters out uncertain and redundant information yielding a more accurate and economical surface representation. Then we present two methods, each of which converts the input point data to a standard shape representation; the first produces an implicit representation while the second yields a triangle mesh.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Silva, C.T.: Point set surfaces. IEEE Visualization 2001 pp. 21–28 (2001)
Amenta, N., Choi, S., Kolluri, R.: The power crust. In: Proceedings of 6th ACM Symposium on Solid Modeling, pp. 249–260 (2001)
Angelidis, A., Cani, M.P.: Adaptive implicit modeling using subdivision curves and surfaces as skeletons. In: SMA ’02: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 45–52. ACM, Boston (2002)
Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)
Arya, S., Mount, D., Netanyahu, N., Silverman, R., Wu, A.: An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. J. ACM 45(6), 891–923 (1998)
Barhak, J., Fischer, A.: Adaptive reconstruction of freeform objects with 3D SOM neural network grids. In: PG ’01: Proceedings of the 9th Pacific Conference on Computer Graphics and Applications, p. 97 (2001)
Barhak, J., Fischer, A.: Parameterization and reconstruction from 3D scattered points based on neural network and PDE techniques. IEEE Trans. Visual. Comput. Graph. 7(1), 1–16 (2001)
Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, New York (1995)
Blanz, V., Mehl, A., Vetter, T., Seidel, H.P.: A statistical method for robust 3D surface reconstruction from sparse data. In: Y. Aloimonos, G. Taubin (eds.) 2nd International Symposium on 3D Data Processing, Visualization, and Transmission, 3DPVT 2004, pp. 293–300. IEEE (2004)
Bloomenthal, J., Wyvill, B. (eds.): Introduction to Implicit Surfaces. Kaufmann, San Francisco (1997)
Botsch, M., Kobbelt, L.: High-quality point-based rendering on modern GPUs. In: PG ’03: Proceedings of the 11th Pacific Conference on Computer Graphics and Applications, pp. 335–343. IEEE Computer Society, Washington, DC (2003)
Botsch, M., Kobbelt, L.: Real-time shape editing using radial basis functions. Comput. Graph. Forum 24(3), 611–621 (2005)
Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: SIGGRAPH ’01, pp. 67–76. ACM, Boston (2001)
Chen, S., Wigger, J.: Fast orthogonal least squares algorithm for efficient subset model selection. IEEE Trans. Signal Process. 43(7), 1713–1715 (1995)
Cheng, Y.: Mean shift, mode seeking, and clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17, 790–799 (1995)
Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optimiz. 6, 418–445 (1996)
Comaniciu, D., Meer, P.: Mean shift: A robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 24(5), 603–619 (2002)
Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)
Dey, T.K., Goswami, S.: Tight Cocone: A water-tight surface reconstructor. In: Proceedings of 8th ACM Symposium Solid Modeling Applications, pp. 127–134 (2003)
Dey, T.K., Sun, J.: Adaptive MLS surfaces for reconstruction with guarantees. In: Eurographics Symposium on Geometry Processing 2005, pp. 43–52 (2005)
Fenn, M., Steidl, G.: Robust local approximation of scattered data. In: R. Klette, R. Kozera, L. Noakes, J. Weickert (eds.) Geometric Properties from Incomplete Data, pp. 317–334. Springer, Berlin Heidelberg New York (2005)
Fleishman, S., Cohen-Or, D., Silva, C.T.: Robust moving least-squares fitting with sharp features. ACM Trans. Graph. 24(3), 544–552 (2005)
Fukunaga, K., Hostetler, L.D.: The estimation of the gradient of a density function with applications in pattern recognition. IEEE Trans. Inform. Theory 21, 32–40 (1975)
Girosi, F.: An equivalence between sparse approximation and support vector machines. Neural Comput. 10(6), 1455–1480 (1998)
Golub, G., VanLoan, G.: Matrix Computations, 2nd edn. John Hopkins University Press, Baltimore, MD (1989)
Gu, P., Yan, X.: Neural network approach to the reconstruction of freeform surfaces for reverse engineering. Comput. Aided Des. 27(1), 59–64 (1995)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, Berlin Heidelberg New York (2001)
Hoffmann, M., Várady, L.: Free-form surfaces for scattered data by neural networks. J. Geom. Graph. 2, 1–6 (1998)
Iglesias, A., Echevarría, G., Gálvez, A.: Functional networks for B-spline surface reconstruction. Future Gener. Comput. Syst. 20(8), 1337–1353 (2004)
Ivrissimtzis, I., Lee, Y., Lee, S., Jeong, W.K., Seidel, H.P.: Neural mesh ensembles. In: Y. Aloimonos, G. Taubin (eds.) 2nd International Symposium on 3D Data Processing, Visualization, and Transmission, 3DPVT 2004, pp. 308–315. IEEE (2004)
Ivrissimtzis, I.P., Jeong, W.K., Seidel, H.P.: Using growing cell structures for surface reconstruction. In: Shape Modeling International, pp. 78–88, 288. IEEE Computer Society, Washington, DC (2003)
Jeong, W.K., Ivrissimtzis, I.P., Seidel, H.P.: Neural meshes: Statistical learning based on normals. In: Pacific Conference on Computer Graphics and Applications, pp. 404–408. IEEE Computer Society, Washington, DC (2003)
Jolliffe, I.T.: Principal component analysis. In: Principal Component Analysis. Springer, Berlin Heidelberg New York (1986)
Kanai, T., Ohtake, Y., Kase, K.: Hierarchical error-driven approximation of impplicit surfaces from polygonal meshes. In: Proceedings of Geometry Processing, pp. 21–30 (2006)
Kazhdan, M.M.: Reconstruction of solid models from oriented point sets. In: Symposium on Geometry Processing, pp. 73–82 (2005)
Knopf, G.K., Sangole, A.: Interpolating scattered data using 2D self-organizing feature maps. Graph. Models 66(1), 50–69 (2004)
Lange, C., Polthier, K.: Anisotropic fairing of point sets. Comput. Aided Geom. Des. 22(7), 680–692 (2005)
Levin, D.: The approximation power of moving least-squares. Math. Comput 67(224), 1517–1531 (1998)
Linsen, L.: Point cloud representation. Tech. Rep. 2001-3, Fakultät für Informatik, Universität Karlsruhe (2001)
Lloyd, S.: An algorithm for vector quantizer design. IEEE Trans. Commun. 28(7), 84–95 (1982)
Mederos, B., Velho, L., de Figueiredo, L.H.: Smooth surface reconstruction from noisy clouds. J. Brazilian Comput. Soc. (2004)
Mitra, N.J., Guibas, L., Giesen, J., Pauly, M.: Probabilistic fingerprints for shapes. In: Symposium on Geometry Processing, pp. 121–130 (2006)
Morse, B.S., Yoo, T.S., Chen, D.T., Rheingans, P., Subramanian, K.R.: Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. In: Proc. of SMI, pp. 89–98. IEEE Computer Society, Washington, DC (2001)
Mostafa, M.G.H., Yamany, S.M., Farag, A.A.: Integrating shape from shading and range data using neural networks. CVPR 02, 2015–2020 (1999)
Mumford, D.: The dawning of the age of stochasticity. In: V.I. Arnold, M. Atiyah, P. Lax, B. Mazur (eds.) Mathematics: Frontiers and Perspectives 2000, pp. 197–218. American Mathematical Society, Providence, RI (1999)
Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.P.: Multi-level partition of unity implicits. ACM Trans. Graph. 22(3), 463–470 (2003)
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)
Ohtake, Y., Belyaev, A.G., Alexa, M.: Sparse low-degree implicits with applications to high quality rendering, feature extraction, and smoothing. In: Symposium on Geometry Processing, pp. 149–158 (2005)
Ohtake, Y., Belyaev, A.G., Seidel, H.P.: An integrating approach to meshing scattered point data. In: ACM Symposium on Solid and Physical Modeling (2005)
Parzen, E.: On the estimation of a probability density function and the mode. Ann. Math. Stat. 33, 1065–1076 (1962)
Patané, G.: SIMS: a multi-level approach to surface reconstruction with sparse implicits. In: IEEE International Conference on Shape Modeling and Applications, pp. 222–233 (2006)
Pauly, M., Keiser, R., Kobbelt, L.P., Gross, M.: Shape modeling with point-sampled geometry. In: Proceedings of SIGGRAPH 2003 22, 641–650 (2003)
Pauly, M., Mitra, N.J., Guibas, L.J.: Uncertainty and variability in point cloud surface data. In: Eurographics Symposium on Point-Based Graphics, pp. 77–84. Zurich, Switzerland (2004)
Pfister, H., Zwicker, M., Baar, J.V., Gross, M.: Surfels: Surface elements as rendering primitives. In: Proceedings of ACM SIGGRAPH 2000, pp. 335–342 (2000)
Poggio, T., Smale, S.: The mathematics of learning: dealing with data. Amer. Math. Soc. Notice 50(5), 537–544 (2003)
Rosenblatt, M.: Remarks on some non-para-metric estimates of a density function. Ann. Math. Stat. 27, 832–837 (1956)
Rusinkiewicz, S., Levoy, M.: QSplat: a multiresolution point rendering system for large meshes. In: Proceedings of ACM SIGGRAPH 2000, pp. 343–352 (2000)
Saleem, W.: A flexible framework for learning-based surface reconstruction. Dissertation, Computer Science Department, University of Saarland, Saarbrücken (2004)
Samozino, M., Alexa, M., Alliez, P., Yvinec, M.: Reconstruction with voronoi central radial basis functions. In: Proceedings of Eurographics, pp. 51–60 (2006, in press)
Schall, O., Belyaev, A.G., Seidel, H.P.: Robust filtering of noisy scattered point data. In: M. Pauly, M. Zwicker (eds.) Eurographics Symposium on Point-Based Graphics 2005, pp. 71–77. Stony Brook, NY (2005)
Schölkopf, B., Smola, A.J.: Learning with Kernels. MIT Press, Cambridge, MA (2002)
Schölkopf, B., Steinke, F., Blanz, V.: Object correspondence as a machine learning problem. In: ICML ’05: Proceedings of the 22nd International Conference on Machine Learning, pp. 776–783. ACM, Boston (2005)
Schölkopf, B., Giesen, J., Spalinger, S.: Kernel methods for implicit surface modeling. In: Advances in Neural Information Processing Systems 17, pp. 1193–1200. MIT Press, Cambridge, MA (2005)
Shen, C., O’Brien, J.F., Shewchuk, J.R.: Interpolating and approximating implicit surfaces from polygon soup. ACM Trans. Graph 23(3), 896–904 (2004)
Steinke, F., Schölkopf, B., Blanz, V.: Support vector machines for 3D shape processing. Comput. Graph. Forum 24(3), 285–294 (2005)
Super, B.J.: Learning chance probability functions for shape retrieval or classification. In: CVPRW ’04: Proceedings of the 2004 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW’04) vol. 6, p. 93. IEEE Computer Society, Washington, DC (2004)
Taubin, G.: A signal processing approach to fair surface design. In: SIGGRAPH ’95: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, pp. 351–358 (1995)
Tobor, I., Reuter, P., Schlick, C.: Multi-scale reconstruction of implicit surfaces with attributes from large unorganized point sets. In: Proceedings of SMI, pp. 19–30 (2004)
Turk, G., O’Brien, J.F.: Shape transformation using variational implicit functions. In: Proceedings of SIGGRAPH ’99, pp. 335–342. ACM/Addison-Wesley, Boston (1999)
Turk, G., O’Brien, J.F.: Modelling with implicit surfaces that interpolate. ACM Trans. Graph. 21(4), 855–873 (2002)
Várady, L., Hoffmann, M., Kovács, E.: Improved free-form modeling of scattered data by dynamic neural networks. J. Geom. Graph. 3, 177–181 (1999)
Vázquez, P.P., Feixas, M., Sbert, M., Heidrich, W.: Automatic view selection using viewpoint entropy and its applications to image-based modelling. Comput. Graph. Forum 22(4), 689–700 (2003)
Verri, A., Camastra, F.: A novel kernel method for clustering. IEEE Trans. Pattern Anal. Mach. Intell. 27(5), 801–804 (2005)
Walder, C., Schölkopf, B., Chapelle, O.: Implicit surface modelling with a globally regularised basis of compact support. In: Computer Graphics Forum (Proc. Eurographics). Blackwell, Oxford (2006)
Willis, A., Speicher, J., Cooper, D.B.: Surface sculpting with stochastic deformable 3D surfaces. ICPR 2, 249–252 (2004)
Xie, H., McDonnell, K.T., Qin, H.: Surface reconstruction of noisy and defective data sets. In: IEEE Visualization, pp. 259–266 (2004)
Yang, M., Lee, E.: Improved neural network model for reverse engineering. Int. J. Prod. Res. 38(9), 2067–2078 (2000)
Yu, Y.: Surface reconstruction from unorganized points using self-organizing neural networks. In: IEEE Visualization, Conference Proceedings, pp. 61–64 (1999)
Zwicker, M., Pfister, H., van Baar, J., Gross, M.: Surface splatting. In: Proceedings of SIGGRAPH 2001, pp. 371–378. ACM, New York (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Saleem, W., Schall, O., Patanè, G. et al. On stochastic methods for surface reconstruction. Visual Comput 23, 381–395 (2007). https://doi.org/10.1007/s00371-006-0094-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-006-0094-3