Abstract
Point cloud data arises naturally from 3-dimensional scanners, LiDAR sensors, and industrial computed tomography (i.e. CT scans) among other sources. Most point clouds obtained through experimental means exhibit some level of noise, inhibiting mesh reconstruction algorithms and topological data analysis techniques. To alleviate the problems caused by noise, smoothing algorithms are often employed as a preprocessing step. Moving least squares is one such technique, however, many of these techniques are designed to work on surfaces in \(\mathbb {R}^3\). As interesting point clouds can naturally live in higher dimensions, we seek a method for smoothing higher dimensional point clouds. To this end, we turn to the distance to measure function. In this paper, we provide a theoretical foundation for studying the gradient flow induced by the squared distance to measure function, as introduced by Chazal, Cohen-Steiner, and Mérigot. In particular, we frame the gradient flow as a Filippov system and find a method for solving the squared distance to measure gradient flow, induced by the uniform empirical measure, using higher order Voronoi diagrams. In contrast to some existing techniques, this gradient flow provides a smoothing algorithm which computationally scales with dimensionality.
Similar content being viewed by others
Notes
Available at https://github.com/ponl/GradSmooth.
References
Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Point set surfaces. In: Proceedings of the Conference on Visualization (VIS’01), pp. 21–28. IEEE, Washington, DC (2001)
Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Computing and rendering point set surfaces. IEEE Trans. Vis. Comput. Graphics 9(1), 3–15 (2003)
Amenta, N., Kil, Y.J.: Defining point-set surfaces. ACM Trans. Graphics 23(3), 264–270 (2004)
Biák, M., Hanus, T., Janovská, D.: Some applications of Filippov’s dynamical systems. J. Comput. Appl. Math. 254, 132–143 (2013)
Bobenko, A.I., Schröder, P.: Discrete Willmore flow. In: Proceedings of the 3rd Eurographics Symposium on Geometry Processing (SGP’05), # 101. Eurographics Association, Aire-la-Ville (2005)
Brécheteau, C.: The DTM-signature for a geometric comparison of metric-measure spaces from samples (2017). arXiv:1702.02838
Buchet, M., Dey, T.K., Wang, J., Wang, Y.: Declutter and resample: towards parameter free denoising. In: Aronov, B., Katz, M.J. (eds.) Proceedings of the 33rd International Symposium on Computational Geometry (SoCG’17). Leibniz International Proceedings in Informatics, vol. 77, pp. 23:1–23:16. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2017)
Chazal, F., Chen, D., Guibas, L.J., Jiang, X., Sommer, C.: Data-driven trajectory smoothing. Research Report RR-7754, INRIA (2011)
Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Geometric inference for measures based on distance functions. Research Report RR-6930, INRIA (2010)
Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Geometric inference for probability measures. Found. Comput. Math. 11(6), 733–751 (2011)
Chazal, F., Fasy, B.T., Lecci, F., Michel, B., Rinaldo, A., Wasserman, L.A.: Robust topological inference: Distance to a measure and kernel distance (2014). arXiv:1412.7197
Chazal, F., Massart, P., Michel, B.: Rates of convergence for robust geometric inference. Electron. J. Stat. 10(2), 2243–2286 (2016)
Curless, B., Levoy, M.: A volumetric method for building complex models from range images. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’96), pp. 303–312. ACM, New York (1996)
Dey, T.K.: Curve and surface reconstruction: algorithms with mathematical analysis. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 23. Cambridge University Press, Cambridge (2007)
Dey, T.K., Sun, J.: An adaptive MLS surface for reconstruction with guarantees. In: Proceedings of the 3rd Eurographics Symposium on Geometry Processing (SGP’05), Art. No. 43. Eurographics Association, Aire-la-Ville (2005)
di Bernardo, M., Budd, C.J., Champneys, A., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical Sciences, vol. 163. Springer, London (2008)
di Bernardo, M., Liuzza, D.: Incremental stability of planar Filippov systems. In: Proceedings of the 2013 European Control Conference (ECC), pp. 3706–3711. IEEE, Washington, DC (2013)
Dong, M., Chou, W., Fang, B.: Underwater matching correction navigation based on geometric features using sonar point cloud data. Sci. Program. 2017, 10 (2017)
Filippov, A.F.: Differential equations with discontinuous right-hand side. Mat. Sb. (N.S.) 51(93), 99–128 (1960)
Gevaert, C., Persello, C., Sliuzas, R., Vosselman, G.: Informal settlement classification using point-cloud and image-based features from UAV data. ISPRS J. Photogramm. Remote Sens. 125, 225–236 (2017)
Guibas, L.J., Mérigot, Q., Morozov, D.: Witnessed k-distance. In: Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG’11), pp. 57–64. ACM, New York (2011)
Hu, L., Xu, X., Wang, L., Guo, N., Xie, F.: 3D registration method based on scattered point cloud from B-model ultrasound image. In: Proceedings Volume 10245, International Conference on Innovative Optical Health Science, Art. No. 102450C (2017)
Ito, T.: A Filippov solution of a system of differential equations with discontinuous right-hand sides. Econ. Lett. 4(4), 349–354 (1979)
Kantorovich, L., Rubinshtein, G.: On a space of totally additive functions. Vestn. Leningr. Univ. 13(7), 52–59 (1958) (in Russian)
Kolluri, R., Shewchuk, J.R., O’Brien, J.F.: Spectral surface reconstruction from noisy point clouds. In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (SGP’04), pp. 11–21. ACM, New York (2004)
Levin, D.: Mesh-independent surface interpolation. In: Brunnett, G., Hamann, B., Müller, H., Linsen, L. (eds.) Geometric Modeling for Scientific Visualization, pp. 37–49. Springer, Berlin (2004)
Malihi, S., Valadan Zoej, M.J., Hahn, M., Mokhtarzade, M., Arefi, H.: 3D building reconstruction using dense photogrammetric point cloud. In: Proceedings of the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. XLI-B3, pp. 71–74 (2016)
Mederos, B., Velho, L., de Figueiredo, L.H., de Figueirêdo, H.F.: Robust smoothing of noisy point clouds. In: Proceedings of the SIAM Conference on Geometric Design and Computing (2003)
Morgan, J.W., Tian, G.: Ricci flow and the Poincaré conjecture (2007). arXiv:math/0607607v2
Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40(3), 646–663 (2011)
Steer, P., Lague, D., Gourdon, A., Croissant, T., Crave, A.: 3D granulometry: grain-scale shape and size distribution from point cloud dataset of river environments. In: Proceedings of the EGU General Assembly 2016, vol. 18, EGU2016-8514 (2016)
Turk, G., Levoy, M.: Zippered polygon meshes from range images. In: Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’94), pp. 311–318. ACM, New York (1994)
White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the International Congress of Mathematics, vol. 1, pp. 525–538. Higher Education Press, Beijing (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Rights and permissions
About this article
Cite this article
O’Neil, P., Wanner, T. Analyzing the Squared Distance-to-Measure Gradient Flow System with k-Order Voronoi Diagrams. Discrete Comput Geom 61, 91–119 (2019). https://doi.org/10.1007/s00454-018-0037-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-018-0037-6