Abstract
Estimation of differential geometric properties on a discrete surface is a fundamental work in computer graphics and computer vision. In this paper, we present an accurate and robust method for estimating differential quantities from unorganized point cloud. The principal curvatures and principal directions at each point are computed with the help of partial derivatives of the unit normal vector at that point, where the normal derivatives are estimated by fitting a linear function to each component of the normal vectors in a neighborhood. This method takes into account the normal information of all neighboring points and computes curvatures directly from the variation of unit normal vectors, which improves the accuracy and robustness of curvature estimation on irregular sampled noisy data. The main advantage of our approach is that the estimation of curvatures at a point does not rely on the accuracy of the normal vector at that point, and the normal vectors can be refined in the process of curvature estimation. Compared with the state of the art methods for estimating curvatures and Darboux frames on both synthetic and real point clouds, the approach is shown to be more accurate and robust for noisy and unorganized point cloud data.
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Gross M, Pfister H. Point-Based Graphics. San Fransiso, CA: Morgan Kaufmann, 2007. 1–7
Petitjean S. A survey of methods for recovering quadrics in triangle meshes. ACM Comput Surv, 2002, 34(2): 211–262
Sander P T, Zucker S W. Inferring surface trace and differential structure from 3-D images. IEEE Trans Pattern Anal Mach Intell, 1990, 12(9): 833–854
Rusinkiewicz S. Estimating curvatures and their derivatives on triangle meshes. In: Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium. Washington: IEEE Computer Society, 2004. 486–493
Gatzke T, Grimm C. Estimating curvature on triangular meshes. Int J Shape Model, 2006, 12(1): 1–29
Magid E, Soldea O, Rivlin E. A comparison of Gaussian and mean curvature estimation methods on triangular meshes of range image data. Comput Vis Image Underst, 2007, 107(3): 139–159
Goldfeather J, Interrante V. A novel cubic-order algorithm for approximating principal direction vectors. ACM Trans Graph, 2004, 23(1): 45–63
Theisel H, Rossl C, Zayer R, et al. Normal based estimation of the curvature tensor for triangular meshes. In: PG’04: Proceedings of the Computer Graphics and Applications, 12th Pacific Conference. Washington: IEEE Computer Society, 2004. 288–297
Batagelo H C, Wu S. Estimating curvatures and their derivatives on meshes of arbitrary topology from sampling directions. Vis Comput, 2007, 23(9): 803–812
Kalogerakis E, Simari P, Nowrouzezahrai D, et al. Robust statistical estimation of curvature on discretized surfaces. In: Proceedings of the Fifth Eurographics symposium on Geometry Processing. Aire-la-Ville: Eurographics Association, 2007. 13–22
Taubin G. Estimating the tensor of curvature of a surface from a polyhedral approximation. In: Proceedings of the Fifth International Conference on Computer Vision. Washington: IEEE Computer Society, 1995. 902–907
Do Carmo M. Differential Geometry of Curves and Surfaces. London: Prentice-Hall, 1976. 503
Meek D S, Walton D J. On surface normal and Gaussian curvature approximations given data sampled from a smooth surface. Comput Aided Geom Des, 2000, 17(6): 521–543
Meyer P S M, Desbrun M, Barr A, et al. Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III. London: Springer-Verlag, 2003. 35–57
Cohen-Steiner D, Morvan J M. Restricted delaunay triangulations and normal cycle. In: SCG’03: Proceedings of the Nineteenth Annual Symposium on Computational Geometry. New York: ACM, 2003. 312–321
McIvor A, Valkenburg R. A comparison of local surface geometry estimation methods. Mach Vision Appl, 1997, 10(1): 17–26
Cazals F, Pouget M. Estimating differential quantities using polynomial fitting of osculating jets. In: SGP’03: Proceedings of the 2003 Eurographics/ACM Siggraph Symposium on Geometry Processing. Aire-la-Ville: Eurographics Association, 2003. 177–187
Cazals F, Pouget M. Estimating differential quantities using polynomial fitting of osculating jets. Comput Aided Geom Des, 2005, 22(2): 121–146
Chen X, Schmitt F. Intrinsic surface properties from surface triangulation. In: Proceedings of the Second European Conference on Computer Vision. London: Springer-Verlag, 1992. 739–743
Hameiri E, Shimshoni I. Estimating the principal curvatures and the Darboux frame from real 3D range data. In: Proceedings of the 3D Data Processing, Visualization, and Transmission. Washington: IEEE Computer Society, 2002. 258–267
Hoppe H, DeRose T, Duchamp T, et al. Surface reconstruction from unorganized points. In: Siggraph 1992. New York: ACM, 1992. 71–78
Mitra N, Nguyen A. Estimating surface normals in noisy point cloud data. In: SCG ′03: Proceedings of the Nineteenth Annual Symposium on Computational Geometry. New York: ACM, 2003. 322–328
Shen C, O’Brien J F, Shewchuk J R. Interpolating and approximating implicit surfaces from polygon soup. In: Siggraph 2004. New York: ACM, 2004. 896–904
Adamson A, Alexa M. Point-sampled cell complexes. ACM Trans Graph, 2006, 25(3): 671–680
Guennebaud G, Gross M. Algebraic point set surfaces. ACM Trans Graph, 2007, 26(3): 23
Cheng Z, Zhang X, Fourcaud T. Tree skeleton extraction from a single range image. In: PMA06: Proceedings of Second International Symposium on Plant Growth Modeling, Simulation, Visualization and Applications. Los Alamitos: IEEE Computer Society, 2007. 274–281
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Supported in part by the National Natural Science Foundation of China (Grant Nos. 60672148, 60872120), the National High-Tech Research & Development Program of China (Grant Nos. 2006AA01Z301, 2008AA01Z301), and Beijing Municipal Natural Science Foundation (Grant No. 4062033)
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Cheng, Z., Zhang, X. Estimating differential quantities from point cloud based on a linear fitting of normal vectors. Sci. China Ser. F-Inf. Sci. 52, 431–444 (2009). https://doi.org/10.1007/s11432-009-0061-5
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DOI: https://doi.org/10.1007/s11432-009-0061-5