Skip to main content
SpringerLink
Log in

Efficient reconstruction of non-simple curves

  • Published:
Journal of Zhejiang University SCIENCE C Aims and scope Submit manuscript

Abstract

We present a novel algorithm to reconstruct curves with self-intersections and multiple parts from unorganized strip-shaped points, which may have different local shape scales and sampling densities. We first extract an initial curve, a graph composed of polylines, to model the different structures of the points. Then a least-squares optimization is used to improve the geometric approximation. The initial curve is extracted in three steps: anisotropic farthest point sampling with an adaptable sphere, graph construction followed by non-linear region identification, and edge refinement. Our algorithm produces faithful results for points sampled from non-simple curves without pre-segmenting them. Experiments on many simulated and real data demonstrate the efficiency of our method, and more faithful curves are reconstructed compared to other existing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albou, L.P., Schwarz, B., Poch, O., Wurtz, J.M., Moras, D., 2008. Defining and characterizing protein surface using alpha shapes. Proteins, 76(1):1–12. [doi:10.1002/prot.22301]

    Article  Google Scholar 

  • Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T., 2001. Point Set Surfaces. Proc. Conf. on Visualization, p.21–28.

  • Bernardini, F., Bajaj, C.L., 1997. Sampling and Reconstructing Manifolds Using α-Shapes. Proc. 9th Canadian Conf. on Computational Geometry, p.193–198.

  • Cao, J., Tagliasacchi, A., Olson, M., Zhang, H., Su, Z., 2010. Point Cloud Skeletons via Laplacian Based Contraction. Shape Modeling Int. Conf., p.187–197. [doi:10.1109/SMI.2010.25]

  • Cheng, S., Funke, S., Golin, M., Kumar, P., Poon, S., Ramos, E., 2005. Curve reconstruction from noisy samples. Comput. Geom., 31(1–2):63–100. [doi:10.1016/j.comgeo.2004.07.004]

    Article  MATH  MathSciNet  Google Scholar 

  • De-Alarcón, P.A., Pascual-Montano, A., Gupta, A., Carazo, J.M., 2002. Modeling shape and topology of low-resolution density maps of biological macromolecules. Biophys. J., 83(2):619–632. [doi:10.1016/S0006-3495(02)75196-5]

    Article  Google Scholar 

  • Delicado, P., 2001. Another look at principal curves and surfaces. J. Multivar. Anal., 77(1):84–116. [doi:10.1006/jmva.2000.1917]

    Article  MATH  MathSciNet  Google Scholar 

  • Dey, T.K., Kumar, P., 1999. A Simple Provable Algorithm for Curve Reconstruction. Proc. 10th Annual ACMSIAM Symp. on Discrete Algorithms, p.893–894.

  • Edelsbrunner, H., Mucke, E.P., 1994. Three-dimensional alpha shapes. ACM Trans. Graph., 13(1):43–72. [doi:10.1145/174462.156635]

    Article  MATH  Google Scholar 

  • Einbeck, J., Tutz, G., Evers, L., 2005. Local principal curves. Statist. Comput., 15(4):301–313. [doi:10.1007/s11222-005-4073-8]

    Article  MathSciNet  Google Scholar 

  • Funke, S., Ramos, E.A., 2001. Reconstructing a Collection of Curves with Corners and Endpoints. Proc. 12th Annual ACM-SIAM Symp. on Discrete Algorithms, p.344–353.

  • Gold, C.M., Snoeyink, J., 2001. A one-step crust and skeleton extraction algorithm. Algorithmica, 30(2):144–163. [doi:10.1007/s00453-001-0014-x]

    Article  MATH  MathSciNet  Google Scholar 

  • Hastie, T., Stuetzle, W., 1989. Principal curves. J. Am. Statist. Assoc., 84(406):502–516. [doi:10.2307/2289936]

    Article  MATH  MathSciNet  Google Scholar 

  • Hiyoshi, H., 2006. Closed Curve Reconstruction from Unorganized Sample Points. Proc. 3rd Int. Symp. on Voronoi Diagrams in Science and Engineering, p.122–131. [doi:10.1109/ISVD.2006.14]

  • Kégl, B., Krzyzak, A., 2002. Piecewise linear skeletonization using principal curves. IEEE Trans. Pattern Anal. Mach. Intell., 24(1):59–74. [doi:10.1109/34.982884]

    Article  Google Scholar 

  • Kégl, B., Krzyzak, A., Linder, T., Zeger, K., 2000. Learning and design of principal curves. IEEE Trans. Pattern Anal. Mach. Intell., 22(3):281–297. [doi:10.1109/34.841759]

    Article  Google Scholar 

  • Krasnoshchekov, D.N., Polishchuk, V., 2008. Robust Curve Reconstruction with k-Order alpha-Shapes. IEEE Int. Conf. on Shape Modeling and Applications, p.279–280. [doi:10.1109/SMI.2008.4548006]

  • Lee, I.K., 1999. Curve reconstruction from unorganized points. Comput. Aid. Geom. Des., 17(2):161–177. [doi:10.1016/S0167-8396(99)00044-8]

    Article  MATH  Google Scholar 

  • Levin, D., 1998. The approximation power of moving least squares. Math. Comput., 67(224):1517–1531. [doi:10.1090/S0025-5718-98-00974-0]

    Article  MATH  Google Scholar 

  • Lin, H., Chen, W., Wang, G., 2005. Curve reconstruction based on an interval B-spline curve. Vis. Comput., 21(6):418–427. [doi:10.1007/s00371-005-0304-4]

    Article  Google Scholar 

  • Lipman, Y., Cohen-Or, D., Levin, D., Tal-Ezer, H., 2007. Parameterization-free projection for geometry reconstruction. ACM Trans. Graph., 26(3):22. [doi:10.1145/1276377.1276405]

    Article  Google Scholar 

  • Liu, X., Jia, Y., 2005. A bottom-up algorithm for finding principal curves with applications to image skeletonization. Pattern Recogn., 38(7):1079–1085. [doi:10.1016/j.patcog.2004.11.016]

    Article  MATH  MathSciNet  Google Scholar 

  • Ohbuchi, R., Takei, T., 2003. Shape-Similarity Comparison of 3D Models Using alpha Shapes. Proc. 11th Pacific Conf. on Computer Graphics and Applications, p.293–302. [doi:10.1109/PCCGA.2003.1238271]

  • Pauly, M., Gross, M., Kobbelt, L.P., 2002. Efficient Simplification of Point-Sampled Surfaces. Proc. Conf. on Visualization, p.163–170. [doi:10.1109/VISUAL.2002.1183771]

  • Ruiz, O., Vanegas, C., Cadavid, C., 2007. Principal component and Voronoi skeleton alternatives for curve reconstruction from noisy point sets. J. Eng. Des., 18(5):437–458. [doi:10.1080/09544820701403771]

    Article  Google Scholar 

  • Shapira, L., Shamir, A., Cohen-Or, D., 2008. Consistent mesh partitioning and skeletonisation using the shape diameter function. Vis. Comput., 24(4):249–259. [doi:10.1007/s00371-007-0197-5]

    Article  Google Scholar 

  • Singh, R., Cherkassky, V., Papanikolopoulos, N.P., 2000. Self-organizing maps for the skeletonization of sparse shapes. IEEE Trans. Neur. Networks, 11(1):241–248. [doi:10.1109/72.822527]

    Article  Google Scholar 

  • Song, Y., 2010. Boundary fitting for 2D curve reconstruction. Vis. Comput., 26(3):187–204. [doi:10.1007/s00371-009-0395-4]

    Article  Google Scholar 

  • Sorkine, O., Cohen-Or, D., 2004. Least-Squares Meshes. Proc. Shape Modeling Applications, p.191–199. [doi:10.1109/SMI.2004.1314506]

  • Tagliasacchi, A., Zhang, H., Cohen-Or, D., 2009. Curve skeleton extraction from incomplete point cloud. ACM Trans. Graph., 28(3), Article 71, p.1–9. [doi:10.1145/1531326.1531377]

    Article  Google Scholar 

  • Tharpey, T., Flury, B., 1996. Self-consistency: a fundamental concept in statistics. Statist. Sci., 11(3):229–243. [doi:10.1214/ss/1032280215]

    Article  MathSciNet  Google Scholar 

  • Tibshirani, R., 1992. Principal curves revisited. Statist. Comput., 2(4):183–190. [doi:10.1007/BF01889678]

    Article  Google Scholar 

  • Verbeek, J.J., Vlassis, N., Kröse, B., 2002. A k-segments algorithm for finding principal curves. Pattern Recogn. Lett., 23(8):1009–1017. [doi:10.1016/S0167-8655(02)00032-6]

    Article  MATH  Google Scholar 

  • Wang, H., Lee, T.C.M., 2006. Automatic parameter selection for a k-segments algorithm for computing principal curves. Pattern Recogn. Lett., 27(10):1142–1150. [doi:10.1016/j.patrec.2005.12.005]

    Article  Google Scholar 

  • Wang, W., Pottmann, H., Liu, Y., 2006. Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph., 25(2):214–238. [doi:10.1145/1138450.1138453]

    Article  Google Scholar 

  • Yang, Z., Deng, J., Chen, F., 2005. Fitting unorganized point clouds with active implicit B-spline curves. Vis. Comput., 21(8–10):831–839. [doi:10.1007/s00371-005-0340-0]

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Yuan-di Zhao, Jun-jie Cao, Zhi-xun Su & Zhi-yang Li

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, China

    Jun-jie Cao

Authors
  1. Yuan-di Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-jie Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi-xun Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhi-yang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun-jie Cao.

Additional information

Project supported by the National Natural Science Foundation of China-Guangdong Joint Fund (No. U0935004), the National Natural Science Foundation of China (No. 60873181), and the Fundamental Research Funds for the Central Universities, China

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, Yd., Cao, Jj., Su, Zx. et al. Efficient reconstruction of non-simple curves. J. Zhejiang Univ. - Sci. C 12, 523–532 (2011). https://doi.org/10.1631/jzus.C1000308

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1631/jzus.C1000308

Key words

CLC number

Search

Navigation