Skip to main content
SpringerLink
Log in

Automatic surface reconstruction with alpha-shape method

  • original article
  • Published:
The Visual Computer Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, we present an algorithm for the reconstruction of piecewise linear surfaces from unorganized sample points with an improved α-shape. Alpha-shape provides a nice mathematical definition of the “shape” of a set of points, but the selection of an α value is tricky in surface reconstruction. F or solving this problem and the non-uniform distribution, we scale the α ball according to the point’s density. The method discussed in this paper might be applied for surface reconstruction, and the process is fully automatic.

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

  1. Edelsbrunner H, Mücke EP (1994) Three dimensional alpha shapes. ACM Trans Graph 13(1):43–72

    Article  Google Scholar 

  2. Bernardini F, Bajaj C, Chen J, Schikore D (1999) Automatic Reconstruction of 3D CAD Models form Digital Scans. Int J Comput Geom Appl 9(4&5):327–370

    Google Scholar 

  3. Hoppe H, DeRose T, Duchamp T, McDonald J, Stuelzle W (1992) Surface reconstruction from unorganized points. Comput Graph 26(2):71–78

    Article  Google Scholar 

  4. Bajaj CL, Coyle EJ, Lin K-N (1996) Arbitrary topology shape reconstruction from planar cross sections. Graph Models Image Process 58:524–543

    Article  Google Scholar 

  5. Amenta N, Bern M, Kamvysselis M, (1998) A new Voronoi-based surface reconstruction algorithm. Proc. SIGGRAPH ’98, pp 415–421

  6. Mencl R (1995) A Graph-Theoretic Approach to Surface Reconstruction. Research Report # 568, Department of Computer Science, University of Dortmund, In: Proceedings of the Eurographics Annual Conference 1995, Maastricht, Comput Graph Forum 14(3):445–456

  7. O’Rourke J (1981) Polyhedra of minimal area as 3D object models. In: Proc. Of the International Joint Conference on Artificial Intelligence, pp 664–666

  8. Boissonnat JD (1984) Geometric structures for three-Dimensional shape representation. ACM Trans Graph 3(4):266–286

    Article  Google Scholar 

  9. Veltkamp RC (1992) Closed object boundaries from scattered points. Ph.D Thesis, Center for Mathematics and Computer Science, Amsterdam

  10. Attene M, Spagnuolo M (2000) Automatic Surface Reconstruction from point sets in space. Comput Graph Forum (Procs. of EUROGRAPHICS 2000) 19(3):457–465

    Google Scholar 

  11. Edelsbrunner H, Kirkpatrick DG, Seidel R (1983) On the shape of a set of points in the plane. IEEE Trans Inf Theory IT 29:551–559

    Article  MathSciNet  Google Scholar 

  12. Edelsbrunner H (1992) Weighted alpha shapes. Technical Report UIUC-DCS-R-92-1760. Department of Computer Science, University of Illinois, Urbana Champagne, IL

  13. Melkemi M (1997) A-shapes of a finite point set. In: 13th ACM Sympsium on Computational Geometry, Nice, France, pp 367–369

  14. Melkemi M (2003) Three-dimensional shapes of a finite set of points. Int J Patt Recog Artificial Intel 17(2):301–318

    Article  Google Scholar 

  15. Guo B, Menon J, Willette B (1997) Surface Reconstruction Using Alpha Shapes. Comput Graph Forum 16(4):177–190

    Article  Google Scholar 

  16. Teichmann M, Capps M (1998) Surface Reconstruction with Anisotropic Density-Scaled Alpha Shapes. In: Proceedings of IEEE Visualization, pp 67–72

  17. Rosenblatt M (1956) Remarks on some non-parametric estimates of a density function. Ann Math Stat 27:832–837

    Article  MathSciNet  Google Scholar 

  18. Parzen E (1962) On the estimation of a probability density function and the mode. Ann Math Stat 33:1065–1076

    Article  MathSciNet  Google Scholar 

  19. Jones MC, Marron JS, Sheather SJ (1996) A Brief Survey of Bandwidth Selection for Density Estimation. J Amer Stat Assoc 91:401–407

    Article  Google Scholar 

  20. Jones MC, Marron JS, Sheather SJ (1996) Progress in Data-Based Bandwidth Selection for kernel Density Estimation. Comput Stat 11:337–381

    Google Scholar 

  21. Cheng MY (1997) A bandwidth Selector for Local Linear Density Estimators. Ann Stat 25:1001–1013

    Article  Google Scholar 

  22. Grund B, Polzehl J (1997) Bias Corrected Bootstrap Bandwidth selection. J Nonparametric Stat 8:97–126

    Article  MathSciNet  Google Scholar 

  23. Wand MP, Jones MC (1993) Comparison of smoothing parametrizations in bivariate kernel density estimation. J Amer Stat Assoc 88:520–528

    Article  MathSciNet  Google Scholar 

  24. Ruppert D (1997) Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J Amer Stat Assoc 92:1049–1062

    Article  MathSciNet  Google Scholar 

  25. Härdle W (1991) Smoothing techniques: With Implementation in S. Springer, New York

    Google Scholar 

  26. Scott DW (1992) Multivariate density estimation: Theory, practice, and visualization. John Wiley, New York

    Google Scholar 

  27. Sheather SJ (1992) The performance of six popular bandwidth selection methods on some real data sets. Comput Stat 7:225–250

    Google Scholar 

  28. Park BU, Turlach BA (1992) Practical performance of several data driven bandwidth selectors. Comput Stat 7(3):251–270

    Google Scholar 

  29. Cao R, Cuevas A, González-Manteiga W (1994) A comparative study of several smoothing methods in density estimation. Comput Stat Data Anal 17:153–176

    Article  Google Scholar 

  30. Ciesielski Z (1991) A symptotic nonparametric spline density estimation. Probab Math Stat 12:1–24

    MathSciNet  Google Scholar 

  31. Wand MP, Marron JS, Ruppert D, (1991) Transformations in density estimation. J Amer Stat Assoc 86:343–353

    Article  MathSciNet  Google Scholar 

  32. Ruppert D, Cline DBH, (1994) Bias reduction in kernel density estimation by smoothed empirical transformations. Ann Stat 22:185–210

    Article  MathSciNet  Google Scholar 

  33. Wand MP (1992) Error analysis for general multivariate kernel estimators. J Nonparametric Stat 2:1–15

    Article  MathSciNet  Google Scholar 

  34. Ciesielski Z (1990) Asymptotic non-parametric spline density estimation in several variables. Int Stud Numer Math 94:25–53

    Article  MathSciNet  Google Scholar 

  35. O’Sullivan F, Pawitan Y, (1993) Multidimensional density estimation by tomography. J R Stat Soc B 55:509–521

    MathSciNet  Google Scholar 

  36. Sain SR, Baggerly KA, Scott DW (1994) Cross-validation of multivariate densities. J Amer Stat Assoc 89:807–817

    Article  MathSciNet  Google Scholar 

  37. Wand MP, Jones MC (1994) Multivariate plug-in bandwidth selection. Comput Stat 9:97–116

    MathSciNet  Google Scholar 

  38. Jones MC, (1990) Variable Kernel Density Estimates and Variable Kernel Density Estimates. Austral J Stat 32:361–371

    Article  Google Scholar 

  39. Loftsgaarden DO, Quesenberry CP (1965) A Nonparametric Estimate of a Multivariate Density Function. Ann Math Stat 36:1049–1051

    Article  MathSciNet  Google Scholar 

  40. Terrell GR, Scott DW (1992) Variable Kernel Density Estimation. Ann Stat 20:1236–1265

    Article  MathSciNet  Google Scholar 

  41. Matousek J (1995) On enclosing k points by a circle. Inf Process Lett 53:217–221

    Article  MathSciNet  Google Scholar 

  42. Eppstein D, Erickson J (1994) Iterated nearest neighbors and finding minimal polytopes. Discrete Comput Geom 11:321–350

    Article  MathSciNet  Google Scholar 

  43. Joe B (1991) Construction of three-dimensional Delaunay triangulations using local transformations. Comput Aided Geom Des 8(2):123–142

    Article  MathSciNet  Google Scholar 

  44. Dobkin DP, Laszlo MJ (1989) Primitives for the manipulation of three-dimensional subdivisions. Algorithmica 4(1):3–32

    Article  MathSciNet  Google Scholar 

  45. Dey TK, Goswami S, (2002) Tight Cocone: A water tight surface reconstructor. Technical Report OS-U-CISRC-12/02-TR31, The Ohio State Univ.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Engineering, Hiroshima University, Japan

    Xiaolong Xu

  2. Department of Integrated Arts & Sciences, Hiroshima University, Japan

    Koichi Harada

Authors
  1. Xiaolong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Koichi Harada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaolong Xu .

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu , X., Harada , K. Automatic surface reconstruction with alpha-shape method. Vis Comput 19, 431–443 (2003). https://doi.org/10.1007/s00371-003-0207-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-003-0207-1

Keywords

Search

Navigation