Connect the Dots: The Reconstruction of Region Boundaries from Contour Sampling Points

  • Peer Stelldinger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7346)

Abstract

Twodimensional contour reconstruction from a set of points is a very common problem not only in computer vision. I.e. in graph theory one may ask for the minimal spanning tree or the shortest Hamiltonian graph. In psychology the question arises under which circumstances people are able to recognize certain contours given only a few points. In the context of discrete geometry, there exist a lot of algorithms for 2D contour reconstruction from sampling points. Here a commonly addressed problem is to define an algorithm for which it can be proved that the reconstuction result resembles the original contour if this has been sampled according to certain density criteria. Most of these algorithms can not properly deal with background noise like humans can do. This paper gives an overview of the most important algorithms for contour reconstruction and shows that a relatively new algorithm, called ‘cleaned refinement reduction’ is the most robust one with regard to significant background noise and even shows a reconstruction ability being similar to the one of a child at the age of 4.

Keywords

Background Noise Travel Salesman Problem Delaunay Triangulation Reduction Algorithm Voronoi Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Althaus, E., Mehlhorn, K.: TSP-based curve reconstruction in polynomial time. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 686–695 (2000)Google Scholar
  2. 2.
    Amenta, N., Bern, M., Eppstein, D.: The crust and the β-skeleton: Combinatorial curve reconstruction. Graph. Models and Image Proc. 60(2), 125–135 (1998)CrossRefGoogle Scholar
  3. 3.
    Amenta, N., Bern, M., Kamvysselis, M.: A new Voronoi-based surface reconstruction algorithm. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interact. Techn., pp. 415–421 (1998)Google Scholar
  4. 4.
    Attali, D.: r-Regular shape reconstruction from unorganized points. In: Proceedings of the 13th Annual ACM Symposium on Comput. Geom., pp. 248–253 (1997)Google Scholar
  5. 5.
    Barlow, H.: The efficiency of detecting changes of density in random dot patterns. Vision Research 18(6), 637–650 (1978)CrossRefGoogle Scholar
  6. 6.
    Bernardini, F., Bajaj, C.: Sampling and reconstructing manifolds using alpha-shapes. In: Proc. 9th Canad. Conf. Comput. Geom. (1997)Google Scholar
  7. 7.
    Dey, T., Kumar, P.: A simple provable algorithm for curve reconstruction. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discr. Algorithms, pp. 893–894. Society for Industrial and Applied Mathematics, Philadelphia (1999)Google Scholar
  8. 8.
    Dey, T., Mehlhorn, K., Ramos, E.: Curve reconstruction: Connecting dots with good reason. In: Proceedings of the 15th Annual Symposium on Computational Geometry, pp. 197–206. ACM, New York (1999)Google Scholar
  9. 9.
    Edelsbrunner, H.: The union of balls and its dual shape. Discrete and Computational Geometry 13(1), 415–440 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Jaromczyk, J., Toussaint, G.: Relative neighborhood graphs and their relatives. Proceedings of the IEEE 80(9), 1502–1517 (1992)CrossRefGoogle Scholar
  11. 11.
    Mukhopadhyay, A., Das, A.: An RNG-based heuristic for curve reconstruction. In: 3rd International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2006, pp. 246–251 (2006)Google Scholar
  12. 12.
    O’Rourke, J., Booth, H., Washington, R.: Connect-the-dots: a new heuristic. Computer Vision, Graphics, and Image Processing 39(2), 258–266 (1987)MATHCrossRefGoogle Scholar
  13. 13.
    Rosenberg, B., Langridge, D.: A computational view of perception. Perception 2(4) (1973)Google Scholar
  14. 14.
    Stelldinger, P., Köthe, U., Meine, H.: Topologically Correct Image Segmentation Using Alpha Shapes. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 542–554. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Stelldinger, P., Tcherniavski, L.: Contour Reconstruction for Multiple 2D Regions Based on Adaptive Boundary Samples. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 266–279. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Tcherniavski, L., Bähnisch, C., Meine, H., Stelldinger, P.: How to define a locally adaptive sampling criterion for topologically correct reconstruction of multiple regions. Pattern Recognition Letters 33(11), 1451–1459 (2012)CrossRefGoogle Scholar
  17. 17.
    Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recognition 12(4), 261–268 (1980)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Uttal, D., Gregg, V., Tan, L., Chamberlin, M., Sines, A.: Connecting the dots: Children’s use of a systematic figure to facilitate mapping and search. Developmental Psychology 37(3), 338–350 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peer Stelldinger
    • 1
  1. 1.International Computer Science Institute (ICSI)BerkeleyUSA

Personalised recommendations