Skip to main content
Log in

On finding a shortest isothetic path and its monotonicity inside a digital object

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Algorithms for finding shortest paths and their numerous variants have been studied extensively over several decades in graph theory, computational geometry, and in operations research. Still, design of such algorithms poses newer challenges in various emerging areas of image analysis and computer vision. This mandates further investigation of application-specific shortest-path problems in the framework of digital geometry. We present here an efficient combinatorial algorithm for finding a shortest isothetic path (SIP) between two grid points in a digital object without any hole, such that the SIP lies entirely inside the object. Our algorithm first determines a maximum-area isothetic polygon that is inscribed within the object. Certain combinatorial rules are then used to construct the SIP on the basis of its constituent monotone sub-paths, the number of which is invariant regardless of the choice of SIP between two given grid points. For a given grid size, the proposed algorithm runs in \(O(n\log n)\) time, where n is the number of grid points appearing on the boundary of the inscribed polygon. Experimental results show the effectiveness of the algorithm and its further prospects in shape analysis.

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

Access this article

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. Amini, A.A., Weymouth, T.E., Jain, R.C.: Using dynamic programming for solving variational problems in vision. IEEE Trans. Pattern Anal. Mach. Intell. 12(9), 855–867 (1990)

    Article  Google Scholar 

  2. Arkin, E., Mitchell, J., Piatko, C.: Minimum-link watchman tours. Inf. Process. Lett. 86, 203–207 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. de Berg, M.: On rectilinear link distance. Comput. Geom. Theory Appl. 1(1), 13–34 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry Algorithms And Applications. Springer-Verlag, Heidelberg (2008)

    MATH  Google Scholar 

  5. de Berg, M., van Kreveld, M., Nilsson, B.J., Overmars, M.H.: Finding shortest paths in the presence of orthogonal obstacles using a combined l 1 and link metric. In: Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory, LNCS, vol. 447, pp. 213–224. Springer-Verlag, Bergen (1990)

    Google Scholar 

  6. Berger, A., Razouk, N., Angelides, G.: Distance- and curvature-constrained shortest paths and an application in mission planning. In: Proceedings of the 44th Annual Southeast Regional Conference, ACM-SE 44, pp. 766–767. ACM, New York (2006)

    Book  Google Scholar 

  7. Biswas, A., Bhowmick, P., Bhattacharya, B.B.: TIPS: On Finding a Tight Isothetic Polygonal Shape Covering a 2D Object. In: Proceedings of the 14th Scandinavian Conference on Image Analysis, LNCS, vol. 3540, pp. 930–939. Springer-Verlag, Joensuu (2005)

    Google Scholar 

  8. Biswas, A., Bhowmick, P., Bhattacharya, B.B.: Construction of isothetic covers of a digital object: A combinatorial approach. J. Vis. Commun. Image Represent. 21(4), 295–310 (2010)

    Article  Google Scholar 

  9. Buckley, M., Yang, J.: Regularised shortest-path extraction. Pattern Recog. Lett. 18(7), 621–629 (1997)

    Article  Google Scholar 

  10. Bülow, T., Klette, R.: Approximation of 3d shortest polygons in simple cube curves. In: Digital and Image Geometry, Lecture Notes in Computer Science, pp. 285–298. Springer (2001)

  11. Chin, W.P., Ntafos, S.: The zookeeper route problem. Inf. Sci. 63(3), 245–259 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Clarkson, K.L., Kapoor, S., Vaidya, P.: Rectilinear shortest paths through polygonal obstacles in \(O(n (\log n)^{2})\) time. In: Proceedings of the 3rd Annual Symposium on Computational Geometry, SCG ’87, pp. 251–257. ACM, New York (1987)

    Google Scholar 

  13. Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 252–258 (2004)

    Article  Google Scholar 

  14. Culberson, J.C., Reckhow, R.A.: Orthogonally convex coverings of orthogonal polygons without holes. J. Comput. Syst. Sci. 39(2), 166–204 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dutt, M., Biswas, A., Bhowmick, P., Bhattacharya, B.B.: On finding shortest isothetic path inside a digital object. In: Barneva, R.P. (ed.) Proceedings of the 15th International Workshop on Combinatorial Image Analysis: IWCIA’12, LNCS, vol. 7655, pp. 1–15. Springer-Verlag, Austin (2012)

    Google Scholar 

  17. Farin, G., Hoschek, J., Kim, M.S.: Handbook of Computer Aided Geometric Design. Elsevier Science B. V., North Holland (2002)

    MATH  Google Scholar 

  18. Garey, M., Johnson, D.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)

    Book  MATH  Google Scholar 

  20. Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods and red-blues separation. In: Proceedings of the 5th Annual International Conference on Computing and Combinatorics, COCOON ’99, pp. 473–482. Springer-Verlag, Berlin, Heidelberg (1999)

    Book  Google Scholar 

  21. Huang, T., Li, L., Young, E.F.Y.: On the construction of optimal obstacle-avoiding rectilinear steiner minimum trees. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 30(5), 718–731 (2011)

    Article  Google Scholar 

  22. Inkulu, R., Kapoor, S.: Planar rectilinear shortest path computation using corridors. Comput. Geom. Theory Appl. 42, 873–884 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karpinski, M., Zelikovsky, A.: New approximation algorithms for the steiner tree problem. J. Comb. Optim. 1(4), 47–65 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Picture Analysis. Morgan Kaufmann, San Francisco (2004)

    Google Scholar 

  25. Larson, R.C., Li, V.O.: Finding minimum rectilinear distance paths in the presence of barriers. Networks 11, 285–304 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, F., Klette, R.: Finding the shortest path between two points in a simple polygon by applying a rubberband algorithm. In: Proceedings of the First Pacific Rim conference on Advances in Image and Video Technology, PSIVT’06, vol. 4319, pp. 280–291. Springer-Verlag, Berlin, Heidelberg (2006)

    Book  Google Scholar 

  27. Lin, G.H., Xue, G.: A linear-time heuristic for rectilinear steiner trees. In: Proceedings., First Great Lakes Symposium on VLSI, pp. 152–156. IEEE (1991)

  28. Lin, G.H., Xue, G.: Balancing steiner minimum trees and shortest-path trees in the rectilinear plane. In: Proceedings of the 1999 IEEE International Symposium on Circuits and Systems, ISCAS’99, vol. 6, pp. 117–120. IEEE (1999)

  29. Ling, H., Jacobs, D.W.: Shape classification using the inner-distance. IEEE Trans. Pattern Anal. Mach. Intell. 29, 286–299 (2007)

    Article  Google Scholar 

  30. Lozano-Perez, T., Wesley, M.A.: An algorithm for planning collision-free paths among polyhedral obstacles. Mag. Commun. ACM 22(10), 560–570 (1979)

    Article  Google Scholar 

  31. Ntafos, S.: Watchman routes under limited visibility. Comput. Geom. Theory Appl. 1, 149–170 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  32. de Resende, P.J., Lee, D.T., Wu, Y.F.: Rectilinear shortest paths with rectangular barriers. In: Proceedings of the 1st Annual Symposium on Computational Geometry, SCG ’85, pp. 204–213. ACM, New York (1985)

    Book  Google Scholar 

  33. Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, STOC ’84, vol. 15, pp. 193–215. ACM, New York (1986)

    Google Scholar 

  34. Sheng, Y.B., Ke, W.Y., Ping, L.J.: Finding the shortest path between two points in a simple polygon by applying a rubberband algorithm. In: WRI World Congress on Software Engineering, 2009, WCSE ’09, vol. 1, pp. 239–243 (2009)

  35. Sloboda, F., Zatko, B., Ferianc, P.: Minimum perimeter polygon and its application. In: Klette, R., Kropatsch, W. (eds.) Theoretical Foundations of Computer Vision, vol. 69, pp 59–70. Mathematical Research, Akademie Verlag, Berlin (1992)

    Google Scholar 

  36. Sloboda, F., Zatko, B., Stoer, J.: On approximation of planar one-dimensional continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp 113–160. Springer, Singapore (1998)

    Google Scholar 

  37. Sun, C., Pallottino, S.: Circular shortest path in images. Pattern Recog. 36(3), 709–719 (2003)

    Article  Google Scholar 

  38. Tan, X.: Approximation algorithms for the watchman route and zookeeper’s problems. Discret. Appl. Math. 136(2–3), 363–376 (2004)

    Article  MATH  Google Scholar 

  39. Tan, X., Hirata, T.: Finding shortest safari routes in simple polygons. Inf. Process. Lett. 87(4), 179–186 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wei, X.: Monotone path queries and monotone subdivision problems in polygonal domains, Ph.D. thesis, Hong Kong University of Science and Technology (2010)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Partha Bhowmick.

Additional information

A preliminary version of this paper appeared in the Proc. IWCIA 2012 [16]

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dutt, M., Biswas, A., Bhowmick, P. et al. On finding a shortest isothetic path and its monotonicity inside a digital object. Ann Math Artif Intell 75, 27–51 (2015). https://doi.org/10.1007/s10472-014-9421-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-014-9421-y

Keywords

Navigation