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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 1, pp 71–83 | Cite as

Morphological Decomposition and Compression of Binary Images via a Minimum Set Cover Algorithm

  • Davod Farazmanesh
  • Ali TavakoliEmail author
Article

Abstract

In this paper, by a novel morphological decomposition method, we propose an efficient compression approach. Our presented decomposition arises from solving a minimum set cover problem (MSCP) obtained from the image skeleton data. We first use the skeleton pixels to create a collection of blocks which cover the foreground. The given blocks are both overlapped and too many. Hence, in order to find the minimum number of the blocks that cover the foreground, we form an MCSP. To solve this problem, we present a new algorithm of which accuracy is better than those of the known methods in the literatures. Also, its error analysis is studied to obtain an error bound. In the sequel, we present a fast algorithm which include an extra parameter by which the relation between the accuracy and CPU time can be controlled. Finally, several examples are given to confirm the efficiency of our approach.

Keywords

Decomposition Compression Minimum set cover algorithm 

Mathematics Subject Classification

68U10 94A08 

Notes

Acknowledgements

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

References

  1. 1.
    Quddus, A., Fahmy, M.M.: Binary text image compression using overlapping rectangular partitioning. Pattern Recognit. Lett. 20, 81–88 (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Zakaria, M.F., Vroomen, L.J., Zsombor-Murray, P., van Kessel, J.M.: Fast algorithm for the computation of moment invariants. Pattern Recognit. 20(6), 639–643 (1987)CrossRefGoogle Scholar
  3. 3.
    Dai, M., Baylou, P., Najim, M.: An efficient algorithm for computation of shape moments from run-length codes or chain codes. Pattern Recognit. 25(10), 1119–1128 (1992)CrossRefGoogle Scholar
  4. 4.
    Flusser, J.: Refined moment calculation using image block representation. IEEE Trans. Image Process. 9(11), 1977–1978 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Li, B.C.: A new computation of geometric moments. Pattern Recognit. 26(1), 109–113 (1993)CrossRefGoogle Scholar
  6. 6.
    Spiliotis, I.M., Mertzios, B.G.: Real-time computation of two-dimensional moments on binary images using image block representation. IEEE Trans. Image Process. 7(11), 1515–1609 (1998)CrossRefGoogle Scholar
  7. 7.
    Kawaguchi, E., Endo, T.: On a method of binary-picture representation and its application to data compression. IEEE Trans. Pattern Anal. 2(1), 27–35 (1980)CrossRefzbMATHGoogle Scholar
  8. 8.
    Sossa-Azuela, J.H., Yáñez-Márquez, C., Dáz de León S, J.L.: Computing geometric moments using morphological erosions. Pattern Recognit. 34(2), 271–276 (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Suk, T., Flusser, J.: Refined morphological methods of moment computation. In: 20th International Conference on Pattern Recognition ICPR10, IEEE Computer Society, vol. 30, pp. 966–970 (2010)Google Scholar
  10. 10.
    Eppstein, D.: aph-theoretic solutions to computational geometry problems. In: 35th International Workshop on Graph-Theoretic Concepts in Computer Science WG09. Lecture Notes in Computer Science, vol. 5911, pp. 1–16. Springer (2009)Google Scholar
  11. 11.
    Ferrari, L., Sankar, P.V., Sklansky, J.: Minimal rectangular partitions of digitized blobs. Comput. Vision Graph. Image Process. 28(1), 58–71 (1980)CrossRefzbMATHGoogle Scholar
  12. 12.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM 45(5), 783–797 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Imai, H., Asano, T.: Efficient algorithms for geometric graph search problems. SIAM J. Comput. 15(2), 478–494 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lipski Jr., W., Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On two-dimensional data organization II, Fundamenta Informaticae. Ann. Soc. Math. Pol. 4(2), 254–260 (1979)zbMATHGoogle Scholar
  15. 15.
    Keil, J.M.: Polygon decomposition. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 491–518. Elsevier, New York (2000)CrossRefGoogle Scholar
  16. 16.
    Ohtsuki, T.: Minimum dissection of rectilinear regions. Proc. IEEE Int. Conf. Circuits Syst. ISCAS 82, 1210–1213 (1982)Google Scholar
  17. 17.
    Suk, T., Höschl IV, C., Flusser, J.: Rectangular decomposition of binary images. In: ACIVS, 14th International Conference, Czech Republic (2012 September).  https://doi.org/10.1007/978-3-642-33140-4_19
  18. 18.
    Suk, T., Höschl IV, C., Flusser, J.: Decomposition of binary images: a survey and comparison. Pattern Recognit. 45, 4279–4291 (2012)CrossRefGoogle Scholar
  19. 19.
    Arcelli, C., Di Baja, G.S.: Ridge points in euclidean distance maps. Pattern Recognit. Lett 1, 237–243 (1992)CrossRefGoogle Scholar
  20. 20.
    Hesselink, W.H., Roerdink, J.B.T.M.: Euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform. IEEE Trans. Pattern. Anal. 30(12), 2204–2217 (2008)CrossRefGoogle Scholar
  21. 21.
    Jalba, A.C., Sobiecki, A., Telea, A.C.: An unified multiscale framework for planar, surface, and curve skeletonization. IEEE Trans. Pattern. Anal. 38(1), 30–45 (2016)CrossRefGoogle Scholar
  22. 22.
    Marie, R., Labbani-Igbida, O., Mouaddib, E.M.: The delta medial axis: a fast and robust algorithm for filtered skeleton extraction. Pattern Recognit. 56, 26–39 (2016)CrossRefGoogle Scholar
  23. 23.
    Presti, L.L., Cascia, M.L.: 3D skeleton-based human action classification: a survey. Pattern Recognit. 53, 130–147 (2016)CrossRefGoogle Scholar
  24. 24.
    Wu, Q.J., Bourland, J.D.: Three-dimensional skeletonization for computer-assisted treatment planning in radiosurgery. Comput. Med. Imaging Gr. 24(4), 243–251 (2000)CrossRefGoogle Scholar
  25. 25.
    Borgefors, G.: Distance transformations in digital images. Comput. Vision Graph. 34, 344–371 (1986)CrossRefGoogle Scholar
  26. 26.
    Hirata, T.: A unified linear-time algorithm for computing distance maps. Inform. Process. Lett. 58, 129–133 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Maurer Jr., C.R., Qi, R., Raghavan, V.: A linear time algorithm for computing the euclidean distance transform in arbitrary dimensions. IEEE Trans. Pattern. Anal. 25(2), 265–270 (2003)CrossRefGoogle Scholar
  28. 28.
    Hesselink, W.H.: A linear-time algorithm for euclidean feature transform sets. Inform. Process. Lett. 102, 181–186 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maurer Jr., C.R., Raghavan, V., Qi, R.: A Linear time algorithm for computing the euclidean distance transform in arbitrary dimensions. Inf. Process. Med. Imaging 58, 358–364 (2001)CrossRefzbMATHGoogle Scholar
  30. 30.
    Meijster, A., Roerdink, J.B.T.M., Hesselink, W.H.: A general algorithm for computing distance transforms in linear time. In: Goutsias, J., Vincent, L., Bloomberg, D.S. (eds.) Mathematical Morphology and Its Applications to Image and Signal Processing, pp. 331–340. Kluwer Academic, Dordrecht (2000)Google Scholar
  31. 31.
    Strzodka, R., Telea, A.: Generalized distance transforms and skeletons in graphics hardware. In: Proceedings of the Joint EUROGRAPHICS and IEEE TCVG Symposium on Visualization, pp. 221–230 (2004)Google Scholar
  32. 32.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press, Cambridge (1991)zbMATHGoogle Scholar
  33. 33.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, Raymond E., Thatcher, James W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  34. 34.
    Desai, R., Yang, Q., Wu, Z., Meng, W., Yu, C.: Identifying redundant search engines in a very large scale metasearch engine context. In: ACM WIDM ’06, 8th ACM International Workshop on Web Information and Data Management, November 10, pp. 51–58. Arlington, Virginia, USA (2006)Google Scholar
  35. 35.
    Yang, Q., MacPeek, J., Nofsinger, A.: Efficient and effective practical algorithms for the set-covering problem. In: Conference on Scientific Computing (CSC08), The 2008 World Congress in Computer Science, Computer Engineering and Applied Computing (WORLDCOMP08), Las Vegas, July 14–17 (2008)Google Scholar
  36. 36.
    Yang, Q., Nofsinger, A., McPeek, J., Phinney, J., Knuesel, R.: A complete solution to the set covering problem. In: Proceedings of the International Conference on Scientific Computing (CSC), Las Vegas, NV, USA, 27–30 July, pp. 36–41 (2015)Google Scholar
  37. 37.
    Department of Image Processing: Tree leaf database. http://zoi.utia.cas.cz/tree_leaves

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentVali-e-Asr University of RafsanjanRafsanjanIran
  2. 2.Mathematics DepartmentUniversity of MazandaranBabolsarIran

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