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Binary Tomography on Triangular Grid Involving Hexagonal Grid Approach

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Combinatorial Image Analysis (IWCIA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 11255))

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Abstract

In this paper, we consider the binary tomography reconstruction problem of images on the triangular grid. The reconstruction process is based on three natural directions of projections, defined by the lane directions of the triangular grid (they are analogous to row and column directions on the square grid). The recently proposed shifted projection approach is applied, which allows that the number of delta and nabla shape triangular pixels in each lane to be exactly determined. The structure of the set of the same type pixels coincides with the structure of the hexagonal grid. The proposed new reconstruction process solve separately the task for the delta and nabla shape pixels on two hexagonal grids. Experimental results on a number of test images is presented and analyzed. The new method shows advantage in both aspects, quality of the reconstructions and running times.

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References

  1. Abdalla, M., Nagy, B.: Dilation and erosion on the triangular tessellation: an independent approach. IEEE Access 6, 23108–23119 (2018)

    Article  Google Scholar 

  2. Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG - software for convex-constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001)

    Article  Google Scholar 

  3. Birgin, E., Martínez, J.: A box-constrained optimization algorithm with negative curvature directions and spectral projected gradients. In: Alefeld, G., Chen, X. (eds.) Topics in Numerical Analysis. COMPUTING, vol. 15, pp. 49–60. Springer, Vienna (2001). https://doi.org/10.1007/978-3-7091-6217-0_5

    Chapter  Google Scholar 

  4. Borgefors, G.: Distance transformations on hexagonal grids. Pattern Recogn. Lett. 9(2), 97–105 (1989)

    Article  Google Scholar 

  5. Brimkov, V.E., Barneva, R.P.: “Honeycomb” vs square and cubic models. Electron. Notes Theoret. Comput. Sci. 46, 321–338 (2001)

    Article  Google Scholar 

  6. Brimkov, V.E., Barneva, R.P.: Analytical honeycomb geometry for raster and volume graphics. Comput. J. 48(2), 180–199 (2005)

    Article  Google Scholar 

  7. Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal and triangular arrays. Commun. ACM 15(3), 827–837 (1972)

    Article  MathSciNet  Google Scholar 

  8. Gale, D.: A theorem on flows in networks. Pacific J. Math. 7(2), 1073–1082 (1957)

    Article  MathSciNet  Google Scholar 

  9. Golay, M.: Hexagonal parallel pattern transformations. IEEE Trans. Comput. 18, 733–740 (1969)

    Article  Google Scholar 

  10. Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. Image Process. 4, 1213–1222 (1995)

    Article  Google Scholar 

  11. Herman, G.T., Kuba, A.: Advances in Discrete Tomography and Its Applications. Birkhäuser, Basel (2007)

    Book  Google Scholar 

  12. Kardos, P., Palagyi, K.: Hexagonal parallel thinning algorithms based on sufficient conditions for topology preservation. In: Proceedings of the International Symposium CompIMAGE, pp. 63–68 (2012)

    Google Scholar 

  13. Kardos, P., Palagyi, K.: On topology preservation in triangular. In: Proceedings of the International Symposium on Image and Signal Processing and Analysis (ISPA), pp. 789–794 (2013)

    Google Scholar 

  14. Kardos, P., Palagyi, K.: On topology preservation of mixed operators in triangular, square, and hexagonal grids. Discrete Appl. Math. 216, 441–448 (2017)

    Article  MathSciNet  Google Scholar 

  15. Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Publishers/Elsevier Science B.V., San Francisco/Amsterdam (2004)

    MATH  Google Scholar 

  16. Luczak, E., Rosenfeld, A.: Distance on a hexagonal grid. IEEE Trans. Comput. C–25(5), 532–533 (1976)

    Article  Google Scholar 

  17. Lukić, T., Balázs, P.: Binary tomography reconstruction based on shape orientation. Pattern Recogn. Lett. 79, 18–24 (2016)

    Article  Google Scholar 

  18. Lukić, T., Nagy, B.: Energy-minimization based discrete tomography reconstruction method for images on triangular grid. In: Barneva, R.P., Brimkov, V.E., Aggarwal, J.K. (eds.) IWCIA 2012. LNCS, vol. 7655, pp. 274–284. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34732-0_21

    Chapter  Google Scholar 

  19. Lukić, T., Nagy, B.: Deterministic discrete tomography reconstruction by energy minimization method on the triangular grid. Pattern Recogn. Lett. 49, 11–16 (2014)

    Article  Google Scholar 

  20. Lukić, T., Nagy, B.: Regularized Binary Tomography on The Hexagonal Grid (2018, Submitted)

    Google Scholar 

  21. Matej, S., Herman, G.T., Vardi, A.: Binary tomography on the hexagonal grid using Gibbs priors. Int. J. Imaging Syst. Technol. 9, 126–131 (1998)

    Article  Google Scholar 

  22. Moisi, E., Nagy, B.: Discrete tomography on the triangular grid: a memetic approach. In: Proceedings of 7th International Symposium on Image and Signal Processing and Analysis (ISPA 2011), pp. 579–584, Dubrovnik, Croatia (2011)

    Google Scholar 

  23. Moisi, E., Nagy, B., Cretu, V.: Reconstruction of binary images represented on equilateral triangular grid using evolutionary algorithms. In: Balas, V., Fodor, J., Várkonyi-Kóczy, A., Dombi, J., Jain, L. (eds.) Soft Computing Applications. AISC, vol. 195, pp. 561–571. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-33941-7_49

    Chapter  Google Scholar 

  24. Nagy, B.: Shortest paths in triangular grids with neighbourhood sequences. J. Comput. Inf. Technol. 11, 111–122 (2003)

    Article  Google Scholar 

  25. Nagy, B., Barczi, K.: Isoperimetrically optimal polygons in the triangular grid with jordan-type neighbourhood on the boundary. Int. J. Comput. Math. 90, 1629–1652 (2013)

    Article  Google Scholar 

  26. Nagy, B., Lukić, T.: Dense projection tomography on the triangular tiling. Fundamenta Informaticae 145, 125–141 (2016)

    Article  MathSciNet  Google Scholar 

  27. Nagy, B., Lukić, T.: New projection approach for binary tomography on triangular grid (2018, Submitted)

    Google Scholar 

  28. Prause, G., Onnasch, D.: Binary reconstruction of the heart chambers from biplane angiographic image sequences. IEEE Trans. Med. Imag. 15, 532–46 (1997)

    Article  Google Scholar 

  29. Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Can. J. Math. 9, 371–377 (1957)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

Tibor Lukić acknowledges the Ministry of Education and Sciences of the R. of Serbia for support via projects OI-174008 and III-44006. He also acknowledges support received from the Hungarian Academy of Sciences through the DOMUS project.

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Authors and Affiliations

  1. Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, Mersin-10, Famagusta, Turkey

    Benedek Nagy

  2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Tibor Lukić

Authors
  1. Benedek Nagy
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  2. Tibor Lukić
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Corresponding author

Correspondence to Tibor Lukić .

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Editors and Affiliations

  1. State University of New York at Fredonia, Fredonia, NY, USA

    Reneta P. Barneva

  2. State University of New York, Buffalo, NY, USA

    Valentin E. Brimkov

  3. University of Porto, Porto, Portugal

    João Manuel R.S. Tavares

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Nagy, B., Lukić, T. (2018). Binary Tomography on Triangular Grid Involving Hexagonal Grid Approach. In: Barneva, R., Brimkov, V., Tavares, J. (eds) Combinatorial Image Analysis. IWCIA 2018. Lecture Notes in Computer Science(), vol 11255. Springer, Cham. https://doi.org/10.1007/978-3-030-05288-1_6

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  • DOI: https://doi.org/10.1007/978-3-030-05288-1_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05287-4

  • Online ISBN: 978-3-030-05288-1

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