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Optimum solution and evaluation of rectangular jigsaw puzzles based on branch and bound method and combinatorial accuracy

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Abstract

Automatic solution of rectangular jigsaw puzzles can be broken down into two separate steps of calculating pairwise compatibility metrics and jigsaw assembling algorithm. This work discriminates between two different sources of errors, each corresponding to one of these two steps. In this regard, type I error is defined as the imperfection of the used compatibility metric, and type II error is reserved to measure the imperfection of jigsaw assembling algorithm. Differentiating between these two types of error allows us to tweak and optimize different parts of the algorithm to achieve the best performance. Based upon these defined terms, this study argues that current jigsaw assembling algorithms mainly rely on either greedy methods or metaheuristic algorithms, which may impose a considerable amount of type II error to the final solution. This paper demonstrates that a powerful and perfect (i.e., type II error-free) jigsaw assembling algorithm is achievable by combining branch and bound technique with graph theory. This perfect jigsaw assembling algorithm is then utilized to measure the performances of various compatibility metrics and color models. The superiority of red-green-blue (RGB) color model and Mahalanobis gradient compatibility (MGC) metric in solving rectangular jigsaw puzzles is shown by providing conclusive evidence. Additionally, a mean opinion score (MOS) test is conducted to examine the accuracy of the existing metrics. According to the results from MOS test, we argue that the existing performance criteria are not concise and accurate; thus, a new accuracy metric is proposed on the basis of comparing different sub-blocks of solutions. Finally, the efficiency of jigsaw assembling algorithm is measured by proposing a new performance criterion.

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References

  1. Abdi H (2007) The Kendall rank correlation coefficient. Encyclopedia of measurement and statistics. Sage, thousand oaks, CA:508-510

  2. Adler J, Parmryd I (2010) Quantifying colocalization by correlation: the Pearson correlation coefficient is superior to the Mander's overlap coefficient. Cytometry Part A 77(8):733–742

    Article  Google Scholar 

  3. Arjmandi MK, Pooyan M, Mikaili M, Vali M, Moqarehzadeh A (2011) Identification of voice disorders using long-time features and support vector machine with different feature reduction methods. J Voice 25(6):e275–e289

    Article  Google Scholar 

  4. Brown BJ, Toler-Franklin C, Nehab D, Burns M, Dobkin D, Vlachopoulos A, Doumas C, Rusinkiewicz S, Weyrich TA (2008) System for high-volume acquisition and matching of fresco fragments: reassembling Theran wall paintings. In: ACM transactions on graphics (TOG), vol 3. ACM, p 84

  5. Cho TS, Avidan S, Freeman WT (2010) A probabilistic image jigsaw puzzle solver. In: 2010 IEEE. Conference on computer vision and pattern recognition (CVPR), pp 183–190

  6. Chu Y-J, Liu T-H (1965) On shortest arborescence of a directed graph. Sci Sinica 14(10):1396

    MathSciNet  MATH  Google Scholar 

  7. Chung MG, Fleck MM, Forsyth DA (1998) Jigsaw puzzle solver using shape and color. In: 1998 Fourth International Conference on Signal Processing Proceedings, 1998. ICSP'98. IEEE, pp 877–880

  8. Clausen J (1999) Branch and bound algorithms-principles and examples. University of Copenhagen, Department of Computer Science, pp 1–30, Available from: http://janders.eecg.toronto.edu/1387/readings/b_and_b.pdf. Last accessed (March 2017)

  9. Edmonds J (1967) Optimum branching. J Res Natl Bur Stand 71(4):233–240

  10. Farn EJ, Chen CC (2008) A jigsaw puzzle based secret key exchange scheme. In: International Conference on Machine Learning and Cybernetics Vol. 6, pp. 3067–3071

  11. Farn, EJ, Chen CC (2009) Novel steganographic method based on jig swap puzzle images. J Electron Imaging 18(1):013003. doi:10.1117/1.3073979

  12. Fournier J-C (2010) Graphs theory and applications: with exercises and problems, vol 72. John Wiley & Sons, New York

    Google Scholar 

  13. Freeman H, Garder L (1964) Apictorial jigsaw puzzles: the computer solution of a problem in pattern recognition. IEEE Trans Electron Comput 2:118–127

    Article  Google Scholar 

  14. Gallagher AC (2012) Jigsaw puzzles with pieces of unknown orientation. In: Computer Vision and Pattern Recognition (CVPR), 2012 I.E. Conference on, pp. 382–389

  15. Ghasemzadeh H (2014) A metaheuristic approach for solving jigsaw puzzles. In: Intelligent Systems (ICIS), 2014 Iranian Conference on (pp. 1–6). IEEE

  16. Ghasemzadeh H, Kayvanrad MH (2015) Toward a robust and secure echo steganography method based on parameters hopping. In: Signal Processing and Intelligent Systems Conference (SPIS), 2015, pp. 143–147

  17. Ghasemzadeh H, Mehrara H, Khas MT (2014) Cipher-text only attack on hopping window time domain scramblers. In Computer and Knowledge Engineering (ICCKE), 2014 4th International eConference on, pp. 194–199

  18. Ghasemzadeh H, Khass MT, Arjmandi MK, Pooyan M (2015) Detection of vocal disorders based on phase space parameters and Lyapunov spectrum. Biomed. Signal Process Control 22:135–145

  19. Ghasemzadeh H, Khass MT, Arjmandi MK (2016) Audio steganalysis based on reversed psychoacoustic model of human hearing. Digital Signal Process 51:133–141

  20. Ghasemzadeh H, Tajik Khas M, Mehrara H (2017) (Preprint available at https://arxiv.Org/abs/1701.05601) jigsaw cryptanalysis of audio scrambling systems. The ISC International Journal of Information Security Under revision

  21. Koller D, Levoy M (2006) Computer-aided reconstruction and new matches in the forma urbis romae Bullettino Della Commissione Archeologica Comunale di Roma 2

  22. Lin H-Y, Fan-Chiang W-C (2012) Reconstruction of shredded document based on image feature matching. Expert Syst Appl 39(3):3324–3332

    Article  Google Scholar 

  23. Logeswaran L (2014) Solving jigsaw puzzles using paths and cycles. In: Proceedings of the British Machine Vision Conference (BMVC)

  24. Makridis M, Papamarkos N (2010) A new technique for solving puzzles. IEEE Trans Syst Man Cybern B Cybern 40(3):789–797

  25. Murakami T, Toyama F, Shoji K, Miyamichi J (2008) Assembly of puzzles by connecting between blocks. In Pattern Recognition, 2008. ICPR 2008. 19th International Conference on (pp. 1–4)

  26. Olmos A (2011) Kingdom FAA (2004) McGill calibrated colour image database, http://tabby.vision.mcgill.ca. Last accessed (2011)

  27. Pomeranz D, Shemesh M, Ben-Shahar O (2011) A fully automated greedy square jigsaw puzzle solver. In Computer Vision and Pattern Recognition (CVPR), 2011 I.E. Conference on, pp. 9–16

  28. ITU Recommendation 500-10 (2012) Methodology for the subjective assessment of the quality of television pictures Available from : https://www.itu.int/rec/R-REC-BT.500. Last accessed (March 2017)

  29. Russell S, Norvig P (1995) Artificial intelligence: a modern approach. Prentice-Hall, Englewood Cliffs

  30. Sağıroğlu MŞ, Erçil A (2010) Optimization for automated assembly of puzzles. TOP 18(2):321–338

    Article  MathSciNet  MATH  Google Scholar 

  31. Salzberg SL, Searls DB, Kasif S (1998) Computational methods in molecular biology, vol 32. Elsevier

  32. Sholomon D, David OE, Netanyahu NS (2016) An automatic solver for very large jigsaw puzzles using genetic algorithms. Genet Program Evolvable Mach:1–23

  33. Son K, Hays J, Cooper DB (2016) Solving small-piece jigsaw puzzles by growing consensus. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp 1193–1201

  34. Toyama F, Fujiki Y, Shoji K, Miyamichi J (2002) Assembly of puzzles using a genetic algorithm. In: 16th International Conference on pattern recognition, 2002. Proceedings, IEEE, pp 389–392

    Google Scholar 

  35. Tsai C-W, Tseng S-P, Chiang M-C, Yang C-S (2012) A high performance algorithm for puzzle reconstruction problem. In: 2012 International Conference on machine learning and Cybernetics (ICMLC). IEEE, pp 1698-1703

  36. Weiss-Cohen M, Halevi Y (2005) Knowledge retrieval for automatic solving of jigsaw puzzles. In: International Conference on computational Intelligence for Modelling, control and automation, 2005 and International Conference on Intelligent agents. Web Technologies and Internet Commerce, IEEE, pp 379–383

    Google Scholar 

  37. Yu B, Yuan B (1993) A more efficient branch and bound algorithm for feature selection. Pattern Recogn 26(6):883–889

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Acknowledgements

The authors would like to thank all the participants in the subjective tests. We would also like to thank Tom Almer as a member of Michigan State University volunteer English tutoring program for his assistance with preparing the manuscript and providing us with his helpful comments in improving the English of the manuscript.

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Authors and Affiliations

  1. Department of Communicative Sciences and Disorders, Michigan State University, East Lansing, MI, USA

    Hamzeh Ghasemzadeh & Meisam K. Arjmandi

  2. Department of Computational Mathematics Science and Engineering, Michigan State University, East Lansing, MI, USA

    Hamzeh Ghasemzadeh

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  1. Hamzeh Ghasemzadeh
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  2. Meisam K. Arjmandi
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Correspondence to Hamzeh Ghasemzadeh.

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Ghasemzadeh, H., Arjmandi, M.K. Optimum solution and evaluation of rectangular jigsaw puzzles based on branch and bound method and combinatorial accuracy. Multimed Tools Appl 77, 6837–6861 (2018). https://doi.org/10.1007/s11042-017-4601-5

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  • DOI: https://doi.org/10.1007/s11042-017-4601-5

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