Skip to main content
SpringerLink
Log in

Automatic Solution of Jigsaw Puzzles

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

We present a method for automatically solving apictorial jigsaw puzzles that is based on an extension of the method of differential invariant signatures. Our algorithms are designed to solve challenging puzzles, without having to impose any restrictive assumptions on the shape of the puzzle, the shapes of the individual pieces, or their intrinsic arrangement. As a demonstration, the method was successfully used to solve two commercially available puzzles. Finally we perform some preliminary investigations into scalability of the algorithm for even larger puzzles.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. More generally, one can allow pieces to touch at a finite number of points, although this is uncommon in real world examples.

  2. The older term “classifying submanifold” is used in place of “signature” in [20].

  3. We note that there is a misprint in formula (11) of [16], where the numerator should contain \(p(\sigma ^{i},\widetilde{S} ^{\Delta})\) instead of \(h(\sigma ^{i},\widetilde{S} ^{\Delta})\).

  4. By convention, arg0=0.

  5. This requirement is removed for a round when the parameter sequence (see Sect. 6.3) increments.

References

  1. Boutin, M.: Numerically invariant signature curves. Int. J. Comput. Vis. 40, 235–248 (2000)

    Article  MATH  Google Scholar 

  2. Bruckstein, A.M., Holt, R.J., Netravali, A.N., Richardson, T.J.: Invariant signatures for planar shape recognition under partial occlusion. CVGIP, Image Underst. 58, 49–65 (1993)

    Article  Google Scholar 

  3. Burdea, G.C., Wolfson, H.J.: Solving jigsaw puzzles by a robot. IEEE Trans. Robot. Autom. 5, 752–764 (1989)

    Article  Google Scholar 

  4. Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26, 107–135 (1998)

    Article  Google Scholar 

  5. Cartan, É.: Les problèmes d’équivalence. In: Oeuvres Complètes; Part. II, vol. 2, pp. 1311–1334. Gauthier-Villars, Paris (1953)

    Google Scholar 

  6. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22, 61–79 (1997)

    Article  MATH  Google Scholar 

  7. Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  8. The DARPA Shredder Challenge. http://www.shredderchallenge.com/ (2011)

  9. Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 139–232 (2005)

  10. Freeman, H., Carder, L.: Apictorial jigsaw puzzles: the computer solution of a problem in pattern recognition. IEEE Trans. Electromagn. Compat. 13, 118–127 (1964)

    Article  Google Scholar 

  11. Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)

    MATH  MathSciNet  Google Scholar 

  12. Gallagher, A.C.: Jigsaw puzzles with pieces of unknown orientation. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 382–389. IEEE Comp. Soc. Press, Los Alamitos (2012)

    Chapter  Google Scholar 

  13. Goldberg, D., Malon, C., Bern, M.: A global approach to the automatic solution of jigsaw puzzles. Comput. Geom. 28, 165–174 (2004)

    Article  MathSciNet  Google Scholar 

  14. Grayson, M.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)

    MATH  MathSciNet  Google Scholar 

  15. Guggenheimer, H.W.: Differential Geometry. McGraw-Hill, New York (1963)

    MATH  Google Scholar 

  16. Hoff, D., Olver, P.J.: Extensions of invariant signatures for object recognition. J. Math. Imaging Vis. 45(2), 176–185 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Ration. Mech. Anal. 134, 275–301 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kong, W., Kimia, B.B.: On solving 2D and 3D puzzles using curve matching. In: Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 583–590 (2001)

    Google Scholar 

  19. Lankton, S.: Active Contours. http://www.shawnlankton.com/2007/05/active-contours/ (2007)

  20. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  21. Pham, D.T., Karaboga, D.: Intelligent Optimisation Techniques: Genetic Algorithms, Tabu Search, Simulated Annealing and Neural Networks. Springer, New York (2000)

    Book  Google Scholar 

  22. Radack, G.M., Badler, N.I.: Jigsaw puzzle matching using a boundary-centered polar encoding. Comput. Graph. Image Process. 19, 1–17 (1982)

    Article  Google Scholar 

  23. The Rain Forest Giant Floor Puzzle. Frank Schaffer Publications, Inc., Torrance, (1992)

  24. Sağiroğlu, M.S., Erçil, A.: A texture based approach to reconstruction of archaeological finds. In: Mudge, M., Ryan, N., Scopigno, R. (eds.) Proceedings of the 6th International Conference on Virtual Reality, Archaeology and Intelligent Cultural Heritage (VAST05), Aire-la-Ville, Switzerland, Eurographics Assoc., pp. 137–142 (2005)

    Google Scholar 

  25. Shakiban, C., Lloyd, P.: Signature curves statistics of DNA supercoils. In: Mladenov, I.M., Hirschfeld, A.C. (eds.) Geometry, Integrability and Quantization, vol. 5, pp. 203–210. Sofia, Bulgaria, Softex (2004)

    Google Scholar 

  26. Shakiban, C., Lloyd, R.: Classification of signature curves using latent semantic analysis. In: Li, H., Olver, P.J., Sommer, G. (eds.) Computer Algebra and Geometric Algebra with Applications. Lecture Notes in Computer Science, vol. 3519, pp. 152–162. Springer, New York (2005)

    Chapter  Google Scholar 

  27. Turney, J.L., Mudge, T.N., Volz, R.A.: Recognizing partially occluded parts. IEEE Trans. Pattern Anal. Mach. Intell. 7, 410–421 (1985)

    Article  Google Scholar 

  28. Wang, Y.: Smoothing Splines: Methods and Applications. Monographs Stat. Appl. Probability, vol. 121. CRC Press, Boca Raton (2011)

    Book  Google Scholar 

  29. Wesley, R., Cathedral, product #51015. SunsOut, Inc., Costa Mesa

  30. Wolfson, H., Schonberg, E., Kalvin, A., Lamdan, Y.: Solving jigsaw puzzles by computer. Ann. Oper. Res. 12, 51–64 (1988)

    Article  MathSciNet  Google Scholar 

  31. Wu, K., Levine, M.D.: 3D part segmentation using simulated electrical charge distributions. IEEE Trans. Pattern Anal. Mach. Intell. 19, 1223–1235 (1997)

    Article  Google Scholar 

  32. Yates, C.: The Baffler: The Nonagon. Ceaco, Newton (2010)

    Google Scholar 

  33. Yao, F.-H., Shao, G.-F.: A shape and image merging technique to solve jigsaw puzzles. Pattern Recognit. Lett. 24, 1819–1835 (2003)

    Article  Google Scholar 

  34. Zisserman, A., Forsyth, D.A., Mundy, J.L., Rothwell, C.A.: Recognizing general curved objects efficiently. In: Mundy, J.L., Zisserman, A. (eds.) Geometric Invariance in Computer Vision, pp. 228–251. MIT Press, Cambridge (1992)

    Google Scholar 

Download references

Acknowledgements

Supported in part by NSF Grant DMS 08-07317.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, La Jolla, CA, 92093, USA

    Daniel J. Hoff

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter J. Olver

Authors
  1. Daniel J. Hoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter J. Olver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter J. Olver.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoff, D.J., Olver, P.J. Automatic Solution of Jigsaw Puzzles. J Math Imaging Vis 49, 234–250 (2014). https://doi.org/10.1007/s10851-013-0454-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-013-0454-3

Keywords

Search

Navigation