Advertisement

On the Complexity of Affine Image Matching

  • Christian Hundt
  • Maciej Liśkiewicz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

The problem of image matching is to find for two given digital images A and B an admissible transformation that converts image A as close as possible to B. This problem becomes hard if the space of admissible transformations is too complex. Consequently, in many real applications, like the ones allowing nonlinear elastic transformations, the known algorithms solving the problem either work in exponential worst-case time or can only guarantee to find a local optimum. Recently Keysers and Unger have proved that the image matching problem for this class of transformations is NP-complete, thus giving evidence that the known exponential time algorithms are justified. On the other hand, allowing only such transformations as translations, rotations, or scalings the problem becomes tractable. In this paper we analyse the computational complexity of image matching for a larger space of admissible transformations, namely for all affine transformations. In signal processing there are no efficient algorithms known for this class. Similarly, the research in combinatorial pattern matching does not cover this set of transformations neither providing efficient algorithms nor proving intractability of the problem, although it is a basic one and of high practical importance. The main result of this paper is that the image matching problem can be solved in polynomial time even allowing all affine transformations.

Keywords

Polynomial Time Pattern Match Polynomial Time Algorithm Image Match Digital Watermark 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amir, A., et al.: Two-dimensional pattern matching with rotations. Theor. Comput. Sci. 314(1-2), 173–187 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amir, A., Kapah, O., Tsur, D.: Faster Two dimensional pattern matching with rotations. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 409–419. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Amir, A., Chencinski, E.: Faster two dimensional scaled matching. In: Lewenstein, M., Valiente, G. (eds.) CPM 2006. LNCS, vol. 4009, pp. 200–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Baeza-Yates, R., Valiente, G.: An image similarity measure based on graph matching. In: Proc. SPIRE, pp. 28–38. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  5. 5.
    Brown, L.G.: A survey of image registration techniques. ACM Computing Surveys 24(4), 325–376 (1992)CrossRefGoogle Scholar
  6. 6.
    Claire, K., Rabani, Y., Sinclair, A.: Low distortion maps between point sets. In: Proc. STOC, pp. 272–280 (2004)Google Scholar
  7. 7.
    Cox, I.J., Bloom, J.A., Miller, M.L.: Digital Watermarking, Principles and Practice. Morgan Kaufmann, San Francisco (2001)Google Scholar
  8. 8.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)zbMATHGoogle Scholar
  9. 9.
    de Berg, M., et al.: Computational Geometry, Algorithms and Applications. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  10. 10.
    Deguillaume, F., Voloshynovskiy, S.V., Pun, T.: Method for the estimation and recovering from general affine transforms in digital watermarking applications. Proc. SPIE 4675, 313–322 (2002)CrossRefGoogle Scholar
  11. 11.
    Fredriksson, K., Ukkonen, E.: A rotation invariant filter for two-dimensional string matching. In: Farach-Colton, M. (ed.) CPM 1998. LNCS, vol. 1448, pp. 118–125. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Fredriksson, K., Navarro, G., Ukkonen, E.: Optimal exact and fast approximate two dimensional pattern matching allowing rotations. In: Apostolico, A., Takeda, M. (eds.) CPM 2002. LNCS, vol. 2373, pp. 235–248. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Horn, B.K.P.: Robot Vision. MIT Press, Cambridge (1989)Google Scholar
  14. 14.
    Indyk, P.: Algorithmic aspects of geometric embeddings. In: Proc. FOCS, pp. 10–33 (2001)Google Scholar
  15. 15.
    Indyk, P., Motwani, R., Venkatasubramanian, S.: Geometric matching under noise: Combinatorial bounds and algorithms. In: Proc. SODA, pp. 354–360 (1999)Google Scholar
  16. 16.
    Jensen, J.R.: Introductory Digital Image Processing, A Remote Sensing Perspective. Prentice-Hall, Upper Saddle River (1986)Google Scholar
  17. 17.
    Kasturi, R., Jain, R.C.: Computer Vision: Principles. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  18. 18.
    Keysers, D., Unger, W.: Elastic image matching is NP-complete. Pattern Recognition Letters 24(1-3), 445–453 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Landau, G.M., Vishkin, U.: Pattern matching in a digitized image. Algorithmica 12(3/4), 375–408 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Liśkiewicz, M., Wölfel, U.: On the intractability of inverting geometric distortions in watermarking schemes. In: Barni, M., et al. (eds.) IH 2005. LNCS, vol. 3727, pp. 176–188. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Medical Image Analysis 2(1), 1–36 (1998)CrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C., Safra, S.: The complexity of low-distortion embeddings between point sets. In: Proc. SODA, pp. 112–118 (2005)Google Scholar
  23. 23.
    Schulman, L., Cardoze, D.: Pattern matching for spatial point sets. In: Proc. FOCS, pp. 156–165 (1998)Google Scholar
  24. 24.
    Uchida, S., Sakoe, H.: A monotonic and continuous two-dimensional warping based on dynamic programming. In: Pattern Recognition, vol. 1, pp. 521–524 (1998)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Christian Hundt
    • 1
  • Maciej Liśkiewicz
    • 1
  1. 1.Institut für Theoretische Informatik, Universität zu LübeckGermany

Personalised recommendations