Abstract
The problem of image matching under rotation is to find for two given digital images A and B a rotation that converts image A as close as possible to B. The research in combinatorial pattern matching led to a series of improving algorithms which commonly attack this problem by searching the complete set of all rotations of A. We present the first optimal algorithm of this kind, i.e, one that solves image matching in time for images of size n ×n. Subsequently, for image matching under compositions of rotation and scaling we show a new lower bound Ω(n 6 / logn) on the cardinality of , the set of rotated and scaled transformations of A. This bound almost matches the upper bound O(n 6).
Partially supported by DFG research grant RE 672/5-1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bovik, A. (ed.): Handbook of Image and Video Processing. Academic Press, San Diego (2000)
Amir, A., Butman, A., Crochemore, M., Landau, G., Schaps, M.: Two-dimensional pattern matching with rotations. Theor. Comput. Sci. 314(1-2), 173–187 (2004)
Amir, A., Butman, A., Lewenstein, M., Porat, E.: Real two dimensional scaled matching. Algorithmica 53(3), 314–336 (2009)
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)
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)
Brent, R.P.: Fast multiple-precision evaluation of elementary functions. Journal of the Association for Computing Machinery 23(2), 242–251 (1976)
Brown, L.G.: A survey of image registration techniques. ACM Computing Surveys 24(4), 325–376 (1992)
Cox, I.J., Bloom, J.A., Miller, M.L.: Digital Watermarking, Principles and Practice. Morgan Kaufmann, San Francisco (2001)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)
Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15, 341–363 (1986)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1954)
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)
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)
Hundt, C., Liśkiewicz, M.: On the complexity of affine image matching. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 284–295. Springer, Heidelberg (2007)
Hundt, C., Liśkiewicz, M.: Two-dimensional pattern matching with combined scaling and rotation. In: Ferragina, P., Landau, G.M. (eds.) CPM 2008. LNCS, vol. 5029, pp. 5–17. Springer, Heidelberg (2008)
Hundt, C., Liśkiewicz, M., Nevries, R.: A combinatorial geometric approach to two-dimensional robustly pattern matching with scaling and rotation. To appear in Theor. Comput. Sci.
Indyk, P.: Algorithmic aspects of geometric embeddings. In: Proc. FOCS 2001, pp. 10–33 (2001)
Indyk, P., Motwani, R., Venkatasubramanian, S.: Geometric matching under noise: Combinatorial bounds and algorithms. In: Proc. SODA 1999, pp. 354–360 (1999)
Kasturi, R., Jain, R.C.: Computer Vision: Principles. IEEE Computer Society Press, Los Alamitos (1991)
Kenyon, C., Rabani, Y., Sinclair, A.: Low distortion maps between point sets. In: Proc. STOC 2004, pp. 272–280 (2004)
Kropatsch, W.G., Bischof, H. (eds.): Digital Image Analysis - Selected Techniques and Applications. Springer, Berlin (2001)
Landau, G.M., Vishkin, U.: Pattern matching in a digitized image. Algorithmica 12(3/4), 375–408 (1994)
Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Medical Image Analysis 2(1), 1–36 (1998)
Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)
Nevries, R.: Entwicklung und Analyse eines beschleunigten Image Matching-Algorithmus für natürliche Bilder, Diplomarbeit, Universität Rostock (2008)
Nouvel, B., Rémila, E.: Incremental and transitive discrete rotations. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 199–213. Springer, Heidelberg (2006)
Papadimitriou, C., Safra, S.: The complexity of low-distortion embeddings between point sets. In: Proc. SODA 2005, pp. 112–118 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hundt, C., Liśkiewicz, M. (2009). New Complexity Bounds for Image Matching under Rotation and Scaling. In: Kucherov, G., Ukkonen, E. (eds) Combinatorial Pattern Matching. CPM 2009. Lecture Notes in Computer Science, vol 5577. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02441-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-02441-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02440-5
Online ISBN: 978-3-642-02441-2
eBook Packages: Computer ScienceComputer Science (R0)