Abstract
The power of quantum mechanics has been extensively exploited to meet the high computational requirement of classical image processing. However, existing quantum image models can only represent the images sampled in Cartesian coordinates. In this paper, quantum log-polar image (QUALPI), a novel quantum image representation is proposed for the storage and processing of images sampled in log-polar coordinates. In QUALPI, all the pixels of a QUALPI are stored in a normalized superposition and can be operated on simultaneously. A QUALPI can be constructed from a classical image via a preparation whose complexity is approximately linear in the image size. Some common geometric transformations, such as symmetry transformation, rotation, etc., can be performed conveniently with QUALPI. Based on these geometric transformations, a fast rotation-invariant quantum image registration algorithm is designed for log-polar images. Performance comparison with classical brute-force image registration method reveals that our quantum algorithm can achieve a quartic speedup.
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References
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proeedings of 35th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Almitos, CA, pp. 124–134 (1994)
Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 212–219 (1996)
Childs, A.M., Van Dam, W.: Quantum algorithm for algebraic problems. Rev. Mod. Phys. 82, 1–52 (2010)
Gonzalez, R.C., Woods, R.E., Eddins, S.L.: Digital Image Processing. Publishing House of Electronics Industry, Beijing (2002)
Venegas-Andraca, S.E., Bose, S.: Storing, processing and retrieving an image using quantum mechanics. In: Proceedings of the SPIE Conference on Quantum Information and Computation, pp. 137–147 (2003)
Venegas-Andraca, S.E., Ball, J.L., Burnett, K., Bose, S.: Processing images in entangled quantum systems. Quantum Inf. Process. 9, 1–11 (2010)
Latorre, J.I.: Image compression and entanglement. arXiv:quant-ph/0510031 (2005)
Le, P.Q., Dong, F., Hirota, K.: A flexible representation of quantum images for polynomial preparation, image compression, and processing operations. Quantum Inf. Process. 10(1), 63–84 (2011)
Brown, L.G.: A survey of image registration techniques (abstract). ACM Comput. Surv. (CSUR) Arch. 24(4), 325–376 (1992)
Le, P.Q., Iliyasu, A.M., Dong, F., Hirota, K.: Strategies for designing geometric transformations on quantum images. Theor. Comput. Sci. 412, 1406–1418 (2011)
Le, P.Q., Iliyasu, A.M., Dong, F., Hirota, K.: Efficient color transformations on quantum images. J. Adv. Comput. Intell. Intell. Inf. 15(6), 698–706 (2011)
Sun, B., Le, P.Q., Iliyasu, A.M., et al.: A Multi-channel representation for images on quantum computers using the RGB\(\alpha \) color space. In: Proceedings of the IEEE 7th International Symposium on Intelligent Signal Processing, pp. 160–165 (2011)
Zhang, Y., Lu, K., et al.: NEQR: a novel enhanced quantum representation of digital images. Quantum Inf. Process. (2013). doi:10.1007/s11128-013-0567-z
Iliyasu, A.M., Le, P.Q., Dong, F., Hirota, K.: Watermarking and authentication of quantum images based on restricted geometric transformations. Inf. Sci. 186, 126–149 (2012)
Zhang, W., Gao, F., Liu, B., Wen, Q., Chen, H.: A watermark strategy for quantum images based on quantum Fourier transform. (2012). Quantum Inf. Process. doi:10.1007/s11128-012-0423-6
Araujo, H., Dias, J.M.: An introduction to the log-polar mapping. In: Proceedings of 2nd Workshop on Cybernetic Vision, pp. 139–144 (1996)
Zokai, S., Wolberg, G.: Image registration using log-polar mappings for recovery of large-scale similarity and projective transformations. IEEE Trans. Image Process. 14(10), 1422–1434 (2005)
Matungka, R., Zheng, Y.F., Ewing, R.L.: 2D invariant object recognition using log-polar transform. In: Proceedings of World Congress on Intelligent Control and Automation, pp. 223–228 (2008)
Pun, C.M., Lee, M.C.: Log-polar wavelet energy signatures for rotation and scale invariant texture classification. IEEE Trans. Pattern Anal. Mach. Intell. 25(5), 590–603 (2003)
Matungka, R.: Studies on Log-polar Transform for Image Registration and Improvements Using Adaptive Sampling and Logarithmic Spiral. The Ohio State University, Columbus (2009)
Yang, G.W., Song, X.Y., Hung, W.N.N., et al.: Group theory based synthesis of binary reversible circuits. Lecture Notes in Computer Science, vol. 3959, pp. 365–374 (2006)
Yan, F., Le, P.Q., Iliyasu, A.M., Sun, B., Garcia, J.A., Dong, F., Hirota, K.: Assessing the similarity of quantum images based on probability measurements, pp. 1–6. IEEE World Congress on Computational Intelligence (2012)
Fujimoto Lab. http://www.fujilab.dnj.ynu.ac.jp/study-sub-e.html
Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inf. Transm. 9, 177–183 (1973)
Acknowledgments
The authors appreciate the kind comments and professional criticisms of the anonymous reviewer. This has greatly enhanced the overall quality of the manuscript and opened numerous perspectives geared toward improving the work. This work is partially supported by National High-tech R&D Program of China (863 Program) under Grants 2012AA010901, 2012AA01A301, NCET, and National Science Foundation (NSF) China 61272142, 61103082, 61003075, 61170261, 61103193. Moreover, it is a part of Innovation Fund Sponsor Project of Excellent Postgraduate Student (B120601 and CX2012A002).
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Zhang, Y., Lu, K., Gao, Y. et al. A novel quantum representation for log-polar images. Quantum Inf Process 12, 3103–3126 (2013). https://doi.org/10.1007/s11128-013-0587-8
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DOI: https://doi.org/10.1007/s11128-013-0587-8