Abstract
Quantum median filtering is an important step for many quantum signal processing algorithms. Current quantum median filtering designs show limitations in either computational complexity or incomplete noise detection. We propose a design of quantum median filtering, which uses approximate median filtering with noise tolerance threshold to remove salt-and-pepper noise. Instead of calculating the median, we search an approximate median by sorting four times, which reduces the computational complexity from \(O\left( {21{q^2} + 63q} \right) \) to \(O\left( {12{q^2} + 36q} \right) \). Here, q is the qubit used to represent the gray value. Furthermore, we adopt a two-level threshold to detect the noise points as much as possible. Finally, we design a complete quantum circuit to implement the approximate median filtering. The computational complexity of our proposed circuit is \(O\left( {10{n^2} + 14{q^2}} \right) \) for a NEQR quantum image with a size of \({2^n} \times {2^n}\). The complexity analysis shows that our proposed method significantly speeds up the filtering process compared with the classical filtering methods and the existing quantum filtering methods. In addition, the simulation results prove the proposed approximate median filtering is feasible.
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References
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)
Li, H.S., Chen, X., Xia, H.Y., et al.: A quantum image representation based on bitplanes. IEEE Access 6, 62396–62404 (2018)
Vlasov, A.Y.: Quantum computations and images recognition. arxiv:quant-ph/9703010 (1997). Accessed 20 Oct 2018
Beach, G., Lomont, C., Cohen, C.: Quantum image processing. In: Proceedings of the 32nd IEEE Conference Applied Imagery Pattern Recognition, Bellingham, WA, USA, 39-44 (2003)
Iliyasu, A.M.: Roadmap to talking quantum movies: a contingent inquiry. IEEE Access 7(99), 23864–23913 (2019)
Venegasandraca, S.E.: Storing, processing, and retrieving an image using quantum mechanics. Proc. SPIE Int. Soc. Opt. Eng. 5105(8), 1085–1090 (2003)
Latorre, J.I.: Image compression and entanglement. arxiv:quant-ph/0510031 (2005). Accessed 18 Feb 2019
Venegas-Andraca, S.E., Ball, J.L.: Processing images in entangled quantum systems. Quantum Inf. Process. 9(1), 1–11 (2010)
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)
Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Inf. Process. 15, 1–35 (2016)
Zhang, Y., Lu, K., Gao, Y.: NEQR: a novel enhanced quantum representation of digital images. Quantum Inf. Process. 12(8), 2833–2860 (2013)
Sun, B., Iliyasu, A.M., Yan, F., Dong, F., et al.: An RGB multi-channel representation for images on quantum computers. J. Adv. Comput. Intell. Intell. Info 17(3), 404–417 (2013)
Yuan, S., Mao, X., Xue, Y., et al.: SQR: a simple quantum representation of infrared images. Quantum Inf. Process. 13(6), 1353–1379 (2014)
Li, H., Zhu, Q., Zhou, R., et al.: Multi-dimensional color image storage and retrieval for a normal arbitrary quantum superposition state. Quantum Inf. Process. 13(4), 991–1011 (2014)
Zhang, Y., Lu, K., Gao, Y., et al.: A novel quantum representation for log-polar images. Quantum Inf. Process. 12(9), 3103–3126 (2013)
Le, P.Q., Iliyasu, A.M., Dong, F., et al.: Fast geometric transformations on quantum images. IAENG Int. J. Appl. Math. 40(3), 113–123 (2010)
Le, P.Q., Iliyasu, A.M., Dong, F., et al.: Strategies for designing geometric transformations on quantum images. Theor. Comput. Sci. 412(15), 1406–1418 (2011)
Wang, J., Nan, Jiang, Luo, Wang: Quantum image translation. Quantum Inf. Process. 14(5), 1–16 (2014)
Fan, P., Zhou, R.G., Jing, N., et al.: Geometric transformations of multidimensional color images based on NASS. Inf. Sci. 340, 191–208 (2016)
Zhou, R.G., Tan, C., Ian, H.: Global and local translation designs of quantum image based on FRQI. Int. J. Theor. Phys. 56(4), 1382–1398 (2017)
Iliyasu, A.M., Le, P.Q., Dong, F., et al.: Watermarking and authentication of quantum images based on restricted geometric transformations. Inf. Sci. 186(1), 126–149 (2012)
Zhang, Y., Lu, K., Xu, K., et al.: Local feature point extraction for quantum images. Quantum Inf. Process. 14(5), 1573–1588 (2015)
Zhang, Y., Lu, K., Gao, Y.H.: QSobel: a novel quantum image edge extraction algorithm. Sci. China Inf. Sci. 58(1), 1–13 (2015)
Jiang, N., Yijie, Dang, Wang, J.: Quantum image matching. Quantum Inf. Process. 15(9), 3543–3572 (2016)
Dang, Y., Jiang, N., Hu, H., et al.: Analysis and improvement of the quantum image matching. Quantum Inf. Process. 16(11), 269 (2017)
Li, H.S., Qingxin, Z., Lan, S., et al.: Image storage, retrieval, compression and segmentation in a quantum system. Quantum Inf. Process. 12(6), 2269–2290 (2013)
Caraiman, S., Manta, V.I.: Image segmentation on a quantum computer. Quantum Inf. Process. 14(5), 1693–1715 (2015)
Iliyasu, A.M., Le, P.Q., Dong, F., et al.: Watermarking and authentication of quantum images based on restricted geometric transformations. Inf. Sci. 186, 126–149 (2012)
Zhang, W.W., Gao, F., Liu, B., et al.: A watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(4), 793–803 (2013)
Yang, Y.G., Jia, X., Xu, P., et al.: Analysis and improvement of the watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(8), 2765–2769 (2013)
Song, X.H., Wang, S., Liu, S., et al.: A dynamic watermarking scheme for quantum images using quantum wavelet transform. Quantum Inf. Process. 12(12), 3689–3706 (2013)
Song, X.H., Niu, X.M.: Comment on: novel image encryption/decryption based on quantum fourier transform and double phase encoding. Quantum Inf. Process. 13(6), 1301–1304 (2014)
Hua, T., Chen, J., Pei, D., et al.: Quantum image encryption algorithm based on image correlation decomposition. Int. J. Theor. Phys. 54(2), 526–537 (2015)
Zhou, R.G., Wu, Q., Zhang, M.Q., et al.: Quantum image encryption and decryption algorithms based on quantum image geometric transformations. Int. J. Theor. Phys. 52, 1802–1817 (2013)
Iliyasu, A.M., Le, P.Q., Dong, F., et al.: A framework for representing and producing movies on quantum computers. Int. J. Quantum Inf. 09(06), 1459–1497 (2011)
Yan, F., Iliyasu, A.M., Yang, H., et al.: Video encryption and decryption on quantum computers. Int. J. Theor. Phys. 54(8), 2893–2904 (2015)
Yan, F., Iliyasu, A.M., Khan, A.R., et al.: Measurements-based moving target detection in quantum video. Int. J. Theor. Phys. 55(4), 2162–2173 (3016)
Yan, F., Le, P.Q., Iliyasu, A.M., et al.: Assessing the similarity of quantum images based on probability measurements. In: 2012 IEEE Congress on Evolutionary Computation. Brisbane, Australia (2012)
Yan, F., Iliyasu, A.M., Abdullah, et al.: A parallel comparison of multiple pairs of images on quantum computers. Int. J. Innov. Comput. Appl. 5, 199–212 (2016)
Iliyasu, A.M., Yan, F., Kaoru, H.: Metric for estimating congruity between quantum images. Entropy 18(10), 360 (2016)
Liu, X.A., Zhou, R.G., El-Rafei, A.: Similarity assessment of quantum images. Quantum Inf. Process. 18(8), 244 (2019)
Zhou, R.G., Liu, X.A., Zhu, C., et al.: Similarity analysis between quantum images. Quantum Inf. Process. 17(6), 121 (2018)
Lomont, C.: Quantum convolution and quantum correlation algorithms are physically impossible. arxiv:quant-ph/0309070 (2005). Accessed 20 Feb 2019
Yuan, S., Lu, Y., Mao, X., Zhou, J., et al.: Quantum image filtering in the spatial domain. Int. J. Theor. Phys. 56(8), 1572–9575 (2017)
Yuan, S., Lu, Y., Mao, X., et al.: Improved quantum image filtering in the spatial domain. Int. J. Theor. Phys. 57(3), 804–813 (2018)
Li, P., Liu, X., Xiao, H.: Quantum image weighted average filtering in spatial domain. Int. J. Theor. Phys. 56(11), 3690–3716 (2017)
Li, P., Liu, X., Xiao, H.: Quantum image median filtering in the spatial domain. Quantum Inf. Process. 17(3), 1573-1332 (2018)
Jiang, S.X., Zhou, R.G., Hu, W.W., et al.: Improved quantum image median filtering in the spatial domain. Int. J. Theor. Phys. 58(7), 2115–2133 (2019)
Fan, P., Zhou, R.G., Hu, W.W., et al.: Quantum image edge extraction based on classical Sobel operator for NEQR. Quantum Inf. Process. 18(1), 1573-1332 (2019)
Dong, W., University, H., Kaifeng, et al.: Design of quantum comparator based on extended general toffoli gates with multiple targets. Comput. Sci. 39(9), 302–306 (2012)
Barenco, A., Bennett, C.H., Cleve, R., et al.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995)
Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant Nos. 61762014 and 61762012 and the Science and Technology Project of Guangxi under Grant Nos. 2018JJA170083 and 2018JJA170089.
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Xia, H., Xiao, Y., Song, S. et al. Quantum circuit design of approximate median filtering with noise tolerance threshold. Quantum Inf Process 19, 183 (2020). https://doi.org/10.1007/s11128-020-02678-6
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DOI: https://doi.org/10.1007/s11128-020-02678-6