Quantum image edge extraction based on classical Sobel operator for NEQR

  • Ping FanEmail author
  • Ri-Gui Zhou
  • Wenwen Hu
  • Naihuan Jing


As the basic problem in image processing and computer vision, the purpose of edge detection is to identify the point where the brightness of the digital image changes obviously. It is an indispensable task in digital image processing that image edge detection significantly reduces the amount of data and eliminates information that can be considered irrelevant, preserving the important structural properties of the image. However, because of the sharp increase in the image data in the actual applications, real-time problem has become a limitation in classical image processing. In this paper, quantum image edge extraction for the novel enhanced quantum representation (NEQR) is designed based on classical Sobel operator. The quantum image model of NEQR utilizes the inherent entanglement and superposition properties of quantum mechanics to store all the pixels of an image in a superposition state, which can realize parallel computation for calculating the gradients of the image intensity of all the pixels simultaneously. Through constructing and analyzing the quantum circuit of realization image edge extraction, we demonstrate that our proposed scheme can extract edges in the computational complexity of \(\mathrm{O}({n^2} + {2^{q + 4}})\) for a NEQR quantum image with a size of \({2^n} \times {2^n}\). Compared with all the classical edge extraction algorithms and some existing quantum edge extraction algorithms, our proposed scheme can reach a significant and exponential speedup. Hence, our proposed scheme would resolve the real-time problem of image edge extraction in practice image processing.


Quantum image processing Edge detection Sobel operator Real-time problem 



This work is supported by the National Natural Science Foundation of China under Grant Nos. 61763014, 61463016, 61462026, and 61762012, the National Key R&D Plan under Grant Nos. 2018YFC1200200 and 2018YFC1200205, the Fund for Distinguished Young Scholars of Jiangxi Province under Grant No. 2018ACB21013, Science and technology research project of Jiangxi Provincial Education Department under Grant No. GJJ170382, Project of International Cooperation and Exchanges of Jiangxi Province under Grant No. 20161BBH80034, Project of Humanities and Social Sciences in colleges and universities of Jiangxi Province under Grant No.JC161023.


  1. 1.
    Yan, F., Iliyasu, A.M., Le, P.Q.: A review of advances in its security technologies. Int. J. Quantum Inf. 15, 1730001 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Inf. Process. 15(1), 1–35 (2016)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Feynman, R.: Simulating physics with computers. Perseus Books 21(6), 467–488 (1999)MathSciNetGoogle Scholar
  4. 4.
    Deutsch, D.: Quantum theory, the church-turing principle and the universal quantum computer. Proc. R. Soc. Lond. 400(1818), 97–117 (1985)ADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Shor, P.: Quantum theory, Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994)Google Scholar
  6. 6.
    Grover, L.: Quantum theory, A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 212–219 (1996)Google Scholar
  7. 7.
    Iliyasu, A.M.: Quantum theory, towards the realisation of secure and efficient image and video processing applications on quantum computers. Entropy 15, 2874–2974 (2013)ADSMathSciNetzbMATHGoogle Scholar
  8. 8.
    Iliyasu, A.M., Le, P.Q., et al.: A two-tier scheme for greyscale quantum image watermarking and recovery. Int. J. Innov. Comput. Appl. 5(2), 85–101 (2013)Google Scholar
  9. 9.
    ILugiato, L.A., Gatti, A., Brambilla, E.: Quantum imaging. J. Opt. B Quantum Semiclassical Opt. 4(3), 176–184 (2002)Google Scholar
  10. 10.
    Eldar, Y.C., Oppenheim, A.V.: Quantum signal processing. IEEE Signal Process. Mag. 19(6), 12–32 (2001)ADSGoogle Scholar
  11. 11.
    Schutzhold, R.: Pattern recognition on a quantum computer. Phys. Rev. A 67(6), 062311 (2002)ADSMathSciNetGoogle Scholar
  12. 12.
    Venegasandraca, S.E.: Storing, processing, and retrieving an image using quantum mechanics. Proc. SPIE Int. Soc. Opt. Eng. 5105(8), 1085–1090 (2003)Google Scholar
  13. 13.
    Venegasandraca, S.E., Ball, J.: Processing images in entangled quantum systems. Quantum Inf. Process. 9(1), 1–11 (2010)MathSciNetGoogle Scholar
  14. 14.
    Latorre, J.I. : Image Compression and Entanglement. arXiv:quant-ph/0510031 (2005)
  15. 15.
    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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhang, Y., Lu, K., Gao, Y., et al.: NEQR: a novel enhanced quantum representation of digital images. Quantum Inf. Process. 12(8), 2833–2860 (2013)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhang, Y., Lu, K., Gao, Y., Gao, Y., et al.: A novel quantum representation for log-polar images. Quantum Inf. Process. 12(9), 3103–3126 (2013)ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Li, H.S., Zhu, Q., Song, L., et al.: Image storage, retrieval, compression and segmentation in a quantum system. Quantum Inf. Process. 12(6), 2269–2290 (2013)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Li, H.S., 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)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Yuan, S., Mao, X., Xue, Y., et al.: SQR: a simple quantum representation of infrared images. Quantum Inf. Process. 13(6), 1353–1379 (2014)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Sang, J., Wang, S., Li, Q.: A novel quantum representation of color digital images. Quantum Inf. Process. 16(2), 42 (2017)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, H., Fan, P., Xia, H., et al.: Quantum implementation circuits of quantum signal representation and type conversion. IEEE Trans. Circuits Syst. I Regul. Pap. PP(99), 1–14 (2018)Google Scholar
  23. 23.
    Le, P.Q., Iliyasu, A.M., Dong, F., et al.: Fast geometric transformations on quantum images. Int. J. Appl. Math. 40(3), 113–123 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fan, P., Zhou, R., Jing, N., et al.: Geometric transformations of multidimensional color images based on NASS. Inf. Sci. 340, 191–208 (2016)Google Scholar
  25. 25.
    Wang, J., Jiang, N., Wang, L.: Quantum image translation. Quantum Inf. Process. 14(5), 1589–1604 (2015)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Jiang, N., Wang, L.: Quantum image scaling using nearest neighbor interpolation. Quantum Inf. Process. 14(5), 1559–1571 (2015)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Sang, J., Wang, S., Niu, X.: Quantum realization of the nearest-neighbor interpolation method for FRQI and NEQR. Quantum Inf. Process. 15(1), 37–64 (2016)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhou, R.G., Hu, W., Fan, P., et al.: Quantum realization of the bilinear interpolation method for NEQR. Sci. Rep. 7, 2511 (2017)ADSGoogle Scholar
  30. 30.
    Jiang, N., Wu, W.Y., Wang, L.: The quantum realization of Arnold and Fibonacci image scrambling. Quantum Inf. Process. 13(5), 1223–1236 (2014)ADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Jiang, N., Wang, L., Wu, W.Y.: Quantum Hilbert image scrambling. Int. J. Theor. Phys. 53(7), 2463–2484 (2014)zbMATHGoogle Scholar
  32. 32.
    Zhou, R.G., Sun, Y.J., Fan, P.: Quantum image Gray-code and bit-plane scrambling. Quantum Inf. Process. 14(5), 1717–1734 (2015)ADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Mogos, G.: Hiding data in a qimage file. Lect. Notes Eng. Comput. Sci. 2174(1), 448–452 (2009)Google Scholar
  34. 34.
    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)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhang, W.W., Gao, F., Liu, B., et al.: A watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(2), 793–803 (2015)ADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Song, X., Wang, S., El-Latif, A.A.A., et al.: Dynamic watermarking scheme for quantum images based on Hadamard transform. Multimed. Syst. 20(4), 379–388 (2014)Google Scholar
  37. 37.
    Miyake, S., Nakamae, K.: A quantum watermarking scheme using simple and small-scale quantum circuits. Quantum Inf. Process. 15(5), 1849–1864 (2016)ADSMathSciNetzbMATHGoogle Scholar
  38. 38.
    Jiang, N., Zhao, N., Wang, L.: LSB based quantum image steganography algorithm. Int. J. Theor. Phys. 55(1), 107–123 (2016)zbMATHGoogle Scholar
  39. 39.
    Shahrokh, H., Mosayeb, N.: A novel LSB based quantum watermarking. Int. J. Theor. Phys. 55(10), 1–14 (2016)zbMATHGoogle Scholar
  40. 40.
    Tseng, C., Hwang, T.: Quantum digital image processing algorithms. In: Proceedings of the 16th IPPR Conference on Computer Vision, Graphics and Image Processing pp. 827–834 (2003)Google Scholar
  41. 41.
    Fu, X., Ding, M., Sun, Y., et al.: A new quantum edge detection algorithm for medical images. Proc. SPIE Int. Soc. Opt. Eng. 7497, 749724 (2009)Google Scholar
  42. 42.
    Zhang, Y., Lu, K., Gao, Y.: Q Sobel: a novel quantum image edge extraction algorithm. Sci. China Inf. Sci. 58(1), 1–13 (2015)Google Scholar
  43. 43.
    Zhang, Y., Lu, K., Xu, K., et al.: Local feature point extraction for quantum images. Quantum Inf. Process. 14(5), 1573–1588 (2015)ADSMathSciNetzbMATHGoogle Scholar
  44. 44.
    Jiang, N., Dang, K.Y., Wang, J.: Quantum image matching. Quantum Inf. Process. 15(9), 3543–3572 (2016)ADSMathSciNetzbMATHGoogle Scholar
  45. 45.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Publishing House of Electronics Industry, Prentice Hall (2002)Google Scholar
  46. 46.
    Gao, W.: Technique of Multimedia Data Compression. Publishing House of Electronics Industry, Prentice Hall (1994)Google Scholar
  47. 47.
    Kirsch, R.A.: Computer determination of the constituent structure of biological images. Comput. Biol. Med. 4(3), 315–328 (1971)Google Scholar
  48. 48.
    Canny, J.: A computational approach to edge detection. IEEE TPAMI. 8(6), 679–697 (1986)Google Scholar
  49. 49.
    Islam, M.S., Rahman, M.M., Begum, Z., et al.: Low cost quantum realization of reversible multiplier circuit. Inf. Technol. J. 8(2), 208–213 (2009)Google Scholar
  50. 50.
    Thapliyal, H., Ranganathan, N.: Design of efficient reversible binary subtractors based on a new reversible gate. In: Proceedings of the IEEE Computer Society Annual Symposium on VLSI, Tampa, Florida, pp. 229–234 (2009)Google Scholar
  51. 51.
    Thapliyal, H., Ranganathan, N.: A new design of the reversible subtractor circuit. Nanotechnology 117, 1430–1435 (2011)Google Scholar
  52. 52.
    Barenco, A., Bennett, C.H., Cleve, R., et al.: Elementary gates for quantum computation. Phys. Rev. A. 52(5), 3457–3467 (1995)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ping Fan
    • 1
    Email author
  • Ri-Gui Zhou
    • 2
  • Wenwen Hu
    • 2
  • Naihuan Jing
    • 3
  1. 1.School of Information EngineeringEast China Jiaotong UniversityNanchangChina
  2. 2.College of Information EngineeringShanghai Maritime UniversityShanghaiChina
  3. 3.Department of mathematicsNorth Carolina State UniversityRaleighUSA

Personalised recommendations