Effective quaternion radial harmonic Fourier moments for color image representation


Image moments based on orthogonal manner play an important role in image processing and image analysis tasks. Although many variants have been proposed, there is still big room to reduce the geometrical errors and numerical errors. To address this issue, we propose an effective quaternion radial harmonic Fourier moments (Q-RHFMs) for color image representation. We compare the Q-RHFMs with other image moments and deep data-driven features, on multiple tasks including image reconstruction, watermarking and retrieval. Experimental results show the priorities of Q-RHFMs over other moments and deep features. The codes are available at https://github.com/ZlyaoNjust/Q-RHFMs.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Chun-peng, W., Xing-yuan, W., Zhi-qiu, X.: Geometrically invariant image watermarking based on fast radial harmonic Fourier moments. Sig. Process. Image Commun. 45, 10–23 (2016)

    Article  Google Scholar 

  2. 2.

    You, X., Du, L., Cheung, Y., Chen, Q.: A blind watermarking scheme using new nontensor product wavelet filter banks. IEEE Trans. Image Process. 19(12), 3271–3284 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Shao, Z., Shu, H., Wu, J., Chen, B., Coatrieux, J.L.: Quaternion Bessel–Fourier moments and their invariant descriptors for object reconstruction and recognition. Pattern Recognit. 47(2), 603–611 (2014)

    Article  Google Scholar 

  4. 4.

    Chevtchenko, S.F., Vale, R.F., Macario, V.: Multi-objective optimization for hand posture recognition. Expert Syst. Appl. 92, 170–181 (2018)

    Article  Google Scholar 

  5. 5.

    Bolourchi, P., Demirel, H., Uysal, S.: Target recognition in sar images using radial Chebyshev moments. SIViP 11(6), 1033–1040 (2017)

    Article  Google Scholar 

  6. 6.

    Di Ruberto, C., Putzu, L., Rodriguez, G.: Fast and accurate computation of orthogonal moments for texture analysis. Pattern Recognit. 83, 498–510 (2018)

    Article  Google Scholar 

  7. 7.

    Tran, T.-T., Pham, V.-T., Shyu, K.-K.: Zernike moment and local distribution fitting fuzzy energy-based active contours for image segmentation. SIViP 8(1), 11–25 (2014)

    Article  Google Scholar 

  8. 8.

    Batioua, I., Benouini, R., Zenkouar, K., Zahi, A., et al.: 3d image analysis by separable discrete orthogonal moments based on Krawtchouk and Tchebichef polynomials. Pattern Recognit. 71, 264–277 (2017)

    Article  Google Scholar 

  9. 9.

    Liu, X., Li, C., Tian, L.: Hand gesture recognition based on wavelet invariant moments. In: 2017 IEEE International Symposium on Multimedia (ISM), pp. 459–464. IEEE (2017)

  10. 10.

    Teague, M.R.: Image analysis via the general theory of moments. JOSA 70(8), 920–930 (1980)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Teh, C.-H., Chin, R.T.: On image analysis by the methods of moments. In: Proceedings CVPR’88: The Computer Society Conference on Computer Vision and Pattern Recognition, pp. 556–561. IEEE (1988)

  12. 12.

    Zhi, R., Cao, L., Cao, G.: Translation and scale invariants of Krawtchouk moments. Inf. Process. Lett. 130, 30–35 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gishkori, S., Mulgrew, B.: Pseudo-zernike moments based sparse representations for SAR image classification. IEEE Trans. Aerosp. Electron. Syst. 55, 1037–1044 (2018)

    Article  Google Scholar 

  14. 14.

    Ping, Z.L., Wu, R.G., Sheng, Y.L.: Image description with Chebyshev–Fourier moments. JOSA A 19(9), 1748–1754 (2002)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Uchaev, D.V., Uchaev, D.V., Malinnikov, V.A.: Orthogonal wavelet moments and their multifractal invariants. In: Seventh International Conference on Machine Vision (ICMV 2014), vol. 9445, p. 94450U. International Society for Optics and Photonics (2015)

  16. 16.

    Xiao, B., Ma, J.-F., Wang, X.: Image analysis by Bessel–Fourier moments. Pattern Recognit. 43(8), 2620–2629 (2010)

    Article  Google Scholar 

  17. 17.

    Sheng, Y., Shen, L.: Orthogonal Fourier–Mellin moments for invariant pattern recognition. JOSA A 11(6), 1748–1757 (1994)

    Article  Google Scholar 

  18. 18.

    Yang, J., Zhang, L., Tang, Y.Y.: Mellin polar coordinate moment and its affine invariance. Pattern Recognit. 85, 37–49 (2019)

    Article  Google Scholar 

  19. 19.

    Hu, H., Zhang, Y., Shao, C., Ju, Q.: Orthogonal moments based on exponent functions: exponent-Fourier moments. Pattern Recognit. 47(8), 2596–2606 (2014)

    Article  Google Scholar 

  20. 20.

    Upneja, R., Pawlak, M., Sahan, A.M.: An accurate approach for the computation of polar harmonic transforms. Optik 158, 623–633 (2018)

    Article  Google Scholar 

  21. 21.

    Ren, H., Ping, Z., Bo, W., Wu, W., Sheng, Y.: Multidistortion-invariant image recognition with radial harmonic fourier moments. JOSA A 20(4), 631–637 (2003)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Deng, A.-W., Wei, C.-H., Gwo, C.-Y.: Stable, fast computation of high-order Zernike moments using a recursive method. Pattern Recognit. 56, 16–25 (2016)

    Article  Google Scholar 

  23. 23.

    Upneja, R.: Accurate and fast Jacobi–Fourier moments for invariant image recognition. Optik 127(19), 7925–7940 (2016)

    Article  Google Scholar 

  24. 24.

    Singh, C., Ranade, S.K.: A high capacity image adaptive watermarking scheme with radial harmonic fourier moments. Digit. Signal Proc. 23(5), 1470–1482 (2013)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Singh, C., Upneja, R.: A computational model for enhanced accuracy of radial harmonic Fourier moments. In: World Congress of Engineering, London, UK, pp. 1189–1194 (2012)

  26. 26.

    Singh, C., Upneja, R.: Error analysis in the computation of orthogonal rotation invariant moments. J. Math. Imaging Vis. 49(1), 251–271 (2014)

    Article  Google Scholar 

  27. 27.

    Kantor, I.L., Kantor, I.L., Solodovnikov, A.S.: Hypercomplex Numbers: An Elementary Introduction to Algebras. Springer, Berlin (1989)

    Google Scholar 

  28. 28.

    Wang, C., Wang, X., Li, Y., Xia, Z., Zhang, C.: Quaternion polar harmonic fourier moments for color images. Inf. Sci. 450, 141–156 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Niu, P., Wang, P., Liu, Y., Yang, H., Wang, X.: Invariant color image watermarking approach using quaternion radial harmonic Fourier moments. Multimedia Tools Appl. 75(13), 7655–7679 (2016)

    Article  Google Scholar 

  30. 30.

    Yang, H.-Y., Wang, X.-Y., Niu, P.-P., Wang, A.-L.: Robust color image watermarking using geometric invariant quaternion polar harmonic transform. ACM Trans. Multimed. Comput. Commun. Appl. 11(3), 40 (2015)

    Google Scholar 

  31. 31.

    Hosny, K.M., Darwish, M.M.: Robust color image watermarking using invariant quaternion Legendre–Fourier moments. Multimed. Tools Appl. 77(19), 24727–24750 (2018)

    Article  Google Scholar 

  32. 32.

    He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778 (2016)

  33. 33.

    Philbin, J., Chum, O., Isard, M., Sivic, J., Zisserman, A.: Object retrieval with large vocabularies and fast spatial matching. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE (2007)

  34. 34.

    Krizhevsky, A., Hinton, G., et al.: Learning multiple layers of features from tiny images (2009)

  35. 35.

    Li, Y.N.: Quaternion polar harmonic transforms for color images. IEEE Signal Process. Lett. 20(8), 803–806 (2013)

    Article  Google Scholar 

Download references


The authors would like to thank the editor and the anonymous reviewers for their critical and constructive comments and suggestions. This work has been supported by the National Natural Science Foundation of China (Grant No. 61702262), Funds for International Cooperation and Exchange of the National Natural Science Foundation of China (Grant No. 61861136011), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20181299), Science and Technology on Parallel and Distributed Processing Laboratory (PDL) Open Fund (WDZC20195500106), Young Elite Scientists Sponsorship Program by CAST (2018QNRC001), the Fundamental Research Funds for the Central University (Grant No. 30920032201), National Science Fund of China (Grand No. U1713208), Program for Changing Scholars.

Author information



Corresponding author

Correspondence to Shanshan Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yao, Z., Liu, Y., Zhang, S. et al. Effective quaternion radial harmonic Fourier moments for color image representation. SIViP 15, 93–101 (2021). https://doi.org/10.1007/s11760-020-01726-z

Download citation


  • Color image representation
  • Orthogonal moments
  • Fourier transform
  • Image processing