Skip to main content
SpringerLink
Log in

Eigenvalues of Quaternion Tensors with Applications to Color Video Processing

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper studies the eigenvalues of quaternion tensors and their applications. Firstly, we investigate the definitions and properties of the left and right eigenvalues of the quaternion tensors. Then we establish the quaternion tensor Lanczos method to calculate the eigenvalues and eigentensors of a Hermitian quaternion tensor and the principal component analysis for the quaternion tensor. Moreover, we present the k-means method for quaternion tensor and use it to color video clustering. Finally, we conducted experiments to prove the effectiveness of the proposed theory and algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data Availibility

Enquiries about data availability should be directed to the authors.

References

  1. Bow, S.T.: Pattern Recognition and Image Preprocessing, Marcel Dekker, Inc. (2002)

  2. Brazell, M., Li, N., Navasca, C., Tamon, C.: Solving multilinear systems via tensor inversion. SIAM J. Matrix Anal. Appl. 34, 542–570 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, K.C., Qi, L.Q., Zhang, T.: A survey on the spectral theory of nonnegative tensors. Numer. Linear Algebra Appl. 20(6), 891–912 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, B., Shu, H., Coatrieux, G., Chen, G., Sun, X., et al.: Color image analysis by quaternion-type moments. J. Math. Imaging Vis. 51(1), 124–144 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Xiao, X., Zhou, Y.: Low-rank quaternion approximation for color image processing. IEEE Trans. Image Process. 29, 1426–1439 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z., Lu, L.Z.: A projection method and Kronecker product preconditioner for solving Sylvester tensor equations. Sci. China Math. 55(6), 1281–1292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Comon, P., G.H. G., L.H. Lim, B. Mourrain,: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30(3), 1254–1279 (2008)

  8. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. De Moor, B., Moonen, M. (eds.): SVD and Signal Processing. III. Algorithms, Applications and Architectures. Elsevier, Amsterdam (1995)

  10. Ding, W.Y., Wei, Y.M.: Solving multi-linear systems with \({\cal{M} }\)-tensors. J. Sci. Comput. 68(2), 689–715 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Einstein, A.: The foundation of the general theory of relativity. In: Kox, A.J., Klein, M.J., Schulmann, R. (eds.) The Collected Papers of Albert Einstein 6, pp. 146–200. Princeton University Press, Princeton (NJ) (2007)

  12. Farid, F.O., Wang, Q.W., Zhang, F.: On the eigenvalues of quaternion matrices. Linear Multilinear Algebra 59(4), 451–473 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gaidhane, V.H., Hote, Y.V., Singh, V.: An efficient approach for face recognition based on common eigenvalues. Pattern Recognit. 47(5), 1869–1879 (2014)

    Article  Google Scholar 

  14. Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins Press (1989)

  15. Hartigan, J.A., Wong, M.A.: A \(k\)-means clustering algorithm. J. R. Stat. Soc. 28(1), 100–108 (1979)

    MATH  Google Scholar 

  16. He, Z.H., Wang, Q.W., Zhang, Y.: A system of quaternary coupled Sylvester-type real quaternion matrix equations. Automatica 87, 25–31 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. He, Z.H., Liu, J., Tam, T.Y.: The general \(\phi \)-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations. Electron. J. Linear Algebra 32, 475–499 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, Z.H.: Some new results on a system of Sylvester-type quaternion matrix equations. Linear Multilinear Algebra 69(16), 3069–3091 (2021)

  19. He, Z.H., Chen, C., Wang, X.X.: A simultaneous decomposition for three quaternion tensors with applications in color video signal processing. Anal. Appl. (Singap.) 19(03), 529–549 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. He, Z.H., Ng, M.K., Zeng, C.: Generalized singular value decompositions for tensors and their applications. Numer. Math. Theory Methods Appl. 14(3), 692–713 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jia, Z., Ng, M.K., Song, G.J.: Robust quaternion matrix completion with applications to image inpainting. Numer. Linear Algebra Appl. 26, e2245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jia, Z., Ng, M.K., Song, G.J.: Lanczos method for large-scale quaternion singular value decomposition. Numer. Algorithms. 82, 699–717 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jia, Z., Ng, M.K.: Structure preserving quaternion generalized minimal residual method. SIAM J. Matrix Anal. Appl. 42, 616–634 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jolliffe, I.: Principal component analysis. Encyclopedia of Statistics in Behavioral Science (2005)

  25. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kyrchei, I.: Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl. 438, 136–152 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kyrchei, I.: Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl. Math. Comput. 238, 193–207 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Le, N., Bihan, S.J. Sangwine, Quaternion principal component analysis of color images, In: Proceedings 2003 international conference on image processing, vol 1, pp. I–809 (2003)

  29. Le Bihan, N., Mars, J.: Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Signal Process. 84(7), 1177–1199 (2004)

    Article  MATH  Google Scholar 

  30. Kanungo, T., Mount, D.M., Netanyahu, N., Piatko, C., Silverman, R., Wu, A.Y.: An efficient k-means clustering algorithm, In: Analysis and Implementation Proceedings of IEEE conference on computer vision and pattern recognition, pp. 881–892 (2002)

  31. Liang, M.L., Zheng, B., Zhao, R.J.: Tensor inversion and its application to the tensor equations with Einstein product. Linear Multilinear Algebra 67(4), 843–870 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Qi, L.Q., Yu, G.H., Wu, E.X.: Higher order positive semidefinite diffusion tensor imaging. SIAM J. Imaging Sci. 3(3), 416–433 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rodman, L.: Topics in Quaternion Linear Algebra. Princeton University Press (2014)

  34. Sun, L.Z., Zheng, B.D., Bu, C.J., Wei, Y.M.: Moore–Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64, 686–698 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thodi, D.M., Rodriguez, J.J.: Expansion embedding techniques for reversible watermarking. IEEE Trans. Image Process. 16(3), 721–730 (2007)

    Article  MathSciNet  Google Scholar 

  36. Tian, H., Cai, H., Lai, J. H., Xu, X.: Effective image noise removal based on difference eigenvalue. In: IEEE international conference on image processing. IEEE, pp. 3357–3360 (2011)

  37. Took, C.C., Mandic, D.P.: Augmented second-order statistics of quaternion random signals. Signal Process. 91, 214–224 (2011)

    Article  MATH  Google Scholar 

  38. Took, C.C., Mandic, D.P.: The quaternion LMS algorithm for adaptive filtering of hypercomplex real world processes. IEEE Trans. Signal Process. 57, 1316–1327 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Took, C.C., Mandic, D.P.: Quaternion-valued stochastic gradient-based adaptive IIR filtering. IEEE Trans. Signal Process. 58(7), 3895–3901 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, Q.W., Wang, X.X.: Arnoldi method for large quaternion right eigenvalue problem. J. Sci. Comput. 82, 58 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xu, M., Wei, C.: Remotely sensed image classification by complex network eigenvalue and connected degree. Comput. Math. Methods Med. 2012, 143–156 (2012)

    Article  MATH  Google Scholar 

  42. Yang, Y.N., Yang, Q.Z.: Further results for Perron Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31(5), 2517–2530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu, S.W., He, Z.H., Qi, T.C., Wang, X.X.: The equivalence canonical form of five quaternion matrices with applications to imaging and Sylvester-type equations. J. Comput. Appl. Math. 393, 113494 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research was supported by the National Natural Science Foundation of China (Grant Nos. 12271338 and 11971294).

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Zhuo-Heng He

  2. Department of Mathematics and Statistics, University of Nevada, Reno, NV, 89557, USA

    Xiang-Xiang Wang

  3. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095, USA

    Yun-Fan Zhao

Authors
  1. Zhuo-Heng He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang-Xiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yun-Fan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuo-Heng He.

Ethics declarations

Conflicts of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

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

This research was supported by the National Natural Science Foundation of China (Grant Nos. 12271338 and 11971294).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, ZH., Wang, XX. & Zhao, YF. Eigenvalues of Quaternion Tensors with Applications to Color Video Processing. J Sci Comput 94, 1 (2023). https://doi.org/10.1007/s10915-022-02058-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-02058-5

Keywords

AMS Subject Classification

Search

Navigation