A New Hybrid form of Krawtchouk and Tchebichef Polynomials: Design and Application

  • Sadiq H. Abdulhussain
  • Abd Rahman Ramli
  • Basheera M. Mahmmod
  • M. Iqbal Saripan
  • S. A. R. Al-Haddad
  • Wissam A. Jassim


In the past decades, orthogonal moments (OMs) have received a significant attention and have widely been applied in various applications. OMs are considered beneficial and effective tools in different digital processing fields. In this paper, a new hybrid set of orthogonal polynomials (OPs) is presented. The new set of OPs is termed as squared Krawtchouk–Tchebichef polynomial (SKTP). SKTP is formed based on two existing hybrid OPs which are originated from Krawtchouk and Tchebichef polynomials. The mathematical design of the proposed OP is presented. The performance of the SKTP is evaluated and compared with the existing hybrid OPs in terms of signal representation, energy compaction (EC) property, and localization property. The achieved results show that SKTP outperforms the existing hybrid OPs. In addition, face recognition system is employed using a well-known database under clean and different noisy environments to evaluate SKTP capabilities. Particularly, SKTP is utilized to transform face images into moment (transform) domain to extract features. The performance of SKTP is compared with existing hybrid OPs. The comparison results confirm that SKTP displays remarkable and stable results for face recognition system.


Orthogonal polynomials Discrete orthogonal moments Energy compaction Localization property Face recognition system 

List of Abbreviation


Continuous orthogonal moment


Discrete cosine transform


Discrete Krawtchouk–Tchebichef transform


Discrete Tchebichef–Krawtchouk transform


Energy compaction


Face Recognition


Geometric moment


Krawtchouk Polynomial


Krawtchouk–Tchebichef polynomial


Orthogonal moment


Orthogonal polynomial


Squared Krawtchouk–Tchebichef polynomial


Squared discrete Krawtchouk–Tchebichef transform


Support vector machine


Tchebichef–Krawtchouk polynomial


Tchebichef polynomial


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hmimid, A., Sayyouri, M., Qjidaa, H.: Fast computation of separable two-dimensional discrete invariant moments for image classification. Pattern Recognit. 48(2), 509–521 (2015)CrossRefGoogle Scholar
  2. 2.
    Pee, C.-Y., Ong, S.H., Raveendran, P.: Numerically efficient algorithms for anisotropic scale and translation Tchebichef moment invariants. Pattern Recognit. Lett. 92, 68–74 (2017)CrossRefGoogle Scholar
  3. 3.
    Mahmmod, B.M., Ramli, A.R., Abdulhussain, S.H., Al-Haddad, S.A.R., Jassim, W.A., Abdulhussian, S.H., Al-Haddad, S.A.R., Jassim, W.A.: Low-distortion MMSE speech enhancement estimator based on laplacian prior. IEEE Access 5(1), 9866–9881 (2017)CrossRefGoogle Scholar
  4. 4.
    Abdulhussain, S.H., Ramli, A.R., Saripan, M.I., Mahmmod, B.M., Al-Haddad, S., Jassim, W.A.: Methods and challenges in shot boundary detection: a review. Entropy 20(4), 214 (2018)CrossRefGoogle Scholar
  5. 5.
    Hu, M.-K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179–187 (1962)CrossRefGoogle Scholar
  6. 6.
    Sheng, Y., Shen, L.: Orthogonal FourierMellin moments for invariant pattern recognition. JOSA A 11(6), 1748–1757 (1994)CrossRefGoogle Scholar
  7. 7.
    Chong, C.-W., Raveendran, P., Mukundan, R.: Translation and scale invariants of Legendre moments. Pattern Recognit. 37(1), 119–129 (2004)CrossRefGoogle Scholar
  8. 8.
    Khotanzad, A., Hong, Y.H.: Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Mach. Intell. 12(5), 489–497 (1990)CrossRefGoogle Scholar
  9. 9.
    Mukundan, R., Ong, S.H., Lee, P.A.: Image analysis by Tchebichef moments. IEEE Trans. Image Process. 10(9), 1357–1364 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Yap, P.-T., Paramesran, R., Ong, S.-H.: Image analysis by Krawtchouk moments. IEEE Trans. Image Process. 12(11), 1367–1377 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shao, Z., Shu, H., Wu, J., Chen, B., Coatrieux, J.L.: Quaternion BesselFourier moments and their invariant descriptors for object reconstruction and recognition. Pattern Recognit. 47(2), 603–611 (2014)CrossRefGoogle Scholar
  12. 12.
    Chen, B., Shu, H., Coatrieux, G., Chen, G., Sun, X., Coatrieux, J.L.: Color image analysis by quaternion-type moments. J. Math. Imaging Vis. 51(1), 124–144 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jassim, W.A., Raveendran, P., Mukundan, R.: New orthogonal polynomials for speech signal and image processing. IET Signal Process. 6(8), 713–723 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Foncannon, J.J.: Irresistible integrals: symbolics, analysis and experiments in the evaluation of integrals. Math. Intell. 28(3), 65–68 (2006)CrossRefGoogle Scholar
  15. 15.
    Jassim, W.A., Raveendran, P.: Face recognition using discrete Tchebichef–Krawtchouk transform. In: IEEE International Symposium on Multimedia (ISM), 2012 , pp. 120–127 (2012)Google Scholar
  16. 16.
    Rivero-Castillo, D., Pijeira, H., Assunçao, P.: Edge detection based on Krawtchouk polynomials. J. Comput. Appl. Math. 284, 244–250 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Abdulhussain, S.H., Ramli, A.R., Mahmmod, B.M., Al-Haddad, S.A.R., Jassim, W.A.: Image edge detection operators based on orthogonal polynomials. Int. J. Image Data Fusion 8(3), 293–308 (2017)Google Scholar
  18. 18.
    Yap, P.-T., Paramesran, R.: Local watermarks based on Krawtchouk moments. In: TENCON: 2004 IEEE region 10 conference IEEE 2004, pp. 73–76 (2004)Google Scholar
  19. 19.
    Mahmmod, B.M., bin Ramli, A.R., Abdulhussain, S.H., Al-Haddad, S.A.R., Jassim, W.A.: Signal compression and enhancement using a new orthogonal-polynomial-based discrete transform. IET Signal Process. 12(1), 129–142 (2018)Google Scholar
  20. 20.
    Xiao, B., Zhang, Y., Li, L., Li, W., Wang, G.: Explicit Krawtchouk moment invariants for invariant image recognition. J. Electron. Imag. 25(2), 23002 (2016)CrossRefGoogle Scholar
  21. 21.
    Nakagaki, K., Mukundan, R.: A fast 4 x 4 forward discrete tchebichef transform algorithm. IEEE Signal Process. Lett. 14(10), 684–687 (2007)CrossRefGoogle Scholar
  22. 22.
    Mukundan, R.: Some computational aspects of discrete orthonormal moments. IEEE Trans. Image Process. 13(8), 1055–1059 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Abdulhussain, S.H., Ramli, A.R., Al-Haddad, S.A.R., Mahmmod, B.M., Jassim, W.A.: On computational aspects of tchebichef polynomials for higher polynomial order. IEEE Access 5(1), 2470–2478 (2017)CrossRefGoogle Scholar
  24. 24.
    Abdulhussain, S.H., Ramli, A.R., Al-Haddad, S.A.R., Mahmmod, B.M., Jassim, w A: Fast recursive computation of krawtchouk polynomials. J. Math. Imag. Vis. 60(3), 285–303 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, G., Luo, Z., Fu, B., Li, B., Liao, J., Fan, X., Xi, Z.: A symmetry and bi-recursive algorithm of accurately computing Krawtchouk moments. Pattern Recognit. Lett. 31(7), 548–554 (2010)CrossRefGoogle Scholar
  26. 26.
    Thung, K.-H., Paramesran, R., Lim, C.-L.: Content-based image quality metric using similarity measure of moment vectors. Pattern Recognit. 45(6), 2193–2204 (2012)CrossRefGoogle Scholar
  27. 27.
    Hu, B., Liao, S.: Local feature extraction property of Krawtchouk moment. Lecture Notes Softw. Eng. 1(4), 356–359 (2013)CrossRefGoogle Scholar
  28. 28.
    Zhu, H., Liu, M., Shu, H., Zhang, H., Luo, L.: General form for obtaining discrete orthogonal moments. IET Image Process. 4(5), 335–352 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jain, A .K.: Fundamentals of Digital Image Processing. Prentice-Hall, Inc., Englewood (1989)zbMATHGoogle Scholar
  30. 30.
  31. 31.
    Chang, C.-C., Lin, C.-J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Computer EngineeringUniversity of BaghdadBaghdadIraq
  2. 2.Department of Computer and Communication System EngineeringUniversiti Putra MalaysiaSerdangMalaysia
  3. 3.ADAPT Center, School of Engineering, Trinity College DublinUniversity of DublinDublin 2Ireland

Personalised recommendations