Journal of Mathematical Imaging and Vision

, Volume 61, Issue 4, pp 555–570 | Cite as

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

  • Sadiq H. AbdulhussainEmail author
  • 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)zbMATHGoogle 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)Google 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)Google 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)Google Scholar
  5. 5.
    Hu, M.-K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179–187 (1962)zbMATHGoogle Scholar
  6. 6.
    Sheng, Y., Shen, L.: Orthogonal FourierMellin moments for invariant pattern recognition. JOSA A 11(6), 1748–1757 (1994)Google Scholar
  7. 7.
    Chong, C.-W., Raveendran, P., Mukundan, R.: Translation and scale invariants of Legendre moments. Pattern Recognit. 37(1), 119–129 (2004)zbMATHGoogle 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)Google 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)MathSciNetzbMATHGoogle 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)MathSciNetGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetGoogle Scholar
  14. 14.
    Foncannon, J.J.: Irresistible integrals: symbolics, analysis and experiments in the evaluation of integrals. Math. Intell. 28(3), 65–68 (2006)Google 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)MathSciNetzbMATHGoogle 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)Google 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)Google Scholar
  22. 22.
    Mukundan, R.: Some computational aspects of discrete orthonormal moments. IEEE Trans. Image Process. 13(8), 1055–1059 (2004)MathSciNetGoogle 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)Google 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)MathSciNetzbMATHGoogle 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)Google 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)zbMATHGoogle Scholar
  27. 27.
    Hu, B., Liao, S.: Local feature extraction property of Krawtchouk moment. Lecture Notes Softw. Eng. 1(4), 356–359 (2013)Google 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)MathSciNetGoogle 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)Google 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