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

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Abstract

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.

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Abbreviations

COM:

Continuous orthogonal moment

DCT:

Discrete cosine transform

DKTT:

Discrete Krawtchouk–Tchebichef transform

DTKT:

Discrete Tchebichef–Krawtchouk transform

EC:

Energy compaction

FR:

Face Recognition

GM:

Geometric moment

KP:

Krawtchouk Polynomial

KTP:

Krawtchouk–Tchebichef polynomial

OM:

Orthogonal moment

OP:

Orthogonal polynomial

SKTP:

Squared Krawtchouk–Tchebichef polynomial

SKTT:

Squared discrete Krawtchouk–Tchebichef transform

SVM:

Support vector machine

TKP:

Tchebichef–Krawtchouk polynomial

TP:

Tchebichef polynomial

References

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  5. Hu, M.-K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179–187 (1962)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  7. Chong, C.-W., Raveendran, P., Mukundan, R.: Translation and scale invariants of Legendre moments. Pattern Recognit. 37(1), 119–129 (2004)

    Article  MATH  Google Scholar 

  8. Khotanzad, A., Hong, Y.H.: Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Mach. Intell. 12(5), 489–497 (1990)

    Article  Google Scholar 

  9. Mukundan, R., Ong, S.H., Lee, P.A.: Image analysis by Tchebichef moments. IEEE Trans. Image Process. 10(9), 1357–1364 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Yap, P.-T., Paramesran, R., Ong, S.-H.: Image analysis by Krawtchouk moments. IEEE Trans. Image Process. 12(11), 1367–1377 (2003)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Foncannon, J.J.: Irresistible integrals: symbolics, analysis and experiments in the evaluation of integrals. Math. Intell. 28(3), 65–68 (2006)

    Article  Google Scholar 

  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)

  16. Rivero-Castillo, D., Pijeira, H., Assunçao, P.: Edge detection based on Krawtchouk polynomials. J. Comput. Appl. Math. 284, 244–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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. Yap, P.-T., Paramesran, R.: Local watermarks based on Krawtchouk moments. In: TENCON: 2004 IEEE region 10 conference IEEE 2004, pp. 73–76 (2004)

  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)

  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)

    Article  Google Scholar 

  21. Nakagaki, K., Mukundan, R.: A fast 4 x 4 forward discrete tchebichef transform algorithm. IEEE Signal Process. Lett. 14(10), 684–687 (2007)

    Article  Google Scholar 

  22. Mukundan, R.: Some computational aspects of discrete orthonormal moments. IEEE Trans. Image Process. 13(8), 1055–1059 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  27. Hu, B., Liao, S.: Local feature extraction property of Krawtchouk moment. Lecture Notes Softw. Eng. 1(4), 356–359 (2013)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  29. Jain, A .K.: Fundamentals of Digital Image Processing. Prentice-Hall, Inc., Englewood (1989)

    MATH  Google Scholar 

  30. AT&T Corp.: The Database of Faces (2016). http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html

  31. Chang, C.-C., Lin, C.-J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Engineering, University of Baghdad, Baghdad, Iraq

    Sadiq H. Abdulhussain & Basheera M. Mahmmod

  2. Department of Computer and Communication System Engineering, Universiti Putra Malaysia, Serdang, Selangor, Malaysia

    Sadiq H. Abdulhussain, Abd Rahman Ramli, Basheera M. Mahmmod, M. Iqbal Saripan & S. A. R. Al-Haddad

  3. ADAPT Center, School of Engineering, Trinity College Dublin, University of Dublin, Dublin 2, Ireland

    Wissam A. Jassim

Authors
  1. Sadiq H. Abdulhussain
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  2. Abd Rahman Ramli
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  3. Basheera M. Mahmmod
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  4. M. Iqbal Saripan
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  5. S. A. R. Al-Haddad
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  6. Wissam A. Jassim
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Correspondence to Sadiq H. Abdulhussain.

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Abdulhussain, S.H., Ramli, A.R., Mahmmod, B.M. et al. A New Hybrid form of Krawtchouk and Tchebichef Polynomials: Design and Application. J Math Imaging Vis 61, 555–570 (2019). https://doi.org/10.1007/s10851-018-0863-4

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