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Fast Recursive Computation of Krawtchouk Polynomials

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Abstract

Krawtchouk polynomials (KPs) and their moments are used widely in the field of signal processing for their superior discriminatory properties. This study proposes a new fast recursive algorithm to compute Krawtchouk polynomial coefficients (KPCs). This algorithm is based on the symmetry property of KPCs along the primary and secondary diagonals of the polynomial array. The \(n-x\) plane of the KP array is partitioned into four triangles, which are symmetrical across the primary and secondary diagonals. The proposed algorithm computes the KPCs for only one triangle (partition), while the coefficients of the other three triangles (partitions) can be computed using the derived symmetry properties of the KP. Therefore, only N / 4 recursion times are required. The proposed algorithm can also be used to compute polynomial coefficients for different values of the parameter p in interval (0, 1). The performance of the proposed algorithm is compared with that in previous literature in terms of image reconstruction error, polynomial size, and computation cost. Moreover, the proposed algorithm is applied in a face recognition system to determine the impact of parameter p on feature extraction ability. Simulation results show that the proposed algorithm has a remarkable advantage over other existing algorithms for a wide range of parameters p and polynomial size N, especially in reducing the computation time and the number of operations utilized.

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References

  1. Lau, T.S.: Theory of canonical moments and its applications in polynomial regression—part I. Purdue University Technical Reports, pp. 1–73 (1983)

  2. Sadjang, P.N., Koepf, W., Foupouagnigni, M.: On moments of classical orthogonal polynomials. J. Math. Anal. Appl. 424(1), 122–151 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Alt, F.L.: Digital pattern recognition by moments. J. ACM 9(2), 240–258 (1962)

    Article  MATH  Google Scholar 

  5. Jassim, W.A., Raveendran, P.: Face recognition using discrete Tchebichef–Krawtchouk transform. In: 2012 IEEE International Symposium on Multimedia (ISM), pp. 120–127 (2012)

  6. Jassim, W.A., Paramesran, R., Zilany, M.S.A.: Enhancing noisy speech signals using orthogonal moments. IET Signal Process. 8(8), 891–905 (2014)

    Article  Google Scholar 

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

  8. 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 

  9. 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)

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

    Article  MathSciNet  Google Scholar 

  11. Yap, P.-T., Paramesran, R., Ong, S.-H.: Image analysis using Hahn moments. Pattern Anal. Mach. Intell. IEEE Trans. 29(11), 2057–2062 (2007)

    Article  Google Scholar 

  12. 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, 2470–2478 (2017)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  15. Nikiforov, A.F., Uvarov, V.B., Suslov, S.K.: Classical orthogonal polynomials of a discrete variable. In: Fletcher, C.A.J., Glowinski, R., Hillebrandt, W., Holt, M., Hut, P., Keller, H.B., Killeen, J., Orszag, S.A., Rusanov, V.V. (eds.) Classical Orthogonal Polynomials of a Discrete Variable, pp. 18–54. Springer-Verlag, Berlin, Heidelberg (1991)

  16. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  17. Yap, P.T., Raveendran, P., Ong, S.H.: Krawtchouk moments as a new set of discrete orthogonal moments for image reconstruction. In: Proceedings of the 2002 International Joint Conference on Neural Networks (IJCNN ’02), vol. 1, pp. 908–912 (2002). doi:10.1109/IJCNN.2002.1005595

  18. 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 

  19. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  21. 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 

  22. 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 

  23. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1964)

  24. Chihara, L., Stanton, D.: Zeros of generalized Krawtchouk polynomials. J. Approx. Theory 60(1), 43–57 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fienup, J.R.: Invariant error metrics for image reconstruction. Appl. Opt. 36(32), 8352–8357 (1997)

    Article  Google Scholar 

  26. AT&T Corp: The database of faces (2016). http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html. Accessed 01 Jan 2016

  27. Peng, H., Long, F., Ding, C.: Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy. IEEE Trans. Pattern Anal. Mach. Intell. 27(8), 1226–1238 (2005)

    Article  Google Scholar 

  28. 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 and Communication Systems Engineering, Universiti Putra Malaysia, Serdang, Malaysia

    Sadiq H. Abdulhussain, Abd Rahman Ramli, Syed Abdul Rahman Al-Haddad & Basheera M. Mahmmod

  2. Sigmedia, ADAPT Center, Department of Electronic and Electrical Engineering, Trinity College Dublin, Dublin 2, Ireland

    Wissam A. Jassim

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

    Sadiq H. Abdulhussain & Syed Abdul Rahman Al-Haddad

  4. 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. Syed Abdul Rahman Al-Haddad
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  4. Basheera M. Mahmmod
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  5. Wissam A. Jassim
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Correspondence to Sadiq H. Abdulhussain.

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Abdulhussain, S.H., Ramli, A.R., Al-Haddad, S.A.R. et al. Fast Recursive Computation of Krawtchouk Polynomials. J Math Imaging Vis 60, 285–303 (2018). https://doi.org/10.1007/s10851-017-0758-9

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  • DOI: https://doi.org/10.1007/s10851-017-0758-9

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