Fast Recursive Computation of Krawtchouk Polynomials

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 $$n-x$$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.

[1]  Raveendran Paramesran,et al.  Enhancing noisy speech signals using orthogonal moments , 2014, IET Signal Process..

[2]  V. B. Uvarov,et al.  Classical Orthogonal Polynomials of a Discrete Variable , 1991 .

[3]  Sim Heng Ong,et al.  Image Analysis by Tchebichef Moments , 2001, IEEE Trans. Image Process..

[4]  Dennis Stanton,et al.  Zeros of generalized Krawtchouk polynomials , 1990 .

[5]  Raveendran Paramesran,et al.  Image Analysis Using Hahn Moments , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  H. Shu,et al.  General form for obtaining discrete orthogonal moments , 2010 .

[7]  Héctor Pijeira Cabrera,et al.  Edge detection based on Krawtchouk polynomials , 2015, J. Comput. Appl. Math..

[8]  Kim-Han Thung,et al.  Content-based image quality metric using similarity measure of moment vectors , 2012, Pattern Recognit..

[9]  Paramesran Raveendran,et al.  Krawtchouk moments as a new set of discrete orthogonal moments for image reconstruction , 2002, Proceedings of the 2002 International Joint Conference on Neural Networks. IJCNN'02 (Cat. No.02CH37290).

[10]  Raveendran Paramesran,et al.  Image analysis by Krawtchouk moments , 2003, IEEE Trans. Image Process..

[11]  M. Teague.  Image analysis via the general theory of moments , 1980 .

[12]  J. Fienup.  Invariant error metrics for image reconstruction. , 1997, Applied optics.

[13]  P. N. Sadjang,et al.  On moments of classical orthogonal polynomials , 2015 .

[14]  Raveendran Paramesran,et al.  Face Recognition Using Discrete Tchebichef-Krawtchouk Transform , 2012, 2012 IEEE International Symposium on Multimedia.

[15]  Victor H. Moll,et al.  Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals , 2004 .

[16]  J. J. Foncannon.  Irresistible integrals: symbolics, analysis and experiments in the evaluation of integrals , 2006 .

[17]  Syed Abdul Rahman Al-Haddad,et al.  On Computational Aspects of Tchebichef Polynomials for Higher Polynomial Order , 2017, IEEE Access.

[18]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[19]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[20]  P. Yap,et al.  Local watermarks based on Krawtchouk moments , 2004, 2004 IEEE Region 10 Conference TENCON 2004..

[21]  Abd. Rahman Ramli,et al.  Image edge detection operators based on orthogonal polynomials , 2017 .

[22]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[23]  Jane Labadin,et al.  Feature selection based on mutual information , 2015, 2015 9th International Conference on IT in Asia (CITA).

[24]  Bo Fu,et al.  A symmetry and bi-recursive algorithm of accurately computing Krawtchouk moments , 2010, Pattern Recognit. Lett..

[25]  Chih-Jen Lin,et al.  LIBSVM: A library for support vector machines , 2011, TIST.

[26]  Raveendran Paramesran,et al.  New orthogonal polynomials for speech signal and image processing , 2012, IET Signal Process..

[27]  Franz L. Alt,et al.  Digital Pattern Recognition by Moments , 1962, JACM.

[28]  Fuhui Long,et al.  Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy , 2003, IEEE Transactions on Pattern Analysis and Machine Intelligence.