Bivariate Hahn moments for image reconstruction

Haiyong Wu and Senlin Yan 2
  • 1 Lab of Image Science and Technology School of Computer Science and Engineering, Southeast University, 210096 Nanjing, China
  • 2 School of Mathematics and Information Technology Xiaozhuang University, 211171 Nanjing, China


This paper presents a new set of bivariate discrete orthogonal moments which are based on bivariate Hahn polynomials with non-separable basis. The polynomials are scaled to ensure numerical stability. Their computational aspects are discussed in detail. The principle of parameter selection is established by analyzing several plots of polynomials with different kinds of parameters. Appropriate parameters of binary images and a grayscale image are obtained through experimental results. The performance of the proposed moments in describing images is investigated through several image reconstruction experiments, including noisy and noise-free conditions. Comparisons with existing discrete orthogonal moments are also presented. The experimental results show that the proposed moments outperform slightly separable Hahn moments for higher orders.

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  • Campisi, P., Neri, A., Panci, G. and Scarano, G. (2004). Robust rotation-invariant texture classification using a model based approach, IEEE Transactions on Image Processing13(6): 782–791.

  • Dai, X., Shu, H., Luo, L., Han, G.N. and Coatrieux, J.L. (2010). Reconstruction of tomographic images from limited range projections using discrete Radon transform and Tchebichef moments, Pattern Recognition43(3): 1152–1164.

  • Dunkl, C.F. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge.

  • Fujarewicz, K. (2010). Planning identification experiments for cell signaling pathways: An NFκB case study, International Journal of Applied Mathematics and Computer Science20(4): 773–780, DOI: 10.2478/v10006-010-0059-6.

  • Hu, M. (1962). Visual pattern recognition by moment invariants, IRE Transactions on Information Theory8(2): 179–187.

  • Iliev, P. and Xu, Y. (2007). Discrete orthogonal polynomials and difference equations of several variables, Advances in Mathematics212(1): 1–36.

  • Ismail, M., Foncannon, J. and Pekonen, O. (2008). Classical and quantum orthogonal polynomials in one variable, The Mathematical Intelligencer30(1): 54–60.

  • Mukundan, R., Ong, S. and Lee, P.A. (2001). Image analysis by Tchebichef moments, IEEE Transactions on Image Processing10(9): 1357–1364.

  • Nene, S., Nayar, S. and Murase, H. (1988). Columbia Object Image Library (coil 100) 1996, Ph.D. thesis, Columbia University, New York, NY.

  • Papakostas, G., Karakasis, E. and Koulouriotis, D. (2010). Novel moment invariants for improved classification performance in computer vision applications, Pattern Recognition43(1): 58–68.

  • See, K., Loke, K., Lee, P. and Loe, K. (2007). Image reconstruction using various discrete orthogonal polynomials in comparison with DCT, Applied Mathematics and Computation193(2): 346–359.

  • Sroubek, F., Cristóbal, G. and Flusser, J. (2007). A unified approach to superresolution and multichannel blind deconvolution, IEEE Transactions on Image Processing16(9): 2322–2332.

  • Teague, M.R. (1980). Image analysis via the general theory of moments, Journal of the Optical Society of America A70(8): 920–930.

  • Wang, J.Z., Wiederhold, G., Firschein, O. and Wei, S.X. (1997). Wavelet-based image indexing techniques with partial sketch retrieval capability, Proceedings of the IEEE International Forum on Research and Technology Advances in Digital Libraries, ADL 1997, Washington, DC, USA, pp. 13–24.

  • Wang, Z. and Bovik, A.C. (2009). Mean squared error: Love it or leave it? A new look at signal fidelity measures, IEEE Signal Processing Magazine26(1): 98–117.

  • Wang, Z., Bovik, A.C., Sheikh, H.R. and Simoncelli, E.P. (2004). Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing13(4): 600–612.

  • Hunek, W.P and Latawiec, K.J. (2011). A study on new right/left inverses of nonsquare polynomial matrices, International Journal of Applied Mathematics and Computer Science21(2): 331–348, DOI: 10.2478/v10006-011-0025-y.

  • Xu, Y. (2004). On discrete orthogonal polynomials of several variables, Advances in Applied Mathematics33(3): 615–632.

  • Xu, Y. (2005). Second-order difference equations and discrete orthogonal polynomials of two variables, International Mathematics Research Notices2005(8): 449–475.

  • Yap, P.T., Paramesran, R. and Ong, S.H. (2003). Image analysis by Krawtchouk moments, IEEE Transactions on Image Processing12(11): 1367–1377.

  • Yap, P. T., Paramesran, R. and Ong, S. H. (2007). Image analysis using Hahn moments, IEEE Transactions on Pattern Analysis and Machine Intelligence29(11): 2057–2062.

  • Zhang, D. and Lu, G. (2001). Content-based shape retrieval using different shape descriptors: A comparative study, Proceedings of the International Conference on Intelligent Multimedia and Distance Education, ICIMADE01, Fargo, ND, USA, pp. 1–9.

  • Zhou, J., Shu, H., Zhu, H., Toumoulin, C. and Luo, L. (2005). Image analysis by discrete orthogonal Hahn moments, in J.S. Marques, N. P´erez de la Blanca and P. Pina (Eds.), Image Analysis and Recognition, Springer, Berlin/Heidelberg, pp. 524–531.

  • Zhu, H. (2012). Image representation using separable two-dimensional continuous and discrete orthogonal moments, Pattern Recognition45(4): 1540–1558.

  • Zhu, H., Liu, M., Li, Y., Shu, H. and Zhang, H. (2011). Image description with nonseparable two-dimensional Charlier and Meixner moments, International Journal of Pattern Recognition and Artificial Intelligence25(1): 37–55.

  • Zhu, H., Liu, M., Shu, H., Zhang, H. and Luo, L. (2010). General form for obtaining discrete orthogonal moments, IET Image Processing4(5): 335–352.

  • Zhu, H., Shu, H., Zhou, J., Luo, L. and Coatrieux, J.L. (2007). Image analysis by discrete orthogonal dual Hahn moments, Pattern Recognition Letters28(13): 1688–1704.

  • Zunić, J., Hirota, K. and Rosin, P.L. (2010). A Hu moment invariant as a shape circularity measure, Pattern Recognition43(1): 47–57.


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