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Accurate Computation of Orthogonal Fourier-Mellin Moments

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Abstract

Orthogonal Fourier-Mellin moments (OFMMs) suffer from geometric error and the numerical integration error. The geometric error arises when the square image is mapped into a unit disk and the mapping does not become perfect. The numerical integration error arises when the double integration is approximated by the zeroth order summation. In this paper, we propose methods which reduce these errors. The geometric error is reduced by considering the arc-grids lying on the boundary of the unit disk and the square grids lying completely inside the disk. The numerical integration error is reduced by Gaussian numerical integration, for which a simple computational framework is provided. The relative contributions of geometric error and numerical integration error to the total error are also analyzed. It is observed that the geometric error is significant only for the small images whereas the magnitude of numerical integration is significantly high for all image sizes, which increases with the order of moments. A simple computational framework which is similar to the conventional zeroth order approximation is also proposed which not only reduces numerical integration error but also reduces geometric error without considering arc-grids. The improved accuracy of OFMMs are shown to provide better image reconstruction, numerical stability and rotation and scale invariance. Exhaustive experimental results on a variety of real images have shown the efficacy of the proposed methods.

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References

  1. Andrew, T.B.J., David, N.C.L.: Integrated wavelet and Fourier-Mellin invariant feature in fingerprint verification system. In: Proceedings of the 2003 ACM SIGMM Workshop on Biometrics Methods and Applications, pp. 82–88 (2003)

    Chapter  Google Scholar 

  2. Bailey, R.R., Srinath, M.D.: Orthogonal moment features for use with parametric and non-parametric classifiers. IEEE Trans. Pattern Anal. Mach. Intell. 18(4), 389–398 (1996)

    Article  Google Scholar 

  3. Bin, T.J., Jiwen, C., Wenjing, K., Dandan, L.: Subpixel edge location based on orthogonal Fourier-Mellin moments. Image Vis. Comput. 26, 563–569 (2008)

    Article  Google Scholar 

  4. Chen, Y.M., Chiang, J.-H.: Face recognition using combined multiple feature extraction based on Fourier-Mellin approach for single example image per person. Pattern Recogn. Lett. 31, 1833–1841 (2010)

    Article  MathSciNet  Google Scholar 

  5. Chong, C.-W., Raveendran, P., Mukundan, R.: The scale invariants of pseudo-Zernike moments. PAA Pattern Anal. Appl. 6, 176–184 (2003)

    Article  MathSciNet  Google Scholar 

  6. Farajzadeh, N., Faez, K., Pan, G.: Study on the performance of moments as invariant descriptors for practical face recognition systems. IET Comput. Vis. 4(4), 272–285 (2010)

    Article  Google Scholar 

  7. Hosny, K.M.: Fast and accurate method for radial moment’s computation. Pattern Recognit. Lett. 31(2), 143–150 (2010)

    Article  Google Scholar 

  8. Hosny, K.M., Shouman, M.A., Abdel Salam, H.M.: Fast computation of orthogonal Fourier-Mellin moments in polar coordinates. J. Real-Time Image Process. 6(2), 73–80 (2010)

    Article  Google Scholar 

  9. Kan, C., Srinath, M.D.: Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments. Pattern Recognit. 35(1), 143–154 (2002)

    Article  MATH  Google Scholar 

  10. Liao, S.X., Pawlak, M.: On the accuracy of Zernike moments for image analysis. IEEE Trans. Pattern Anal. Mach. Intell. 20(12), 1358–1364 (1998)

    Article  Google Scholar 

  11. Lin, H., Si, J., Abouseleman, G.P.: Orthogonal Rotation-Invariant moments for digital image processing. IEEE Trans. Image Process. 17(3), 272–282 (2008)

    Article  MathSciNet  Google Scholar 

  12. Papakostas, G.A., Mertzios, B.G., Karras, D.A.: Performance of the orthogonal moments in reconstructing biomedical images. In: Proceedings of the 16th International Conference on Systems, Signals and Image Processing (IWSSIP) (2009)

    Google Scholar 

  13. Rajaraman, V.: Computer Oriented Numerical Methods, 3rd edn. Prentice Hall of India, New Delhi (2004)

    Google Scholar 

  14. Sheng, Y., Shen, L.: Orthogonal Fourier-Mellin moments for invariant pattern recognition. J. Opt. Soc. Am. 11(6), 1748–1757 (1994)

    Article  Google Scholar 

  15. Singh, C., Walia, E.: Algorithms for fast computation of Zernike moments and their numerical stability. Image Vis. Comput. 29, 251–259 (2011)

    Article  Google Scholar 

  16. Singh, C., Walia, E.: Fast and numerically stable methods for the computation of Zernike moments. Pattern Recognit. 43, 2497–2506 (2010)

    Article  MATH  Google Scholar 

  17. Walia, E., Singh, C., Goyal, A.: On the fast computation of orthogonal Fourier-Mellin moments with improved numerical stability. J. Real-Time Image Process. (2010). doi:10.1007/s11554-010-0172-7

    MATH  Google Scholar 

  18. Wang, X., Xiao, B., Jian-Feng, M., Xiu-Li, B.: Scaling and rotation invariant analysis approach to object recognition based on radon and Fourier-Mellin transforms. Pattern Recognit. 40(12), 3503–3508 (2007)

    Article  MATH  Google Scholar 

  19. Wee, C.Y., Paramesran, R.: On the computational aspects of Zernike moments. Image Vis. Comput. 25, 967–980 (2007)

    Article  Google Scholar 

  20. Xin, Y., Pawlak, M., Liao, S.: Accurate computation of Zernike moments in polar coordinates. IEEE Trans. Image Process. 16, 581–587 (2007)

    Article  MathSciNet  Google Scholar 

  21. Zhang, H., Shu, H.Z., Haigron, P., Li, B.S., Luo, L.M.: Construction of a complete set of orthogonal Fourier-Mellin moment invariants for pattern recognition applications. Image Vis. Comput. 28, 38–40 (2010)

    Article  Google Scholar 

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Acknowledgements

We are thankful to the anonymous reviewers for their suggestions for raising the standard of the paper. The alternative recursive relations given in Sect. 3.3 for trigonometric functions are suggested by one of the reviewers which is highly appreciated.

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  1. Department of Computer Science, Punjabi University, Patiala, Punjab, India

    Chandan Singh & Rahul Upneja

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  1. Chandan Singh
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  2. Rahul Upneja
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Correspondence to Rahul Upneja.

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Singh, C., Upneja, R. Accurate Computation of Orthogonal Fourier-Mellin Moments. J Math Imaging Vis 44, 411–431 (2012). https://doi.org/10.1007/s10851-012-0335-1

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