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Orthogonal Affine Invariants from Gaussian-Hermite Moments

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Computer Analysis of Images and Patterns (CAIP 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11679))

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Abstract

We propose a new kind of moment invariants with respect to an affine transformation. The new invariants are constructed in two steps. First, the affine transformation is decomposed into scaling, stretching and two rotations. The image is partially normalized up to the second rotation, and then rotation invariants from Gaussian-Hermite moments are applied. Comparing to the existing approaches – traditional direct affine invariants and complete image normalization – the proposed method is more numerically stable. The stability is achieved thanks to the use of orthogonal Gaussian-Hermite moments and also due to the partial normalization, which is more robust to small changes of the object than the complete normalization. Both effects are documented in the paper by experiments. Better stability opens the possibility of calculating affine invariants of higher orders with better discrimination power. This might be useful namely when different classes contain similar objects and cannot be separated by low-order invariants.

This work has been supported by the Czech Science Foundation (Grant No. GA18-07247S) and by the Praemium Academiae, awarded by the Czech Academy of Sciences. Bo Yang has been supported by the Fundamental Research Funds for the Central Universities (No. 3102018ZY025) and the Fund Program for the Scientific Activities of the Selected Returned Overseas Professionals in Shaanxi Province (No. 2018024).

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Notes

  1. 1.

    The reader may recall the so-called “indirect approach” to constructing rotation invariants from Legendre [5, 15] and Krawtchouk [37] moments. The authors basically expressed geometric moments in terms of the respective OG moments and substituted into (9). They ended up with clumsy formulas of questionable numerical properties.

References

  1. Amayeh, G., Kasaei, S., Bebis, G., Tavakkoli, A., Veropoulos, K.: Improvement of Zernike moment descriptors on affine transformed shapes. In: 9th International Symposium on Signal Processing and Its Applications ISSPA 2007, pp. 1–4. IEEE (2007)

    Google Scholar 

  2. Belghini, N., Zarghili, A., Kharroubi, J.: 3D face recognition using Gaussian Hermite moments. Int. J. Comput. Appl. Spec. Issue Softw. Eng., Databases Expert. Syst. SEDEX(1), 1–4 (2012)

    Google Scholar 

  3. Canterakis, N.: 3D Zernike moments and Zernike affine invariants for 3D image analysis and recognition. In: Ersbøll, B.K., Johansen, P. (eds.) Proceedings of the 11th Scandinavian Conference on Image Analysis SCIA 1999. DSAGM (1999)

    Google Scholar 

  4. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)

    MATH  Google Scholar 

  5. Deepika, C.L., Kandaswamy, A., Vimal, C., Satish, B.: Palmprint authentication using modified Legendre moments. Procedia Comput. Sci. 2, 164–172 (2010). Proceedings of the International Conference and Exhibition on Biometrics Technology ICEBT’10

    Article  Google Scholar 

  6. Farokhi, S., Sheikh, U.U., Flusser, J., Yang, B.: Near infrared face recognition using Zernike moments and Hermite kernels. Inf. Sci. 316, 234–245 (2015)

    Article  Google Scholar 

  7. Flusser, J.: On the independence of rotation moment invariants. Pattern Recognit. 33(9), 1405–1410 (2000)

    Article  Google Scholar 

  8. Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognit. 26(1), 167–174 (1993)

    Article  MathSciNet  Google Scholar 

  9. Flusser, J., Suk, T., Zitová, B.: 2D and 3D Image Analysis by Moments. Wiley, Chichester (2016)

    Book  MATH  Google Scholar 

  10. Grace, J.H., Young, A.: The Algebra of Invariants. Cambridge University Press, Cambridge (1903)

    MATH  Google Scholar 

  11. Gurevich, G.B.: Foundations of the Theory of Algebraic Invariants. Nordhoff, Groningen (1964)

    MATH  Google Scholar 

  12. Heikkilä, J.: Pattern matching with affine moment descriptors. Pattern Recognit. 37, 1825–1834 (2004)

    Article  MATH  Google Scholar 

  13. Hickman, M.S.: Geometric moments and their invariants. J. Math. Imaging Vis. 44(3), 223–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hilbert, D.: Theory of Algebraic Invariants. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  15. Hosny, K.M.: New set of rotationally Legendre moment invariants. Int. J. Electr. Comput. Syst. Eng. 4(3), 176–180 (2010)

    Google Scholar 

  16. Liu, Q., Zhu, H., Li, Q.: Image recognition by affine Tchebichef moment invariants. In: Deng, H., Miao, D., Lei, J., Wang, F.L. (eds.) AICI 2011, Part III. LNCS (LNAI), vol. 7004, pp. 472–480. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23896-3_58

    Chapter  Google Scholar 

  17. Mei, Y., Androutsos, D.: Robust affine invariant shape image retrieval using the ICA Zernike moment shape descriptor. In: 16th IEEE International Conference on Image Processing (ICIP), pp. 1065–1068, November 2009

    Google Scholar 

  18. Pei, S.C., Lin, C.N.: Image normalization for pattern recognition. Image Vis. Comput. 13(10), 711–723 (1995)

    Article  Google Scholar 

  19. Reiss, T.H.: The revised fundamental theorem of moment invariants. IEEE Trans. Pattern Anal. Mach. Intell. 13(8), 830–834 (1991)

    Article  Google Scholar 

  20. Rothe, I., Süsse, H., Voss, K.: The method of normalization to determine invariants. IEEE Trans. Pattern Anal. Mach. Intell. 18(4), 366–376 (1996)

    Article  Google Scholar 

  21. Schur, I.: Vorlesungen über Invariantentheorie. Springer, Berlin (1968). https://doi.org/10.1007/978-3-642-95032-2. in German

    Book  MATH  Google Scholar 

  22. Shen, D., Ip, H.H.S.: Generalized affine invariant image normalization. IEEE Trans. Pattern Anal. Mach. Intell. 19(5), 431–440 (1997)

    Article  Google Scholar 

  23. Shen, J.: Orthogonal Gaussian-Hermite moments for image characterization. In: Casasent, D.P. (ed.) Intelligent Robots and Computer Vision XVI: Algorithms, Techniques, Active Vision, and Materials Handling, vol. 3208, pp. 224–233. SPIE (1997)

    Google Scholar 

  24. Shen, J., Shen, W., Shen, D.: On geometric and orthogonal moments. Int. J. Pattern Recognit. Artif. Intell. 14(7), 875–894 (2000)

    Article  MATH  Google Scholar 

  25. Suk, T., Flusser, J.: Graph method for generating affine moment invariants. In: Proceedings of the 17th International Conference on Pattern Recognition ICPR 2004, pp. 192–195. IEEE Computer Society (2004)

    Google Scholar 

  26. Suk, T., Flusser, J.: Affine normalization of symmetric objects. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2005. LNCS, vol. 3708, pp. 100–107. Springer, Heidelberg (2005). https://doi.org/10.1007/11558484_13

    Chapter  Google Scholar 

  27. Suk, T., Flusser, J.: Affine moment invariants generated by graph method. Pattern Recognit. 44(9), 2047–2056 (2011)

    Article  Google Scholar 

  28. Sylvester assisted by F. Franklin, J.J.: Tables of the generating functions and groundforms for simultaneous binary quantics of the first four orders taken two and two together. Am. J. Math. 2, 293–306, 324–329 (1879)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sylvester assisted by F. Franklin, J.J.: Tables of the generating functions and groundforms for the binary quantics of the first ten orders. Am. J. Math. 2, 223–251 (1879)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, B., Dai, M.: Image analysis by Gaussian-Hermite moments. Signal Process. 91(10), 2290–2303 (2011)

    Article  MATH  Google Scholar 

  31. Yang, B., Flusser, J., Kautsky, J.: Rotation of 2D orthogonal polynomials. Pattern Recognit. Lett. 102(1), 44–49 (2018)

    Article  Google Scholar 

  32. Yang, B., Flusser, J., Suk, T.: Steerability of Hermite kernel. Int. J. Pattern Recognit. Artif. Intell. 27(4), 1–25 (2013)

    Article  Google Scholar 

  33. Yang, B., Flusser, J., Suk, T.: 3D rotation invariants of Gaussian-Hermite moments. Pattern Recognit. Lett. 54(1), 18–26 (2015)

    Article  Google Scholar 

  34. Yang, B., Flusser, J., Suk, T.: Design of high-order rotation invariants from Gaussian-Hermite moments. Signal Process. 113(1), 61–67 (2015)

    Article  Google Scholar 

  35. Yang, B., Kostková, J., Flusser, J., Suk, T.: Scale invariants from Gaussian-Hermite moments. Signal Process. 132, 77–84 (2017)

    Article  Google Scholar 

  36. Yang, B., Li, G., Zhang, H., Dai, M.: Rotation and translation invariants of Gaussian-Hermite moments. Pattern Recognit. Lett. 32(2), 1283–1298 (2011)

    Article  Google Scholar 

  37. 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 

  38. Zhang, H., Shu, H., Coatrieux, G., Zhu, J., Wu, J., Zhang, Y., Zhu, H., Luo, L.: Affine Legendre moment invariants for image watermarking robust to geometric distortions. IEEE Trans. Image Process. 20(8), 2189–2199 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, Y., Wen, C., Zhang, Y., Soh, Y.C.: On the choice of consistent canonical form during moment normalization. Pattern Recognit. Lett. 24(16), 3205–3215 (2003)

    Article  Google Scholar 

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Authors and Affiliations

  1. The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 182 08, Praha 8, Czech Republic

    Jan Flusser & Tomáš Suk

  2. School of Automation, Northwestern Polytechnical University, 127 West Youyi Road, Xi’an, 710 072, Shaanxi, People’s Republic of China

    Bo Yang

Authors
  1. Jan Flusser
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  2. Tomáš Suk
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  3. Bo Yang
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Correspondence to Tomáš Suk .

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Editors and Affiliations

  1. Department of Computer and Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

    Mario Vento

  2. Department of Computer and Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

    Gennaro Percannella

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Flusser, J., Suk, T., Yang, B. (2019). Orthogonal Affine Invariants from Gaussian-Hermite Moments. In: Vento, M., Percannella, G. (eds) Computer Analysis of Images and Patterns. CAIP 2019. Lecture Notes in Computer Science(), vol 11679. Springer, Cham. https://doi.org/10.1007/978-3-030-29891-3_36

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  • DOI: https://doi.org/10.1007/978-3-030-29891-3_36

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  • Online ISBN: 978-3-030-29891-3

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