Journal of Mathematical Imaging and Vision

, Volume 51, Issue 2, pp 248–259 | Cite as

Projective Invariants of D-moments of 2D Grayscale Images

  • YuanBin WangEmail author
  • XingWei Wang
  • Bin Zhang
  • Ying Wang


This paper presents a novel method to derive invariants of 2D grayscale images under projective transformation. Invariants of images are good features for object recognition and have attracted extensive attention. Although geometric invariants of point locations such as cross ratios are well known for centuries, we have found no reported invariants for grayscale images that remain the same under projective transformation. It has even been proven that projective invariants of images cannot be derived from the standard geometric moments of images. However, this does not mean that there is no projective invariant of images in other forms. We will prove in this paper that projective invariants of images do exist as functions of the generalized moments of images. We first derive some projective invariant relations between an image function and its derivative functions. Next, we extend the traditional definition of moments by considering both the image function and its derivative functions. Then we derive a set of functions of the generalized moments that are projective invariant. Experimental results indicate that the proposed invariants have certain discriminating power for object recognition.


Derivative image Invariant Moment Projective transformation 



The authors wish to thank the editor and the reviewers for their comments which helped to improve the quality of our manuscript. This work is supported by the National Science Foundation for Distinguished Young Scholars of China under Grant no. 61225012 and No. 71325002; the National Natural Science Foundation of China under Grant No. 61073062; the Specialized Research Fund of the Doctoral Program of Higher Education for the Priority Development Areas under Grant No. 20120042130003; the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20110042110024; the Fundamental Research Funds for the Central Universities under Grant No. N110204003, No. N120804001 and No. N120104001


  1. 1.
    Hilbert, D.: Theory of Algebraic Invariants. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  2. 2.
    Mundy, J.L., Zisserman, A.: Geometric Invariance in Computer Vision. Cambridge MIT Press, Massachusetts (1992)Google Scholar
  3. 3.
    Flusser, J., Suk, T., Zitová, B.: Moments and Moment Invariants in Pattern Recognition. Wiley, Chichester (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Laseby, J., Bayro-Corrochano, E.: Analysis and computation of projective invariants from multiple views in the geometric algebra framework. Int. J. Pattern Recognit. Artif. Intell. 13, 1105–1121 (1999)CrossRefGoogle Scholar
  5. 5.
    Bayro-Corrochano, E., Banarer, V.: A geometric approach for the theory and applications of 3D projective invariants. J. Math. Imaging Vis. 16, 131–154 (2001)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bayro-Corrochano, E., Vallejo, R.: Pattern Recognit. Geometric preprocessing and neurocomputing for pattern recognition and pose estimation 36(12), 2909–2926 (2002)Google Scholar
  7. 7.
    Bayro-Corrochano, E., Vallejo-Aguilar, R., Arana-Daniel, N.: Applications of Clifford support vector machines and Clifford moments for classification. In: Proceedings of the International Joint Conference on Neural Networks, pp. 3003–3008. Budapest (2004)Google Scholar
  8. 8.
    Van Gool, L., Kempenaers, P., Oosterlinck, A.: Recognition and semi-differential invariants. In: Proceeding of Computer Vision and Pattern Recognition, pp. 454–460. Hawaii (1991)Google Scholar
  9. 9.
    Weiss, I.: Projective invariants of shapes. In: Proceeding of Computer Vision and Pattern Recognition, pp. 1125–34. Ann Arbor (1988)Google Scholar
  10. 10.
    Weiss, I.: Differential invariants without derivatives. In: Proceeding of the 11th International Conference on Pattern Recognition, pp. 394–398. Quebec (1992)Google Scholar
  11. 11.
    Wilczynski, E.: In: Projective Differential Geometry of Curves and Ruled Surfaces. Teubner, B.G. (ed.) Chelsea Publishing Co., Leipzig (1906)Google Scholar
  12. 12.
    Lenz, R., Meer, P.: Point configuration invariants under simultaneous projective and permutation transformations. Pattern Recognit. 27(11), 1523–1532 (1994)CrossRefGoogle Scholar
  13. 13.
    Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognit. 26, 167–174 (1993)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Suk, T., Flusser, J.: Affine moment invariants generated by automated solution of the equations. In: Proceeding of the 19th International Conference on Pattern Recognition, pp. 1–4. (2008)Google Scholar
  15. 15.
    Hu, M.K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8, 179–187 (1962)zbMATHGoogle Scholar
  16. 16.
    Belkasim, S.O., Shridhar, M., Ahmadi, M.: Pattern recognition with moment invariants: a comparative study and new results. Pattern Recognit. 24, 1117–1138 (1991)CrossRefGoogle Scholar
  17. 17.
    Boldyš, J., Flusser, J.: Extension of moment features’ invariance to blur. J. Math. Imaging Vis. 32, 227–238 (2008)CrossRefGoogle Scholar
  18. 18.
    Flusser, J., Boldys, J., Zitová, B.: Moment forms invariant to rotation and blur in arbitrary number of dimensions. IEEE Trans. Pattern Analy. Mach. Intell. 25(2), 234–246 (2003)CrossRefGoogle Scholar
  19. 19.
    Fu, B., Zhou, J.Z., Li, Y.H., Zhang, G.J., Wang, C.: Image analysis by modified Legendre moments. Pattern Recognit. 40(2), 691–704 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Galvez, J.M., Canton, M.: Normalization and shape recognition of three dimensional objects by 3-D moments. Pattern Recognit. 26, 667–681 (1993)CrossRefGoogle Scholar
  21. 21.
    Gruber, M., Hsu, K.Y.: Moment-based image normalization with high noise-tolerance. Pattern Recognit. 19, 136–139 (1997)Google Scholar
  22. 22.
    Guo, X.: 3-D moment invariants under rigid transformation, Proceedings of the 5th International Conference on Computer Analysis of Images and Pattern, 518–522(1993)Google Scholar
  23. 23.
    Hosny, K.M.: Exact Legendre moment computation for gray level images. Pattern Recognit. 40(12), 3597–3605 (2007)CrossRefzbMATHGoogle Scholar
  24. 24.
    Hupkens, T.M., de Clippeleir, J.: Noise and intensity invariant moments. Pattern Recognit. 16, 371–376 (1995)CrossRefGoogle Scholar
  25. 25.
    Khotanzad, A., Hong, Y.H.: Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990)CrossRefGoogle Scholar
  26. 26.
    Li, Y.: Reforming the theory of invariant moments for pattern recognition. Pattern Recognit. 25, 723–730 (1992)CrossRefGoogle Scholar
  27. 27.
    Liao, S.X., Pawlak, M.: On image analysis by moments. IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996)CrossRefGoogle Scholar
  28. 28.
    Lo, C.H., Don, H.S.: 3-D moment forms: their construction and application to object identification and positioning. IEEE Trans. Pattern Anal. Mach. Intell. 11, 1053–1064 (1989)CrossRefGoogle Scholar
  29. 29.
    Lu, J., Yoshida, Y.: Blurred image recognition based on phase invariants. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E82A, 1450–1455 (1999)Google Scholar
  30. 30.
    Mamistvalov, A.G.: N-dimensional moment invariants and conceptual mathematical theory of recognition n-dimensional solids. IEEE Trans. Pattern Anal. Mach. Intell. 20, 819–831 (1998)CrossRefGoogle Scholar
  31. 31.
    Markandey, V.: R.J.P. deFigueiredo, Robot sensing techniques based on high-dimensional moment invariants and tensors. IEEE Transactions on Robotics and Automation 8, 186–195 (1992)CrossRefGoogle Scholar
  32. 32.
    Mindru, F., Tuytelaars, T., Van Gool, L., Moons, T.: Moment invariants for recognition under changing viewpoint and illumination. Comput. Vis. Image Underst. 94(1–3), 3–27 (2004)CrossRefGoogle Scholar
  33. 33.
    Papakostas, G.A., Karakasis, E.G., Koulouriotis, D.E.: Efficient and accurate computation of geometric moments on gray-scale Images. Pattern Recognit. 41(6), 1895–1904 (2008)CrossRefzbMATHGoogle Scholar
  34. 34.
    Paschalakis, S., Lee, P.: Combined geometric transformation and illumination invariant object recognition. In: Proceeding of the 15th International Conference on Pattern Recognition, pp. 588–591 (2000)Google Scholar
  35. 35.
    Reiss, T.H.: The revised fundamental theorem of moment invariants. IEEE Trans. Pattern Anal. Mach. Intell. 13, 830–834 (1991)Google Scholar
  36. 36.
    Revaud, J., Lavoué, G., Baskurt, A.: Improving Zernike moments comparison for optimal similarity and rotation angle retrieval. IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 627–636 (2009)CrossRefGoogle Scholar
  37. 37.
    Sadjadi, F.A., Hall, E.L.: Three dimensional moment invariants. IEEE Trans. Pattern Anal. Mach. Intell. 2, 127–136 (1980)CrossRefzbMATHGoogle Scholar
  38. 38.
    Singh, C., Walia, E.: Fast and numerically stable methods for the computation of Zernike moments. Pattern Recognit. 43(7), 2497–2506 (2010)CrossRefzbMATHGoogle Scholar
  39. 39.
    Teague, M.R.: Image analysis via the general theory of moments. J. Opt. Soc. Am. 70, 920–930 (1980)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Wallin, A., Kubler, O.: Complete sets of complex Zernike moment invariants and the role of the pseudoinvariants. IEEE Trans. Pattern Anal. Mach. Intell. 17, 1106–1110 (1995)CrossRefGoogle Scholar
  41. 41.
    Wang, L., Healey, G.: Using Zernike moments for the illumination and geometry invariant classification of multispectral texture. IEEE Trans. Image Process. 7, 196–203 (1998)CrossRefGoogle Scholar
  42. 42.
    Wee, C.Y., Paramesran, R., Mukundan, R.: Fast computation of geometric moments using a symmetric kernel. Pattern Recognit. 41(7), 2369–2380 (2008)CrossRefzbMATHGoogle Scholar
  43. 43.
    Xu, D., Li, H.: 3-D Affine moment invariants generated by geometric primitives. In: Proceeding of the 18th International Conference on Pattern Recognition, pp. 544–547. (2006)Google Scholar
  44. 44.
    Xu, D., Li, H.: Geometric moment invariants. Pattern Recognit. 41(1), 240–249 (2008)CrossRefzbMATHGoogle Scholar
  45. 45.
    Yang, G.Y., Shu, H.Z., Toumoulin, C., Han, G.N., Luo, L.M.: Efficient Legendre moment computation for grey level images. Pattern Recognit. 39(1), 74–80 (2006)CrossRefGoogle Scholar
  46. 46.
    Zhu, H.Q., Shu, H.Z., Xia, T., Luo, L.M., Coatrieux, J.L.: Translation and scale invariants of Tchebichef moments. Pattern Recognit. 40(9), 2530–2542 (2007)CrossRefzbMATHGoogle Scholar
  47. 47.
    Zunic, J., Hirota, K., Rosin, P.L.: A Hu moment invariant as a shape circularity measure. Pattern Recognit. 43(1), 47–57 (2010)CrossRefzbMATHGoogle Scholar
  48. 48.
    Van Gool, L., Moons, T., Pauwels, E., Oosterlinck, A.: Vision and Lie’s approach to invariance. Image Vis. Comput. 13(4), 259–277 (1995)CrossRefGoogle Scholar
  49. 49.
    Voss, K., Susse, H.: Adaptive Modelle und Invarianten für zweidimensionale Bilder. Shaker Verlag, Aachen (1995)Google Scholar
  50. 50.
    Wang, Y.B., Zhang, B., Yao, T.S.: Moment invariants of restricted projective transformations. In: 2008 International Symposium on Information Science and Engineering, vol. 1, pp. 249–253. (2008)Google Scholar
  51. 51.
    Wang, Y.B., Zhang, B., Yao, T.S.: Projective invariants of co-moments of 2D images. Pattern Recognit. 43(10), 3233–3242 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • YuanBin Wang
    • 1
    Email author
  • XingWei Wang
    • 1
  • Bin Zhang
    • 1
  • Ying Wang
    • 2
  1. 1.College of Information Science and EngineeringNortheastern UniversityShenYang China
  2. 2.Department of Computer ScienceWorcester Polytechnic InstituteWorcesterUSA

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