Journal of Mathematical Imaging and Vision

, Volume 56, Issue 1, pp 125–136 | Cite as

Shape Interpretation of Second-Order Moment Invariants



This paper deals with the following problem: What can be said about the shape of an object if a certain invariant of it is known? Such a, herein called, shape invariant interpretation problem has not been studied/solved for the most of invariants, but also it is not known to which extent the shape interpretation of certain invariants does exist. In this paper, we consider a well-known second-order affine moment invariant. This invariant has been expressed recently Xu and Li (2008) as the average square area of triangles whose one vertex is the shape centroid while the remaining two vertices vary through the shape considered. The main results of the paper are (i) the ellipses are shapes which minimize such an average square triangle area, i.e., which minimize the affine invariant considered; (ii) this minimum is \(1/(16\pi ^2)\) and is reached by the ellipses only. As by-products, we obtain several results including the expression of the second Hu moment invariant in terms of one shape compactness measure and one shape ellipticity measure. This expression further leads to the shape interpretation of the second Hu moment invariant, which is also given in the paper.


Moments Moment invariants Shape Pattern recognition Image processing 



This work is partially supported by the Serbian Ministry of Science—projects OI174008/OI174026.


  1. 1.
    Aktaş, M.A., Žunić, J.: A family of shape ellipticity measures for galaxy classification. SIAM J. Imaging Sci. 6, 765–781 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andersen, D.R., Bershady, M.A., Sparke, L.S., Gallagher III, J.S., Wilcots, E.M.: A measurement of disk ellipticity in nearby spiral galaxies. Astrophys. J. 551, 131–134 (2001)CrossRefGoogle Scholar
  3. 3.
    Bazell, D., Peng, Y.: A comparison of neural network algorithms and preprocessing methods for star-galaxy discrimination. Astrophys. J. Suppl. Ser. 116, 47–55 (1998)CrossRefGoogle Scholar
  4. 4.
    Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recogn. 26, 167–174 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Flusser, J., Kautsky, J., Sroubek, F.: Implicit moment invariants. Int. J. Comput. Vis. 86, 72–86 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frei, Z., Guhathakurta, P., Gunn, J.E., Tyson, J.A.: A catalog of digital images of 113 nearby galaxies. Astronom. J. 111, 174–181 (1996)CrossRefGoogle Scholar
  7. 7.
    Goderya, S.N., Lolling, S.M.: Morphological classification of galaxies using computer vision and artificial neural networks: a computational scheme. Astrophys. Space Sci. 279, 377–387 (2002)CrossRefMATHGoogle Scholar
  8. 8.
    Hu, M.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8, 179–187 (1961)MATHGoogle Scholar
  9. 9.
    Klette, R., Žunić, J.: Multigrid convergence of calculated features in image analysis. J. Math. Imaging Vis. 13, 173–191 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lahav, O., Naim, A., Sodré Jr., L., Storrie-Lombardi, M.C.: Neural computation as a tool for galaxy classification: methods and examples. Mon. Not. R. Astron. Soc. 283, 1–29 (1996)CrossRefGoogle Scholar
  11. 11.
    Galaxy classification using fractal signature: Lekshmi, S., Revathy, K., Prabhakaran Nayar, S.R. Astronomy Astrophys 405, 1163–1167 (2003)CrossRefGoogle Scholar
  12. 12.
    Mähönen, P., Frantti, T.: Fuzzy classifier for star-galaxy separation. Astrophys. J. 541, 261–263 (2000)CrossRefGoogle Scholar
  13. 13.
    Misztal, K., Jacek Tabor, J.: Ellipticity and circularity measuring via Kullback-Leibler divergence. J. Math. Imaging Vis. Accepted. doi: 10.1007/s10851-015-0618-4
  14. 14.
    Otsu, N.: A threshold selection method from gray level histograms. IEEE Trans. Syst. Man Cybernet. 9, 62–66 (1979)CrossRefGoogle Scholar
  15. 15.
    Rosin, P.L.: Measuring shape: ellipticity, rectangularity, and triangularity. Machine Vision Appl. 14, 172–184 (2003)CrossRefGoogle Scholar
  16. 16.
    Rosin, P.L.: Computing global shape measures. In: Chen, C.H., Wang, P.S.P. (eds.) Handbook of Pattern Recognition and Computer Vision, pp. 177–196. World Scientific (2005)Google Scholar
  17. 17.
    Rosin, P.L., Žunić, J.: Measuring squareness and orientation of shapes. J. Math. Imaging Vision 39, 13–27 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rosin, P.L., Žunić, J.: Orientation and anisotropy of multi-component shapes from boundary information. Pattern Recogn. 44, 2147–2160 (2011)Google Scholar
  19. 19.
    Tool, A.Q.: A method for measuring ellipticity and the determination of optical constants of metals. Phys. Rev. (Series I) 31, 1–25 (1910)CrossRefGoogle Scholar
  20. 20.
    Xu, D., Li, H.: Geometric moment invariants. Pattern Recogn. 41, 240–249 (2008)CrossRefMATHGoogle Scholar
  21. 21.
    Zhang, H., Huazhong, S., Han, G.N., Coatrieux, G., Limin, L., Coatrieux, J.L.: Blurred image recognition by Legendre moment invariants. IEEE Trans. Image Process. 19, 596–611 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, L., Xiao, W.W., Ji, Z.: Local affine transform invariant image watermarking by Krawtchouk moment invariants. IET Inf. Secur. 1, 97–105 (2007)CrossRefGoogle Scholar
  23. 23.
    Žunić, D., Žunić, J.: Shape ellipticity from Hu moment invariants. Appl. Math. Comput. 226, 406–414 (2014)MathSciNetMATHGoogle Scholar
  24. 24.
    Žunić, J., Hirota, K., Rosin, P.L.: A Hu invariant as a shape circularity measure. Pattern Recogn. 43, 47–57 (2010)CrossRefMATHGoogle Scholar
  25. 25.
    Žunić, J., Kopanja, L., Fieldsend, J.E.: Notes on shape orientation where the standard method does not work. Pattern Recogn. 39, 856–865 (2006)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia
  2. 2.Faculty of Computer ScienceBelgradeSerbia

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