Mathematics in Computer Science

, Volume 10, Issue 1, pp 179–205 | Cite as

Detecting Curves of Symmetry in Images Via Hough Transform

  • Giorgio Ricca
  • Mauro C. Beltrametti
  • Anna Maria Massone


The Hough transform is a standard pattern recognition technique introduced between the 1960s and the 1970s for the detection of straight lines, circles, and ellipses with several applications including the detection of symmetries in images. Recently, based on algebraic geometry arguments, the procedure has been extended to the automated recognition of special classes of algebraic plane curves. This allows us to detect curves of symmetry present in images, that is, curves that recognize midpoints maps of various shapes extracted by an ad hoc symmetry algorithm, here proposed. Further, in the case of straight lines, the detection of lines of symmetry allows us, by a pre-processing step of the image, to improve the efficiency of the recognition algorithm on which the Hough transform technique is founded, without loss of generality and additional computational costs.


Symmetry detection Algebraic plane curves Pattern recognition Hough transform 

Mathematics Subject Classification

Primary 68D18 68U10 Secondary 14H50 92C50 


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  1. 1.
    Beltrametti, M.C., Carletti, E., Gallarati, D., Monti Bragadin, G.: Lectures on Curves, Surfaces and Projective Varieties—A Classical View of Algebraic Geometry, European Mathematical Society, Textbooks in Mathematics, 9. Translated by F. Sullivan. Zurich (2009)Google Scholar
  2. 2.
    Beltrametti M.C., Massone A.M., Piana M.: Hough transform of special classes of curves. SIAM J. Imaging Sci. 6(1), 391–412 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beltrametti M.C., Robbiano L.: An algebraic approach to Hough transforms. J. Algebra 371, 669–681 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bochnak J., Coste M., Roy M.-F.: Real Algebraic Geometry, Ergeb. Math. Grenzgeb. 36. Springer, Berlin (1998)CrossRefGoogle Scholar
  5. 5.
    Cailliere, D., Denis, F., Pele, D., Baskurt, A.: 3D mirror symmetry detection using Hough transform, Image Processing, 2008. ICIP 2008. 15-th IEEE International Conference on Environment and Electrical Engineering, June 10–13, Rome, Italy, pp. 1772–1775 (2015)Google Scholar
  6. 6.
    Campi, C., Perasso, A., Beltrametti, M.C., Massone, A.M.: The Hough transform and a novel prognostic index for chronic leukemia. Preprint (2015)Google Scholar
  7. 7.
    Cham T.-J., Cipolla R.: Symmetry detection through local skewed symmetries. Image Vis. Comput. 13(5), 439–450 (1995)CrossRefGoogle Scholar
  8. 8.
    de la Puente M.J.: Real plane algebraic curves. Expo. Math. 20(4), 291–314 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duda R.O., Hart P.E.: Use of the Hough transformation to detect lines and curves in pictures. Commun. ACM 15(1), 11–15 (1972)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fiz F., Marini C., Piva R., Miglino M., Massollo M., Bongiovanni F., Morbelli S., Bottoni G., Campi C., Bacigalupo A., Bruzzi P., Frassoni F., Piana M., Sambuceti G.: Adult advanced chronic lymphocytic leukemia: computational analysis of whole-body CT documents a bone structure alteration. Radiology 271(3), 805–813 (2014)CrossRefGoogle Scholar
  11. 11.
    Godement, R.: Cours d’Algèbre, Deuxième Edition, Enseignements de Sciences. Hermann, Paris (1966)Google Scholar
  12. 12.
    Golub G.H., Van Loan C.F.: Matrix Computations. 2nd edn. The Johns Hopkins University Press, Baltimore (1989)zbMATHGoogle Scholar
  13. 13.
    Gonzalez R.C., Woods R.E.: Digital Image Processing. 2nd edn. Prentice Hall, Upper Saddle River (2002)Google Scholar
  14. 14.
    Hassanein, A.S., Mohamed, S., Sameer, M., Ragab, M.E.: A survey on Hough transform, theory, techniques and applications. arXiv:1502.02160v1 [cs.CV] (2015)
  15. 15.
    Hough, P.V.C.: Method and means for recognizing complex patterns, US Patent 3069654, December 18 (1962)Google Scholar
  16. 16.
    Loy, G., Eklundh, J.-O.: Detecting symmetry and symmetric constellations of features. ECCV’06 Proceedings of the 9-th European conference on Computer Vision, Part II, Springer-Verlag, Berlin, Heidelberg, pp. 508–521 (2006)Google Scholar
  17. 17.
    Massone A.M., Perasso A., Campi C., Beltrametti M.C.: Profile detection in medical and astronomical imaging by means of the Hough transform of special classes of curves. J. Math. Imaging Vis. 51(2), 296–310 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Perasso, A., Campi, C., Massone, A.M., Beltrametti, M.C.: Spinal canal and spinal marrow segmentation by means of the Hough Transform of special classes of curves. In: Murino V., Puppo E. (eds.) ICIAP 2015, Part I, LNCS,vol. 9279, pp. 590–600 (2015)Google Scholar
  19. 19.
    Princen J., Illingworth J., Kittler J.: A formal definition of the Hough transform: properties and relationships. J. Math. Imaging Vis. 1, 153–168 (1992)CrossRefGoogle Scholar
  20. 20.
    Ricca, G., Beltrametti, M.C., Massone, A.M.: An iterative approach to Hough transform without re-voting. arXiv:1407.3969v1 [cs.CV] (2014)
  21. 21.
    Ricca, G., Beltrametti, M.C., Massone, A.M.: Piecewise recognition of bone skeleton profiles via an iterative Hough transform approach without re-voting. Proc. SPIE 9413, Medical Imaging 2015: Image Processing, vol. 9413, p. 94132M (2015)Google Scholar
  22. 22.
    Robbiano L.: Hyperplane sections, Gröbner bases, and Hough transforms. J. Pure Appl. Algebra 219, 2434–2448 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sendra, J.R., Winkler, F., Pérez-Diaz, S.: Rational Algebraic Curves—A Computer Algebra Approach, Algorithms and Computation in Mathematics, vol. 22. Springer, Berlin (2008)Google Scholar
  24. 24.
    Shikin E.V.: Handbook and Atlas of Curves. CRC Press, Inc., Boca Raton (1995)zbMATHGoogle Scholar
  25. 25.
    Torrente M.-L., Beltrametti M.C.: Almost-vanishing polynomials and an application to the Hough transform. J. Algebra Appl. 13(8), 1450057 (2014) (39 pp.)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yip R.K.K.: A Hough transform technique for the detection of parallel projected rotational symmetry. Pattern Recognit. Lett. 20, 991–1004 (1999)CrossRefzbMATHGoogle Scholar
  27. 27.
    Yuen K.S.Y., Chan W.W.: Two methods for detecting symmetries. Pattern Recognit. Lett. 15(3), 279–286 (1994)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Giorgio Ricca
    • 1
  • Mauro C. Beltrametti
    • 1
  • Anna Maria Massone
    • 2
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenoaItaly
  2. 2.CNR-SPINGenoaItaly

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