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Extracting Meaningful Curves from Images

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Abstract

Since the beginning, Mathematical Morphology has proposed to extract shapesfrom images as connected components of level sets. These methods have proved veryefficient in shape recognition and shape analysis. In this paper, we present an improved method to select the most meaningful level lines (boundaries of level sets) from an image. This extraction can be based on statistical arguments, leading to a parameter free algorithm. It permits to roughly extract all pieces of level lines of an image, that coincide with pieces of edges. By this method, the numberof encoded level lines is reduced by a factor 100, without any loss of shape contents. In contrast to edge detection algorithms or snakes methods, such a level lines selection method delivers accurate shape elements, without user parameter since selection parameters can be computed by the Helmholtz Principle. The paper aims at improving the original method proposed in [10]. We give a mathematicalinterpretation of the model, which explains why some pieces of curve are overdetected. We introduce a multiscale approach that makes the method more robust to noise. A more local algorithm is introduced, taking local contrast variations into account. Finally, we empirically prove that regularity makes detection more robust but does not qualitatively change the results.

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References

  1. A. Alexandrov and Y. Reshetnyak,General Theory of Irregular Curves, vol. 29 ofMathematics and Its Applications: Soviet Series. Kluwer Academic Publishers, 1989.

  2. E.J. Breen and R. Jones, “Attribute openings, thinnings and granulometries”Computer Vision and Image Understanding, Vol. 64, No. 3, pp. 377–389, 1996.

    Google Scholar 

  3. J. Canny, “A computational approach to edge detection” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 8, No. 6, pp. 679–698, 1986.

    Google Scholar 

  4. F. Cao, “Good continuation in digital images” inProceeding of ICCV 03, Nice, Vol. 1, 2003, pp. 440–447.

  5. V. Caselles, B. Coll, and J.M. Morel, “A kanizsa Program” inProgress in Nonlinear Differential Equations and their Applications, Vol. 25, 1996, pp. 35–55.

    Google Scholar 

  6. V. Caselles, T. Coll, and J.M. Morel, “Topographic maps and local contrast changes in natural images” International Journal of Computer Vision, Vol. 33, No. 1, pp. 5–27, 1999.

    Google Scholar 

  7. V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours” International Journal of Computer Vision, Vol. 22, No. 1, pp. 61–79, 1997.

    Google Scholar 

  8. T. Chan and L. Vese, “Active contours without edges” IEEE Transactions on Image Processing, Vol. 10, No. 2, pp. 266–277, 2001.

    Google Scholar 

  9. A. Desolneux, L. Moisan, and J.M. Morel, “Meaningful alignments,’ International Journal of Computer Vision, Vol. 40, No. 1, pp. 7–23, 2000.

    Google Scholar 

  10. A. Desolneux, L. Moisan, and J.M. Morel, “Edge detection by Helmholtz principle” Journal of Mathematical Imaging and Vision, Vol. 14, No. 3, pp. 271–284,2001.

    Google Scholar 

  11. A. Desolneux, L. Moisan, and J.M. Morel, “A grouping principle and four applications” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 4, pp. 508–513, 2003.

    Google Scholar 

  12. A. Desolneux, L. Moisan, and J.M. Morel, “Variational snake theory” inGeometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios (Eds.), Springer Verlag, 2003.

  13. L.C. Evans and R.Gariepy,Measure Theory and Fine Properties of Functions, CRC Press. Ann Harbor, 1992.

    Google Scholar 

  14. P. Felzenszwalb and D. Huttenlocher, “Image segmentation using local variation” inProceedings IEEE Conference on Computer Vision and Pattern Recognition, 1998, pp. 98–104.

  15. R. Haralick, “Digital step edges from zero crossing of second directional derivatives” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6, pp. 58–68, 1984.

    Google Scholar 

  16. G. Kanizsa,La Grammaire du Voir. Diderot, 1996. Original title:Grammatica del vedere. French translation from Italian.

  17. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models” International Journal of Computer Vision, Vol. 1, pp. 321–331, 1987.

    Google Scholar 

  18. R. Kimmel and A.M. Bruckstein, “On regularized laplacian Zero Crossings and Other Optimal Edge Integrators” International Journal of Computer Vision, Vol. 53, No. 3, pp. 225–243, 2003.

    Google Scholar 

  19. J.J. Koenderink, “The structure of images” Biol. Cybern., Vol. 50, pp. 363–370, 1984.

    Google Scholar 

  20. T. Lindeberg, “Feature detection with automatic scale selection” International Journal of Computer Vision, Vol. 30, No. 2, pp. 77–116, 1998.

    Google Scholar 

  21. J.L. Lisani, L. Moisan, P. Monasse, and J.M. Morel, “On the theory of planar shape” SIAM Multiscale Modeling and Simulation, Vol. 1, No. 1, pp. 1–24, 2003.

    Google Scholar 

  22. J.L. Lisani, P. Monasse, and L. Rudin, “Fast shape extraction and application” Preprint 16, CMLA, ENS-Cachan. Available at http://www.cmla.ens-cachan.fr, 2001.

  23. S. Mallat,A Wavelet Tour in Signal Processing, 2nd edition. Academic Press, 1999.

  24. D. Marr,Vision. W.H. and Co: N.York, 1982.

  25. D. Marr and E. Hildreth, “Theory of edge detection” inProceeding of Royal Society of London, Vol. 207, 1980, pp. 187–207.

    Google Scholar 

  26. G. Matheron,Random Sets and Integral Geometry, John Wiley: N.Y., 1975.

    Google Scholar 

  27. F. Meyer and P. Maragos, “Nonlinear scale-space representation with morphological levelings” J. of Visual Comm. and Image Representation, Vol. 11, pp. 245–265, 2000.

    Google Scholar 

  28. P. Monasse, “Morphological representation of digital images and application to registration” Ph.D. thesis, Université Paris IX Dauphine, 2000.

    Google Scholar 

  29. P. Monasse and F. Guichard, “Fast computation of a contrast invariant representation” IEEE Transactions on Image Processing, Vol. 9, No. 5, pp. 860–872, 2000.

    Google Scholar 

  30. D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems” Communication on Pure and Applied Mathematics, Vol. XLII, No. 4, 1989.

  31. P. Musé, F. Sur, and J.M. Morel, “Sur les seuils de reconnaissance de formes”Traitement du Signal, Vol. 19, Nos. 5/6, 2003.

  32. P. Musé, F. Sur, F. Cao, and Y. Gousseau, “Unsupervised thresholds for shape matching” inIEEE Int. Conf. on Image Processing, ICIP, 2003.

  33. P. Musé, F. Sur, F. Cao, Y. Gousseau, and J.M. Morel, “Accurate estimates of false alarm number in shape recognition” Technical Report 5086, INRIA, 2004. submitted.

  34. N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation” International Journal of Computer Vision, Vol. 46, No. 3, pp. 223–247, 2002.

    Google Scholar 

  35. E. Le Pennec and S. Mallat, “Sparse geometrical image approximation with bandelets” accepted for publication in IEEE Trans. on Image Proc., 2003.

  36. P. Salembier and L. Garrido, “Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval” IEEE Transactions on Image Processing, Vol. 9, No. 4, pp. 561–576, 2000.

    Google Scholar 

  37. P. Salembier and J. Serra, “Flat zones filtering, connected operators, and filters by reconstruction,’ IEEE Transactions on Image Processing, Vol. 4, No. 8, pp. 1153–1160, 1995.

    Google Scholar 

  38. J. Serra,Image Analysis and Mathematical Morphology, Academic Press, 1982.

  39. J. Shi and J. Malik, “Normalized cuts and image segmentation” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 22, No. 8, pp. 888–905, 2000.

    Google Scholar 

  40. J.L. Starck, E.J. Candès, and D.L. Donoho, “The Curvelet Transform for Image Denoising” IEEE Transactions on Image Processing, Vol. 11, pp. 670–684, 2000.

    Google Scholar 

  41. M. Wertheimer, “Untersuchungen zur Lehre der Gestalt, II”Psychologische Forschung, Vol. 4, No. 301–350, 1923.

    Google Scholar 

  42. A.P. Witkin, “Scale space filtering” inProc. of IJCAI, Karlsruhe, 1983, pp. 1019–1021.

  43. S.C. Zhu, “Embedding gestalt Laws in Markov Random Fields” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 21, No. 11, pp. 1170–1187, 1999.

    Google Scholar 

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Authors and Affiliations

  1. IRISA/INRIA, Campus Universitaire de Beaulieu, 35042, Rennes, Cedex, France

    Frédéric Cao

  2. École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235, Cachan, Cedex, France

    Pablo Musé & Frédéric Sur

Authors
  1. Frédéric Cao
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  2. Pablo Musé
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  3. Frédéric Sur
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Corresponding authors

Correspondence to Frédéric Cao, Pablo Musé or Frédéric Sur.

Additional information

Frédéric Cao graduated from the École Polytechnique (France), and obtained his Ph.D. in 2000 at École Normale Supérieure de Cachan. From 2001, he has been with Irisa (Inria Rennes). His research interests are still image and video analysis by geometrical methods, including partial differential equations and statistical methods.

Pablo Musé was born in Montevideo, Uruguay, in 1975. He received the Electrical Engineer degree from the Universidad de la Repblica, Uruguay, in 1999, and the DEA (M.Sc.) in Mathematics, Vision and Learning from the École Normale Supérieure de Cachan, France, in 2001. He has obtained a Ph.D. in Applied Mathematics in 2004 in ENS Cachan, where he currently has a researcher position.

Frédéric Sur was born in 1976. He studied mathematics at École Normale Supérieure de Cachan from 1997 to 2001 and received the DEA Mathématiques, Vision, Apprentissage. He obtained his Ph.D. thesis in 2004 and now he has a post doc position in LORIA/CNRS.

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Cao, F., Musé, P. & Sur, F. Extracting Meaningful Curves from Images. J Math Imaging Vis 22, 159–181 (2005). https://doi.org/10.1007/s10851-005-4888-0

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