Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Grain Filters

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Motivated by operators simplifying the topographic map of a function, we study the theoretical properties of two kinds of “grain” filters. The first category, discovered by L. Vincent, defines grains as connected components of level sets and removes those of small area. This category is composed of two filters, the maxima filter and the minima filter. However, they do not commute. The second kind of filter, introduced by Masnou, works on “shapes”, which are based on connected components of level sets. This filter has the additional property that it acts in the same manner on upper and lower level sets, that is, it commutes with an inversion of contrast. We discuss the relations of Masnou's filter with other classes of connected operators introduced in the literature. We display some experiments to show the main properties of the filters discussed above and compare them.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Alvarez, F. Guichard, P. Lions, and J. Morel, “Axioms and fundamental equations of image processing: Multiscale analysis and P.D.E,” Archive for Rational Mechanics and Analysis, Vol. 16, No. 9, pp. 200–257, 1993.

    Google Scholar 

  2. L. Ambrosio, V. Caselles, S. Masnou, and J. Morel, “The connected components of sets of finite perimeter and applications to image processing,” European Journal of Mathematics, Vol. 3, pp. 39–92, 2001.

    Google Scholar 

  3. C. Ballester and V. Caselles, “The M-components of level sets of continuous functions in WBV,” Publicacions Matem´atiques, Vol. 45, pp. 477–527, 2001.

    Google Scholar 

  4. C. Ballester, V. Caselles, and P. Monasse, “The tree of shapes of an image,” Preprint CMLA, 2001.

  5. V. Caselles, B. Coll, and J. Morel, “A Kanisza program,” Progress in Nonlinear Differential Equations and their Applications, Vol. 25, pp. 35–55, 1996.

    Google Scholar 

  6. V. Caselles, B. Coll, and J. Morel, “Is scale-space possible?” in Proc. of the 1stWorkshop on Scale-Space Theories in Computer Vision, Utrecht, The Netherlands, 1997. Grain Filters 269

  7. J. Froment, “A compact and multiscale image model based on level sets,” in Proc. of the 2ndWorkshop on Scale-Space Theories in Computer Vision, Corfu, Greece, pp. 152–163, 1999.

  8. J. Goutsias and H. Heijmans, “Fundamenta morphologicae mathematicae,” Fundamenta Informaticae, Vol. 41, No. 1/2, pp. 1–31, 2000.

    Google Scholar 

  9. F. Guichard and J. Morel, “Image iterative smoothing and P.D.E.'s,” Book in preparation, 2000.

  10. H. Heijmans, Morphological Image Operators, Academic Press, Boston, Mass., 1994.

    Google Scholar 

  11. H.Heijmans, “Self-dual morphological operators and filters,” J. Math. Imaging and Vision, Vol. 6, No. 1, pp. 15–36, 1996.

    Google Scholar 

  12. H. Heijmans, “Connected morphological operators for binary images,” in IEEE Int. Conference on Image Processing, ICIP'97, Vol. 2, Santa barbara (CA), USA, 1997, pp. 211–214.

    Google Scholar 

  13. C. Kuratowski, Topologie I, II, Editions J. Gabay: Paris, 1992.

  14. C. Lantuéjoul and S. Beucher, “On the use of geodesic metric in image analysis,” Journal of Microscopy, Vol. 121, pp. 39–49, 1981.

    Google Scholar 

  15. C. Lantuéjoul and F. Maisonneuve, “Geodesic methods in image analysis,” Pattern Recognition, Vol. 17, pp. 117–187, 1984.

    Google Scholar 

  16. P. Maragos and R. Schafer, “Morphological filters. Part I: their set-theoretic analysis and relations to linear shift-invariant filters,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 35, pp. 1153–1169, 1987a.

    Google Scholar 

  17. P. Maragos and R. Schafer, “Morphological filters. Part II: their relations to median, order-statistic, and stack filters,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 35, pp. 1170–1184, 1987b.

    Google Scholar 

  18. S. Masnou, “Image restoration involving connectedness,” in Proceedings of the 6th International Workshop on Digital Image Processing and Computer Graphics, Vol. 3346, Vienna, Austria, 1998.

  19. G. Matheron, Random Sets and Integral Geometry, JohnWiley: N.Y., 1975.

    Google Scholar 

  20. F. Meyer, Mathematical Morphology and Its Application to Signal and Image Processing, Chapt. From Connected Operators to Levelings. Kluwer Academic Publishers, H. Heijmans and J. Roerdink edition, 1998a.

  21. F. Meyer, Mathematical Morphology and Its Application to Signal and Image Processing, chapter The Levelings. Kluwer Academic Publishers, H.J.A.M. Heijmans and J. Roerdink edition, 1998b.

  22. F. Meyer and S. Beucher, “Morphological segmentation,” Journal of Visual Communication and Image Representation, Vol. 1, No. 1, pp. 21–46, 1990.

    Google Scholar 

  23. L. Moisan, “Affine plane curve evolution: A fully consistent scheme. IEEE Transactions on Image Processing, Vol. 7, No. 3, pp. 411–420, 1998.

    Google Scholar 

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

    Google Scholar 

  25. P. Monasse and F. Guichard, “Scale-space from a level lines tree,” Journal of Visual Communication and Image Representation, Vol. 11, pp. 224–236, 2000b.

    Google Scholar 

  26. M. Newman, Elements of the Topology of Plane Sets of Points. Dover Publications: New York, 1992.

    Google Scholar 

  27. P. Salembier, “Morphological multiscale segmentation for image coding,” Signal Processing, Vol. 38, No. 3, pp. 359–386, 1994.

    Google Scholar 

  28. P. Salembier, P. Brigger, J. Casas, and M. Pardas, “Morphological operators for image and video compression,” IEEE Transactions on Image Processing, Vol. 5, No. 6, pp. 881–898, 1996a.

    Google Scholar 

  29. P. Salembier, F. Meyer, P. Brigger, and L. Bouchard, “Morphological operators for very low bit rate video coding,” in Proceedings of International Conference of Image Processing, 1996b, p. 19P1.

  30. P. Salembier, A. Oliveras, and L. Garrido, “Antiextensive connected operators for image and sequence processing,” IEEE Transactions on Image Processing, Vol. 7, No. 4, pp. 555–570, 1998.

    Google Scholar 

  31. P. Salembier and J. Serra, “Flat zones filtering, connected operators and filters by reconstruction,” IEEE Transactions on Image Processing, Vol. 4, pp. 1153–1160, 1995.

    Google Scholar 

  32. G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,” International Journal of Computer Vision,Vol. 11, No. 1, pp. 25–44, 1993.

    Google Scholar 

  33. J. Serra, Image Analysis and Mathematical Morphology, Academic Press: New York, 1982.

    Google Scholar 

  34. J. Serra, “Introduction to mathematical morphology,” Computer Vision, Graphics and Image Processing, Vol. 35, No. 3, pp. 283–305, 1986.

    Google Scholar 

  35. J. Serra, “Connections for sets and functions,” Preprint, É,cole des Mines de Paris, Centre de Morphologie Mathématique, Fontainebleau, France, 1999.

  36. J. Serra and P. Salembier, “Connected operators and pyramids,” in Proceedings of SPIE Conference on Image Algebra and Mathematical Morphology, Vol. 2030, San Diego, California, pp. 65–76, 1993.

    Google Scholar 

  37. L. Vincent, “Grayscale area openings and closings, their efficient implementation and applications,” in Proceedings of the 1st Workshop on Mathematical Morphology and its Applications to Signal Processing, J. Serra and P. Salembrier (Eds.), Barcelona, Spain, pp. 22–27, 1993.

  38. L. Vincent, “Morphological area openings and closings for greyscale images,” in Proceedings of the Workshop Shape in Picture: Mathematical Description of Shape in Gray-Level Images, Driebergen, The Netherlands, pp. 197–208, 1994.

  39. L. Vincent and P. Soille, “Watersheds in digital spaces: An effi-cient algorithm based on immersion simulations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 6, pp. 583–598, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department de Tecnologia, Universitat Pompeu-Fabra, Passeig de Circumvalació 8, 08003, Barcelona, Spain

    Vicent Caselles

  2. CMLA, ENS Cachan, 61 Av. du President Wilson, 94235, Cachan Cedex, France

    Pascal Monasse

Authors
  1. Vicent Caselles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pascal Monasse
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caselles, V., Monasse, P. Grain Filters. Journal of Mathematical Imaging and Vision 17, 249–270 (2002). https://doi.org/10.1023/A:1020715626538

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020715626538

Search

Navigation