Advertisement

Non-adaptive and Amoeba Quantile Filters for Colour Images

  • Martin WelkEmail author
  • Andreas Kleefeld
  • Michael Breuß
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9082)

Abstract

Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2×2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these. Using amoeba structuring elements, we devise image-adaptive versions of both filter classes. Experiments demonstrate the favourable properties of the filters.

Keywords

Quantile Rank-order filter Color image Matrix field Amoebas 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agoston, M.K.: Computer Graphics and Geometric Modeling: Implementation and Algorithms. Springer, London (2005)Google Scholar
  2. 2.
    Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34, 344–371 (1986)CrossRefGoogle Scholar
  3. 3.
    Burgeth, B., Kleefeld, A.: Morphology for color images via loewner order for matrix fields. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 243–254. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Burgeth, B., Kleefeld, A.: An approach to color-morphology based on Einstein addition and Loewner order. Pattern Recognition Letters 47, 29–39 (2014)CrossRefGoogle Scholar
  5. 5.
    Burgeth, B., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology on tensor data using the Loewner ordering. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 357–368. Springer, Berlin (2006)CrossRefGoogle Scholar
  6. 6.
    Chaudhuri, P.: On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association 91(434), 862–872 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fabbri, R., Da F. Costa, L., Torelli, J.C., Bruno, O.M.: 2D Euclidean distance transform algorithms: A comparative survey. ACM Computing Surveys, 40(1), art. 2 (2008)Google Scholar
  8. 8.
    Guichard, F., Morel, J.-M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. IMA Conference Series (New Series), vol. 63, pp. 525–562. Clarendon Press, Oxford (1997)Google Scholar
  9. 9.
    Hayford, J.F.: What is the center of an area, or the center of a population? Journal of the American Statistical Association 8(58), 47–58 (1902)CrossRefGoogle Scholar
  10. 10.
    Ikonen, L., Toivanen, P.: Shortest routes on varying height surfaces using gray-level distance transforms. Image and Vision Computing 23(2), 133–141 (2005)CrossRefGoogle Scholar
  11. 11.
    Kleefeld, A., Breuß, M., Welk, M., Burgeth, B.: Adaptive filters for color images: Median filtering and its extensions. In: Trémeau, A., Schettini, R., Tominaga, S. (eds.) CCIW 2015. LNCS, vol. 9016, pp. 149–158. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  12. 12.
    Lerallut, R., Decencière, É., Meyer, F.: Image processing using morphological amoebas. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol. 30, pp. 13–22. Springer, Dordrecht (2005)CrossRefGoogle Scholar
  13. 13.
    Lerallut, R., Decencière, É., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)CrossRefGoogle Scholar
  14. 14.
    Löwner, K.: Über monotone Matrixfunktionen. Mathematische Zeitschrift 38, 177–216 (1934)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Meyer, F., Maragos, P.: Nonlinear scale-space representation with morphological levelings. Journal of Visual Communication and Image Representation 11, 245–265 (2000)CrossRefGoogle Scholar
  16. 16.
    Rivest, J.-F., Soille, P., Beucher, S.: Morphological gradients. Journal of Electronic Imaging 2(4), 326–336 (1993)CrossRefGoogle Scholar
  17. 17.
    Sapiro, G.: Vector (self) snakes: a geometric framework for color, texture and multiscale image segmentation. In: Proc. 1996 IEEE International Conference on Image Processing, Lausanne, Switzerland, vol. 1, pp. 817–820 (September 1996)Google Scholar
  18. 18.
    Spence, C., Fancourt, C.: An iterative method for vector median filtering. In: Proc. of 2007 IEEE International Conference on Image Processing, vol. 5, pp. 265–268 (2007)Google Scholar
  19. 19.
    Tukey, J.W.: Exploratory Data Analysis. Addison–Wesley, Menlo Park (1971)Google Scholar
  20. 20.
    Weber, A.: Über den Standort der Industrien. Mohr, Tübingen (1909)Google Scholar
  21. 21.
    Welk, M.: Amoeba techniques for shape and texture analysis. Technical Report cs:1411.3285, arXiv.org (2014)Google Scholar
  22. 22.
    Welk, M., Breuß, M.: Morphological amoebas and partial differential equations. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 185, pp. 139–212. Elsevier Academic Press (2014)Google Scholar
  23. 23.
    Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. Journal of Mathematical Imaging and Vision 39, 87–99 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Welk, M., Feddern, C., Burgeth, B., Weickert, J.: Median filtering of tensor-valued images. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 17–24. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  25. 25.
    Welk, M., Weickert, J., Becker, F., Schnörr, C., Feddern, C., Burgeth, B.: Median and related local filters for tensor-valued images. Signal Processing 87, 291–308 (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Biomedical Computer Science and TechnologyUniversity of Health Sciences, Medical Informatics and Technology (UMIT)Hall/TyrolAustria
  2. 2.Faculty of Mathematics, Natural Sciences and Computer ScienceBrandenburg Technical University Cottbus-SenftenbergCottbusGermany

Personalised recommendations