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Mathematical Morphology for Vector Images Using Statistical Depth

  • Conference paper
Mathematical Morphology and Its Applications to Image and Signal Processing (ISMM 2011)

Abstract

The open problem of the generalization of mathematical morphology to vector images is handled in this paper using the paradigm of depth functions. Statistical depth functions provide from the “deepest” point a “center-outward ordering” of a multidimensional data distribution and they can be therefore used to construct morphological operators. The fundamental assumption of this data-driven approach is the existence of “background/foreground” image representation. Examples in real color and hyperspectral images illustrate the results.

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References

  1. Al-Otum, H.M.: Morphological operators for color image processing based on mahalanobis distance. Optical Engineering 42(9), 2595–2606 (2003)

    Article  Google Scholar 

  2. Angulo, J.: Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis. Comput. Vis. Image Underst. 107(1-2), 56–73 (2007)

    Article  Google Scholar 

  3. Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognition 40(11), 2914–2929 (2007)

    Article  MATH  Google Scholar 

  4. Barnett, V.: The ordering of multivariate data (with discussion). Journal of the Royal Statistical Society Series A 139(3), 318–354 (1976)

    Article  MathSciNet  Google Scholar 

  5. Cuesta-Albertos, J.A., Nieto-Reyes, A.: The random tukey depth. Computational Statistics and Data Analysis 52, 4979–4988 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Donoho, D.L., Gasko, M.: Breakdown properties of location estimates based on halfspace depth and projected outlyingness. The Annals of Statistics 20(4), 1803–1827 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garcia, A., Vachier, C., Vallée, J.P.: Multivariate mathematical morphology and bayesian classifier application to colour and medical images. In: Image Processing: Algorithms and Systems, p. 681203. SPIE, San Jose (2008)

    Google Scholar 

  8. Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological operators for image sequences. Comput. Vis. Image Underst. 62(3), 326–346 (1995)

    Article  Google Scholar 

  9. Lezoray, O., Elmoataz, A., Meurie, C.: Mathematical morphology in any color space. In: ICIAPW 2007: Proceedings of the 14th International Conference of Image Analysis and Processing - Workshops, pp. 183–187 (2007)

    Google Scholar 

  10. Liu, R.Y.: On a notion of data depth based on random simplices. Annals of Statistics 18, 405–414 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mosler, K., Hoberg, R.: Data analysis and classification with the zonoid depth. DIMACS Series, pp. 49–59 (2006)

    Google Scholar 

  12. Najman, L., Talbot, H.: Mathematical morphology: from theory to applications. ISTE-Wiley (June 2010)

    Google Scholar 

  13. Pitas, I., Tsakalides, P.: Multivariate ordering in color image processing. IEEE Transactions on Circuits Systems Video Technology 1(3), 247–256 (1991)

    Article  Google Scholar 

  14. Plaza, A., Martinez, P., Perez, R., Plaza, J.: A new approach to mixed pixel classification of hyperspectral imagery based on extended morphological profiles. Pattern Recognition 37(6), 1097–1116 (2004)

    Article  Google Scholar 

  15. Serfling, R.: A depth function and a scale curve based on spatial quantiles. Statistical Data Analysis Based on the L 1 norm and Related Methods, 25–38 (2002)

    Google Scholar 

  16. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Inc., Orlando (1983)

    Google Scholar 

  17. Serra, J.: The “False colour” problem. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 13–23. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Soille, P.: Morphological Image Analysis. Springer, Heidelberg (1999), http://web.ukonline.co.uk/soille

    Book  MATH  Google Scholar 

  19. Tukey, J.W.: Mathematics and picturing data. In: Proceeding of the International Congress on Mathematics, pp. 523–531 (1975)

    Google Scholar 

  20. Velasco-Forero, S., Angulo, J.: Hit-or-miss transform in multivariate images. In: Blanc-Talon, J., Bone, D., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2010, Part I. LNCS, vol. 6474, pp. 452–462. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  21. Velasco, S., Angulo, J.: Morphological processing of hyperspectral images using kriging-based supervised ordering. In: 17th IEEE International Conference on Image Processing (ICIP), pp. 1409–1412 (2010)

    Google Scholar 

  22. Vardi, Y., Zhang, C.H.: The multivariate l 1 median and associated data depth. Proceeding of the Nacional Academy of Sciences 97(4), 1423–1436 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zuo, Y., Serfling, R.: General notions of statistical depth function. Annals of Statistics 28(2), 461–482 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Velasco-Forero, S., Angulo, J. (2011). Mathematical Morphology for Vector Images Using Statistical Depth. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds) Mathematical Morphology and Its Applications to Image and Signal Processing. ISMM 2011. Lecture Notes in Computer Science, vol 6671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21569-8_31

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  • DOI: https://doi.org/10.1007/978-3-642-21569-8_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21568-1

  • Online ISBN: 978-3-642-21569-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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