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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 18, 2016

N-ary Mathematical Morphology

  • Emmanuel Chevallier , Augustin Chevallier and Jesús Angulo


Mathematical morphology on binary images can be fully described by set theory. However, it is not sufficient to formulate mathematical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the definition of sup and inf operators. More generally, mathematical morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the notion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose first an extension of binary mathematical morphology to an intermediary situation: images composed of a finite number of independent unordered labels. We propose then an second extension to a continuous case.


[1] Arehart A.B., Vincent L., Kimia B.B., Mathematical morphology: The Hamilton-Jacobi connection, Springer-Verlag, Berlin, 1999. Search in Google Scholar

[2] Bloch. I, Duality vs. adjuntion for fuzzy mathematical morphology and general form of fuzzy erosions and dilations. Fuzzy Sets and Systems, 2009, 160: 1858–1867. 10.1016/j.fss.2009.01.006Search in Google Scholar

[3] Busch C., Eberle M, Morphological Operations for Color-Coded Images. In Proc. of the EUROGRAPHICS’95, 1995, Vol. 14, No. 3, C193–C204. 10.1111/j.1467-8659.1995.cgf143_0193.xSearch in Google Scholar

[4] Carlinet E., Thierry G., A Color Tree of Shapeswith Illustrations on Filtering, Simplification, and Segmentation,Mathematical Morphology and Its Applications to Signal and Image Processing, 2015, 363–374. 10.1007/978-3-319-18720-4_31Search in Google Scholar

[5] Chevallier E., Chevallier A., Angulo J., N-ary Mathematical Morphology. In Proc. of 12th International Symposium on Mathematical Morphology (ISMM 2015), LNCS 9082, 2015, pp. 339–350. 10.1007/978-3-319-18720-4_29Search in Google Scholar

[6] Deng T.-Q., Heijmans H.J.A.M., Grey-Scale Morphology Based on Fuzzy Logic. Journal of Mathematical Imaging and Vision, 2002, 16(2): 155–171. 10.1023/A:1013999431844Search in Google Scholar

[7] Franchi G., Angulo J., Ordering on the Probability Simplex of Endmembers for Hyperspectral Morphological Image Processing. In Proc. of 12th International Symposium on Mathematical Morphology (ISMM 2015), LNCS 9082, 2015, pp. 410–421. 10.1007/978-3-319-18720-4_35Search in Google Scholar

[8] Gronde J. , Roerdink J., Sponges for Generalized Morphology, Mathematical Morphology and Its Applications to Signal and Image Processing, 2015, 351–362. 10.1007/978-3-319-18720-4_30Search in Google Scholar

[9] Hanbury A., Serra J., Morphological operators on the unit circle. IEEE Trans. Image Processing, 2001, 10(12):1842–50. 10.1109/83.974569Search in Google Scholar PubMed

[10] Heijmans H.J.A.M., Morphological image operators. Academic Press, Boston, 1994. Search in Google Scholar

[11] Matheron G., Random Sets and Integral Geometry.Wiley, New York, 1975. Search in Google Scholar

[12] Meyer F., Adjunctions on the lattice of hierarchies. HAL, hal-00566714, 2011. Search in Google Scholar

[13] Ronse Ch., Agnus. V., Morphology on label images: Flat-type operators and connections. Journal of Mathematical Imaging and Vision, 2005, 22(2-3), 283-307. 10.1007/s10851-005-4895-1Search in Google Scholar

[14] Ch. Ronse. Ordering Partial Partition for Image Segmentation and Filtering: Merging, Creating and Inflating Blocks. Journal of Mathematical Imaging and Vision, 2014, 49(1):202–233. 10.1007/s10851-013-0455-2Search in Google Scholar

[15] Salembier P., Serra J., Flat Zones Filtering, Connected Operators, and Filters by reconstruction. IEEE Trans. on Images Processing, 2014, 4(8): 1153–60. 10.1109/83.403422Search in Google Scholar

[16] Sapiro G., Kimmel R., Shaked D., Kimia B., Bruckstein A., Implementing continuous-scale morphology via curve evolution, Pattern Recognition, 1993, 26: 1363–1372. 10.1016/0031-3203(93)90142-JSearch in Google Scholar

[17] Serra J., Image Analysis and Mathematical Morphology. Academic Press, London, 1982. Search in Google Scholar

[18] Serra J., Image Analysis and Mathematical Morphology. Vol II: Theoretical Advances, Academic Press, London, 1988. Search in Google Scholar

[19] Soille P, Morphological Image Analysis, Springer-Verlag, Berlin, 1999. 10.1007/978-3-662-03939-7Search in Google Scholar

Received: 2015-7-10
Accepted: 2016-1-30
Published Online: 2016-3-18

© 2016 Emmanuel Chevallier et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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