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.
 Arehart A.B., Vincent L., Kimia B.B., Mathematical morphology: The Hamilton-Jacobi connection, Springer-Verlag, Berlin, 1999. Search in Google Scholar
 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
 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
 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
 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
 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
 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
 Heijmans H.J.A.M., Morphological image operators. Academic Press, Boston, 1994. Search in Google Scholar
 Matheron G., Random Sets and Integral Geometry.Wiley, New York, 1975. Search in Google Scholar
 Meyer F., Adjunctions on the lattice of hierarchies. HAL, hal-00566714, 2011. Search in Google Scholar
 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
 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
 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
 Serra J., Image Analysis and Mathematical Morphology. Academic Press, London, 1982. Search in Google Scholar
 Serra J., Image Analysis and Mathematical Morphology. Vol II: Theoretical Advances, Academic Press, London, 1988. Search in Google Scholar
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