J. P. Barthélemy, F. Brucker, and C. Osswald, Combinatorial optimization and hierarchical classifications, Quarterly Journal of the Belgian, French and Italian Operations Research Societies, vol.2, issue.3, pp.179-219, 2004.
DOI : 10.1007/s10288-004-0051-9

L. Najman, Ultrametric Watersheds, 2009.
DOI : 10.1007/s10851-005-4892-4

URL : https://hal.archives-ouvertes.fr/hal-00622405

J. Benzécri, Analyse des données: la Taxinomie, Dunod, vol.1, 1973.

S. Johnson, Hierarchical clustering schemes, Psychometrika, vol.58, issue.4, pp.241-254, 1967.
DOI : 10.1007/BF02289588

T. Pavlidis, Hierarchies in structural pattern recognition, Proceedings of the IEEE, vol.67, issue.5, pp.737-744, 1979.
DOI : 10.1109/PROC.1979.11323

P. Soille, Constrained connectivity for hierarchical image decomposition and simplification . Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol.30, issue.7, pp.1132-1145, 2008.

L. Najman and M. Schmitt, Geodesic saliency of watershed contours and hierarchical segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.18, issue.12, pp.1163-1173, 1996.
DOI : 10.1109/34.546254

URL : https://hal.archives-ouvertes.fr/hal-00622128

L. Guigues, J. P. Cocquerez, and H. L. Men, Scale-Sets Image Analysis, International Journal of Computer Vision, vol.20, issue.6, pp.289-317, 2006.
DOI : 10.1007/s11263-005-6299-0

URL : https://hal.archives-ouvertes.fr/hal-00705364

P. A. Arbeláez and L. D. Cohen, A Metric Approach to Vector-Valued Image Segmentation, International Journal of Computer Vision, vol.133, issue.2, pp.119-126, 2006.
DOI : 10.1007/s11263-006-6857-5

T. Pavlidis and . In, Structural Pattern Recognition, pp.90-123, 1977.
DOI : 10.1007/978-3-642-88304-0

F. Meyer and S. Beucher, Morphological segmentation, Journal of Visual Communication and Image Representation, vol.1, issue.1, pp.21-46, 1990.
DOI : 10.1016/1047-3203(90)90014-M

F. Meyer, Morphological segmentation revisited, Space, Structure and Randomness, pp.315-347, 2005.
DOI : 10.1007/0-387-29115-6_13

F. Meyer and L. Najman, Segmentation, arbre de poids minimum et hiérarchies, Morphologie mathématique 1: approches déterministes. Lavoisier, pp.201-233, 2008.

J. B. Roerdink and A. Meijster, The watershed transform: Definitions, algorithms and parallelization strategies, Fundamenta Informaticae, vol.41, issue.12, pp.187-228, 2001.

G. Bertrand, On Topological Watersheds, Journal of Mathematical Imaging and Vision, vol.34, issue.6, pp.217-230, 2005.
DOI : 10.1007/s10851-005-4891-5

URL : https://hal.archives-ouvertes.fr/hal-00622398

L. Najman, M. Couprie, and G. Bertrand, Watersheds, mosaics, and the emergence paradigm, Discrete Applied Mathematics, vol.147, issue.2-3, pp.301-324, 2005.
DOI : 10.1016/j.dam.2004.09.017

URL : https://hal.archives-ouvertes.fr/hal-00622113

J. Cousty, G. Bertrand, M. Couprie, and L. Najman, Fusion Graphs: Merging Properties and Watersheds, Journal of Mathematical Imaging and Vision, vol.147, issue.2???3, pp.87-104, 2008.
DOI : 10.1007/s10851-007-0047-0

URL : https://hal.archives-ouvertes.fr/hal-00622473

J. Cousty, L. Najman, G. Bertrand, and M. Couprie, Weighted fusion graphs: Merging properties and watersheds, Discrete Applied Mathematics, vol.156, issue.15, pp.3011-3027, 2008.
DOI : 10.1016/j.dam.2008.01.005

URL : https://hal.archives-ouvertes.fr/hal-00622473

J. Cousty, G. Bertrand, L. Najman, and M. Couprie, Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.31, issue.8, 2009.
DOI : 10.1109/TPAMI.2008.173

URL : https://hal.archives-ouvertes.fr/hal-00622410

R. Diestel, Graph Theory. Graduate Texts in Mathematics, 1997.

T. Kong and A. Rosenfeld, Digital Topology, Image Process, vol.48, issue.3, pp.357-393, 1989.
DOI : 10.1007/978-1-4615-1529-6_3

J. Cousty, L. Najman, and J. Serra, Some Morphological Operators in Graph Spaces, 2009.
DOI : 10.1016/j.patrec.2008.03.019

URL : https://hal.archives-ouvertes.fr/hal-00622403

J. Cousty, G. Bertrand, L. Najman, and M. Couprie, Watershed Cuts: Thinnings, Shortest Path Forests, and Topological Watersheds, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.32, issue.5, 2009.
DOI : 10.1109/TPAMI.2009.71

URL : https://hal.archives-ouvertes.fr/hal-00729346

P. Salembier, A. Oliveras, and L. Garrido, Antiextensive connected operators for image and sequence processing, IEEE Transactions on Image Processing, vol.7, issue.4, pp.555-570, 1998.
DOI : 10.1109/83.663500

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.9217

L. Najman and M. Couprie, Building the Component Tree in Quasi-Linear Time, IEEE Transactions on Image Processing, vol.15, issue.11, pp.3531-3539, 2006.
DOI : 10.1109/TIP.2006.877518

URL : https://hal.archives-ouvertes.fr/hal-00622110

M. Krasner, Espaces ultramétrique, C.R. Acad. Sci. Paris, vol.219, pp.433-435, 1944.

B. Leclerc, Description combinatoire des ultramétriques, Mathématique et sciences humaines, vol.73, pp.5-37, 1981.

J. Gower and G. Ross, Minimum Spanning Trees and Single Linkage Cluster Analysis, Applied Statistics, vol.18, issue.1, pp.54-64, 1969.
DOI : 10.2307/2346439

J. B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem, Proceedings of the American Mathematical Society, vol.7, issue.1, pp.48-50, 1956.
DOI : 10.1090/S0002-9939-1956-0078686-7

M. Couprie, L. Najman, and G. Bertrand, Quasi-Linear Algorithms for the Topological Watershed, Journal of Mathematical Imaging and Vision, vol.13, issue.6, pp.231-249, 2005.
DOI : 10.1007/s10851-005-4892-4

URL : https://hal.archives-ouvertes.fr/hal-00622399

J. Cousty, L. Najman, and J. Serra, Raising in watershed lattices (a) Original image (b) Ultrametric watershed (c) Cross section of (b), 15th IEEE ICIP'08, pp.2196-2199, 2008.

. Fig, Zoom on a comparison between two watersheds of a filtered version of the image 4.a. Morphological filtering tends to create large plateaus, and both watersheds (a) and (b) are possible, but only (a) is a subset of a watershed of 4.a. No hierarchical scheme will ever give a result as (b)