L. Alvarez, F. Guichard, P. Lions, and J. Morel, Axioms and fundamental equations of image processing, Archive for Rational Mechanics and Analysis, vol.11, issue.3, pp.199-257, 1993.
DOI : 10.1007/BF00375127

O. Alvarez, E. N. Barron, and H. Ishii, Hopf-Lax formulas for semicontinuous data, Indiana University Mathematics Journal, vol.48, issue.3, pp.993-1035, 1999.
DOI : 10.1512/iumj.1999.48.1648

J. Angulo, Generalised Morphological Image Diffusion. HAL preprint, p.789162, 2013.
DOI : 10.1016/j.na.2015.12.015

URL : https://hal.archives-ouvertes.fr/hal-00789162

J. Angulo, Morphological Bilateral Filtering, SIAM Journal on Imaging Sciences, vol.6, issue.3, pp.1790-1822, 2013.
DOI : 10.1137/110844258

URL : https://hal.archives-ouvertes.fr/hal-00789160

J. Angulo and S. Velasco-forero, Stochastic Morphological Filtering and Bellman-Maslov Chains, Proc. of ISMM'13 (11th International Symposium on Mathematical Morphology ), pp.171-182, 2013.
DOI : 10.1007/978-3-642-38294-9_15

URL : https://hal.archives-ouvertes.fr/hal-00834635

J. Angulo and S. Velasco-forero, Riemannian mathematical morphology, Pattern Recognition Letters, vol.47, pp.93-101, 2014.
DOI : 10.1016/j.patrec.2014.05.015

URL : https://hal.archives-ouvertes.fr/hal-01075759

A. B. Arehart, L. Vincent, and B. B. Kimia, Mathematical morphology: The Hamilton-Jacobi connection, 1993 (4th) International Conference on Computer Vision, pp.215-219, 1993.
DOI : 10.1109/ICCV.1993.378217

D. Attouch and D. Aze, Approximation and regularization of arbitray functions in Hilbert spaces by the Lasry-Lions method. Annales de l'I.H.P., section C, pp.289-312, 1993.

M. Bardi and L. C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Analysis: Theory, Methods & Applications, vol.8, issue.11, pp.1373-1381, 1984.
DOI : 10.1016/0362-546X(84)90020-8

E. N. Barron, R. Jensen, and W. Liu, Hopf???Lax-Type Formula forut+H(u,??Du)=0, Journal of Differential Equations, vol.126, issue.1, pp.48-61, 1996.
DOI : 10.1006/jdeq.1996.0043

E. N. Barron, R. Jensen, and W. Liu, Hopf-Lax formula for u t = H(u, Du) = 0, II, Comm. Partial Differential Equations, vol.22, pp.1141-1160, 1997.

S. Beucher, About a problem of definition of the geodesic erosion. CMM/Mines ParisTech Technical Report, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01403947

I. Bloch, Duality vs. adjuntion for fuzzy mathematical morphology and general form of fuzzy erosions and dilations. Fuzzy Sets and Systems, pp.1858-1867, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00650338

R. Van-den-boomgaard and L. Dorst, The Morphological Equivalent of Gaussian Scale-Space
DOI : 10.1007/978-94-015-8802-7_15

M. Breuß and J. Weickert, Highly accurate PDE-based morphology for general structuring elements, Proc. of Scale Space and Variational Methods in Computer Vision, pp.758-769, 2009.

R. W. Brockett and P. Maragos, Evolution equations for continuous-scale morphology, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.3377-3386, 1994.
DOI : 10.1109/ICASSP.1992.226260

F. H. Clarke and R. , Nonlinear Analysis, Differential Equations and Control, NATO Science Series C: Mathematical and Physical Sciences, vol.528, 1999.
DOI : 10.1007/978-94-011-4560-2

M. G. Crandall, H. Ishii, and P. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-67, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

T. Deng and H. J. Heijmans, Grey-Scale Morphology Based on Fuzzy Logic, Journal of Mathematical Imaging and Vision, vol.16, issue.2, pp.155-171, 2002.
DOI : 10.1023/A:1013999431844

E. H. Diop and J. Angulo, MULTISCALE IMAGE ANALYSIS BASED ON ROBUST AND ADAPTIVE MORPHOLOGICAL SCALE-SPACES, Image Analysis & Stereology, vol.33, issue.2, p.975728, 2014.
DOI : 10.5566/ias.993

URL : https://hal.archives-ouvertes.fr/hal-00975728

L. Dorst, R. Van-den, and . Boomgaard, Morphological signal processing and the slope transform, Signal Processing, vol.38, issue.1, pp.79-98, 1994.
DOI : 10.1016/0165-1684(94)90058-2

E. A. Engbers, R. Van-den, A. W. Boomgaard, and . Smeulders, Decomposition of Separable Concave Structuring Functions, Journal of Mathematical Imaging and Vision, vol.15, issue.3, pp.181-195, 2001.
DOI : 10.1023/A:1012224405930

J. Flachs and M. A. Pollatschek, Duality theorems for certain programs involving minimum or maximum operations, Mathematical Programming, vol.8, issue.1, pp.348-370, 1979.
DOI : 10.1007/BF01582120

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove et al., A Compendium of Continuous Lattices, 1980.
DOI : 10.1007/978-3-642-67678-9

M. Gondran and M. Minoux, Graphs, Dioids and Semirings: New Models and Algorithms, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01304880

J. Goutsias, Morphological analysis of discrete random shapes, Journal of Mathematical Imaging and Vision, vol.48, issue.suppl. 1, pp.193-215, 1992.
DOI : 10.1007/BF00118590

H. J. Heijmans, Morphological image operators, 1994.

P. T. Jackway and M. Deriche, Scale-space properties of the multiscale morphological dilation-erosion, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.18, issue.1, pp.38-51, 1996.
DOI : 10.1109/34.476009

D. Jeulin and P. Jeulin, Synthesis of Rough Surfaces by Random Morphological Model, Proc. of 3rd European Symposium of Stereology, pp.239-246, 1981.

D. Jeulin, Modèles Morphologiques de Structures Aléatoires et de Changement d'Echelle, Thèse d'EtatèsEtatès Sciences Physiques, 1991.

D. Jeulin, Boolean Random Functions Stochastic Geometry, Spatial Statistics and Random Fields, Chapter 5, Series: Lecture Notes in Mathematics, vol.2120, 2014.

V. Kolokoltsov and V. P. Maslov, Idempotent Analysis and Its Applications, 1997.
DOI : 10.1007/978-94-015-8901-7

E. J. Kraus, H. J. Heijmans, and E. R. Dougherty, Gray-scale granulometries compatible with spatial scalings, Signal Processing, vol.34, issue.1, pp.1-17, 1993.
DOI : 10.1016/0165-1684(93)90023-4

J. M. Lasry, P. Lions, and S. Beucher, A remark on regularization in Hilbert spaces, Israel Journal of Mathematics, vol.92, issue.134, pp.257-266, 1981.
DOI : 10.1007/BF02765025

. Ch, . Lantuéjoul, and . Cours, Les ensembles aléatoires " , (Lecture notes) Ecole des Mines de Paris, 1993.

D. T. Luc and M. Volle, Levels Sets Infimal Convolution and Level Addition, Journal of Optimization Theory and Applications, vol.90, issue.3, pp.695-714, 1997.
DOI : 10.1023/A:1022657102069

P. Maragos and R. W. Schafer, Morphological filters--Part I: Their set-theoretic analysis and relations to linear shift-invariant filters, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.35, issue.8, pp.1153-1169, 1987.
DOI : 10.1109/TASSP.1987.1165259

P. Maragos, Slope transforms: theory and application to nonlinear signal processing, IEEE Transactions on Signal Processing, vol.43, issue.4, pp.864-877, 1995.
DOI : 10.1109/78.376839

P. Maragos, Differential morphology and image processing, IEEE Transactions on Image Processing, vol.5, issue.6, pp.922-937, 1996.
DOI : 10.1109/83.503909

URL : http://dspace.lib.ntua.gr/handle/123456789/24330

P. Maragos, Algebraic and PDE Approaches for Lattice Scale-Spaces with Global Constraints, International Journal of Computer Vision, vol.523, issue.2, pp.121-137, 2003.

P. Maragos, Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems, Journal of Mathematical Imaging and Vision, vol.8, issue.3, pp.333-353, 2005.
DOI : 10.1007/s10851-005-4897-z

P. Maragos and C. Vachier, A PDE formulation for viscous morphological operators with extensions to intensity-adaptive operators, 2008 15th IEEE International Conference on Image Processing, pp.2200-2203, 2008.
DOI : 10.1109/ICIP.2008.4712226

G. Matheron, Random Sets and Integral Geometry, 1975.

F. Meyer, Inondation par des fluides visqueux, 1993.

F. Meyer, Operateurs morphologiques visqueux, 2008.

J. J. Moreau, Inf-convolution, convexité des fonctions numériques, J. Math. Pures Appl, vol.49, pp.109-154, 1970.

M. Nachtegael and E. E. Kerre, Connections between binary, gray-scale and fuzzy mathematical morphologies, Fuzzy Sets and Systems, vol.124, issue.1, pp.73-85, 2001.
DOI : 10.1016/S0165-0114(01)00013-6

H. T. Nguyen, Y. Wang, and G. Wei, On Choquet theorem for random upper semicontinuous functions, International Journal of Approximate Reasoning, vol.46, issue.1, pp.3-16, 2007.
DOI : 10.1016/j.ijar.2006.12.004

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-49, 1988.
DOI : 10.1016/0021-9991(88)90002-2

J. Penot and C. , Approximation of Functions and Sets, Approximation, Optimization and Mathematical Economics, pp.255-274, 2001.
DOI : 10.1007/978-3-642-57592-1_23

F. Préteux and M. Schmitt, Boolean Texture Analysis and Synthesis, In (J. Serra Image Analysis and Mathematical Morphology, vol.2, issue.18, 1988.

R. T. Rockafellar, Convex analysis, 1970.
DOI : 10.1515/9781400873173

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, SIAM Journal on Numerical Analysis, vol.29, issue.3, pp.867-884, 1992.
DOI : 10.1137/0729053

A. Seeger and M. Volle, On a convolution operation obtained by adding level sets : classical and new results, RAIRO - Operations Research, vol.29, issue.2, pp.131-154, 1995.
DOI : 10.1051/ro/1995290201311

J. Serra, Image Analysis and Mathematical Morphology, 1982.

J. Serra, Boolean random functions, Journal of Microscopy, vol.2, issue.Suppl 1, pp.41-63
DOI : 10.1111/j.1365-2818.1989.tb02905.x

M. Schmitt, Support Function and Minkowski Addition of Non-Convex Sets Mathematical Morphology and its Applications to image and signal processing, pp.15-22, 1996.

P. Soille, Morphological Image Analysis, 1999.

C. Vachier and F. Meyer, The Viscous Watershed Transform, Journal of Mathematical Imaging and Vision, vol.7, issue.3, pp.251-267, 2005.
DOI : 10.1007/s10851-005-4893-3

C. Vachier and F. Meyer, News from Viscous Land, Proc. of 8th Internation Symposium on Mathematical Morphology (ISMM'07), pp.189-200, 2007.

T. D. Van and N. D. Son, Hopf-Lax-Oleinik-Type Estimates for Viscosity Solutions to Hamilton-Jacobi Equations with Concave-Convex Data, Vietnam Journal of Mathematics, vol.34, issue.2, pp.209-239, 2006.

L. Vincent, Morphological grayscale reconstruction in image analysis: applications and efficient algorithms, IEEE Transactions on Image Processing, vol.2, issue.2, pp.176-201, 1993.
DOI : 10.1109/83.217222

M. Volle, Duality for the Level Sum of Quasiconvex Functions and Applications. ESAIM: Control, Optimisation and Calculus of Variations, pp.329-343, 1998.

G. Wei and Y. Wang, On metrization of the hit-or-miss topology using Alexandroff compactification, International Journal of Approximate Reasoning, vol.46, issue.1, pp.47-64, 2007.
DOI : 10.1016/j.ijar.2006.12.007