What shape is your conjugate? A survey of computational convex analysis and its applications

SIAM Review

Volumn 52, Issue 3, 2010, Pages 505-542

  Lucet, Yves ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Convex analysis; Convex envelope; Distance transform; Fast transforms; Fenchel conjugate; Infconvolution; Lax hopf formula; Legendre fenchel transform; Mathematical morphology; Max plus algebra; Moreau envelope; Piecewise linear quadratic

Indexed keywords

CONVEX ANALYSIS; CONVEX ENVELOPE; DISTANCE TRANSFORMS; FAST TRANSFORM; FENCHEL CONJUGATE; HOPF FORMULA; INF-CONVOLUTION; LEGENDRE-FENCHEL TRANSFORMS; MAX-PLUS ALGEBRA; MOREAU ENVELOPE; PIECEWISE LINEAR;

ALGEBRA; ALGORITHMS; COMPUTER VISION; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); ELECTRIC NETWORK ANALYSIS; FAST FOURIER TRANSFORMS; MATHEMATICAL MORPHOLOGY; MATHEMATICAL OPERATORS; MEDICAL IMAGING; MORPHOLOGY; PIECEWISE LINEAR TECHNIQUES; THERMODYNAMICS;

MATHEMATICAL TRANSFORMATIONS;

EID: 78549293580     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/100788458     Document Type: Article

References (264)

1. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1993.
2. S. M. Aji and R. J. Mceliece, The generalized distributive law, IEEE Trans. Inform. Theory, 46 (2000), pp. 325-343.
3. S. M. Aji and R. J. McEliece, The generalized distributive law and free energy minimization, in Proceedings of the 39th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, University of Illinois at Urbana-Champaign, 2001, pp. 672-681.
4. M. Akian, J.-P. Quadrat, and M. Viot, Bellman processes, in Proceedings of the 11th International Conference on Analysis and Optimization of Systems: Discrete Event Systems, Lecture Notes in Control and Inform. Sci. 199, Springer, New York, 1994, pp. 302-311.
5. M. Akian, J.-P. Quadrat, and M. Viot, Duality between probability and optimization, in Idempotency (Bristol, 1994), Publ. Newton Inst. 11, Cambridge University Press, Cambridge, 1998, pp. 331-353.
6. W. N. Anderson, Jr., and R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl., 26 (1969), pp. 576-594.
7. W. N. Anderson, Jr., R. J. Duffin, and G. E. Trapp, Parallel subtraction of matrices (Hermitian semidefinite), Proc. Nat. Acad. Sci. USA, 69 (1972), pp. 2530-2531.
8. W. N. Anderson, Jr., T. D. Morley, and G. E . Trapp, Characterization of parallel subtraction, Proc. Nat. Acad. Sci. USA, 76 (1979), pp. 3599-3601.
9. W. N. Anderson, Jr., and G. E. Trapp, Inequalities for the parallel connection of resistive n-port networks, J. Franklin Inst., 209 (1975), pp. 305-313.
10. J. Antezana, G. Corach, and D. Stojanoff, Bilateral shorted operators and parallel sums, Linear Algebra Appl., 414 (2006), pp. 570-588.
11. A. B. Arehart, L. Vincent, and B. B. Kimia, Mathematical morphology: The Hamilton-Jacobi connection, in Proceedings of the Fourth IEEE International Conference on Computer Vision, Berlin, 1993, pp. 215-219.
12. M. Atteia and M. Ra?issouli, Self dual operators on convex functionals; geometric mean and square root of convex functionals, J. Convex Anal., 8 (2001), pp. 223-240.
13. H. Attouch and D. Azê, Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Ann. Inst. H. Poincarê. Anal. Non Linêaire, 10 (1993), pp. 289-312.
14. H. Attouch and R. J.-B. Wets, A convergence theory for saddle functions, Trans. Amer. Math. Soc., 280 (1983), pp. 1-41.
15. E. Aurell, S.-G. N. Gurbatov, and S.-G. I. Simdyankin, Numerical Proof of Self-similarity in Burgers' Turbulence, technical report, PDC, Stockholm, 1996; also available online from http://arxiv.org/abs/patt-sol/9602005.
16. F. Baccelli and D. Hong, TCP is max-plus linear and what it tells us on its throughput, in SIGCOMM '00: Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, ACM Press, New York, 2000, pp. 219-230.
17. F. L. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Ser. Probab. Math. Stat., John Wiley & Sons, Chichester, UK, 1992.
18. H. Bae and Y. Lim, Parallel addition and fixed points of compressions on symmetric cones, Math. Nachr., 246/247 (2002), pp. 20-30.
19. M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations, SIAM J. Math. Anal., 29 (1998), pp. 1067-1086.
20. E. N. Barron, R. Jensen, and W. Liu, Explicit solution of some first-order PDE's, J. Dyn. Control Syst., 3 (1997), pp. 149-164.
21. H. H. Bauschke, P. L. Combettes, and D. Noll, Joint minimization with alternating Bregman proximity operators, Pac. J. Optim., 2 (2006), pp. 401-424.
22. H. H. Bauschke, Y. Lucet, and M. Trienis, How to transform one convex function continuously into another, SIAM Rev., 50 (2008), pp. 115-132.
23. H. H. Bauschke, Y. Lucet, and X. Wang, Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators, SIAM J. Control Optim., 46 (2007), pp. 2031-2051.
24. H. H. Bauschke, E. Matoušková, and S. Reich, Projection and proximal point methods: Convergence results and counterexamples, Nonlinear Anal., 56 (2004), pp. 715-738.
25. H. H. Bauschke and M. von Mohrenschildt, Symbolic computation of Fenchel conjugates, SIGSAM Bull., 40 (2006), pp. 18-28.
26. H. H. Bauschke and X. Wang, The kernel average of two convex functions and its application to the extension and representation of monotone operators, Trans. Amer. Math. Soc., 361 (2009), pp. 5947-5965.
27. J. Bec, U. Frisch, and K. Khanin, Kicked Burgers turbulence, J. Fluid Mech., 416 (2000), pp. 239-267.
28. G. Beliakov, A review of applications of the cutting angle methods, in Continuous Optimization, V. Jeyakumar and P. Rubinov, eds., Appl. Optim. 99, Springer, New York, 2005, pp. 209-248.
29. R. Bellman and W. Karush, On a new functional transform in analysis: The maximal transform, Bull. Amer. Math. Soc., 67 (1961), pp. 501-503.
30. R. Bellman and W. Karush, Mathematical programming and the maximum transform, J. Soc. Indust. Appl. Math., 10 (1962), pp. 550-567.
31. R. Bellman and W. Karush, Functional equations in the theory of dynamic programming. XII. An application of the maximum transform, J. Math. Anal. Appl., 6 (1963), pp. 155-157.
32. R. Bellman and W. Karush, On the maximum transform, J. Math. Anal. Appl., 6 (1963), pp. 67-74.
33. F. Bernard and L. Thibault, Prox-regularity of functions and sets in Banach spaces, Set-Valued Anal., 12 (2004), pp. 25-47.
34. F. Bernard and L. Thibault, Prox-regular functions in Hilbert spaces, J. Math. Anal. Appl., 303 (2005), pp. 1-14.
35. F. Bernard and L. Thibault, Uniform prox-regularity of functions and epigraphs in Hilbert spaces, Nonlinear Anal., 60 (2005), pp. 187-207.
36. G. Bianchi, F. Vieira, and L. L. Ling, A novel network traffic predictor based on multifractal traffic characteristic, in Proceedings of the Global Telecommunications Conference, GLOBECOM, 2004, Dallas, TX, IEEE, 2004, pp. 680-684.
37. R. V. D. Boomgaard and H. J. A. M. Heijmans, Morphological scale-space operators: An algebraic framework, in Mathematical Morphology and Its Applications to Image and Signal Processing, J. Goutsias, L. Vincent, and D. S. Bloomberg, eds., Comput. Imaging Vision 18, Springer, The Netherlands, 2002, pp. 282-290.
38. G. Borgefors, Distance transformations on hexagonal grids, Pattern Recognition Lett., 9 (1989), pp. 97-105.
39. J. M. Borwein and C. H. Hamilton, Symbolic Fenchel conjugation, Math. Program., 116 (2007), pp. 17-35.
40. J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, CMS Books Math./Ouvrages Math. SMC 3, Springer, New York, 2000.
41. M. Bougeard, J.-P. Penot, and A. Pommellet, Towards minimal assumptions for the infimal convolution regularization, J. Approx. Theory, 64 (1991), pp. 245-270.
42. L. M. Bregman, A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, Žh. Vyčhisl. Mat. Mat. Fiz., 7 (1967), pp. 620-631.
43. D. Bremner, T. M. Chan, E. D. Demaine, J. Erickson, F. Hurtado, J. Iacono, S. Langerman, and P. Taslakian, Necklaces, convolutions, and X+Y , in Algorithms-ESA 2006, Lecture Notes in Comput. Sci. 4168, Springer, Berlin, 2006, pp. 160-171.
44. Y. Brenier, Un algorithme rapide pour le calcul de transformêes de Legendre-Fenchel discrètes, C. R. Acad. Sci. Paris Sêr. I Math., 308 (1989), pp. 587-589.
45. H. Breu, J. Gil, D. Kirkpatrick, and M. Werman, Linear time Euclidean distance transform algorithms, IEEE Trans. Pattern Anal. Mach. Intelligence, 17 (1995), pp. 529-533.
46. B. Brighi and M. Chipot, Approximated convex envelope of a function, SIAM J. Numer. Anal., 31 (1994), pp. 128-148.
47. B. Burgeth and J. Weickert, An explanation for the logarithmic connection between linear and morphological system theory, Internat. J. Comput. Vision, 64 (2005), pp. 157-169.
48. J. V. Burke and M. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Math. Program., 88 (2000), pp. 157-181.
49. Y. Caron, P. Makris, and N. Vincent, A method for detecting objects using Legendre transform, in RFAI Team Publication, Maghrebian Conference on Computer Science MCSEAI, Annaba, Algeria, Signal Image Parole, 2002, pp. 219-225.
50. B. A. Carrê, An algebra for network routing problems, J. Inst. Math. Appl., 7 (1971), pp. 273-294.
51. C. Chaux, P. L. Combettes, J.-C. Pesquet, and V. R.Wajs, A forward-backward algorithm for image restoration with sparse representations, in Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS'05), Rennes, France, IRISA, 2005, pp. 49-52.
52. J.-C. Chen, Y.-C. Wang, and J.-T. Chen, A novel broadcast scheduling strategy using factor graphs and the sum-product algorithm, IEEE Trans. Wireless Commun., 5 (2006), pp. 1241-1249.
53. L. Chen and H. Y. H. Chuang, A fast algorithm for Euclidean distance maps of a 2-d binary image, Inform. Process. Lett., 51 (1994), pp. 25-29.
54. Y. Chen, Y. Dong, D. Lu, and Y. Pan, The multi-fractal nature of worm and normal traffic at individual source level, in Intelligence and Security Informatics, Lecture Notes in Comput. Sci. 3495, Springer, Berlin, Heidelberg, 2005, pp. 505-510.
55. Y. T. Chin, H. Wang, L. P. Tay, H. Wang, and W. Y. C. Soh, Vision guided AGV using distance transform, in Proceedings of the 32nd International Symposium on Robotics (ISR), International Federation of Robotics, Seoul, Korea, 2001, pp. 1416-1421.
56. F. H. Clarke, R. J. Stern, and P. R. Wolenski, Proximal smoothness and the lower-C2 property, J. Convex Anal., 2 (1995), pp. 117-144.
57. G. Cohen, S. Gaubert, and J.-P. Quadrat, Duality and separation theorems in idempotent semimodules, Linear Algebra Appl., 379 (2004), pp. 395-422.
58. G. Cohen, S. Gaubert, and J.-P. Quadrat, Projection and aggregation in maxplus algebra, in Current Trends in Nonlinear Systems and Control, L. Menini, L. Zaccarian, and C. J. Abdallah, eds., Systems Control Found. Appl., Birkháuser Boston, Boston, MA, 2006, pp. 443-454.
59. G. Cohen, S. Gaubert, J.-P. Quadrat, and I. Singer, Max-plus convex sets and functions, in Idempotent Mathematics and Mathematical Physics, G. L. Litvin and V. P. Maslov, eds., Contemp. Math. 377, AMS, Providence, RI, 2005, pp. 105-129.
60. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168-1200.
61. L. Corrias, Fast Legendre-Fenchel transform and applications to Hamilton-Jacobi equations and conservation laws, SIAM J. Numer. Anal., 33 (1996), pp. 1534-1558.
62. D. Crandall, P. F. Felzenszwalb, and D. P. Huttenlocher, Object recognition by combining appearance and geometry, in Toward Category-Level Object Recognition, Lecture Notes in Comput. Sci. 4170, Springer, Berlin, 2006, pp. 462-482.
63. O. Cuisenaire, Distance Transformations: Fast Algorithms and Applications to Medical Image Processing, Ph.D. thesis, Universitê Catholique de Louvain, Louvain-la-Neuve, Belgium, 1999.
64. P.-E. Danielsson, Euclidean distance mapping, Comput. Graph. Image Process., 14 (1980), pp. 227-248.
65. J. L. Davidson and F. Hummer, Morphology neural networks: An introduction with applications, Circuits Systems Signal Process., 12 (1993), pp. 177-210.
66. P. Del Moral and M. Doisy, Maslov idempotent probability calculus. I, Teor. Veroyatnost. i Primenen., 43 (1998), pp. 735-751.
67. P. Del Moral and M. Doisy, Maslov idempotent probability calculus. II, Teor. Veroyatnost. i Primenen., 44 (1999), pp. 384-400.
68. L. Deniau, Proposition d'un Opérateur Géométrique Pour l'Analyse et l'Identification de Signaux et Images, Ph.D. thesis, Universitê de Paris-Sud, Centre d'Orsay, France, 1997.
69. L. Deniau and J. Blanc-Talon, Fractal Analysis with Hausdorff Distance Under Affine Transformations, technical report, ETCA-CREA-SP, Arcueil, France, 1995.
70. L. Dorst and R. van den Boomgaard, Morphological signal processing and the slope transform, Signal Process., 38 (1994), pp. 79-98.
71. K. R. Driessel and A. Gerisch, Zero-preserving iso-spectral flows based on parallel sums, Linear Algebra Appl., 421 (2007), pp. 69-84.
72. M. V. Droogenbroeckb and H. Talbot, Fast computation of morphological operations with arbitrary structuring elements, Pattern Recognition Lett., 17 (1996), pp. 1451-1460.
73. J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming, Math. Oper. Res., 18 (1993), pp. 202-226.
74. H. Edelsbrunner, Algorithms in Combinatorial Geometry, EATCS Monogr. Theoret. Comput. Sci., Springer, Berlin, 1987.
75. R. Ellaia and J.-B. Hiriart-Urruty, The conjugate of the difference of convex functions, J. Optim. Theory Appl., 49 (1986), pp. 493-498.
76. S.-L. Eriksson-Bique and H. Leutwiler, A generalization of parallel addition, Aequationes Math., 38 (1989), pp. 99-110.
77. S.-L. Eriksson-Bique and H. Leutwiler, Minimal operators from a potential-theoretic viewpoint, in ICPT '91 International Conference on Potential Theory (Amersfoort, 1991), Kluwer Academic, Dordrecht, The Netherlands, 1994, pp. 47-59.
78. R. Fabbri, O. M. Bruno, J. C. Torelli, and L. da F. Costa, 2D Euclidean distance transform algorithms: A comparative survey, ACM Comput. Surv., 40 (2007), pp. 2:1-2:44.
79. M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), pp. 559-575.
80. P. F. Felzenszwalb and D. P. Huttenlocher, Distance Transforms of Sampled Functions, Technical report TR2004-1963, Cornell Computing and Information Science, Cornell University, Ithaca, NY, 2004.
81. P. F. Felzenszwalb and D. P. Huttenlocher, Pictorial structures for object recognition, Int. J. Comput. Vision, 61 (2005), pp. 55-79.
82. P. F. Felzenszwalb and D. P. Huttenlocher, Efficient belief propagation for early vision, Int. J. Comput. Vision, 70 (2006), pp. 41-54.
83. M. Fidler and S. Recker, Conjugate network calculus: A dual approach applying the Legendre transform, Comput. Networks, 50 (2006), pp. 1026-1039.
84. J. A. Fill and D. E. Fishkind, The Moore-Penrose generalized inverse for sums of matrices, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 629-635.
85. Fraclab: A Fractal Analysis Toolbox for Signal and Image Processing, http://fraclab.saclay. inria.fr/.
86. U. Frisch and J. Bec, Burgulence, in Les Houches 2000: New Trends in Turbulence, A. M. Lesieur and E. F. David, eds., Springer/EDP Sciences, Berlin, 2001, pp. 341-383.
87. U. Frisch, J. Bec, and B. Villone, Singularities and the distribution of density in the Burgers/adhesion model, Phys. D, 152-153 (2001), pp. 620-635.
88. B. Gardiner and Y. Lucet, Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis, technical report, University of British Columbia, 2010.
89. B. Gardiner and Y. Lucet, Graph-Matrix Calculus for Computational Convex Analysis, technical report, University of British Columbia, 2010.
90. M. Gavrilova and M. H. Alsuwaiyel, Two Algorithms for Computing the Euclidean Distance Transform, Technical report 2000-661-13, Computer Science Technical Reports, University of Calgary, Calgary, 2000.
91. A. Gheondea and S. Gudder, Sequential product of quantum effects, Proc. Amer. Math. Soc., 132 (2004), pp. 503-512.
92. N. Ghoussoub, A theory of anti-selfdual Lagrangians: Stationary case, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 245-250. (Pubitemid 40176127)
93. N. Ghoussoub, Antisymmetric Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions, Comm. Pure Appl. Math., 60 (2007), pp. 619-653.
94. N. Ghoussoub and A. Moameni, Selfdual variational principles for periodic solutions of Hamiltonian and other dynamical systems, Comm. Partial Differential Equations, 32 (2007), pp. 771-795.
95. J. Gil and R. Kimmel, Efficient dilation, erosion, opening, and closing algorithms, IEEE Trans. Pattern Anal. Mach. Intelligence, 24 (2002), pp. 1606-1617.
96. K. Głazek, A Guide to the Literature on Semirings and Their Applications in Mathematics and Information Sciences, Kluwer Academic, Dordrecht, The Netherlands, 2002.
97. R. Goebel, Self-dual smoothing of convex and saddle functions, J. Convex Anal., 15 (2008), pp. 179-190.
98. J. S. Golan, Power Algebras Over Semirings: With Applications in Mathematics and Computer Science, Math. Appl. 488, Kluwer Academic, Dordrecht, The Netherlands, 1999, pp. 1606-1617.
99. J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999.
100. J. S. Golan, Some recent applications of semiring theory, in Proceedings of the International Conference on Algebra in Memory of Kostia Beidar, National Cheng Kung University, Tainan, Taiwan, 2005.
101. O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), pp. 403-419.
102. J. Gunawardena, An introduction to idempotency, in Idempotency (Bristol, 1994), Publ. Newton Inst. 11, Cambridge University Press, Cambridge, UK, 1998, pp. 1-49.
103. J. Gunawardena, From max-plus algebra to nonexpansive mappings: A nonlinear theory for discrete event systems, Theoret. Comput. Sci., 293 (2003), pp. 141-167.
104. S. N. Gurbatov, S. I. Simdyankin, E. Aurell, U. Frisch, and G. Tóth, On the decay of Burgers turbulence, J. Fluid Mech., 344 (1997), pp. 339-374.
105. S. N. Gurbatov and A. V. Troussov, The decay of multiscale signals-deterministic model of the Burgers turbulence, Phys. D, 145 (2000), pp. 47-64.
106. A. Hamdi, A Moreau-Yosida regularization of a difference of two convex functions, Appl. Math. E-Notes, 5 (2005), pp. 164-170.
107. W. Hare and C. Sagastizábal, Benchmark of some nonsmooth optimization solvers for computing nonconvex proximal points, Pac. J. Optim., 2 (2006), pp. 545-573.
108. H. J. Heijmans and P. Maragos, Lattice calculus of the morphological slope transform, Signal Process., 59 (1997), pp. 17-42.
109. H. J. Heijmans and I. S. Molchanov, Morphology on convolution lattices with applications to the slope transform and random set theory, J. Math. Imaging Vision, 8 (1998), pp. 199-214.
110. P. Helluy and N. Seguin, Relaxation models of phase transition flows, M2AN Math. Model. Numer. Anal., 40 (2006), pp. 331-352.
111. B. J. Hendrey and R. A. Jarvis, Robot manipulator path planning, in Proceedings of TENCON '92, IEEE, Melbourne, Australia, 1992, pp. 865-870.
112. B. J. Hendrey and R. A. Jarvis, A path planning heuristic for robotic manipulators, in Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, Yokohama, Japan, 1993, pp. 1417-1423.
113. W. H. Hesselink, A linear-time algorithm for Euclidean feature transform sets, Inform. Process. Lett., 102 (2007), pp. 181-186.
114. W. Hinterberger and O. Scherzer, Variational methods on the space of functions of bounded Hessian for convexification and denoising, Computing, 76 (2006), pp. 109-133.
115. J.-B. Hiriart-Urruty, Extension of Lipschitz functions, J. Math. Anal. Appl., 77 (1980), pp. 539-554.
116. J.-B. Hiriart-Urruty, Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand., 47 (1980), pp. 123-134.
117. J.-B. Hiriart-Urruty, The deconvolution operation in convex analysis: An introduction, Kibernet. Sistem. Anal., 190 (1994), pp. 97-104.
118. J.-B. Hiriart-Urruty and C. Lemarêchal, Convex Analysis and Minimization Algorithms, Vol. I: Fundamentals, Vol. II: Advanced Theory and Bundle Methods, Grundlehren Math. Wiss. 305-306, Springer, Berlin, 1993.
119. J.-B. Hiriart-Urruty and Y. Lucet, Parametric computation of the Legendre-Fenchel conjugate, J. Convex Anal., 14 (2007), pp. 657-666.
120. J.-B. Hiriart-Urruty and M.-L. Mazure, Formulations variationnelles de l'addition parallèle et de la soustraction parallèlle d'opêrateurs semi-dêfinis positifs, C. R. Acad. Sci. Paris Sêr. I Math., 302 (1986), pp. 527-530.
121. T. Hisakado, K. Okumura, V. Vukadinovic, and L. Trajkovic, Characterization of a simple communication network using Legendre transform, in Proceedings of the IEEE International Symposium on Circuits and Systems, Piscataway, NJ, 2003, pp. 738-741.
122. S. H. Luo, X. G. Liu, and J. S, Zeng, Identification of multi-fractal characteristics of silicon content in blast furnace hot metal, ISIJ Int., 8 (2007), pp. 1102-1107.
123. C. Imbert, Analyse Non Lisse: Fonction d'Appui de la Jacobienne Gênêralisêe de Clarke et de son Enveloppe Plêniè re-Quelques Applications aux ? Equations de Hamilton-Jacobi du Premier Ordre (Fonctions de Hopf-Lax, Hamiltoniens Diff. Convexes, Solutions Sci), Ph.D. thesis, Laboratoire de Mathêmatiques pour l'Industrie et la Physique, Universitê Paul Sabatier-Toulouse III, France, 2000.
124. C. Imbert, Convex analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations, J. Nonlinear Convex Anal., 2 (2001), pp. 333-343.
125. A. N. Iusem, B. F. Svaiter, and M. Teboulle, Entropy-like proximal methods in convex programming, Math. Oper. Res., 19 (1994), pp. 790-814.
126. R. Jarvis, Distance transform based path planning for robot navigation, in Recent Trends in Mobile Robots, Robot. Automat. Syst. 11, Y. Zheng, ed., World Scientific, Singapore, 1993, pp. 3-31.
127. R. Jarvis, An articulated six wheel drive robot for very rough terrain navigation, in Proceedings of the 2002 Australian Conference on Robotics and Automation, Auckland, Australian Robotics and Automation Association, 2002, pp. 133-139.
128. R. Jarvis, Distance transform based visibility measures for covert path planning in known but dynamic environments, in Proceedings of the 2nd International Conference on Autonomous Robots and Agents, Massey University, Palmerston North, New Zealand, 2004, pp. 396-400.
129. R. A. Jarvis, Collision-free trajectory planning using distance transforms, Trans. Inst. Eng. Aust. Mech. Eng., ME10 (1985), pp. 187-191.
130. R. A. Jarvis, Collision-free path planning in time-varying environments, in Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Systems '89, Tsukuba, Japan, 1989, pp. 99-106.
131. J. Johnstone, V. Koch, and Y. Lucet, Convexity of the Proximal Average, technical report, University of British Columbia, 2009.
132. F. Kadhi and A. Trad, Characterization and approximation of the convex envelope of a function, J. Optim. Theory Appl., 110 (2001), pp. 457-466.
133. C.-Y. Kao, S. Osher, and Y.-H. Tsai, Fast sweeping methods for static Hamilton-Jacobi equations, SIAM J. Numer. Anal., 42 (2005), pp. 2612-2632.
134. L. Karimovaa, Y. Kuandykova, N. Makarenkoa, M. Novakb, and S. Helamac, Fractal and topological dynamics for the analysis of paleoclimatic records, Phys. A, 373 (2007), pp. 737-746.
135. W. Karush, A general algorithm for the optimal distribution of effort, Management Sci., 9 (1962), pp. 50-72.
136. C. M. Klein, Conjugate duality and its implications in dynamic programming, Math. Comput. Modelling, 14 (1990), pp. 151-154.
137. C. M. Klein and T. L. Morin, Conjugate duality and the curse of dimensionality, European J. Oper. Res., 50 (1991), pp. 220-228.
138. B. Koopen, Contact of Bodies in 2D-space: Implementing the Discrete Legendre Transform, Master's thesis, Intelligent Autonomous Systems Group, University of Amsterdam, Amsterdam, 2002.
139. D. Kozinska, O. J. Tretiak, J. Nissanov, and C. Ozturk, Multidimensional alignment using the Euclidean distance transform, Graph. Models Image Process., 59 (1997), pp. 373-387.
140. F. R. Kschischang, B. Frey, and H.-A. Loeliger, Factor graphs and the sum-product algorithm, IEEE Trans. Inform. Theory, 47 (2001), pp. 498-519.
141. F. Kubo, Conditional expectations and operations derived from network connections, J.Math. Anal. Appl., 80 (1981), pp. 477-489.
142. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1979/80), pp. 205-224.
143. T. Lachand-Robert and É . Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), pp. 368-379.
144. D. Lambert, J.-P. Crouzeix, V. H. Nguyen, and J.-J. Strodiot, Finite convex integration, J. Convex Anal., 11 (2004), pp. 131-146.
145. J.-M. Lasry and P.-L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), pp. 257-266.
146. M. S. K. Lau and C. P. Kwong, A smoothing method of global optimization that preserves global minima, J. Global Optim., 34 (2006), pp. 369-398.
147. J.-Y. Le Boudec and P. Thiran, Network Calculus, A Theory of Deterministic Queuing Systems for the Internet, Lecture Notes in Comput. Sci. 2050, Springer, Berlin, 2001.
148. B. Legras, B. Joseph, and F. Lefèvre, Vertical diffusivity in the lower stratosphere from Lagrangian back-trajectory reconstructions of ozone profiles, J. Geophys. Research, 108 (2003), pp. 148-227.
149. B. Legras, I. Pisso, G. Berthet, and F. Lefèvre, Variability of the Lagrangian turbulent diffusion in the lower stratosphere, Atmos. Chem. Phys., 5 (2005), pp. 1605-1622.
150. B. Lemaire, The proximal algorithm, in New Methods in Optimization and Their Industrial Uses (Pau/Paris, 1987), Internat. Schriftenreihe Numer. Math. 87, Birkháuser, Basel, 1989, pp. 73-87.
151. C. Lemarêchal and F. Oustry, Growth conditions and U-Lagrangians, Set-Valued Anal., 9 (2001), pp. 123-129.
152. C. Lemarêchal, F. Oustry, and C. Sagastizábal, The U-Lagrangian of a convex function, Trans. Amer. Math. Soc., 352 (2000), pp. 711-729.
153. C. Lemarêchal and C. Sagastizábal, More than first-order developments of convex functions: Primal-dual relations, J. Convex Anal., 3 (1996), pp. 255-268.
154. C. Lemarêchal and C. Sagastizábal, Practical aspects of the Moreau-Yosida regularization: Theoretical preliminaries, SIAM J. Optim., 7 (1997), pp. 367-385.
155. C. Lemarêchal and C. Sagastizábal, Variable metric bundle methods: From conceptual to implementable forms, Math. Program., 76 (1997), pp. 393-410.
156. S. Letovsky and S. Kasif, Predicting protein function from protein/protein interaction data: A probabilistic approach, Bioinform., 19 Suppl 1 (2003), pp. 197-204.
157. J. Lêvy Vêhel and B. Sikdar, A multiplicative multifractal model for TCP traffic, in Proceedings of the 6th IEEE Symposium on Computers and Communications (ISCC), Hammamet, Tunisa, 2001, p. 714.
158. G. L. Litvinov, Maslov dequantization, idempotent and tropical mathematics: A brief introduction, J. Math. Sci. (N.Y.), 140 (2007), pp. 426-444.
159. P. Lotito, E. Mancinelli, and J.-P. Quadrat, Traffic assignment and Gibbs-Maslov semirings, in Idempotent Mathematics and Mathematical Physics, Contemp. Math. 377, AMS, Providence, RI, 2005, pp. 209-219.
160. P. A. Lotito, E. M. Mancinelli, and J.-P. Quadrat, A min-plus derivation of the fundamental car-traffic law, IEEE Trans. Automat. Control, 50 (2005), pp. 699-705.
161. Y. Lucet, A fast computational algorithm for the Legendre-Fenchel transform, Comput. Optim. Appl., 6 (1996), pp. 27-57.
162. Y. Lucet, Faster than the fast Legendre transform, the linear-time Legendre transform, Numer. Algorithms, 16 (1997), pp. 171-185.
163. Y. Lucet, A linear Euclidean distance transform algorithm based on the linear-time Legendre Transform, in Proceedings of the Second Canadian Conference on Computer and Robot Vision (CRV 2005), Victoria, BC, IEEE Computer Society Press, 2005, pp. 262-267.
164. Y. Lucet, Fast Moreau envelope computation I: Numerical algorithms, Numer. Algorithms, 43 (2006), pp. 235-249.
165. Y. Lucet, New sequential exact Euclidean distance transform algorithms based on convex analysis, Image Vision Comput., 27 (2006), pp. 37-44.
166. Y. Lucet, H. H. Bauschke, and M. Trienis, The piecewise linear-quadratic model for computational convex analysis, Comput. Optim. Appl., 43 (2006), pp. 95-118.
167. P. Maragos, Morphological systems: Slope transforms and max-min difference and differential equations, Signal Process., 38 (1994), pp. 57-77.
168. P. Maragos, Slope transforms: Theory and application to nonlinear signal processing, IEEE Trans. Signal Process., 43 (1995), pp. 864-877.
169. P. Maragos, Differential morphology and image processing, IEEE Trans. Image Process., 5 (1996), pp. 922-927.
170. P. Maragos, Differential morphology, in Nonlinear Image Processing, Communications, Networking, and Multimedia, S. K. Mitra and G. L. Sicuranza, eds., Academic Press, San Diego, CA, 2001, pp. 289-329.
171. P. Maragos, PDEs for morphological scale-spaces and Eikonal applications, in Handbook of Image and Video Processing, 2nd ed., Elsevier, London, 2005, pp. 587-612.
172. P. Maragos and M. A. Butt, Curve evolution, differential morphology, and distance transforms applied to multiscale and Eikonal problems, Fund. Inform., 41 (2000), pp. 91-129.
173. M. Marzouqi and R. A. Jarvis, Covert robotics: Hiding in known environments, in Proceedings of the 2004 IEEE Conference on Robotics, Automation and Mechatronics, Singapore, 2004, pp. 804-809.
174. C. R. Maurer. Jr., R. Qi, and V. V. Raghavan, A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions, IEEE Trans. Pattern Anal. Mach. Intelligence, 25 (2003), pp. 265-270.
175. M.-L. Mazure, L'addition parallèle d'opêrateurs interprêtêe comme inf-convolution de formes quadratiques convexes, RAIRO Modêl. Math. Anal. Numêr., 20 (1986), pp. 497-515.
176. M.-L. Mazure, La soustraction parallèle d'opérateurs interprétée comme déconvolution de formes quadratiques convexes, Optimization, 18 (1987), pp. 465-484.
177. M.-L. Mazure, Shorted operators through convex analysis, in Trends in Mathematical Optimization (Irsee, 1986), Internat. Schriftenreihe Numer. Math. 84, Birkháuser, Basel, 1988, pp. 139-152.
178. M.-L. Mazure, Équations de convolution et formes quadratiques, Ann. Mat. Pura Appl. (4), 158 (1991), pp. 75-97.
179. M.-L. Mazure and M. Volle, Équations inf-convolutives et conjugaison de Moreau-Fenchel, Ann. Fac. Sci. Toulouse Math. (5), 12 (1991), pp. 103-126.
180. W. M. McEneaney, Combining Legendre/Fenchel transformed operators on max-plus spaces for nonlinear control solution, in Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, pp. 1152-1157.
181. K. McKinnon and M. Mongeau, A generic global optimization algorithm for the chemical and phase equilibrium problem, J. Global Optim., 12 (1998), pp. 325-351.
182. F. Meng, D. Sun, and G. Zhao, Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization, Math. Program., 104 (2005), pp. 561-581.
183. F. Meng and G. Zhao, On second-order properties of the Moreau-Yosida regularization for constrained nonsmooth convex programs, Numer. Funct. Anal. Optim., 25 (2004), pp. 515-529.
184. F. W. Meng and Y. Hao, Piecewise smoothness for Moreau-Yosida approximation to a piecewise C2 convex function, Adv. Math. (China), 30 (2001), pp. 354-358.
185. R. J. Meziat, D. P. No, and P. Pedregal, An alternative approach for non-linear optimal control problems based on the method of moments, Comput. Optim. Appl., 38 (2007), pp. 147-171.
186. R. J. Meziat and J. Villalobos, Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization, Struct. Multidiscip. Optim., 32 (2006), pp. 507-519.
187. R. Mifflin, A quasi-second-order proximal bundle algorithm, Math. Program., 73 (1996), pp. 51-72.
188. R. Mifflin, L. Qi, and D. Sun, Properties of the Moreau-Yosida regularization of a piecewise C2 convex function, Math. Program., 84 (1999), pp. 269-281.
189. R. Mifflin and C. Sagastizábal, VU-decomposition derivatives for convex max-functions, in Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Econom. and Math. Systems 477, Springer, Berlin, 1999, pp. 167-186.
190. R. Mifflin and C. Sagastizábal, Functions with primal-dual gradient structure and UHessians, in Nonlinear Optimization and Related Topics (Erice, 1998), Appl. Optim. 36, Kluwer Academic, Dordrecht, The Netherlands, 2000, pp. 219-233.
191. R. Mifflin and C. Sagastizábal, On VU-theory for functions with primal-dual gradient structure, SIAM J. Optim., 11 (2000), pp. 547-571.
192. R. Mifflin and C. Sagastizábal, Primal-dual gradient structured functions: Second-order results; links to epi-derivatives and partly smooth functions, SIAM J. Optim., 13 (2003), pp. 1174-1194.
193. R. Mifflin and C. Sagastizábal, On the relation between U-Hessians and second-order epi-derivatives, European J. Oper. Res., 157 (2004), pp. 28-38.
194. R. Mifflin and C. Sagastizábal, VU-smoothness and proximal point results for some nonconvex functions, Optim. Methods Softw., 19 (2004), pp. 463-478.
195. R. Mifflin and C. Sagastizábal, Relating U-Lagrangians to second-order epi-derivatives and proximal-tracks, J. Convex Anal., 12 (2005), pp. 81-93.
196. R. Mifflin and C. Sagastizábal, A VU-algorithm for convex minimization, Math. Program., 104 (2005), pp. 583-608.
197. S. K. Mitra and K. Manjunatha Prasad, The regular shorted matrix and the hybrid sum, Adv. Appl. Math., 18 (1997), pp. 403-422.
198. J.-J. Moreau, Inf-convolution des fonctions numériques sur un
199. J.-J. Moreau, Propriétés des applications "prox," C. R. Acad. Sci. Paris, 256 (1963), pp. 1069-1071.
200. J.-J. Moreau, Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273-299.
201. J.-J. Moreau, Convexity and duality, in Functional Analysis and Optimization, E. R. Caianello, ed., Academic Press, New York, 1966, pp. 145-169.
202. J.-J. Moreau, Inf-convolution, sous-additivité, convexité des fonctions numériques, J.Math. Pures Appl. (9), 49 (1970), pp. 109-154.
203. T. L. Morin and R. E. Marsten, An algorithm for nonlinear knapsack problems, Management Sci., 22 (1976), pp. 1147-1158.
204. T. D. Morley, Parallel summation, Maxwell's principle and the infimum of projections, J. Math. Anal. Appl., 70 (1979), pp. 33-41.
205. N. Nikopoulos and I. Pitas, A fast implementation of 3-D binary morphological transformations, IEEE Trans. Image Process., 9 (2000), pp. 283-286.
206. A. Noullez, S. N. Gurbatov, E. Aurell, and S. I. Simdyankin, The global picture of self-similar and not self-similar decay in Burgers turbulence, Phys. Rev., E71 (2005), article 056305.
207. A. Noullez and M. Vergassola, A fast Legendre transform algorithm and applications to the adhesion model, J. Sci. Comput., 9 (1994), pp. 259-281.
208. A. M. Oberman, Computing the convex envelope using a nonlinear partial differential equation, Math. Models Methods Appl. Sci., 18 (2008), pp. 759-780.
209. A. M. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), pp. 1689-1694.
210. È. L. Pekarev and J. L. Šmuljan, Parallel addition and parallel subtraction of operators, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), pp. 366-387, 470.
211. J.-P. Penot, Proximal mappings, J. Approx. Theory, 94 (1998), pp. 203-221.
212. J.-P. Penot and R. Ratsimahalo, On the Yosida approximation of operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 945-966.
213. R. Poliquin and R. T. Rockafellar, Generalized Hessian properties of regularized nonsmooth functions, SIAM J. Optim., 6 (1996), pp. 1121-1137.
214. R. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), pp. 1805-1838.
215. F. P. Preparata and M. I. Shamos, Computational Geometry, 3rd ed., Texts Monogr. Comput. Sci., Springer, New York, 1990.
216. S. Prohaska and H. C. Hege, Fast visualization of plane-like structures in voxel data, inVIS '02: Proceedings of the Conference on Visualization '02 (Boston, MA), IEEE Computer Society, 2002, pp. 29-36.
217. L. Qi, Second-order analysis of the Moreau-Yosida regularization, in Nonlinear Analysis and Convex Analysis (Niigata, 1998), World Scientific, River Edge, NJ, 1999, pp. 16-25.
218. P. J. Rabier and A. Griewank, Generic aspects of convexification with applications to thermodynamic equilibrium, Arch. Ration. Mech. Anal., 118 (1992), pp. 349-397.
219. A. G. Ramm, A. M. Steinberg, and A. I. Zaslavsky, Stable calculation of the Legendre transform, J. Math. Anal. Appl., 178 (1993), pp. 592-602.
220. A. G. Ramm and A. I. Zaslavsky, Reconstructing singularities of a function from its Radon transform, Math. Comput. Modelling, 18 (1993), pp. 109-138.
221. A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc. (N.S.), 28 (1993), pp. 109-115.
222. A. G. Ramm and A. I. Zaslavsky, X-ray transform, the Legendre transform, and envelopes, J. Math. Anal. Appl., 183 (1994), pp. 528-546.
223. J. P. Revalski and M. Thêra, Enlargements and sums of monotone operators, Nonlinear Anal., 48 (2002), pp. 505-519.
224. R. H. Riedi, Multifractal processes, in Theory and Applications of Long-Range Dependence, P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds., Birkháuser Boston, Boston, MA, 2003, pp. 625-716.
225. R. H. Riedi, M. S. Crouse, V. J. Ribeiro, and R. G. Baraniuk, A multifractal wavelet model with application to network traffic, IEEE Trans. Inform. Theory, 45 (1999), pp. 992-1018.
226. R. H. Riedi and J. L. Vhel, Multifractal Properties of TCP Traffic: A Numerical Study, Technical report 3129, INRIA, Rocquencourt, France, 1997.
227. R. M. Rifkin and R. A. Lippert, Value regularization and Fenchel duality, J. Mach. Learn. Res., 8 (2007), pp. 441-479.
228. G. Ritter and P. Sussner, An introduction to morphological neural networks, in Proceedings of the International Conference on Pattern Recognition (ICPR '96), Washington, DC, IEEE Computer Society Press, 1996, pp. 709-717.
229. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
230. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
231. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.
232. R. T. Rockafellar and P. R. Wolenski, Convexity in Hamilton-Jacobi theory II: Envelope representations, SIAM J. Control Optim., 39 (2000), pp. 1351-1372.
233. A. Rosenfeld and J. L. Pfaltz, Sequential operations in digital picture processing, J. ACM, 13 (1966), pp. 471-494.
234. Scilab, http://www.scilab.org, 1994.
235. J. A. Sethian, Fast marching methods, SIAM Rev., 41 (1999), pp. 199-235.
236. J. A. Sethian, Level Set Methods and Fast Marching Methods, Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 2nd ed., Cambridge Monogr. Appl. Comput. Math. 3, Cambridge University Press, Cambridge, UK, 1999.
237. Z.-S. She, E. Aurell, and U. Frisch, The inviscid Burgers equation with initial data of Brownian type, Comm. Math. Phys., 148 (1992), pp. 623-641.
238. B. Shi, B. Vidakovic, and G. G. Katul, Quantifying the Effects of Atmospheric Stability on the Multifractal Spectrum of Turbulence, technical report, Georgia Institute of Technology, School of Industrial and Systems Engineering, Statistics Group, Atlanta, GA, 2004.
239. F. Y. Shih and Y.-T. Wu, Fast Euclidean distance transformation in two scans using a 3 ? 3 neighborhood, Comput. Vis. Image Underst., 93 (2004), pp. 195-205.
240. K. Siddiqi, S. Bouix, A. Tannenbaum, and S. W. Zucker, The Hamilton-Jacobi skeleton, in Proceedings of the Seventh IEEE International Conference on Computer Vision, Kerkyra, Greece, 1999, p. 828.
241. K. Siddiqi, S. Bouix, A. Tannenbaum, and S. W. Zucker, Hamilton-Jacobi skeletons, Internat. J. Comput. Vision, 48 (2002), pp. 215-231.
242. T. Strómberg, Representation formulae for infimal convolution with applications, in Analysis, Algebra, and Computers in Mathematical Research (Luleå, 1992), Lecture Notes in Pure and Appl. Math. 156, Dekker, NY, 1994, pp. 319-334.
243. T. Strómberg, The operation of infimal convolution, Dissertationes Math. (Rozprawy Mat.), 352 (1996), pp. 1-61.
244. S. Sundar and Z. Shiller, Optimal obstacle avoidance based on the Hamilton-Jacobi-Bellman equation, IEEE Trans. Robot. Automat., 13 (1997), pp. 305-310.
245. K. W. Tang and R. A. Jarvis, A fast algorithm to plan a collision-free path in cluttered 2D environments, in Proceedings of the 2004 IEEE Conference on Robotics, Automation and Mechatronics, Singapore, 2004, pp. 786-791.
246. M. Teboulle, Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res., 17 (1992), pp. 670-690.
247. Y. Tian, How to express a parallel sum of k matrices, J. Math. Anal. Appl., 266 (2002), pp. 333-341.
248. A. N. Tikhonov and V. A. Morozov, Methods for the regularization of ill-posed problems, Vychisl. Metody i Programmirovanie, 35 (1981), pp. 3-34.
249. J. Toriwaki and K. Mori, Distance transformation and skeletonization of 3D pictures and their applications to medical images, in Digital and Image Geometry: Advanced Lectures, Lecture Notes in Comput. Sci. 2243, G. Bertrand, A. Imiya, and R. Klette, eds., Springer, Heidelberg, 2001, pp. 412-429.
250. M. Trienis, Computational Convex Analysis: From Continuous Deformation to Finite Convex Integration, Master's thesis, University of British Columbia, Vancouver, 2007.
251. Y.-H. R. Tsai and S. Osher, Total variation and level set methods in image science, Acta Numer., 14 (2005), pp. 509-573.
252. P. Tseng and Z.-Q. Luo, On computing the nested sums and infimal convolutions of convex piecewise-linear functions, J. Algorithms, 21 (1996), pp. 240-266.
253. A. V. Tuzikov, G. L. Margolin, and A. I. Grenov, Convex set symmetry measurement via Minkowski addition, J. Math. Imaging Vision, 7 (1997), pp. 53-68.
254. M. Vergassola, B. Dubrulle, U. Frisch, and A. Noullez, Burgers' equation, devil's staircases and the mass distribution function for large-scale structures, Astronom. Astrophys., 289 (1994), pp. 325-356.
255. L. A. Vese, A method to convexify functions via curve evolution, Comm. Partial Differential Equations, 24 (1999), pp. 1573-1591.
256. C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2008.
257. L. Vincent, Morphological transformations of binary images with arbitrary structuring elements, Signal Process., 22 (1991), pp. 3-23.
258. L. Vincent, Morphological algorithms, E. Dougherty, ed., Mathematical Morphology in Image Processing, Marcel Dekker, New York, 1992, pp. 255-288.
259. E . T. P. Wong and R. Jarvis, Real time obstacle detection and navigation planning for a humanoid robot in an indoor environment, in Proceedings of the 2004 IEEE Conference on Robotics, Automation and Mechatronics, Singapore, 2004, pp. 693-699.
260. K. Yosida, Functional Analysis, Springer Classics Math., Springer, Berlin, 1995.
261. A. I. Zaslavsky, Legendre transforms and dual varieties, J. Math. Anal. Appl., 183 (1994), pp. 670-684.
262. A. Zelinsky, A mobile robot exploration algorithm, IEEE Trans. Robot. Automat., 8 (1992), pp. 707-717.
263. A. Zelinsky, A navigation algorithm for industrial mobile robots, in IEEE International Workshop on Emerging Technologies and Factory Automation, 1992, pp. 525-530.
264. A. Zelinsky, Using path transforms to guide the search for findpath in 2D, Internat. J. Robot. Res., 13 (1994), pp. 315-325.