Shock capturing, level sets, and PDE based methods in computer vision and image processing: A review of Osher's contributions

Journal of Computational Physics

Volumn 185, Issue 2, 2003, Pages 309-341

  Fedkiw, Ronald P ^a   Sapiro, Guillermo ^b   Shu, Chi Wang ^c  

^a Stanford University   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

^c BROWN UNIVERSITY   (United States)

Author keywords

Computer vision; Image processing; Level set method; Shock capturing method

Indexed keywords

COMPUTER VISION; DROP BREAKUP; IMAGE DENOISING; LEVEL MEASUREMENT; OPTICAL DATA PROCESSING; VARIATIONAL TECHNIQUES;

ALGORITHMS DEVELOPMENT; DEVELOPMENT AND APPLICATIONS; HIGH RESOLUTION; IMAGES PROCESSING; LEVEL SET; LEVEL SET METHOD; OSHER SCHEME; SHOCK-CAPTURING; SHOCK-CAPTURING METHOD; VISION PROCESSING;

NUMERICAL METHODS;

EID: 0037337788     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(02)00016-5     Document Type: Article

References (199)

1. D. Adalsteinsson, J. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys. 118 (1995) 269-277.
2. L. Alvarez, F. Guichard, P.L. Lions, J.M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. 123 (1993) 199-257.
3. L. Alvarez, P.L. Lions, J.M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29 (1992) 845-866.
4. F. Andreu, C. Ballester, V. Caselles, J.M. Mazon, The Dirichlet problem for the total variation flow, J. Funct. Anal. 180 (2001) 347-403.
5. F. Andreu, C. Ballester, V. Caselles, J.M. Mazon, Minimizing total variation flow, Differential Integral Equations 14 (2001). 321-360.
6. D. Balsara, C.-W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys. 160 (2000) 405-452.
7. M. Bardi, S. Osher, The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 22 (1991) 344-351.
8. G. Barles, C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal. 32 (1995) 484-500.
9. M. Berger, A. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82 (1989) 64-84.
10. M. Bertalmio, L.T. Cheng, S. Osher, G. Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2001) 759-780.
11. M. Black, G. Sapiro, D. Marimont, D. Heeger, Robust anisotropic diffusion, IEEE Trans. Image Process. 7 (3) (1998) 421-432.
12. J. Bloomenthal, Introduction to Implicit Surfaces, Morgan Kaufmann, San Francisco, California, 1997.
13. J.P. Boris, D.L. Book, Flux corrected transport I, SHASTA, a fluid transport algorithm that works, J. Comput. Phys. 11 (1973) 38-69.
14. Y. Brenier, S. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988) 8-23.
15. A.M. Bruckstein, On shape from shading, Comput. Vision Graph. Image Process. 44 (1988) 139-154.
16. P. Burchard, L.-T. Cheng, B. Merriman, S. Osher, Motion of curves in three spatial dimensions using a level set approach, J. Comput. Phys. 165 (2001) 463-502.
17. B. Cabral, C. Leedom, Imaging vector fields using line integral convolution, ACM Comput. Graph. (SIGGRAPH '93) 27 (4) (1993) 263-272.
18. R. Caiden, R. Fedkiw, C. Anderson, A numerical method for two-phase flow consisting of separate compressible and in compressible regions, J. Comput. Phys. 166 (2001) 1-27.
19. V. Caselles, F. Catte, T. Coll, F. Dibos, A geometric model for active contours, Numer. Math. 66 (1993) 1-31.
20. V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, Int. J. Comput. Vision 22 (1) (1997) 61-79.
21. V. Caselles, R. Kimmel, G. Sapiro, C. Sbert, Minimal surfaces based object segmentation, IEEE-PAMI 19 (4) (1997) 394-398.
22. V. Caselles, J.M. Morel, G. Sapiro, A. Tannenbaum (Eds.), Special Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Processing and Analysis, IEEE Trans. Image Process. 7 (1998).
23. C. Cercignani, I. Gamba, J. Jerome, C.-W. Shu, Device benchmark comparisons via kinetic, hydrodynamic, and high-field models, Comput. Methods Appl. Mech. Eng. 181 (2000) 381-392.
24. S. Chakravarthy, S. Osher, Numerical experiments with the Osher upwind scheme for the Euler equations, AIAA J. 21 (1983) 1241-1248.
25. T.F. Chan, G.H. Golub, P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput. 20 (1999) 1964-1977.
26. T. Chan, L. Vese, Active contour model without edges, in: M. Neilsen, P. Johansen, O.F. Olson, J. Weickert (Eds.), Lecture Notes in C.S., vol. 1687, Springer-Verlag, Berlin/New York, 1999, p. 141.
27. T. Chan, L. Vese, A level set algorithm for minimizing the Mumford-Shah functional in image processing, in: Proceedings of the 1st IEEE Workshop on "Variational and Level Set Methods in Computer Vision", 2001, pp. 161-168.
28. T. Chan, C.K. Wong, Total variation blind deconvolution, IEEE Trans. Image Process. 7 (1998) 370-375.
29. Y. Chang, T. Hou, B. Merriman, S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys. 124 (1996) 449-464.
30. S. Chen, B. Merriman, M. Kang, R. Caflisch, C. Ratsch, M. Guyre, R. Fedkiw, C. Anderson, S. Osher, A level set method for thin film epitaxial growth, J. Comput. Phys. 167 (2001) 475-500.
31. S. Chen, B. Merriman, S. Osher, P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys. 135 (1997) 8-29.
32. Y.G. Chen, Y. Giga, S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991) 749-786.
33. L.-T. Cheng, P. Burchard, B. Merriman, S. Osher, Motion of curves constrained on surfaces using a level set approach, J. Comput. Phys. 175 (2002) 604-644.
34. D. Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106 (1993) 77-91.
35. A. Cohen, W. Dahmen, I. Daubechies, R. DeVore, Harmonic analysis of the space BV, Acta Math. (2000), submitted.
36. P. Colella, A. Majda, V. Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Comput. 7 (1986) 1059-1080.
37. M. Crandall, P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) 1-42.
38. M. Crandall, P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984) 1-19.
39. M. Crandall, A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comput. 34 (1980) 1-21.
40. U. Diewald, T. Preufer, M. Rumpf, Anisotropic diffusion in vector field visualization on Euclidean domains and surfaces, IEEE Trans. Visual. Comput. Graph. 6 (2000) 139-149.
41. J. Dorsey, P. Hanrahan, Digital materials and virtual weathering, Sci. Amer. 282 (2) (2000) 46-53.
42. L.J. Durlofsky, B. Engquist, S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comput. Phys. 98 (1992) 64-73.
43. M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, W. Stuetzle, Multi-resolution analysis of arbitrary meshes, Comput. Graph. (SIGGRAPH '95 Proc.) (1995) 173-182.
44. M. Eck, H. Hoppe, Automatic reconstruction of B-spline surfaces of arbitrary topological type, Comput. Graph. (1996).
45. J. Eells, L. Lemarie, A report on harmonic maps, Bull. London Math. Soc. 10 (1) (1978) 1-68.
46. B. Engquist, E. Fatemi, S. Osher, Numerical solution of the high frequency asymptotic expansion for hyperbolic equations, in: Proc. 10th Ann. Review of Progress in Applied Comp. Electromagnetics, Monterey, CA, 1994, pp. 32-45.
47. B. Engquist, S. Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980) 45-57.
48. B. Engquist, S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981) 321-351.
49. B. Engquist, S. Osher, S. Zhong, Fast wavelet based algorithms for linear evolution equations, SIAM J. Sci. Statist. Comput. 15 (4) (1994) 755-775.
50. B. Engquist, O. Runborg, A.-K. Tornberg, High frequency wave propagation by the segment projection method, J. Comput. Phys. 178 (2002) 373-390.
51. D. Enright, R. Fedkiw, J. Ferziger, I. Mitchell, A hybrid particle level set method for improved interface capturing, J. Comput. Phys. 183 (2002) 83-116.
52. D. Enright, S. Marschner, R. Fedkiw, Animation and rendering of complex water surfaces, in: Siggraph 2002 Annual Conference, ACM TOG, 21, 2002, pp. 736-744.
53. L.C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J. 42 (1993) 553-557.
54. L.C. Evans, J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom. 33 (1991) 635-681.
55. E. Fatemi, B. Engquist, S. Osher, Numerical solution of the high frequency asymptotic expansion of the scalar wave equation, J. Comput. Phys. 120 (1995) 145-155.
56. E. Fatemi, B. Engquist, S. Osher, Finite difference methods for the nonlinear equations of perturbed geometric optics, ACES J. 11 (1996) 90-98.
57. E. Fatemi, J. Jerome, S. Osher, Solution of the hydrodynamic device model using high order nonoscillatory shock capturing algorithms, IEEE Trans. Comput.-Aided Design Integrated Circ. Syst. 10 (1991) 232-243.
58. O. Faugeras, F. Clément, R. Deriche, R. Keriven, T. Papadopoulo, J. Gomes, G. Hermosillo, P. Kornprobst, D. Lingrad, J. Roberts, T. Viéville, F. Devernay, The inverse EEG and MEG problems: the adjoint state approach I - The continuous case, INRIA Research Report 3673, June 1999.
59. R. Fedkiw, Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method, J. Comput. Phys. 175 (2002) 200-224.
60. R. Fedkiw, T. Aslam, B. Merriman, S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys. 152 (1999) 457-492.
61. R. Fedkiw, T. Aslam, S. Xu, The ghost fluid method for deflagration and detonation discontinuities, J. Comput. Phys. 154 (1999) 393-427.
62. R. Fedkiw, B. Merriman, S. Osher, High accuracy numerical methods for thermally perfect gas flows with chemistry, J. Comput. Phys. 132 (1997) 175-190.
63. R. Fedkiw, B. Merriman, S. Osher, Efficient characteristic projection in upwind difference schemes for hyperbolic systems; the complementary projection method, J. Comput. Phys. 141 (1998) 22-36.
64. N. Foster, R. Fedkiw, Practical animation of liquids, in: Siggraph 2001 Annual Conference, 2001, pp. 15-22.
65. S.F. Frisken, R.N. Perry, A. Rockwood, T. Jones, Adaptively sampled fields: A general representation of shape for computer graphics, in: Computer Graphics (SIGGRAPH), New Orleans, July 2000.
66. D. Gabor, Information theory in electron microscopy, Lab. Invest. 14 (1965) 801-807.
67. Y. Giga, M.-H. Sato, A level set approach to semicontinuous solutions for Cauchy problems, Commun. Partial Differential Equations 26 (2001) 813-839.
68. J. Glimm, L. Xia, Y. Liu, N. Zhao, Conservative front tracking and level set algorithms, PNAS 98 (2001) 14198-14201.
69. S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability preserving high order time discretization methods, SIAM Rev. 43 (2001) 89-112.
70. M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987) 285-314.
71. M. Green, S. Osher, Steiner polynomials, Wulff flows and some new isoperimetric inequalities for convex plane curves, Asian J. Math. 3 (1999) 659-676.
72. F. Guichard, J.M. Morel, Introduction to Partial Differential Equations in Image Processing, Tutorial Notes, IEEE Int. Conf. Image Proc., Washington, DC, October 1995.
73. B. Gustafsson, H.-O. Kreiss, A. Sundstrom, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972) 649-686.
74. M. Gyure, C. Ratsch, B. Merriman, R. Caflisch, S. Osher, J. Zinck, D. Vvedensky, Level set methods for the simulation of epitaxial phenomena, Phys. Rev. E. 58 (1998) 6927-6930.
75. E. Harabetian, S. Osher, Regularization of ill-posed problems via the level set approach, SIAM J. Appl. Math. 58 (1998) 1689-1706.
76. E. Harabetian, S. Osher, C.-W. Shu, An Eulerian approach for vortex motion using a level set regularization procedure, J. Comput. Phys. 127 (1996) 15-26.
77. A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983) 357-393.
78. A. Harten, B. Engquist, S. Osher, S. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231-303.
79. A. Harten, S. Osher, Uniformly high-order accurate non-oscillatory schemes, I, SIAM J. Numer. Anal. 24 (1987) 279-304.
80. A. Harten, S. Osher, B. Engquist, S. Chakravarthy, Some results on uniformly high order accurate essentially non-oscillatory schemes, Appl. Numer. Math. 2 (1986) 347-377.
81. J. Helmsen, E. Puckett, P. Colella, M. Dorr, Two new methods for simulating photolithography development in 3D, Proc. SPIE 2726 (1996) 253-261.
82. C. Hu, C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys. 150 (1999) 97-127.
83. G. Huiskamp, Difference formulas for the surface Laplacian on a triangulated surface, J. Comput. Phys. 95 (1991) 477-496.
84. R.A. Hummel, Representations based on zero-crossings in scale-space, Proc. IEEE CVPR (1986) 204-209.
85. H. Ishii, A generalization of Bence, Merriman, and Osher algorithm for motion by mean curvature, in: A. Damlamian, J. Spruck, A. Visintin (Eds.), Curvature Flows and Related Topics, Gakkôtosho, Tokyo, 1995, pp. 111-127.
86. H. Ishii, G.E. Pires, P.E. Souganidis, Threshold dynamics type schemes for propagating fronts, J. Math. Soc. Jpn. 51 (2) (1999) 267-308.
87. A.K. Jain, Partial differential equations and finite-difference methods in image processing, Part 1: Image representation, J. Optim. Theory Appl. 23 (1977) 65-91.
88. J. Jerome, C.-W. Shu, Transport effects and characteristic modes in the modeling and simulation of submicron devices, IEEE Trans. Comput.-Aided Design Integrated Circ. Syst. 14 (1995) 917-923.
89. G.-S. Jiang, D. Peng, Weighted ENO schemes for Hamilton Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126-2143.
90. G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202-228.
91. M. Kang, R. Fedkiw, X.-D. Liu, A boundary condition capturing method for multiphase incompressible flow, J. Sci. Comput. 15 (2000) 323-360.
92. S. Karni, Multicomponent flow calculations by a consistent primitive algorithm, J. Comput. Phys. 112 (1994) 31-43.
93. M. Kass, A. Witkin, D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vision 1 (1988) 321-331.
94. S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, A. Yezzi, Conformal curvature flows: from phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996) 275-301.
95. Y.-T. Kim, N. Goldenfeld, J. Dantzig, Computation of dendritic microstructures using a level set method, Phys. Rev. E 62 (2000) 2471-2474.
96. B.B. Kimia, A. Tannenbaum, S.W. Zucker, Toward a computational theory of shape: an overview, in: Lecture Notes in Computer Science, 427, Springer-Verlag, New York, 1990, pp. 402-407.
97. R. Kimmel, Numerical geometry of images: theory, algorithms, and applications, Technion CIS Report 9910, October 1999.
98. R. Kimmel, A.M. Bruckstein, Tracking level sets by level sets: a method for solving the shape from shading problem, CVIU 62 (1) (1995) 47-58.
99. J.J. Koenderink, The structure of images, Biol. Cybernet. 50 (1984) 363-370.
100. H.-O. Kreiss, Difference approximations for the initial-boundary value problem for hyperbolic differential equations, in: D. Greenspan (Ed.), Numerical Solutions of Nonlinear Differential Equations, Wiley, New York, 1966, pp. 141-166.
101. F. Lafon, S. Osher, High order two dimensional nonoscillatory methods for solving Hamilton-Jacobi equations, J. Comput. Phys. 123 (1996) 235-253.
102. R. Lagnado, S. Osher, Reconciling differences, Risk Mag. 10 (1997) 79-83.
103. R. Lagnado, S. Osher, A technique for calibrating derivative security pricing models, numerical solution of an inverse problem, J. Comput. Finance 1 (1997) 13-25.
104. T. Lindeberg, Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, Dordrecht, 1994.
105. X.-D. Liu, R. Fedkiw, M. Kang, A boundary condition capturing method for Poisson's equation on irregular domains, J. Comput. Phys. 160 (2000) 151-178.
106. X.-D. Liu, S. Osher, Nonoscillatory high order accurate self similar maximum principle satisfying shock capturing schemes, SIAM J. Numer. Anal. 33 (1996) 760-779.
107. X.-D. Liu, S. Osher, Convex ENO high-order multidimensional schemes without field by field projection or staggered grids, J. Comput. Phys. 142 (1998) 304-330.
108. X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994) 200-212.
109. A. Majda, J. McDonough, S. Osher, The Fourier method for nonsmooth initial data, Math. Comp. 32 (1978) 1041-1081.
110. A. Majda, S. Osher, Reflections of singularities at the boundary, Commun. Pure Appl. Math. 28 (1975) 479-499.
111. A. Majda, S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Commun. Pure Appl. Math. 28 (1975) 607-675.
112. A. Majda, S. Osher, Propagation of error into regions of smoothness for accurate difference approximations to hyperbolic equations, Commun. Pure Appl. Math. 30 (1977) 671-705.
113. A. Majda, S. Osher, A systematic approach for correcting nonlinear instabilities: the Lax-Wendroff scheme for scalar conservation laws, Numer. Math. 30 (1978) 429-452.
114. R. Malladi, J.A. Sethian, B.C. Vemuri, Evolutionary fronts for topology independent shape modeling and recovery, in: Proc. of the 3rd ECCV, Stockholm, Sweden, 1994, pp. 3-13.
115. R. Malladi, J.A. Sethian, B.C. Vemuri, Shape modeling with front propagation: a level set approach, IEEE Trans. PAMI 17 (1995) 158-175.
116. A. Marquina, S. Osher, Explicit algorithms for a new time dependent model based on level set motion for nonlinear debluring and noise removal, SIAM J. Sci. Comput. 22 (2000) 387-405.
117. T. McInerney, D. Terzopoulos, Topologically adaptable snakes, in: Proc. ICCV, Cambridge, MA, June 1995.
118. B. Merriman, J. Bence, S. Osher, Diffusion generated motion by mean curvature, in: J.E. Taylor (Ed.), Computational Crystal Growers Workshop, American Mathematical Society, Providence, RI, 1992, pp. 73-83.
119. B. Merriman, J. Bence, S. Osher, Motion of multiple junctions: a level-set approach, J. Comput. Phys. 112 (1994) 334-363.
120. B. Merriman, R. Caflisch, S. Osher, Level set methods with an application to modeling the growth of thin films, in: I. Athanasopoulos, G. Makreakis, J. Rodrigues (Eds.), Free Boundary Value Problems, Theory and Applications, CRC Press, Boca Raton, 1999, pp. 51-70.
121. W. Mulder, S. Osher, J. Sethian, Computing interface motion in compressible gas dynamics, J. Comput. Phys. 100 (1992) 209-228.
122. D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Commun. Pure Appl. Math. 42 (1989) 577-685.
123. D. Nguyen, R. Fedkiw, H. Jensen, Physically based modeling and animation of fire, in: Siggraph 2002 Annual Conference, ACM TOG, 21, 2002, pp. 721-728.
124. D. Nguyen, R. Fedkiw, M. Kang, A boundary condition capturing method for incompressible flame discontinuities, J. Comput. Phys. 172 (2001) 71-98.
125. D. Nguyen, F. Gibou, R. Fedkiw, A fully conservative ghost fluid method and stiff detonation waves, 12th International Detonation Symposium, San Diego, CA, 2002.
126. J. Oliensis, P. Dupuis, Direct method for reconstructing shape from shading, in: Proceedings SPIE Conf. 1570 on Geometric Methods in Computer Vision, 1991, pp. 116-128.
127. S. Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc. 137 (1969) 177-201.
128. S. Osher, Stability of parabolic difference approximations to certain mixed initial boundary value problems, Math. Comp. 26 (1972) 13-39.
129. S. Osher, Numerical solution of singular perturbation problems and one-sided difference schemes, North-Holland Math. Stud. 47 (1981).
130. S. Osher, The Riemann problem for nonconvex scalar conservation laws and the Hamilton-Jacobi equations, Proc. Amer. Math. Soc. 89 (1983) 641-646.
131. S. Osher, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal. 21 (1984) 217-235.
132. S. Osher, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985) 947-961.
133. S. Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM J. Anal. 24 (1993) 1145-1152.
134. S. Osher, S. Chakravarthy, Upwind schemes and boundary conditions with applications to Euler equations in general geometries, J. Comput. Phys. 50 (1983) 447-481.
135. S. Osher, S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984) 955-984.
136. S. Osher, S. Chakravarthy, Very high order accurate TVD schemes, in: IMA Volumes in Mathematics and Its Applications, 2, Springer-Verlag, New York, 1986, pp. 229-274.
137. S. Osher, L.-T. Cheng, M. Kang, H. Shim, Y.-H. Tsai, Geometric optics in a phase space and Eulerian framework, J. Comput. Phys., to appear.
138. S. Osher, R. Fedkiw, Level set methods: an overview and some recent results, J. Comput. Phys. 169 (2001) 463-502.
139. S. Osher, R. Fedkiw, Level sets and dynamic implicit surfaces, Springer-Verlag, New York, 2002.
140. S. Osher, M. Hafez, W. Whitlow Jr., Entropy condition satisfying approximations for the full potential equation of transonic flow, Math. Comp. 44 (1985) 1-29.
141. S. Osher, B. Merriman, The Wulff shape as the asymptotic limit of a growing crystalline interface, Asian J. Math. 1 (1997) 560-571.
142. S. Osher, R.T. Rudin, Feature-oriented image enhancement using shock filters, SIAM J. Numer. Anal. 27 (1990) 919-940.
143. S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp. 41 (1983) 321-336.
144. S. Osher, F. Santosa, Level set method for optimization problems involving geometry and constraints, I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys. 171 (2001) 277-288.
145. S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.
146. S. Osher, C.-W. Shu, High order essentially non-oscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28 (1991) 902-921.
147. S. Osher, F. Solomon, Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982) 339-374.
148. S. Osher, E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988) 19-51.
149. N. Paragios, R. Deriche, A PDE-based level-set approach for detection and tracking of moving objects, in: Proc. Int. Conf. Comp. Vision '98, Bombay, India, January 1998.
150. D. Peng, B. Merriman, S. Osher, H.-K. Zhao, M. Kang, A PDE-based fast local level set method, J. Comput. Phys. 155 (1999) 410-438.
151. D. Peng, S. Osher, B. Merriman, H.-K. Zhao, The geometry of Wulff crystal shapes and its relations with Riemann problems, in: G.Q. Chen, E. DeBenedetto (Eds.), Contemp. Math., vol. 238, American Mathematical Society, Providence, RI, 1999, p. 251.
152. P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell. 12 (1990) 629-639.
153. E. Praun, A. Finkelstein, H. Hoppe, Lapped textures, ACM Computer Graphics (SIGGRAPH), New Orleans, July 2000.
154. C.B. Price, P. Wambacq, A. Oosterlink, Image enhancement and analysis with reaction-diffusion paradigm, IEE Proc. 137 (1990) 136-145.
155. B. Romeny (Ed.), Geometry Driven Diffusion in Computer Vision, Kluwer Academic Publishers, Dordrecht, 1994.
156. E. Rouy, A. Tourin, A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal. 29 (1992) 867-884.
157. L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259-268.
158. S. Ruuth, B. Merriman, S. Osher, Convolution generated motion as a link between cellular automata and continuum pattern dynamics, J. Comput. Phys. 151 (1999) 836-861.
159. S. Ruuth, B. Merriman, S. Osher, A fixed grid method for capturing the motion of self-intersecting interfaces and related PDEs, J. Comput. Phys. 163 (2000) 1-21.
160. S. Ruuth, B. Merriman, J. Xin, S. Osher, A diffusion generated approach to the curvature motion of filaments, J. Nonlinear Sci. 61 (2001) 473-494.
161. G. Sapiro, Geometric Partial Differential Equations and Image Processing, Cambridge University Press, New York, 2001.
162. J. Serra, Image Analysis and Mathematical Morphology, vol. 2: Theoretical Advances, Academic Press, New York, 1988.
163. J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Sciences, Cambridge University Press, Cambridge, UK, 1996.
164. J.A. Sethian, A fast marching method for three dimensional photolithography development, in: Proc. SPIE, 2726, 1996, p. 261.
165. J. Shah, A common framework for curve evolution, segmentation, and anisotropic diffusion, in: Proc. CVPR, San Francisco, June 1996.
166. J. Shi, C. Hu, C.-W. Shu, A technique of treating negative weights in WENO schemes, J. Comput. Phys. 175 (2002) 108-127.
167. C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor (Editor: A. Quarteroni), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325-432.
168. C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77 (1988) 439-471.
169. C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys. 83 (1989) 32-78.
170. C.-W. Shu, T.A. Zang, G. Erlebacher, D. Whitaker, S. Osher, High-order ENO schemes applied to two- and three-dimensional compressible flow, Appl. Numer. Math. 9 (1992) 45-71.
171. L. Simon, Lectures on Geometric Measure Theory, Australian National University, Australia, 1984.
172. J. Steinhoff, M. Fang, L. Wang, A new Eulerian method for the computation of propagating short acoustic and electromagnetic pulses, J. Comput. Phys. 157 (2000) 683-706.
173. M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985) 558-581.
174. M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146-159.
175. B. Tang, G. Sapiro, V. Caselles, Diffusion of general data on non-flat manifolds via harmonic maps theory: the direction diffusion case, Int. J. Comput. Vision 36 (2) (2000) 149-161.
176. G. Taubin, Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation, IEEE Trans. PAMI 13 (11) (1991) 1115-1138.
177. H. Tek, B.B. Kimia, Image segmentation by reaction-diffusion bubbles, in: Proc. ICCV'95, Cambridge, June, 1995, pp. 156-162.
178. A.W. Toga, Brain Warping, Academic Press, New York, 1998.
179. V.T. Ton, A.R. Karagozian, S. Osher, B. Engquist, Numerical simulation of high speed chemically reactive flow, Theoret. Comput. Fluid Dyn. 6 (1994) 161-179.
180. Y.-H. Tsai, Y. Giga, S. Osher, A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations, Math. Comp., to appear.
181. J. Tsitsiklis, Efficient algorithms for globally optimal trajectories, in: Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, LF, December, 1994, pp. 1368-1373.
182. J. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control 40 (1995) 1528-1538.
183. A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. B 237 (1952) 37-72.
184. G. Turk, Generating textures on arbitrary surfaces using reaction-diffusion, Comput. Graph. 25 (4) (1991) 289-298.
185. B. van Leer, Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method, J. Comput. Phys. 32 (1979) 101-136.
186. J. Varah, Stability of difference approximations to the mixed initial boundary value problem for parabolic systems, SIAM J. Numer. Anal. 8 (1971) 598-615.
187. L.A. Vese, T.F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comp. Vision 50 (3) (2002) 271-293.
188. J. Weickert, Anisotropic Diffusion in Image Processing, ECMI Series, Teubner, Stuttgart, Germany, 1998.
189. R.T. Whitaker, Algorithms for implicit deformable models, in: Proc. ICCV'95, Cambridge, June, 1995, pp. 822-827.
190. G. Winkenbach, D.H. Salesin, Rendering parametric surfaces in pen and ink, Comput. Graph. (SIGGRAPH 96) (1996) 469-476.
191. A.P. Witkin, Scale-space filtering, Int. Joint Conf. Artif. Intell. 2 (1983) 1019-1021.
192. A. Witkin, P. Heckbert, Using particles to sample and control implicit surfaces, Comput. Graph. (SIGGRAPH) (1994) 269-278.
193. A. Witkin, M. Kass, Reaction-diffusion textures, Comput. Graph. (SIGGRAPH) 25 (4) (1991) 299-308.
194. A. Yezzi, S. Soatto, Stereoscopic segmentation, in: IEEE Intl. Conf. on Computer Vision, Vancouver, July, 2001, pp. 59-66.
195. Y.-T. Zhang, C.-W. Shu, High order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM J. Sci. Comput., to appear.
196. H.-K. Zhao, T. Chan, B. Merriman, S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys. 127 (1996) 179-195.
197. H.-K. Zhao, B. Merriman, S. Osher, C. Wang, Capturing the behavior of bubbles and drops using the variational level set approach, J. Comput. Phys. 143 (1998) 495.
198. H.-K. Zhao, S. Osher, R. Fedkiw, Fast surface reconstruction using the level set method, in: 1st IEEE Workshop on Variational and Level Set Methods, in conjunction with the 8th International Conference on Computer Vision (ICCV), Vancouver, Canada, 2001, pp. 194-202.
199. H.-K. Zhao, S. Osher, B. Merriman, M. Kang, Implicit nonparametric shape reconstruction from unorganized points using a variational level set method, Comput. Vision Image Understanding 80 (2000) 295-314.