A Survey in Mathematics for Industry A survey on level set methods for inverse problems and optimal design

European Journal of Applied Mathematics

Volumn 16, Issue 2, 2005, Pages 263-301

  Burger, Martin ^a   Osher, Stanley J ^b  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23844524180     PISSN: 09567925     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0956792505006182     Document Type: Review

References (152)

1. ACAR, R. & VOGEL, C. R. (1994) Analysis of total variation penalty methods. Inverse Problems, 10, 1217-1229.
2. ADALSTEINSSON, B. & SETHIAN, J. A. (1999) The fast construction of extension velocities in level set methods. J. Comp. Phys. 148, 2-22.
3. ALESSANDRINI, G. & RONDI, L. (2001) Optimal stability for the inverse problem of multiple cavities. J. Diff. Eq. 176, 356-386.
4. ALESSANDRINI, G., MORASSI, A. & ROSSET, E. (2002) Detecting an inclusion in an elastic body by boundary measurements. SIAM J. Math. Anal. 33, 1247-1268.
5. ALEXANDROV, O. (2003) Optimal design of microstructured optical fibers. Preprint, University Minnesota.
6. ALEXANDROV, O. & SANTOSA, F. (2003) A topology preserving level set method. Preprint, University Minnesota.
7. ALLAIRE, G. (2002) Shape Optimization by the Homogenization Method. Springer, New York.
8. ALLAIRE, G., JOUVE, F. & TOADER, A. M. (2002) A level-set method for shape optimization. C.R. Acad. Sci. Paris, Ser. I, 334, 1125-1130.
9. ALLAIRE, G., JOUVE, F. & TOADER, A. M. (2004) Structural optimization using sensitivity analysis and a level-set method. J. Comp. Phys. 194, 363-393.
10. AMBROSIO, L. & MASNOU, S. (2003) A direct variational approach to a problem arising in image reconstruction. Interfaces & Free Boundaries, 5, 63-81.
11. BABUSKA, I. & MELENK, J. M. (1997) The partition of unity method. Int. J. Numer. Meth. Eng. 40, 727-758.
12. BARLES, G., SONER, H. M. & SOUGANIDIS, P. E. (1993) Front propagation and phase field theory. SIAM J. Control Optim. 31, 439-469.
13. BENABDA, A., BENAMEUR, H. & JAOUA, M. (1999) Identification of 2D cracks by elastic boundary measurements. Inverse Problems, 15, 67-90.
14. BENAMEUR, H., BURGER, M. & HACKL, B. (2004) Level set methods for geometric inverse problems in linear elasticity. Inverse Problems, 20, 673-697.
15. BENAMEUR, H., BURGER, M. & HACKL, B. (2003) On some geometric inverse problems in linear elasticity. CAM-Report 03-55, UCLA.
16. BENDSØE, M. P. & SIGMUND, O. (2002) Topology Optimization. Springer, Berlin.
17. BERTALMIO, M., CHENG, L. T., OSHER, S. & SAPIRO, G. (2001) Variational problems and partial differential equations on implicit surfaces: The framework and examples in image processing and pattern formation. J. Comput. Phys. 174, 759-780.
18. BERTALMIO, M., MEMOLI, F., CHENG, L. T., SAPIRO, G. & OSHER, S. (2003) Variational problems and partial differential equations on implicit surfaces: Bye bye triangulated surfaces?. In: S. Osher and N. Paragios, editors, Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 381-398. Springer, New York.
19. BOURDIN, B. & CHAMBOLLE, A. (2003) Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9, 19-48.
20. BOURDIN, B. & BURGER, M. (2005) Phase-field methods for inverse obstacle problems. Preparation.
21. BRAIDES, A. (2002) Gamma-Convergence for Beginners. Oxford University Press.
22. BREZZI, F. & FORTIN, F. (1991) Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, 15.
23. BRÜHL, M., HANKE, M. & PIDCOCK, M. (2001) Crack detection using electrostatic measurements. M2AN Math. Model. Numer. Anal. 35, 595-605.
24. BURGER, M. (2001) A level set method for inverse problems. Inverse Problems, 17, 1327-1356.
25. BURGER, M. (2003) A framework for the construction of level-set methods for shape optimization and reconstruction. Interfaces & Free Boundaries, 5, 301-329.
26. BURGER, M. (2004) Levenberg-Marquardt level set methods for inverse obstacle problems. Inverse Problems, 20, 259-282.
27. BURGER, M. (2005) Elliptic partial differential equations on implicit curves and surfaces: Variational formulation and finite element approximation. In preparation.
28. BURGER, M., HACKL, B. & RING, W. (2004) Incorporating topological derivatives into level set methods. J. Comp. Phys. 194, 344-362.
29. BURGER, M. & MÜHLHUBER, W. (2002) Iterative regularization of parameter identification problems by SQP-methods. Inverse Problems, 18, 943-970.
30. BURGER, M. & MÜHLHUBER, W. (2002) Numerical approximation of an SQP-type method for parameter identification problems. SIAM J. Numer. Anal. 40, 1775-1797.
31. BURGER, M., OSHER, S. & YABLONOVITCH, E. (2004) Inverse problem techniques for the design of photonic crystals. IEICE Trans. Electr. 87, 258-265.
32. CARTER, W. C., TAYLOR, J. E. & CAHN, J. W. (1997) Variational methods for microstructural evolution. JOM, 49, 30-36.
33. CAPUZZO-DOLCETTA, I., VITA, S. F. & MARCH, R. (2002) Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Boundaries, 4, 325-343.
34. CHAMBOLLE, A. & LIONS, P. L. (1997) Image recovery via total variational minimization and related problems. Numer. Math. 76, 167-188.
35. CHAN, T. F., HEIMSUND, B. O., NILSSEN, T. K. & TAI, X. C. (2003) Level set methods and augmented Lagrangian for parameter identification. In: V. Barbu, I. Lasiecka, D. Tiba and C. Varsan, editors, Analysis and Optimization of Differential Systems, pp. 189-200. Kluwer, Boston.
36. CHAN, T. F., KANG, S. & SHEN, J. (2002) Euler's elastica and curvature based inpaintings. SIAM J. Appl. Math. 63, 564-592.
37. CHAN, T. F. & TAI, X. C. (2004) Level set and total variation regularization For elliptic inverse problems with discontinuous coefficients. J. Comp. Phys. 193, 40-66.
38. CHAN, T. F. & VESE, L. A. (2001) A level set algorithm for minimizing the Mumford-Shah functional in image processing. In: N. Paragios, editor, Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 161-168. IEEE/Computer Vision Society.
39. CHAN, T. F. & VESE, L. A. (2001) Active contours without edges. IEEE Trans. Image Process. 10, 266-277.
40. COLTON, D. & KIRSCH, A. (1996) A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12, 383-393.
41. COX, S. J. & DOBSON, D. C. (1999) Maximizing band gaps in two-dimensional photonic crystals. SIAM J. Appl. Math. 59, 2108-2120.
42. CRANDALL, M., ISHII, H. & LIONS, P. L. (1992) User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1-67.
43. DAL MASO, G. (1993) An Introduction to Gamma-Convergence. Birkhäuser.
44. DELFOUR, M. C. & ZOLÉSIO, J. P. (2001) Shapes and geometries. Analysis, differential calculus, and optimization. SIAM, Philadelphia, dans les problèmes d'identification de domaine. C.R. Acad. Sci. Paris, serie I, 1761-1764.
45. CHUNG, E. T., CHAN, T. F. & TAI, X. C. (2003) Electrical impedance tomography using level set representation and total variational regularization, CAM-Report 03-64, UCLA.
46. DECKELNICK, K. & DZIUK, G. (2002) A fully discrete numerical scheme for weighted mean curvature flow. Numer. Math. 91, 423-452.
47. DORN, O. (2000) Shape reconstruction in 2D from limited-view multifrequency electromagnetic data. Preprint.
48. DORN, O., MILLER, E. M. & RAPPAPORT, C. M. (2000) A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Problems, 16, 1119-1156.
49. DORN, O., MILLER, E. M. & RAPPAPORT, C. M. (2001) Shape reconstruction in 2D from limited-view multifrequency electromagnetic data. Radon Transform and Tomography, (AMS series Contemporary Mathematics, 278), 97-122.
50. DROSKE, M., MEYER, B., RUMPF, M. & SCHALLER, C. (2001) An adaptive level set method for medical image segmentation. In: R. Leahy and M. Insana, editors, Proceedings of the Annual Symposium on Information Processing in Medical Imaging, pp. 416-422. Springer.
51. DROSKE, M. & RUMPF, M. (2004) A variational approach to non-rigid morphological registration. SIAM Appl. Math. 64, 668-687.
52. DROSKE, M. & RUMPF, M. (2004) A level set formulation for Willmore flow. Interfaces & Free Boundaries, 6, to appear.
53. ENGL, H. W., HANKE, M. & NEUBAUER, A. (1996) Regularization of Inverse Problems. Kluwer, Dordrecht.
54. ENGQUIST, B., TORNBERG, A. K. & TSAI, R. (2004) Discretization of Dirac delta functions in level set methods. CAM-Report 04-16, UCLA.
55. FANG, W. & ITO, K. (2003) Identification of contact regions in semiconductor transistors by level-set methods. J. Comp. Appl. Math. 159, 399-410.
56. FAUGERAS, O., CLÉMENT, F., DERICHE, R., KERIVEN, R., PAPADOPOULO, T., ROBERTS, J., VIÉVILLE, T., DEVERNAY, F., GOMES, J., HERMOSILLO, G., KORNPROBST, P. & LINGRAND, D. (1999) The inverse EEG and MEG problems, the adjoint state approach, I: The continuous case. Research Report RR-3673, INRIA.
57. FEDERER, H. (1969) Geometric Measure Theory. Springer.
58. FREMIOT, G. & SOKOLOWSKI, K. (2000) The structure theorem for the Eulerian derivative of shape functional defined in domains with cracks. Siberian Math. J. 41, 1181-1202.
59. FRÜHAUF, F., LEITAO, A. & SCHERZER, O. (2003) Analysis of regularization methods for the solution of ill-posed problems involving unbounded operators and a relation to constraint optimization. Preprint, University Innsbruck.
60. GIGA, Y. & SATO, M. H. (2001) A level set approach to semicontinuous viscosity solutions for Cauchy problems. Comm. Partial Diff. Eq. 26, 813-839.
61. GRIESMAIER, R. (2004) Level Set Methoden und Distanzmaβe in der Elektrischen Impedanztomographie. Master Thesis, University Innsbruck.
62. GURTIN, M. E. (1995) The nature of configurational forces. Arch. Rat. Mech. Anal. 131, 67-100.
63. GURTIN, M. E. & JABBOUR, M. E. (2002) Interface evolution in three-dimensions with curvature-dependent energy and surface diffusion: Interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Rat. Mech. Anal. 163, 171-208.
64. HABER, E. (2004) A multilevel, level-set method for optimizing eigenvalues in shape design problems. J. Comp. Phys. 198, 518-538.
65. HABER, E. & ASCHER, U. (2001) Preconditioned all-at-once methods for large, sparse parameter estimation problems. Inverse Problems, 17, 1847-1864.
66. HABER, E., ASCHER, U. & OLDENBURG, O. (2000) On optimization techniques for solving nonlinear inverse problems. Inverse Problems, 16, 1263-1280.
67. HARTEN, A., ENGQUIST, B., OSHER, S. & CHAKRAVARTHY, S. (1987) Uniformly high order accurate essentially non-oscillatory schemes III. J. Comp. Phys. 71, 231-303.
68. HETTLICH, F. & RUNDELL, W. (1996) Iterative methods for the reconstraction of an inverse potential problems. Inverse Problems, 12, 251-266.
69. HETTLICH, F. & RUNDELL, W. (1997) Recovery of the support of source term in elliptic differential equation. Inverse Problems, 13, 959-976.
70. HETTLICH, F. & RUNDELL, W. (1998) The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems, 14, 67-82.
71. HINTERMÜLLER, M. & RING, W. (2003) A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64, 442-467.
72. HINTERMÜLLER, M. & RING, W. (2004) An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imag. Vision, 20, 19-42.
73. HINTERMÜLLER, M. & RING, W. (2004) A level set approach for the solution of a state-constrained optimal control problem. Numer. Math. 98, 135-166.
74. HINTERMÜLLER, M. & RING, W. (2003) Numerical aspects of a level set based algorithm for state constrained optimal control problems. Comput. Assisted Mech. & Eng. Sci. 10, 149-161.
75. HO, S., KIM, Y., BULLITT, E. & GERIG, G. (2002) Medical image segmentation by 3-D level set evolution. Technical Report, UNC Chapel Hill.
76. HOU, S., SOLNA, K. & ZHAO, H. (2004) Imaging of location and geometry of extended targets using the response matrix. J. Comp. Phys. 199, 317-338.
77. HU, C. & SHU, C. W. (1999) A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J. Sci. Comp. 21, 666-690.
78. HUANG, H. & LI, Z. (1999) Convergence analysis of the immersed interface method. IMA J. Numer. Anal. 19, 583-608.
79. ITO, K., KUNISCH, K. & LI, Z. (2001) Level-set function approach to an inverse interface problem. Inverse Problems, 17, 1225-1242.
80. JIANG, G. S. & PENG, D. (2000) Weighted ENO-schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126-2143.
81. JIANG, G. S. & SHU, C. W. (1996) Efficient implementation of weighted ENO schemes. J. Comp. Phys. 126, 202-228.
82. 1 spline implicitization of planar curves. In: F. Winkler, editor, Automated Deduction in Geometry. Springer.
83. KINDERMANN, S. (2003) A new iterative regularization method using an equation of Hamilton-Jacobi type. SFB Report 2003-17, SFB F 013, University Linz.
84. KINDERMANN, S. & NEUBAUER, A. (2001) Regularization for surface representations of discontinuous solutions of linear ill-posed problems. Numer. Funct. Anal. Optim. 22, 79-105.
85. KINDERMANN, S. & NEUBAUER, A. (2001) Estimation of discontinuous parameters of elliptic partial differential equations by regularization for surface representations. Inverse Problems, 17, 789-803.
86. KIRSCH, A. & RITTER, S. (2000) A linear sampling method for inverse scattering from an open arc. Inverse Problems, 16, 89-105.
87. LEITAO, A. & SCHERZER, O. (2003) On the relation between constraint regularization, level sets, and shape optimization. Inverse Problems, 19, L1-L11.
88. LEVEQUE, R. & LI, Z. (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019-1044.
89. LI, Z. (1998) A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35(1), 230-254.
90. LIONS, J. L. & MAGENES, E. (1972) Non-Homogeneous Boundary Value Problems and Applications. Springer.
91. LIONS, P. L. (1992) Generalized solutions of Hamilton-Jacobi equations. Pitman.
92. LITMAN, A., LESSELIER, D. & SANTOSA, F. (1998) Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Problems, 14, 685-706.
93. LIU, X. D., OSHER, S. J. & CHAN, T. F. (1996) Weighted essentially non-oscillatory schemes. J. Comp. Phys. 126, 202-212.
94. MALLADI, R. & SETHIAN, J. A. (1997) Level set methods for curvature flow, image enhancement, and shape recovery in medical images. In: H. C. Hege and K. Polthier, editors, Visualization and Mathematics. Experiments, Simulation and Environments, pp. 329-345. Springer.
95. MASNOU, S. & MOREL, J. M. (1998) Level lines based disocclusion. Proceedings of IEEE International Conference on Image Processing 1998, Vol. 3, pp. 259-263. IEEE Computer Society.
96. MERRIMAN, B., BENCE, J. K. & OSHER, S. J. (1994) Motion of multiple junctions: A level set approach. J. Com. Phys. 112, 334-363.
97. MILLER, E. (n.d.) Parametric level set methods. In preparation.
98. MOHAMADI, B. & PIRONNEAU, O. (2001) Applied Shape Optimization for Fluids. Clarendon Press.
99. MOREL, J. M. & SOLIMINI, S. (1994) Variational Methods in Image Segmentation with 7 Image Processing Experiments. Birkhäuser.
100. MOËS, N, SUKUMAR, N., MORAN, B. & BELYTSCHKO, T. (2000) An extended finite element method (x-fem) for two- and three-dimensional crack modeling. ECCOMAS 2000, pp. 11-14.
101. MUMFORD, D. & SHAH, J. (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577-685.
102. MURAT, F. & SIMON, J. (1976) Sur le contrôle par un domaine géometrique. Publication du Laboratoire d'Analyse Numerique, Université de Paris VI.
103. NEUBAUER, A. & SCHERZER, O. (1998) Regularization for curve representations: Uniform convergence for discontinuous solutions of ill-posed problems. SIAM J. Appl. Math. 58, 1891-1900.
104. NOCEDAL, J. & WRIGHT, S. J. (1999) Numerical Optimization. Springer.
105. NOVRUZI, A. & PIERRE, M. (2002) Structure of shape derivatives. J. Evolution Equations, 3, 365-382.
106. OSHER, S. J. & FEDKIW, R. P. (2002) The Level Set Method and Dynamic Implicit Surfaces. Springer.
107. OSHER, S. J. & PARAGIOS, N. (EDITORS) (2002) Geometric Level Set Methods in Imaging, Vision and Graphics, Springer.
108. OSHER, S. & SANTOSA, F. (2001) Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum. J. Comp. Phys. 171, 272-288.
109. OSHER, S. J. & SETHIAN, J. A. (1988) Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 79, 12-49.
110. OSHER, S. J. & SHU, C. W. (1991) Higher-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907-922.
111. PARAGIOS, N. (2003) A level set approach for shape-driven segmentation and tracking of the left ventricle. IEEE Trans. Medical Imaging, 22, 773-776.
112. RAMANANJAONA, C., LAMBERT, M. & LESSELIER, D. (2001) Shape inversion from TM and TE real data by controlled evolution of level sets. Inverse Problems, 17, 1585-1595.
113. RAMANANJAONA, C., LAMBERT, M., LESSELIER, D. & ZOLESIO, J. P. (2001) Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation. Inverse Problems, 17, 1087-1112.
114. RAMANANJAONA, C., LAMBERT, M., LESSELIER, D. & ZOLESIO, J. P. (2002) On novel developments of controlled evolution of level sets in the field of inverse shape problems. Radio Sci. 38, 242-254.
115. RONDI, L. & SANTOSA, F. (2001) Enhanced electrical impedance tomography via the Mumford-Shah functional. ESAIM: Control, Optimisation and Calculus of Variations, 6, 517-538.
116. RUDIN, L. I., OSHER, S. J. & FATEMI, E. (1992) Nonlinear total variation based noise removal algorithms. Physica D, 60, 259-268.
117. RUSSO, G. & SMEREKA, P. (2000) A remark on computing distance functions. J. Comput. Phys. 163, 51-67.
118. SANTOSA, F. (1996) A level-set approach for inverse problems involving obstacles. ESAIM: Control, Optimisation and Calculus of Variations, 1, 17-33.
119. SETHIAN, J. A. & WIEGMANN, A. (2000) Structural boundary design via level set and immersed interface methods. J. Comp. Phys. 163, 489-528.
120. SHU, C. W. & OSHER, S. J. (1988) Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comp. Phys. 77, 439-471.
121. SHU, C. W. & OSHER, S. J. (1989) Implementation of essentially non-oscillatory shock capturing schemes II. J. Comp. Phys. 83, 32-78.
122. SMEREKA, P. (2003) Semi-implicit level set methods for motion by mean curvature and surface diffusion. J. Sci. Comput. 19, 439-456.
123. SOKOLOWSKI, J. & ZOCHOWSKI, A. (1999) On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 1251-1272.
124. SOKOLOWSKI, J. & ZOCHOWSKI, A. (1999) Topological derivatives for elliptic problems. Inverse Problems, 15, 123-134.
125. SOKOLOWSKI, J. & ZOLESIO, J. P. (1992) Introduction to Shape Optimization. Springer.
126. SPITERI, R. J. & RUUTH, S. J. (2002) A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469-491.
127. STOLARSKA, M., CHOPP, D. L., MOES, N. & BELYTSCHKO, T. (2001) Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Meth. Eng. 51, 943-960.
128. STRAIN, J. (1999) Fast tree-based redistancing for level set computations. J. Comput. Phys. 152, 648-666.
129. STRZODKA, R., DROSKE, M. & RUMPF, M. (2003) Image registration by a regularized gradient flow - A Streaming Implementation in DX9 Graphics Hardware. Preprint, University Duisburg-Essen.
130. SUKUMAR, N., CHOPP, D. L., MOES, N. & BELYTSCHKO, T. (2001) Modeling holes and inclusions by level sets in the extended finite element method. Comput. Meth. Appl. Mech. Eng. 190, 6183-6200.
131. SUSSMAN, M. & FATEMI, E. (1999) An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20, 1165-1191.
132. SUSSMAN, M., SMEREKA, P. & OSHER, S. (1994) A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146-159.
133. TAUTENHAHN, U. (1994) On the asymptotical regularization of nonlinear ill-posed problems. Inverse Problems, 10, 1405-1418.
134. TAYLOR, J. E. (1996) Surface motion due to crystalline surface energy gradient flows. In: A. K. Peters, editor, Elliptic and Parabolic Methods in Geometry, pp. 145-162. Wellesley.
135. TAYLOR, J. E., CAHN, J. W. & HANDWERKER, C. A. (1992) Geometric models of crystal growth. Acta Metall. Mater. 40, 1443-1472.
136. TORNBERG, A. C. & ENGQUIST, B. (2000) Regularization techniques for numerical approximation of PDEs with singularities. J. Sci. Comput. 19, 527-552.
137. TORNBERG, A. C. & ENGQUIST, B. (2000) A finite element based level set method for multiphase flow applications. Computing & Visualization in Sci. 3, 93-101.
138. TORNBERG, A. C. & ENGQUIST, B. (2003) Numerical approximations of singular source terms in differential equations. CAM Report 03-62, UCLA.
139. TSAI, A., YEZZI, A., WELLS, W., TEMPANY, C., TUCKER, D., FAN, A., GRIMSON, W. E. & WILLSKY, A. (2003) A shape-based approach to the segmentation of medical imagery using level sets. IEEE Trans. Med. Imag. 22, 137-154.
140. TSAI, R., GIGA, Y. & OSHER, S. J. (2003) A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Comp. 72, 159-181.
141. TSAI, R. & OSHER, S. J. (2004) Level set methods and their applications in image science. Communications in Math. Sciences (2004), Comm. Math. Sci. 1, 623-656.
142. VEMURI, B. C., YE, J., CHEN, Y. & LEONARD, C. M. (2003) Image registration via level-set motion: Applications to atlas-based segmentation. Medical Image Anal. 7, 1-20.
143. VESE, L. A. & CHAN, T. F. (2002) A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vision, 50, 271-293.
144. WANG, M. Y., WANG, X. M. & GUO, D. M. (2003) A level set method for structural topology optimization. Comp. Meth. Appl. Mech. Eng. 192, 227-246.
145. WANG, M. Y., WANG, X. M. & GUO, D. M. (2004) Structural shape and topology optimization in a level-set based framework of region representation. Struct. Multidisc. Optim. 27, 1-19.
146. WANG, M. Y. & WANG, X. M. (2003) Color level sets: A multi-phase level set method for structural topology optimization with multiple materials. Comp. Meth. Appl. Mech. Eng. 193, 469-496.
147. WANG, M. Y. & WANG, X. M. (2003) A level-set based variational method for design and optimization of heterogeneous objects. Preprint, Chinese University of Hong Kong.
148. WANG, M. Y. & WANG, X. M. (2003) PDE-driven level sets, velocity fields, and perimeter penalization for structural topology optimization. Preprint, Chinese University of Hong Kong.
149. WANG, M. Y. & WANG, X. M. (2003) A multi-phase level set model for multi-material structural optimization. Proc. of 5th World Congress of Structural and Multidisciplinary Optimization (WCSMO5).
150. WANG, M. Y. & ZHOU, S. W. (2003) Phase transition: A variational method for structural topology optimization. Preprint, Chinese University of Hong Kong.
151. ZHAO, H. K., CHAN, T. F., MERRIMAN, B. & OSHER, S. J. (1998) A variational level set approach to multiphase motion. J. Comp. Phys. 127, 179-195.
152. ZHAO, H. K., MERRIMAN, B., OSHER, S. J. & WANG, L. (1998) Capturing the behaviour of bubbles and drops using a variational level set approach. J. Comp. Phys. 143, 495-518.