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Theory, algorithms, and applications of level set methods for propagating interfaces

Published online by Cambridge University Press:  07 November 2008

James A. Sethian
James A. Sethian
Affiliation:
Department of MathematicsUniversity of CaliforniaBerkeley, CA 94720, USAE-mail:sethian@math.berkeley.edu
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Abstract

We review recent work on level set methods for following the evolution of complex interfaces. These techniques are based on solving initial value partial differential equations for level set functions, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The methodology results in robust, accurate, and efficient numerical algorithms for propagating interfaces in highly complex settings. We review the basic theory and approximations, describe a hierarchy of fast methods, including an extremely fast marching level set scheme for monotonically advancing fronts, based on a stationary formulation of the problem, and discuss extensions to multiple interfaces and triple points. Finally, we demonstrate the technique applied to a series of examples from geometry, material science and computer vision, including mean curvature flow, minimal surfaces, grid generation, fluid mechanics, combustion, image processing, computer vision, and etching, deposition and lithography in the microfabrication of electronic components.

Type
Research Article
Information
Acta Numerica , Volume 5 , January 1996 , pp. 309 - 395
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Adalsteinsson, D. and Sethian, J. A. (1995 a), ‘A fast level set method for propagating interfaces’, J. Comp. Phys. 118, 269–277.CrossRefGoogle Scholar
Adalsteinsson, D. and Sethian, J. A. (1995 b), ‘A unified level set approach to etching, deposition and lithography I: Algorithms and two-dimensional simulations’, J. Comp. Phys. 120, 128–144.CrossRefGoogle Scholar
Adalsteinsson, D. and Sethian, J. A. (1995 c), ‘A unified level set approach to etching, deposition and lithography II: Three-dimensional simulations’, J. Comp. Phys. 122, 348–366.CrossRefGoogle Scholar
Adalsteinsson, D. and Sethian, J. A. (1996), ‘A unified level set approach to etching, deposition and lithography III: Complex simulations and multiple effects’, J. Comp. Phys. To be submitted.Google Scholar
Alvarez, L., Lions, P.-L. and Morel, M. (1992), ‘Image selective smoothing and edge detection by nonlinear diffusion. II’, SIAM J. Num. Anal. 29, 845–866.CrossRefGoogle Scholar
Ambrosio, L. and Soner, H. M. (1994), Level set approach to mean curvature flow in arbitrary codimension. Preprint.Google Scholar
Angenent, S. (1992), Shrinking doughnuts, in Proceedings of Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Lloyd, N. G. et al. , eds), Birkhäuser, Boston.Google Scholar
Bardi, M. and Falcone, M. (1990), ‘An approximation scheme for the minimum time function’, SIAM J. Control Optim. 28, 950–965.CrossRefGoogle Scholar
Barles, G. (1985), Remarks on a flame propagation model, report 464, INRIA.Google Scholar
Barles, G. (1993), ‘Discontinuous viscosity solutions of first order Hamilton–Jacobi equations: A guided visit’, Non-linear Analysis: Theory, Methods, and Applications 20, 1123–1134.CrossRefGoogle Scholar
Barles, G. and Souganidis, P. E. (1991), ‘Convergence of approximation schemes for fully non-linear second order equations’, Asymptotic Anal. 4, 271–283.CrossRefGoogle Scholar
Bell, J. B., Colella, P. and Glaz, H. M. (1989), ‘A second-order projection method for the incompressible Navier–Stokes equations’, J. Comp. Phys. 85, 257–283.CrossRefGoogle Scholar
Berger, M. and Colella, P. (1989), ‘Local adaptive mesh refinement for shock hydrodynamics’, J. Comp. Phys. 82, 62–84.CrossRefGoogle Scholar
Brackbill, J. U., Kothe, D. B. and Zemach, C. (1992), ‘A continuum method for modeling surface tension’, J. Comp. Phys. 100, 335–353.CrossRefGoogle Scholar
Brakke, K. A. (1978), The Motion of a Surface by Its Mean Curvature, Princeton University Press, Princeton University.Google Scholar
Brakke, K. A. (1990), Surface evolver program, Technical Report GCC 17, University of Minnesota. Geometry Supercomputer Project.Google Scholar
Bronsard, L. and Wetton, B. (1995), ‘A numerical method for tracking curve networks moving with curvature motion’, J. Comp. Phys. 120, 66–87.CrossRefGoogle Scholar
Buttazzo, G. and Visitin, A. (1994), Motion by mean curvature and related topics, in Proceedings of the International Conference at Trento, 1992, Walter de Gruyter, New York.Google Scholar
Cahn, J. E. and Hilliard, J. E. (1958), ‘Free energy of a non-uniform system. I. Interfacial Free Energy’, J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Cale, T. S. and Raupp, G. B. (1990 a), ‘Free molecular transport and deposition in cylindrical features’, J. Vac. Sci. Tech., B 8, 649–655.CrossRefGoogle Scholar
Cale, T. S. and Raupp, G. B. (1990 b), ‘Free molecular transport and deposition in long rectangular trenches’, J. Appl. Phys. 68, 3645–8652.CrossRefGoogle Scholar
Cale, T. S. and Raupp, G. B. (1990 c), ‘A unified line-of-sight model of deposition in rectangular trenches’, J. Vac. Sci. Tech., B 8, 1242–1248.CrossRefGoogle Scholar
Caselles, V., Catte, F., Coll, T. and Dibos, F. (n.d.), A geometric model for active contours in image processing, Technical Report 9210, CEREMADE, Université de Paris-Dauphiné, France. Internal report.Google Scholar
Castillo, J. E. (1991), Mathematical Aspects of Grid Generation, Frontiers in Applied Mathematics 8, SIAM Publications.CrossRefGoogle Scholar
Chang, Y. C., Hou, T. Y., Merriman, B. and Osher, S. J. (1994), ‘A level set formulation of Eulerian interface capturing methods for incompressible fluid flows’, J. Comp. Phys. Submitted.Google Scholar
Chen, Y., Giga, Y. and Goto, S. (1991), ‘Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations’, J. Diff. Geom. 33, 749–786.Google Scholar
Chopp, D. L. (1993), ‘Computing minimal surfaces via level set curvature flow’, J. Comp. Phys. 106, 77–91.CrossRefGoogle Scholar
Chopp, D. L. (1994), ‘Numerical computation of self-similar solutions for mean curvature flow’, J. Exper. Math. 3, 1–15.CrossRefGoogle Scholar
Chopp, D. L. and Sethian, J. A. (1993), ‘Flow under curvature: Singularity formation, minimal surfaces, and geodesies’, J. Exper. Math. 2, 235–255.CrossRefGoogle Scholar
Chorin, A. J. (1968), ‘Numerical solution of the Navier–Stokes equations’, Math. Comp. 22, 745.CrossRefGoogle Scholar
Chorin, A. J. (1973), ‘Numerical study of slightly viscous flow’, J. Fluid Mech. 57, 785–796.CrossRefGoogle Scholar
Chorin, A. J. (1980), ‘Flame advection and propagation algorithms’, J. Comp. Phys. 35, 1–11.CrossRefGoogle Scholar
Cohen, L. D. (1991), ‘On active contour models and balloons’, Computer Vision, Graphics, and Image Processing 53, 211–218.Google Scholar
Colella, P. and Puckett, E. G. (1994), Modern Numerical Methods for Fluid Flow, Lecture Notes, Department of Mechanical Engineering, University of California, Berkeley.Google Scholar
Crandall, M. G. and Lions, P.-L. (1983), ‘Viscosity solutions of Hamilton–Jacobi equations’, Tran. AMS 277, 1–43.CrossRefGoogle Scholar
Crandall, M. G., Evans, L. C. and Lions, P.-L. (1984), ‘Some properties of viscosity solutions of Hamilton–Jacobi equations’, Tran. AMS 282, 487–502.CrossRefGoogle Scholar
Crandall, M. G., Ishii, H. and Lions, P.-L. (1992), ‘User's guide to viscosity solutions of second order partial differential equations’, Bull. AMS 27, 1–67.CrossRefGoogle Scholar
Ecker, K. and Huisman, G. (1991), ‘Interior estimates for hypersurfaces moving by mean curvature’, Inventiones Mathematica 105, 547–569.CrossRefGoogle Scholar
Engquist, B. and Osher, S. J. (1980), ‘Stable and entropy-satisfying approximations for transonic flow calculations’, Math. Comp. 34, 45.CrossRefGoogle Scholar
Evans, L. C. and Spruck, J. (1991), ‘Motion of level sets by mean curvature I’, J. Diff. Geom. 33, 635–681.Google Scholar
Evans, L. C. and Spruck, J. (1992 a), ‘Motion of level sets by mean curvature II’, Trans. AMS 330, 321–332.CrossRefGoogle Scholar
Evans, L. C. and Spruck, J. (1992 b), ‘Motion of level sets by mean curvature III’, J. Geom. Anal. 2, 121–150.CrossRefGoogle Scholar
Evans, L. C. and Spruck, J. (1995), ‘Motion of level sets by mean curvature IV’, J. Geom. Anal. 5, 77–114.CrossRefGoogle Scholar
Evans, L. C., Soner, H. M. and Souganidis, P. E. (1992), ‘Phase transitions and generalized motion by mean curvature’, Comm. Pure Appl. Math. 45, 1097–1123.CrossRefGoogle Scholar
Falcone, M. (1994), The minimum time problem and its applications to front propagation, in Motion by Mean Curvature and Related Topics. Proceedings of the International Conference at Trento, 1992, Walter de Gruyter, New York.Google Scholar
Falcone, M., Giorgi, T. and Loretti, P. (1994), ‘Level sets of viscosity solutions: Some applications to fronts and rendez-vous problems’, SIAM J. Appl. Math. 54, 1335–1354.CrossRefGoogle Scholar
Fatemi, E., Engquist, B. and Osher, S. J. (1995), ‘Numerical solution of the high frequency asymptotic wave equation for the scalar wave equation’, J. Comp. Phys. 120, 145–155.CrossRefGoogle Scholar
Gage, M. (1984), ‘Curve shortening makes convex curves circular’, Inventiones Mathematica 76, 357.CrossRefGoogle Scholar
Gage, M. and Hamilton, R. (1986), ‘The heat-equation shrinking convex plane-curves’, J. Diff. Geom. 23, 69–96.Google Scholar
Giga, Y. and Goto, S. (1992), ‘Motion of hypersurfaces and geometric equations’, J. Math. Soc. Japan 44, 99–111.CrossRefGoogle Scholar
Giga, Y., Goto, S. and Ishii, H. (1992), ‘Global existence of weak solutions for interface equations coupled with diffusion equations’, SIAM J. Math. Anal. 23, 821–835.CrossRefGoogle Scholar
Grayson, M. A. (1987), ‘The heat equation shrinks embedded plane curves to round points’, J. Diff. Geom. 26, 285–314.Google Scholar
Grayson, M. A. (1989), ‘A short note on the evolution of a surface by its mean-curvature’, Duke Math. J. 58, 555–558.CrossRefGoogle Scholar
Greengard, L. and Strain, J. (1990), ‘A fast algorithm for evaluating heat potentials’, Comm. Pure Appl. Math. 43, 949–963.CrossRefGoogle Scholar
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R. (1987), ‘Uniformly high order accurate essentially non-oscillatory schemes. III’, J. Comp. Phys. 71, 231–303.CrossRefGoogle Scholar
Helmsen, J. J. (1994), A Comparison of Three-Dimensional Photolithography Development Methods, PhD thesis, University of California, Berkeley.Google Scholar
Hirt, C. W. and Nicholls, B. D. (1981), ‘Volume of fluid (COF) method for dynamics of free boundaries’, J. Comp. Phys. 39, 201–225.CrossRefGoogle Scholar
Huisken, G. (1984), ‘Flow by mean curvature of convex surfaces into spheres’, J. Diff. Geom. 20, 237–266.Google Scholar
Ilmanen, T. (1992), ‘Generalized flow of sets by mean curvature on a manifold’, Indiana University Mathematics Journal 41, 671–705.CrossRefGoogle Scholar
Ilmanen, T. (1994), Elliptic Regularization and Partial Regularity for Motion by Mean Curvature, Memoirs of the American Mathematical Society, 108.Google Scholar
Kass, M., Witkin, A. and Terzopoulos, D. (1988), ‘Snakes: Active contour models’, Int. J. Comp. Vision pp. 321–331.CrossRefGoogle Scholar
Katardjiev, I. V., Carter, G. and Nobes, M. J. (1988), ‘Precision modeling of the mask-substrate evolution during ion etching’, J. Vac. Sci. Tech. A 6, 2443–2450.CrossRefGoogle Scholar
Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A. and Yezzi, A. (1995), Gradient Flows and Geometric Active Contours, 1994, ICCV.Google Scholar
Kimmel, R. (1995), Curve Evolution on Surfaces, PhD thesis, Dept. of Electrical Engineering, Technion, Israel.Google Scholar
Kimmel, R. and Bruckstein, A. (1992), Shape from shading via level sets, Technical Report 9209, Technion, Israel. Center for Intelligent Systems.Google Scholar
Kimmel, R. and Bruckstein, A. (1993), ‘Shape offsets via level sets’, Computer-Aided Design 25, 154–161.CrossRefGoogle Scholar
Knupp, P. and Steinberg, S. (1993), The fundamentals of grid generation. Preprint.Google Scholar
Lax, P. D. (1970), Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Reg. Conf. Series, Lectures in Applied Math, 11, SIAM, Philadelphia.Google Scholar
Leon, F. A., Tazawa, S., Saito, K., Yoshi, A. and Scharfetter, D. L. (1993), Numerical algorithms for precise calculation of surface movement in 3-D topography simulation, in 1993 International Workshop on VLSI Process and Device Modeling.Google Scholar
LeVeque, R. J. (1992), Numerical Methods for Conservation Laws, Birkhäuser, Basel.CrossRefGoogle Scholar
Lions, P.-L. (1982), Generalized Solution of Hamilton–Jacobi Equations, Pitman, London.Google Scholar
Majda, A. and Sethian, J. A. (1984), ‘Derivation and numerical solution of the equations of low mach number combustion’, Combustion Science and Technology 42, 185–205.CrossRefGoogle Scholar
Malladi, R., Adalsteinsson, D. and Sethian, J. A. (1995 a), ‘A fast level set algorithm for 3D shape recovery’, IEEE Transactions on PAMI. Submitted.Google Scholar
Malladi, R. and Sethian, J. A. (1994), A unified approach for shape segmentation, representation, and recognition, Technical Report 36069, Lawrence Berkeley Laboratory, University of California, Berkeley.Google Scholar
Malladi, R. and Sethian, J. A. (1995), ‘Image processing via level set curvature flow’, Proc. Natl. Acad. of Sci., USA 92, 7046–7050.CrossRefGoogle ScholarPubMed
Malladi, R. and Sethian, J. A. (1996 a), ‘Image processing: Flows under min/max curvature and mean curvature’, Graphical Models and Image Processing. In press.CrossRefGoogle Scholar
Malladi, R. and Sethian, J. A. (1996 b), ‘A unified approach to noise removal, image enhancement, and shape recovery’, IEEE Image Processing. In press.CrossRefGoogle ScholarPubMed
Malladi, R., Sethian, J. A. and Vemuri, B. C. (1994), Evolutionary fronts for topology-independent shape modeling and recovery, in Proceedings of Third European Conference on Computer Vision, LNCS Vol. 800, Stockholm, pp. 3–13.Google Scholar
Malladi, R., Sethian, J. A. and Vemuri, B. C. (1995 b), ‘Shape modeling with front propagation: A level set approach’, IEEE Trans. on Pattern Analysis and Machine Intelligence 17, 158–175.CrossRefGoogle Scholar
McVittie, J. P., Rey, J. C., Bariya, A. J. et al. (1991), SPEEDIE: A profile simulator for etching and deposition, in Proceedings of the SPIE – The International Society for Optical Engineering, Vol. 1392, pp. 126–38.CrossRefGoogle Scholar
Merriman, B., Bence, J. and Osher, S. J. (1994), ‘Motion of multiple junctions: A level set approach’, J. Comp. Phys. 112, 334–363.CrossRefGoogle Scholar
Milne, B. and Sethian, J. A. (1995), ‘Adaptive mesh refinement for level set methods for propagating interfaces’, J. Comp. Phys. Submitted.Google Scholar
Mulder, W., Osher, S. J. and Sethian, J. A. (1992), ‘Computing interface motion in compressible gas dynamics’, J. Comp. Phys. 100, 209–228.CrossRefGoogle Scholar
Mullins, W. W. and Sekerka, R. F. (1963), ‘Morphological stability of a particle growing by diffusion or heat flow’, J. Appl. Phys. 34, 323–329.CrossRefGoogle Scholar
Noh, W. and Woodward, P. (1976), A simple line interface calculation, in Proceedings, Fifth International Conference on Fluid Dynamics (van de Vooran, A. I. and Zandberger, P. J., eds), Springer, Berlin.Google Scholar
Osher, S. and Rudin, L. I. (1990), ‘Feature-oriented image enhancement using shock filters’, SIAM J. Num. Anal. 27, 919–940.CrossRefGoogle Scholar
Osher, S. and Rudin, L. I. (1992), Rapid convergence of approximate solutions of shape-from-shading. To appear.Google Scholar
Osher, S. and Sethian, J. A. (1988), ‘Fronts propagating with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulation’, J. Comp. Phys. 79, 12–49.CrossRefGoogle Scholar
Osher, S. and Shu, C. (1991), ‘High-order nonoscillatory schemes for Hamilton–Jacobi equations’, J. Comp. Phys. 28, 907–922.Google Scholar
Perona, P. and Malik, J. (1990), ‘Scale-space and edge detection using anisotropic diffusion’, IEEE Trans. Pattern Analysis and Machine Intelligence 12, 629–639.CrossRefGoogle Scholar
Pindera, M. Z. and Talbot, L. (1986), Flame-induced vorticity: The effects of stretch, in Twenty-First Symposium (Int'l) on Combustion, The Combustion Institute, Pittsburgh, pp. 1357–1366.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1988), Numerical Recipes, Cambridge University Press.Google Scholar
Puckett, E. G. (1991), A volume-of-fluid interface tracking algorithm with applications to computing shock wave refraction, in Proceedings of the 4th International Symposium on Computational Computational Fluid Dynamics,Davis,California.Google Scholar
Rey, J. C., Cheng, L.-Y., McVittie, J. P. and Saraswat, K. C. (1991), ‘Monte Carlo low pressure deposition profile simulations’, J. Vac. Sci. Tech. A 9, 1083–7.CrossRefGoogle Scholar
Rhee, C., Talbot, L. and Sethian, J. A. (1995), ‘Dynamical study of a premixed V flame’, J. Fluid Mech. 300, 87–115.CrossRefGoogle Scholar
Rouy, E. and Tourin, A. (1992), ‘A viscosity solutions approach to shape-from-shading’, SIAM J. Num. Anal. 29, 867–884.CrossRefGoogle Scholar
Rudin, L., Osher, S. and Fatemi, E. (1992), Nonlinear total variation-based noise removal algorithms, in Modelisations Mathématiques pour le Traitement d'Images, INRIA, pp. 149–179.Google Scholar
Sapiro, G. and Tannenbaum, A. (1993 a), ‘Affine invariant scale-space’, Int. J. Comp. Vision 11, 25–44.CrossRefGoogle Scholar
Sapiro, G. and Tannenbaum, A. (1993 b), Image smoothing based on affine invariant flow, in Proc. of the Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore.Google Scholar
Sapiro, G. and Tannenbaum, A. (1994), Area and length preserving geometric invariant scale-spaces, in Proc. of Third European Conference on Computer Vision, Vol. 801 of LNCS, Stockholm, pp. 449–458.Google Scholar
Scheckler, E. W. (1991), PhD thesis, EECS, University of California, Berkeley.Google Scholar
Scheckler, E. W., Toh, K. K. H., Hoffstetter, D. M. and Neureuther, A. R. (1991), 3D lithography, etching and deposition simulation, in Symposium on VLSI Technology, Oiso, Japan, pp. 97–98.Google Scholar
Sedgewick, R. (1988), Algorithms, Addison-Wesley, New York.Google Scholar
Sethian, J. A. (1982), An Analysis of Flame Propagation, PhD thesis, Department of Mathematics, University of California, Berkeley.Google Scholar
Sethian, J. A. (1984), ‘Turbulent combustion in open and closed vessels’, J. Comp. Phys. 54, 425–456.CrossRefGoogle Scholar
Sethian, J. A. (1985), ‘Curvature and the evolution of fronts’, Comm. Math. Phys. 101, 487–499.CrossRefGoogle Scholar
Sethian, J. A. (1987), Numerical methods for propagating fronts, in Variational Methods for Free Surface Interfaces (Concus, P. and Finn, R., eds), Springer, New York.Google Scholar
Sethian, J. A. (1989), Parallel level set methods for propagating interfaces on the connection machine. Unpublished manuscript.Google Scholar
Sethian, J. A. (1990), ‘Numerical algorithms for propagating interfaces: Hamilton–Jacobi equations and conservation laws’, J. Diff. Geom. 31, 131–161.Google Scholar
Sethian, J. A. (1991), A brief overview of vortex methods, in Vortex Methods and Vortex Motion (Gustafson, K. and Sethian, J. A., eds), SIAM Publications, Philadelphia.Google Scholar
Sethian, J. A. (1994), ‘Curvature flow and entropy conditions applied to grid generation’, J. Comp. Phys. 115, 440–454.CrossRefGoogle Scholar
Sethian, J. A. (1995 a), ‘Algorithms for tracking interfaces in CFD and material science’, Annual Review of Computational Fluid Mechanics.Google Scholar
Sethian, J. A. (1995 b), Level set techniques for tracking interfaces; fast algorithms, multiple regions, grid generation and shape/character recognition, in Curvature flow and related topics, Gakuto Int. Series, Volume 5, Tokyo.Google Scholar
Sethian, J. A. (1995 c), ‘A marching level set method for monotonically advancing fronts’, Proc. Nat. Acad. Sci. To appear.Google Scholar
Sethian, J. A. (1996), Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press. To appear.Google Scholar
Sethian, J. A., Adalsteinsson, D. and Malladi, R. (1996), ‘Efficient fast marching level set methods’. In progress.Google Scholar
Sethian, J. A. and Strain, J. D. (1992), ‘Crystal growth and dendritic solidification’, J. Comp. Phys. 98, 231–253.CrossRefGoogle Scholar
Souganidis, P. E. (1985), ‘Approximation schemes for viscosity solutions of Hamilton– Jacobi equations’, J. Diff. Eqns. 59, 1–43.CrossRefGoogle Scholar
Strain, J. (1988), ‘Linear stability of planar solidification fronts’, Physica D 30, 297–320.CrossRefGoogle Scholar
Strain, J. (1989), ‘A boundary integral approach to unstable solidification’, J. Comp. Phys. 85, 342–389.CrossRefGoogle Scholar
Strain, J. (1990), ‘Velocity effects in unstable solidification’, SIAM J. Appl. Math. 50, 1–15.CrossRefGoogle Scholar
Sussman, M., Smereka, P. and Osher, S. J. (1994), ‘A level set method for computing solutions to incompressible two-phase flow’, J. Comp. Phys. 114, 146–159.CrossRefGoogle Scholar
Taylor, J. E., Cahn, J. W. and Handwerker, C. A. (1992), ‘Geometric models of crystal growth’, Acta Metallurgica et Materialia 40, 1443–74.CrossRefGoogle Scholar
Terzopoulos, D., Witkin, A. and Kass, M. (1988), ‘Constraints on deformable models: Recovering 3d shape and nonrigid motion’, Artificial Intelligence 36, 91–123.CrossRefGoogle Scholar
Toh, K. K. H. (1990), PhD thesis, EECS, University of California, Berkeley.Google Scholar
Toh, K. K. H. and Neureuther, A. R. (1991), ‘Three-dimensional simulation of optical lithography’, Proceedings SPIE, Optical/Laser Microlithography IV 1463, 356–367.CrossRefGoogle Scholar
Young, M. S., Lee, D., Lee, R.. and Neureuther, A. R. (1993), ‘Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects’, Proceedings SPIE, Optical/Laser Microlithography VI 1927, 452–463.CrossRefGoogle Scholar
Zhu, J. and Ronney, P. D. (1995), ‘Simulation of front propagation at large non-dimensional flow disturbance intensities’, Comb. Sci. Tech. To appear.Google Scholar
Zhu, J. and Sethian, J. A. (1992), ‘Projection methods coupled to level set interface techniques’, J. Comp. Phys. 102, 128–138.CrossRefGoogle Scholar