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Level Set Methods for Curvature Flow, Image Enchancement, and Shape Recovery in Medical Images

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Visualization and Mathematics

Summary

Level set methods are powerful numerical techniques for tracking the evolution of interfaces moving under a variety of complex motions. They are based on computing viscosity solutions to the appropriate equations of motion, using techniques borrowed from hyperbolic conservation laws. In this paper, we review some of the applications of this work to curvature motion, the construction of minimal surfaces, image enhancement, and shape recovery. We introduce new schemes for denoising three-dimensional shapes and images, as well as a fast shape recovery techniques for three-dimensional images.

Supported in part by the Applied Mathematics Subprogram of the Office of Energy Research under contract DE-AC03-76SF00098, and the National Science Foundation and DARPA under grant DMS-8919074.

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References

  1. D. Adalsteinsson and J. A. Sethian, A Fast Level Set Method for Propagating Interfaces, J. Comp. Phys. 118:2 (May 1995), 269–277.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Adalsteinsson and J. A. Sethian, A Unified Level Set Approach to Etching, Deposition and Lithography I: Algorithms and Two-dimensional Simulations, J. Comp. Phys. 120:1 (1995), 128–144.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Adalsteinsson and J. A. Sethian, A Unified Level Set Approach to Etching, Deposition and lithography II: Three-dimensional Simulations,J. Comp. Phys. 122:2 (1995), 348–366.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Alvarez, P. L. Lions, and M. Morel, Image Selective Smoothing and Edge detection by Nonlinear Diffusion II, SIAM J. Num. Analysis 29:3 (1992), 845–866.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Angenent, Shrinking Doughnuts, in: Proceedings of Nonlinear Diffusion Equations and Their Equilibrium States, 3, N.G. Lloyd et. al. (eds.), Birkhauser, Boston, 1992.

    Google Scholar 

  6. Y. Chen, Y. Giga, and S. Goto, Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations, J. Diff. Geom. 33 (1991), 749.

    MathSciNet  MATH  Google Scholar 

  7. D. L. Chopp, Computing Minimal Surfaces via Level Set Curvature Flow, J. Comp. Phys. 106 (1993), 77–91.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. L. Chopp, Numerical Computation of Self-Similar Solutions for Men Curvature Flow Experimental Math. 3:1 (1994), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. L. Chopp and J. A. Sethian, Flow Under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics, Experimental Math. 2:4 (1993), 235–255.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. C. Evans and J. Spruck, Motion of Level sets by Mean Curvature I, J. Diff. Geom. 33 (1991), 635.

    MathSciNet  MATH  Google Scholar 

  11. M. Gage, Curve Shortening Makes Convex Curves Circular, Inventiones Mathematica 76 (1984), 357.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Gage and R. Hamilton, The Equation Shrinking Convex Planes Curves, J. Diff. Geom. 23 (1986), 69.

    MathSciNet  MATH  Google Scholar 

  13. M. Grayson, The Heat Equation Shrinks Embedded Plane Curves to Round Points, J. Diff. Geom. 26 (1987), 285.

    MathSciNet  MATH  Google Scholar 

  14. M. Grayson, A Short Note on the Evolution of Surfaces Via Mean Curvatures, J. Diff. Geom. 58 (1989), 555.

    MathSciNet  MATH  Google Scholar 

  15. G. Huisken, Flow by Mean Curvature of Convex Surfaces into Spheres, J. Diff. Geom. 20 (1984) 237.

    MathSciNet  MATH  Google Scholar 

  16. M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active Contour Models, International J. Computer Vision, 321–331, 1988.

    Google Scholar 

  17. R. Malladi and J. A. Sethian, Image Processing via Level Set Curvature Flow, Proc. Natl. Acad. of Sci., USA 92:15 (July 1995), 7046–7050.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Malladi and J. A. Sethian, Image Processing: Flows under Min/Max Curvature and Mean Curvature, Graphical Models and Image Processing 58:2 (March 1996), in press.

    Article  Google Scholar 

  19. R. Malladi and J. A. Sethian, A Unified Approach to Noise Removal, Image Enhancement, and Shape Recovery, to appear in IEEE Transactions on Image Processing, 1996.

    Google Scholar 

  20. R. Malladi, J. A. Sethian, and B. C. Vemuri, Evolutionary Fronts for Topology-independent shape Modeling and Recovery, in Proceedings of Third European Conference on Computer Vision, LNCS Vol. 800, 3–13, Stockholm, Sweden, May 1994.

    Google Scholar 

  21. R. Malladi, J. A. Sethian, and B. C. Vemuri, Shape Modeling with Front Propagation: A Level Set Approach, IEEE Trans. on Pattern Analysis and Machine Intelligence 17:2 (Febr. 1995), 158–175.

    Article  Google Scholar 

  22. S. Osher, and J. A. Sethian, Fronts Propagating with Curvature Dependent speed: Algorithms Based on Hamilton-Jacobi Formulation,J. Comp. Phys. 79 (1988), 12–49.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. Rhee, L. Talbot, and J. A. Sethian, Dynamical Study of a Premixed V flame, J. Fluid Mech. 300 (1995), 87–115.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. A. Sethian, An Analysis of Flame Propagation, Ph.D. Diss., Dept. of Mathematics, University of California, Berkeley, 1982.

    Google Scholar 

  25. J. A. Sethian, Curvature and the Evolution of Fronts, Comm. Math. Physics 101 (1985), 487–499.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. A. Sethian, Numerical Methods for Propagating Fronts, in: Variational Methods for Free Surface Interfaces, P. Concus and R. Finn (eds.), Springer Verlag, New York, 1987.

    Google Scholar 

  27. J. A. Sethian, Numerical Algorithms for Propagating Interfaces: Hamilton-Jacobi Equations and Conservation Laws, J. Diff. Geom. 31 (1990), 131–161.

    MathSciNet  MATH  Google Scholar 

  28. J. A. Sethian, Curvature Flow and Entropy Conditions Applied to Grid Generation, J. Comp. Phys. 115 (1994), 440–454.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. A. Sethian, Level Set Techniques for Tracking Interfaces; Fast Algorithms, Multiple Regions, Grid Generation and Shape/Character Recognition, Proc. of Second Trento Conf. on Mean Curvature Motion, July 1994.

    Google Scholar 

  30. J. A. Sethian, Algorithms for Tracking Interfaces in CFD and Material Science, Annual Review of Computational Fluid Mechanics, 1995.

    Google Scholar 

  31. J. A. Sethian, A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proceedings of the National Academy of Sciences, 93:4 (1996).

    Article  Google Scholar 

  32. J. A. Sethian, A Review of the Theory, Algorithms, and Applications of Level Set Methods for Propagating Interfaces, to appear, in press, Acta Numerica, 1995.

    Google Scholar 

  33. J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, in press, Cambridge University Press, 1996.

    MATH  Google Scholar 

  34. J. A. Sethian, R. Malladi, D. Adalsteinsson, and R. Kimmel, Fast Marching Methods for Computing Solutions to Static Hamilton-Jacobi Equations, submitted for publication, SIAM Journal of Numerical Analysis, Jan. 1996.

    Google Scholar 

  35. J. A. Sethian and J. D. Strain, Crystal Growth and Dendritic Solidification J. Comp. Phys. 98 (1992), 231–253.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Zhu and J. A. Sethian, Projection Methods Coupled to Level Set Interface Techniques, J. Comp. Phys. 102 (1992), 128–138.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Lawrence Berkeley Laboratory, University of California, Berkeley, USA

    Ravi Malladi

  2. Department of Mathematics, University of California, Berkeley, USA

    James A. Sethian

Authors
  1. Ravi Malladi
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  2. James A. Sethian
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Editors and Affiliations

  1. Wissenschaftliche Visualisierung, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustraße 7, D-14195, Berlin, Germany

    Hans-Christian Hege

  2. Fachbereich 3, Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Konrad Polthier

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© 1997 Springer-Verlag Berlin Heidelberg

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Malladi, R., Sethian, J.A. (1997). Level Set Methods for Curvature Flow, Image Enchancement, and Shape Recovery in Medical Images. In: Hege, HC., Polthier, K. (eds) Visualization and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59195-2_21

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  • DOI: https://doi.org/10.1007/978-3-642-59195-2_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63891-6

  • Online ISBN: 978-3-642-59195-2

  • eBook Packages: Springer Book Archive

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