Advertisement

An Inexact Newton-CG-Type Active Contour Approach for the Minimization of the Mumford-Shah Functional

  • Michael Hintermüller
  • Wolfgang Ring
Article

Abstract

The problem of segmentation of a given gray scale image by minimization of the Mumford-Shah functional is considered. The minimization problem is formulated as a shape optimization problem where the contour which separates homogeneous regions is the (geometric) optimization variable. Expressions for first and second order shape sensitivities are derived using the speed method from classical shape sensitivity calculus. Second order information (the shape Hessian of the cost functional) is used to set up a Newton-type algorithm, where a preconditioning operator is applied to the gradient direction to obtain a better descent direction. The issue of positive definiteness of the shape Hessian is addressed in a heuristic way. It is suggested to use a positive definite approximation of the shape Hessian as a preconditioner for the gradient direction. The descent vector field is used as speed vector field in the level set formulation for the propagating contour. The implementation of the algorithm is discussed in some detail. Numerical experiments comparing gradient and Newton-type flows for different images are presented.

level set method shape sensitivity analysis image segmentation active contours Newton algorithm Mumford-Shah functional 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Adalsteinsson and J.A. Sethian, "The fast construction of extension velocities in level set methods," J. Comput. Phys., Vol. 148, No. 1, pp. 2–22, 1999.Google Scholar
  2. 2.
    G. Allaire, F. Jouve, and A.-M. Toader, "A level-set method for shape optimization," C.R. Acad. Sci. Paris, Ser. I, Vol. 334, pp. 1125–1130, 2002.Google Scholar
  3. 3.
    L. Ambrosio and V.M. Tortorelli, "Approximation of functionals depending on jumps by elliptic functionals via-convergence," Comm. Pure Appl. Math., Vol. 43, No. 8, pp. 999–1036, 1990.Google Scholar
  4. 4.
    G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, "Image segmentation using active contours: Calculus of variations or shape gradients?" I.N.R.I.A. Rapport de Recherche, I.N.R.I.A. Paris, No. 4483, 2002.Google Scholar
  5. 5.
    G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Springer-Verlag, New York, 2002. Partial differential equations and the calculus of variations, With a foreword by Olivier Faugeras.Google Scholar
  6. 6.
    M.P. Bendsoe, "Topology design of structures, materials and mechanisms-status and perspectives," in System modelling and optimization, Proceedings of the 19th IFIP TC7 Conference held in Cambridge, M.J.D. Powell and S. Scholtes (Eds.), July 12-16, 1999, pp. 1–17, Boston, MA, 2000. Kluwer Academic Publishers.Google Scholar
  7. 7.
    A. Blake and A. Zisserman, Visual Reconstruction. MIT Press Series in Artificial Intelligence. MIT Press, Cambridge, MA, 1987.Google Scholar
  8. 8.
    B. Bourdin and A. Chambolle, "Implementation of an adaptive finite-element approximation of the Mumford-Shah functional," Numer. Math., Vol. 85, No. 4, pp. 609–646, 2000.Google Scholar
  9. 9.
    A. Braides and G. Dal Maso, "Non-local approximation of the Mumford-Shah functional," Calc. Var. Partial Differential Equations, Vol. 5, No. 4, pp. 293–322, 1997.Google Scholar
  10. 10.
    A. Braides, Approximation of Free-discontinuity Problems, Vol-ume 1694 of Lecture Notes in Mathematics. Springer-Verlag: Berlin, 1998.Google Scholar
  11. 11.
    V. Caselles, R. Kimmel, G. Sapiro, and C. Sbert, "Minimal sur-faces based object segmentation," IEEE Trans. on Pattern Anal. Machine Intell., Vol. 19, No. 4, pp. 394–398, 1997.Google Scholar
  12. 12.
    A. Chambolle, "Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations," SIAM J. Appl. Math., Vol. 55, No. 3, pp. 827–863, 1995.Google Scholar
  13. 13.
    A. Chambolle, "Finite-differences discretizations of the Mumford-Shah functional," M2AN Math. Model. Numer. Anal., Vol. 33, No. 2, pp. 261–288, 1999.Google Scholar
  14. 14.
    A. Chambolle and G.D. Maso, "Discrete approximation of the Mumford-Shah functional in dimension two," M2AN Math. Model. Numer. Anal., Vol. 33, No. 4, pp. 651–672, 1999.Google Scholar
  15. 15.
    T.F. Chan, B.Y. Sandberg, and L.A. Vese, "Active contours with-out edges for vector-valued images," Journal of Visual Com-munication and Image Representation, Vol. 11, pp. 130–141, 2000.Google Scholar
  16. 16.
    T.F. Chan and L.A. Vese, "Image segmentation using level sets and the piecewise constant Mumford-Shah model," UCLACAM Report 00-14, University of California, Los Angeles, 2000.Google Scholar
  17. 17.
    T.F. Chan and L.A. Vese, "A level set algorithm for minimiz-ing the Mumford-Shah functional in image processing," UCLA CAM Report 00-13, University of California, Los Angeles, 2000.Google Scholar
  18. 18.
    T.F. Chan and L.A. Vese, "Active contours without edges," IEEE Trans. Image Processing, Vol. 10, No. 2, pp. 266–277, 2001.Google Scholar
  19. 19.
    T.F. Chan and L.A. Vese, "A multiphase level set framework for image segmentation using the mumford and shah model," Int. J. Comp. Vision, Vol. 50, No. 3, pp. 271–293, 2002.Google Scholar
  20. 20.
    L.D. Cohen, "Avoiding local minima for deformable curves in image analysis," in Curves and Surfaces with Applications in CAGD,A. Le Méhauté, C. Rabut, and L.L. Schumaker (Eds.), pp. 77–84. Vanderbilt University Press: Nashvill, TN, 1997.Google Scholar
  21. 21.
    L.D. Cohen and R. Kimmel, "Global minimum for active contour models: A minimum path approach," International Journal of Computer Vision, Vol. 24, No. 1, pp. 57–78, 1997.Google Scholar
  22. 22.
    M.C. Delfour and J.-P. Zolésio, Shapes and Geometries. Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 2001. Analysis, differential calculus, and optimization.Google Scholar
  23. 23.
    M.C. Delfour and J.-P. Zolésio, "Tangential calculus and shape derivatives," in Shape Optimization and Optimal Design (Cambridge, 1999), Dekker, New York, 2001, pp. 37–60.Google Scholar
  24. 24.
    L.C. Evans, Partial Differential Equations. American Mathematical Society: Providence, RI, 1998.Google Scholar
  25. 25.
    D. Geman and S. Geman, "Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images," IEEE Trans-actions on Pattern Analysis and Machine Intelligence, Vol. 6, No. 6, pp. 721–741, 1984.Google Scholar
  26. 26.
    W. Hackbusch, Theorie und Numerik elliptischer Differential-gleichungen. Teubner Verlag: Stuttgart, 1986.Google Scholar
  27. 27.
    M. Hintermüller and W. Ring, "A level set approach for the solu-tion of a state constrained optimal control problem," Technical Report 212, Special Research Center on Optimization and Con-trol, University of Graz, Austria, 2001. To appear in Numerische Mathematik.Google Scholar
  28. 28.
    M. Hintermüller and W. Ring, "A second order shape optimiza-tion approach for image segmentation," Report 237, Special Re-search Center on Optimization and Control, University of Graz, Austria, March 2002. To appear in SIAM J. Appl. Math.Google Scholar
  29. 29.
    M. Kass, A. Witkin, and D. Terzopoulos, "Snakes; active contour models," Int. J. of Computer Vision, Vol. 1, pp. 321–331, 1987.Google Scholar
  30. 30.
    F. Mémoli and G. Sapiro, "Fast computation of weighted dis-tance functions and geodesics on implicit hyper-surfaces," J. Comput. Phys., Vol. 173, No. 2, pp. 730–764, 2001.Google Scholar
  31. 31.
    J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation, Volume 14 of Progress in Nonlinear Differen-tial Equations and their Applications. Birkhäuser Boston Inc.: Boston, MA, 1995. With seven image processing experiments.Google Scholar
  32. 32.
    D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Comm. Pure Appl. Math., Vol. 42, No. 5, pp. 577–685, 1989.Google Scholar
  33. 33.
    S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag: New York, 2002.Google Scholar
  34. 34.
    J. Sethian, "A fast marching method for monotonically advancing fronts," Proc. Nat. Acad. Sci., Vol. 93, No. 4, pp. 1591–1595, 1996.Google Scholar
  35. 35.
    J.A. Sethian, "Fast marching methods," SIAM Rev., Vol. 41, No. 2, pp. 199–235 (electronic), 1999.Google Scholar
  36. 36.
    J.A. Sethian, Level Set Methods and Fast Marching Methods. 2nd edition, Cambridge University Press, Cambridge. 1999. Evolv-ing interfaces in computational geometry, fluid mechanics, com-puter vision, and materials science.Google Scholar
  37. 37.
    J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimiza-tion. Springer-Verlag: Berlin, 1992. Shape sensitivity analysis.Google Scholar
  38. 38.
    A. Tsai, A. Yezzi, and A.S. Willsky, "Curve evolution implementation of the Mumford-Shah functional for image sege-mentation, denoising, interpolation, and magnification," IEEE Transactions on Image Processing, Vol. 10, No. 8, pp. 1169–1186, 2001.Google Scholar
  39. 39.
    Y.R. Tsai, L.-T. Cheng, S. Osher, and H.-K. Zhao, "Fast sweep-ing algorithms for a class of Hamilton-Jacobi equations," CAM-Report 01-27, Computational and Applied Mathematics, De-partment of Mathematics University of California, Los Angeles, 2001.Google Scholar
  40. 40.
    Y.-H.R. Tsai, "Rapid and accurate computation of the distance function using grids," J. Comput. Phys., Vol. 178, No. 1, pp. 175–195, 2002.Google Scholar
  41. 41.
    J.N. Tsitsiklis, "Efficient algorithms for globally optimal trajec-tories," IEEE Trans. Automat. Control, Vol. 40, No. 9, pp. 1528–1538, 1995.Google Scholar
  42. 42.
    J. Wloka, Partielle Differentialgleichungen.Teubner: Stuttgart, 1982.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Michael Hintermüller
    • 1
  • Wolfgang Ring
    • 1
  1. 1.Special Research Center on Optimization and ControlUniversity of Graz, Institute of MathematicsGrazAustria

Personalised recommendations