Advertisement

An Active Contour Approach for a Mumford-Shah Model in X-Ray Tomography

  • Elena Hoetzl
  • Wolfgang Ring
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5876)

Abstract

This paper presents an active contour approach for the simultaneous inversion and segmentation of X-ray tomography data from its Radon Transform. The optimality system is found as the necessary optimality condition for a Mumford-Shah like functional over the space of piecewise smooth densities, which may be discontinuous across the contour. In our approach the functional variable is eliminated by solving a classical variational problem for each fixed geometry. The solution is then inserted in the Mumford-Shah cost functional leading to a geometrical optimization problem for the singularity set. The resulting shape optimization problem is solved using shape sensitivity calculus and propagation of shape variables in the level-set form.

As a special feature of this paper, a new, second order accurate, finite difference method based approach for the solution of the optimality system is introduced and numerical experiments are presented.

Keywords

Intersection Point Optimality System Active Contour Taylor Series Expansion Descent Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubert, G., Barlaud, M., Faugeras, O., Jehan-Besson, S.: Image segmentation using active contours: calculus of variations or shape gradients? SIAM J. Appl. Math. 63(6), 2128–2154 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numer. Math. 66(1), 1–31 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chan, T.F., Vese, L.A.: A level set algorithm for minimizing the Mumford-Shah functional in image processing. UCLA CAM Report 00-13, University of California, Los Angeles (2000)Google Scholar
  4. 4.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Processing 10(2), 266–277 (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chan, T.F., Vese, L.A.: A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Comp. Vision 50(3), 271–293 (2002)zbMATHCrossRefGoogle Scholar
  6. 6.
    Cohen, L.D., Kimmel, R.: Global minimum for active contour models: a minimum path approach. Int. J. of Comp. Vision 24(1), 57–78 (1997)CrossRefGoogle Scholar
  7. 7.
    Delfour, M.C., Zolésio, J.-P.: Shapes and geometries. In: Analysis, differential calculus, and optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)Google Scholar
  8. 8.
    Hoetzl, E.: Numerical treatment of a mumford-shah model for x-ray tomography. PhD.thesis, Karl Franzens University Graz, Institute for Mathematics (2009)Google Scholar
  9. 9.
    Hintermüller, M., Ring, W.: An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imag. Vis. 20(1–2), 19–42 (2004)CrossRefGoogle Scholar
  10. 10.
    Hintermüller, M., Ring, W.: A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64(2), 442–467 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jehan-Besson, S., Barlaud, M., Aubert, G.: DREAM2S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation (November 2001)Google Scholar
  12. 12.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes; active contour models. Int. J. of Computer Vision 1, 321–331 (1987)CrossRefGoogle Scholar
  13. 13.
    Litman, A., Lesselier, D., Santosa, F.: Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Problems 14, 685–706 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001); Reprint of the 1986 originalzbMATHGoogle Scholar
  15. 15.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Paragios, N., Deriche, R.: Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Int. J. of Vis. Communication and Image Representation (2001) (to appear)Google Scholar
  17. 17.
    Ramlau, R., Ring, W.: A mumford – shah level – set approach for the inversion and segmentation of x – ray tomography data. Journal of Computational Physics (221), 539–557 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Santosa, F.: A level-set approach for inverse problems involving obstacles. ESAIM: Control, Optimization and Calculus of Variations 1, 17–33 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sokołowski, J., Zolésio, J.-P.: Introduction to shape optimization. Springer, Berlin (1992); Shape sensitivity analysiszbMATHGoogle Scholar
  20. 20.
    Tsai, A., Yezzi, A., Willsky, A.S.: Curve evolution implementation of the Mumford-Shah functional for image segementation, denoising, interpolation, and magnification. IEEE Transactions on image processing 10(8), 1169–1186 (2001)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Elena Hoetzl
    • 1
  • Wolfgang Ring
    • 1
  1. 1.Institute of Mathematics and Scientific ComputingUniversity of Graz 

Personalised recommendations