A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data

Journal of Computational Physics

Volumn 221, Issue 2, 2007, Pages 539-557

  Ramlau, Ronny ^a   Ring, Wolfgang ^b  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b UNIVERSITY OF GRAZ   (Austria)

Author keywords

Active contours; Inverse problems; Level set method; Mumford Shah functional; Shape optimization; Shape sensitivity analysis; X ray tomography

Indexed keywords

COMPUTERIZED TOMOGRAPHY; HEURISTIC METHODS; IMAGE SEGMENTATION; IMAGING SYSTEMS; NUMERICAL METHODS; SENSITIVITY ANALYSIS; SHAPE OPTIMIZATION;

ACTIVE CONTOURS; DESCENT DIRECTIONS; LEVEL SET; LEVEL SET METHOD; MUMFORD-SHAH; MUMFORD-SHAH FUNCTIONALS; PIECE-WISE CONSTANTS; SHAPE SENSITIVITY ANALYSIS; SHAPE-OPTIMIZATION; X-RAY TOMOGRAPHY;

INVERSE PROBLEMS;

EID: 33846386077     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.06.041     Document Type: Article

References (47)

1. Ambrosio L., and Tortorelli V.M. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43 8 (1990) 999-1036
2. Aubert G., Barlaud M., Faugeras O., and Jehan-Besson S. Image segmentation using active contours: calculus of variations or shape gradients?. SIAM J. Appl. Math. 63 6 (2003) 2128-2154 (electronic)
3. Aubert G., and Kornprobst P. Mathematical Problems in Image Processing (2002), Springer-Verlag, New York Partial differential equations and the calculus of variations, With a foreword by Olivier Faugeras
4. Leah Bar, Nir Sochen, Nahum Kiryati, Variational pairing of image segmentation and blind restoration, in: Pajdla, Tomás, et al. (Eds.), Computer Vision - ECCV 2004. 8th European Conference on Computer Vision, Prague, Czech Republic, May 11-14, 2004. Proceedings, Part II. Lecture Notes in Computer Science 3022, Springer, Berlin, 2004, pp. 166-177.
5. Ameur H.B., Burger M., and Hackl B. Level set methods for geometric inverse problems in linear elasticity. Inverse Probl. 20 3 (2004) 673-696
6. Blake A., and Zisserman A. Visual Reconstruction. MIT Press Series in Artificial Intelligence (1987), MIT Press, Cambridge, MA
7. Burger M., Hackl B., and Ring W. Incorporating topological derivatives into level set methods. J. Comp. Phys. 194 1 (2004) 344-362
8. Burger M. Levenberg-Marquardt level set methods for inverse obstacle problems. Inverse Probl. 20 1 (2004) 259-282
9. Caselles V., Catté F., Coll T., and Dibos F. A geometric model for active contours in image processing. Numer. Math. 66 1 (1993) 1-31
10. Chambolle A. Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 3 (1995) 827-863
11. Chan T.F., and Vese L.A. Active contours without edges. IEEE Trans. Image Process. 10 2 (2001) 266-277
12. T.F. Chan, L.A. Vese, A level set algorithm for minimizing the Mumford-Shah functional in image processing. in: IEEE/Computer Society Proceedings of the 1st IEEE Workshop on "Variational and Level Set Methods in Computer Vision", 2001, pp. 161-168.
13. Chan T.F., and Vese L.A. A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vision 50 3 (2002) 271-293
14. Chan T.F., and Tai X.-C. Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193 1 (2004) 40-66
15. Cohen L.D., and Kimmel R. Global minimum for active contour models: a minimum path approach. Int. J. Comput. Vision 24 1 (1997) 57-78
16. Delfour M.C., and Zolésio J.-P. Shapes and Geometries (2001), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA Analysis, differential calculus, and optimization
17. Dorn O. Shape reconstruction in scattering media with voids using a transport model and level sets. Can. Appl. Math. Q. 10 2 (2002) 239-275
18. Dorn O., Miller E.L., and Rappaport C.M. A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Probl. 16 5 (2000) 1119-1156 Electromagnetic imaging and inversion of the Earth's subsurface
19. Engl H.W., Hanke M., and Neubauer A. Regularization of Inverse Problems (1996), Kluwer Academic Publishers, Dortrecht
20. Gordon R., Bender R., and Herman G.T. Algebraic reconstruction techniques (art) for three dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29 (1970) 471-481
21. Harabetian E., and Osher S. Regularization of ill-posed problems via the level set approach. SIAM J. Appl. Math. 58 6 (1998) 1689-1706 (electronic)
22. Hintermüller M., and Ring W. An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Image Vision 20 1-2 (2004) 19-42
23. Hintermüller M., and Ring W. A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64 2 (2003) 442-467
24. Hintermüller M., and Ring W. A level set approach for the solution of a state-constrained optimal control problem. Numer. Math. 98 1 (2004) 135-166
25. Ito K., Kunisch K., and Li Z. Level-set function approach to an inverse interface problem. Inverse Probl. 17 5 (2001) 1225-1242
26. 2S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation. Int. J. Comp. Vision 53 1 (2003) 45-70
27. S. Jehan-Besson, M. Barlaud, G. Aubert, Video object segmentation using Eulerian region-based active contours, in: International Conference on Computer Vision, Vancouver, July 2001.
28. Kass M., Witkin A., and Terzopoulos D. Snakes; active contour models. Int. J. Comput. Vision 1 (1987) 321-331
29. J. Kim, A. Tsai, M. Çetin, A.S. Willsky, A curve evolution-based variational approach to simultaneous image restoration and segmentation. in: IEEE International Conference on Image Processing, Rochester, NY, vol. 1, 2002, pp. 109-112.
30. Litman A., Lesselier D., and Santosa F. Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Probl. 14 (1998) 685-706
31. Ring W. A Newton-type total variation diminishing flow. Proc. of the CMA conference on image processing and related inverse problems (2006), Springer Verlag, Oslo
32. Mumford D., and Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 5 (1989) 577-685
33. Natterer F. The Mathematics of Computerized Tomography. Classics in Applied Mathematics vol. 32 (2001), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA Reprint of the 1986 original
34. Osher S., and Sethian J.A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 1 (1988) 12-49
35. Osher S.J., and Fedkiw R.P. Level Set Methods and Dynamic Implicit Surfaces (2002), Springer Verlag, New York
36. Osher S.J., and Santosa F. Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 1 (2001) 272-288
37. N. Paragios, R. Deriche, Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision, Int. J. Visual Commun. Image Represent. (Special Issue on Partial Differential Equations in Image Processing, Computer Vision and Computer Graphics, 2001), to appear.
38. Ramachandran G.N., and Lakshminarayanan A.V. Three-dimensional reconstructions from radiographs and electron micrographs: application of convolutions instead of fourier transforms. Proc. Natl. Acad. Sci. USA 68 (1974) 2236-2240
39. Rieder A. Keine Probleme mit inversen Problemen (2003), Friedr. Vieweg & Sohn, Braunschweig Eine Einführung in ihre stabile Lösung. [An introduction to their stable solution]
40. Santosa F. A level-set approach for inverse problems involving obstacles. ESAIM: Control Optim. Calc. Variat. 1 (1996) 17-33
41. Sapiro G. Geometric Partial Differential Equations and Image Analysis (2001), Cambridge University Press, Cambridge
42. Sethian J.A. Level Set Methods and Fast Marching Methods. second ed. (1999), Cambridge University Press, Cambridge Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science
43. Shepp L.A., and Logan B.F. The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. NS-21 (1974) 21-43
44. Sokołowski J., and Zolésio J.-P. Introduction to Shape Optimization (1992), Springer-Verlag, Berlin Shape sensitivity analysis
45. Terzopoulos D. Deformable models: classic, topology-adaptive and generalized formulations. Geometric Level Set Methods in Imaging, Vision, and Graphics (2003), Springer, New York 21-40
46. Tsai A., Yezzi A., and Willsky A.S. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10 8 (2001) 1169-1186
47. Weickert J. Anisotropic Diffusion in Image Processing. ECMI Series (1998), Teubner, Stuttgart