Mumford-Shah based registration: A comparison of a level set and a phase field approach

Computing and Visualization in Science

Volumn 12, Issue 3, 2009, Pages 101-114

  Droske, Marc ^a   Ring, Wolfgang ^b   Rumpf, Martin ^a  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF GRAZ   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

IMAGE PROCESSING; IMAGE SEGMENTATION;

EDGE-BASED APPROACHES; FREE DISCONTINUITY PROBLEMS; IMAGE INTENSITIES; INITIAL SEGMENTATIONS; LEVEL SETS; MUMFORD SHAHS; OBJECT CONTOURS; PHASE FIELDS; PHASE-FIELD APPROACHES; SHARP INTERFACES; SPATIAL ALIGNMENTS; VARIATIONAL FORMULATIONS; VARIATIONAL PROBLEMS;

PHASE INTERFACES;

EID: 60349104078     PISSN: 14329360     EISSN: 14330369     Source Type: Journal    
DOI: 10.1007/s00791-008-0084-2     Document Type: Article

References (73)

1. Alvarez, L., Snchez, J., Deriche, R., Weickert, J.: Dense disparity map estimation respecting image discontinuities: a pde and scale-space based approach. IAPR Workshop on Machine Vision no. 28-30, pp. 423-427 (2000)
2. L. Ambrosio L. Faina R. March 2001 Variational approximation of a second order free discontinuity problem in computer vision SIAM J. Math. Anal. 32 6 1171 1197
3. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation And Free Discontinuity Problems. Oxford University Press, Oxford (2000)
4. L. Ambrosio V.M. Tortorelli 1990 Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence Comm. Pure Appl. Math. 43 999 1036
5. L. Ambrosio V.M. Tortorelli 1992 On the approximation of free discontinuity problems Boll. Un. Mat. Ital. B 6 7 105 123
6. J. Ball 1988 Global invertibility of Sobolev functions and the interpenetration of matter Proc. R. Soc. Edin. 88 315 328
7. G. Belletini A. Coscia 1994 Discrete approximation of a free discontinuity problem Numer. Funct. Anal. Optim. 15 201 224
8. A. Bonnet 1996 On the regularity of the edge set of Mumford-Shah minimizers Prog. Nonlinear Diff. Equ. Appl. 25 93 103
9. Bornemann, F., Rasch, C.: Finite-Element discretization of static Hamilton-Jacobi equations based on a local variation principle. (2004). See http://www.springerlink.com/content/v61q331816620gq1
10. B. Bourdin 1999 Image segmentation with a Finite Element method ESIAM: Math. Modell. Numer. Anal. 33 2 229 244
11. B. Bourdin A. Chambolle 2000 Implementation of a Finite-Element approximation of the Mumford-Shah functional Numer. Math. 85 4 609 646
12. Braides, A.: Approximation Of Free Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Heidelberg (1998)
13. A. Brook R. Kimmel N.A. Sochen 2003 Variation restoration and edge detection for color images J. Math. Image Vis. 18 247 268
14. M. Burger 2003 A framework for the construction of level set methods for shape optimization and reconstruction Interfaces Free Bound. 5 301 329
15. M. Burger 2004 Levenberg-Marquardt level set methods for inverse obstacle problems Inv. Prob. 20 259 282
16. M. Burger B. Hackl W. Ring 2004 Incorporating topological derivatives into level set methods J. Comput. Phys. 194 344 362
17. V. Caselles F. Catté T. Coll F. Dibos 1993 A geometric model for active contours in image processing Numer. Math. 66 1 31
18. V. Caselles R. Kimmel G. Sapiro 1997 Geodesic active contours Int. J. Comput. Vis. 22 1 61 79
19. A. Chambolle 1995 Image segmentation by variatonal methods: Mumford-Shah functional and the discrete approximations SIAM J. Appl. Math. 55 3 827 863
20. Chan, T.F., Vese, L.A.: Image segmentation using level sets and the piecewise constant Mumford-Shah model. UCLA CAM Report 00-14, University of California, Los Angeles (2000)
21. 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)
22. T.F. Chan L.A. Vese 2001 Active contours without edges IEEE Trans. Image Process. 10 2 266 277
23. Ciarlet, P.G.: Three-Dimensional Elasticity. Elsevier, New York (1988)
24. Clarenz, U., Droske, M., Rumpf, M.: Towards fast non-rigid registration. In: Inverse Problems, Image Analysis and Medical Imaging. AMS Special Session Interaction of Inverse Problems and Image Analysis, vol. 313, pp. 67-84. AMS, New York (2002)
25. Clarenz, U., Henn, S., Rumpf M., Witsch, K.: Relations between optimization and gradient flow methods with applications to image registration. In: Proceedings of the 18th GAMM Seminar Leipzig on Multigrid and Related Methods for Optimisation Problems, pp. 11-30 (2002)
26. Cremers, D., Kohlberger, T., Schnörr, C.: Nonlinear shape statistics in Mumford-Shah based segmentation. In: Heyden, A. et al. (eds) European Conference on Computer Vision (ECCV), Copenhagen, vol. 2351, pp. 93-108. Springer LNCS, Heidelberg (2002)
27. D. Cremers S. Soatto 2005 Motion competition: a variational framework for piecewise parametric motion segmentation Int. J. Comput. Vis. 62 3 249 265
28. D. Cremers F. Tischhäuser J. Weickert C. Schnörr 2002 Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional Int. J. Comput. Vis. 50 3 1364 1379
29. Dacorogna, B.: Direct Methods in the Calculus of Variations. Appl. Math. Sciences, vol. 78. Springer, Berlin (1989)
30. G. Dal Maso J. Morel S. Solimini 1996 A variational method in image segmentation: existence and approximation results Acta Math. 168 89 151
31. C.A. Davatzikos R.N. Bryan J.L. Prince 1996 Image registration based on boundary mapping IEEE Trans. Med. Image 15 1 112 115
32. E. De Giorgi M. Carriero A. Leaci 1989 Existence theorem for a minimum problem with free discontinuity set Arch. Rat. Mech. Anal. 108 195 218
33. Delfour, M.C., Zolésio, J.: Geometries and Shapes: Analysis, Differential Calculus and Optimization. Adv. Des. Control 4. SIAM, Philadelphia (2001)
34. Dibos, F., Séré: An approximation result for the minimizers of the Mumford-Shah functional. Boll. Un. Math. Ital. 11A(1), (1997)
35. Droske, M.: On variational problems and gradient flows in image processing. Ph.D. thesis, Universität Duisburg-Essen (2005)
36. M. Droske W. Ring 2006 A Mumford-Shah level-set approach for geometric image registration SIAM J. Appl. Math. 66 6 2127 2148
37. M. Droske M. Rumpf 2007 Multi scale joint segmentation and registration of image morphology IEEE Trans. Pattern Anal. Mach. Intell. 29 12 2181 2194
38. M. Droske M. Rumpf 2004 A variational approach to non-rigid morphological registration SIAM Appl. Math. 64 2 668 687
39. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
40. Feng, X., Prohl, A.: Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting. Math. Modell. Numer. Anal. (1999)
41. Féron, O., Mohammad-Djafari, A.: Image fusion and unsupervised joint segmentation using a hmm and mcmc algorithms. J. Electron. Image 14(2) (2005)
42. Fried, M.: Image segmentation using adaptive finite elements. Technical Report, University of Canberra, Australia (2003)
43. U. Grenander M.I. Miller 1998 Computational anatomy: an emerging discipline Q. Appl. Math. LVI 4 617 694
44. Hackbusch, W.: Multigrid Methods and Applications. Springer, Berlin (1985)
45. W. Hackbusch S. Sauter 1997 Composite finite elements for the approximation of pdes on domains with complicated micro-structures Numer. Math. 75 447 472
46. S. Henn 2003 A Levenberg-Marquardt scheme for nonlinear image registration BIT Numer. Math. 43 4 743 759
47. M. Hintermüller W. Ring 2003 A second order shape optimization approach for image segmentation SIAM Appl. Math. 64 2 442 467
48. M. Hintermüller W. Ring 2004 An inexact Newton-cg-type active contour approach for the minimization of the Mumford-Shah functional J. Math. Image Vis. 20 19 42
49. K. Höllig U. Reif J. Wipper 2001 Weighted extended b-spline approximation of dirichlet problems SIAM J. Numer. Anal. 39 2 442 462
50. S.C. Joshi M.I. Miller 2000 Landmark matching via large deformation diffeomorphisms IEEE Trans. Med. Image 9 8 1357 1370
51. Kapur, T., Yezzi, L., Zöllei, L.: A variational framework for joint segmentation and registration. IEEE CVPR-MMBIA (2001)
52. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol. 120. Springer, New York (1996)
53. F. Maes A. Collignon D. Vandermeulen G. Marchal P. Suetens 1997 Multi-modal volume registration by maximization of mutual information IEEE Trans. Med. Image 16 7 187 198
54. Mikula, K., Sarti, A., Sgallari, F.: Handbook of medical image analysis: segmentation and registration models. In: Suri, J. et al. (eds) Co-volume level set method in subjective surface based medical image segmentation, Kluwer Academic/Plenum Publishers, New York (2004)
55. Miller, M., Trouve, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375-405 (2002, in press)
56. Modersitzki, J., Fischer, B.: Fast diffusion registration. Special Issue of Contemporary Mathematics, AMS, New York (2000)
57. Modersitzki, J., Fischer, B.: Curvature based image registration. JMIV 18(1) (2003)
58. Morel, J., Solimini, S.: Variational models in image segmentation. Birkäuser, Basel (1994)
59. D. Mumford J. Shah 1989 Optimal approximation by piecewise smooth functions and associated variational problems Comm. Pure Appl. Math. 42 577 685
60. Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Heidelberg (2002)
61. Osher, S.J., Paragios, N.: Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, Heidelberg (2003)
62. S.J. Osher J.A. Sethian 1988 Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 79 12 49
63. T.J. Richardson S.K. Mitter 1997 A variational formulation based edge focussing algorithm Sadhana Acad. P. Eng. S. 22 4 553 574
64. Sauter, S., Stahn, N.: Composite finite elements and multi-grid. part i: Convergence theory in 1-d. Technical Report, Preprint, 11-01, University of Zürich (2001)
65. Sethian, J.A.: Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge University Press, Cambridge (1996)
66. Sokołowski, J., Zolésio, J.P.: Introduction to shape optimization. In: Shape Sensitivity Analysis. Springer, Berlin (1992)
67. J.P. Thirion 1998 Image matching as a diffusion process: an analogy with maxwell's demon Med. Image Anal. 2 243 260
68. A. Tsai A. Yezzi A. Willsky 2001 Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification IEEE Trans. Image Process. 10 8 1169 1186
69. Unal, G., Slabaugh, G.: Coupled PDEs for non-rigid registration and segmentation. In: IEEE Computer Society International Conference on Computer Vision and Pattern Recognition (CVPR) (2005)
70. B. Vemuri J. Ye Y. Chen C. Leonard 2003 Image registration via level-set motion: applications to atlas-based segmentation Med. Image Anal. 7 1 20
71. Viola, P.A., Wells III, W.: Alignment by maximization of mutual information. Technical Report. AITR-1548 (1995)
72. Warnke, R.: Schnelle Löser für elliptische Randwertprobleme mit springenden Koeffizienten. Dissertation, Zürich (2003)
73. Wells, W., Viola, P., Atsumi, H., Nakajima, S., Kikinis, R.: Multi-modal volume registration by maximization of mutual information (1996)