A piecewise constant level set method for elliptic inverse problems

Applied Numerical Mathematics

Volumn 57, Issue 5-7 SPEC. ISS., 2007, Pages 686-696

  Tai, Xue Cheng ^a   Li, Hongwei ^a  

^a UNIVERSITY OF BERGEN   (Norway)

Author keywords

Inverse problem; Level set method; Operator splitting; Piecewise constant

Indexed keywords

NUMERICAL METHODS; SET THEORY; SPURIOUS SIGNAL NOISE;

LEVEL SET METHOD; OPERATOR SPLITTING; PIECEWISE CONSTANT;

INVERSE PROBLEMS;

EID: 34147142049     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2006.07.010     Document Type: Article

References (31)

1. Ascher U., and Haber E. Grid refinement and scaling for distributed parameter estimation problems. Inverse Problems 17 (2001) 571-590
2. U. Ascher, E. Haber, Computational methods for large distributed parameter estimation problems with possible discontinuities, in: M. Colaco, H. Orlande, G. Dulikravich (Eds.), Symp. Inverse Problems, Design and Optimization, 2004, pp. 201-208. Can be downloaded from: http://www.cs.ubc.ca/spider/ascher/papers/ah3.pdf or http://www.lmt.coppe.ufrj.br/ipdo/papers%5Cipdo-042.pdf
3. Ascher U., Haber E., and Huang H. On effective methods for implicit piecewise smooth surface recovery. SIAM J. Scient. Comput. 28 (2006) 339-368
4. Burger M. A level set method for inverse problems. Inverse Problems 17 (2001) 1327-1355
5. Burger M., and Osher S.J. A survey on level set methods for inverse problems and optimal design. European J. Appl. Math. 16 2 (2005) 263-301
6. Chan T.F., and Tai X.-C. Augmented Lagrangian and total variation methods for recovering discontinuous coefficients from elliptic equations. In: Bristeau M., Etgen G., Fitzgibbon W., Lions J.L., Periaux J., and Wheeler M.F. (Eds). Computational Science for the 21st Century (1997), Willey, New York 597-607
7. Chan T.F., and Tai X.-C. Identification of discontinuous coefficients from elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25 3 (2003) 881-904
8. Chan T.F., and Tai X.-C. Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193 (2003) 40-66
9. Chavent G., and Kunisch K. Regularization of linear least squares problems by total bounded variation. ESAIM Control Optim. Calc. Var. 2 (1997) 359-376 (electronic)
10. Chen Z., and Zou J. An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim. 37 3 (1999) 892-910 (electronic)
11. O. Christiansen, X.-C. Tai, Fast implementation of piecewise constant level set methods, Cam-report-06, UCLA, Applied Mathematics, 2005
12. Chung E.T., Chan T.F., and Tai X.-C. Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205 (2005) 357-372
13. Chung J.T., and Vese L.A. Energy minimization based segmentation and denoising using a multilayer level set approach. In: Rangarajan A., Vemuri B.C., and Yuille A.L. (Eds). Lecture Notes in Computer Science vol. 3757 (2005), Springer, Berlin 439-455
14. Dorn O., Miller E., and Rappaport C. A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Problems 16 (2000) 1119-1156 (special issue on Electromagnetic Imaging and Inversion of the Earth's Subsurface)
15. S. Esedoglu, Y.-H.R. Tsai, Threshold dynamics for the piecewise constant Mumford-shah functional, Tech. Rep. CAM-report-04-63, UCLA Dep. Math., 2004
16. Gibou F., and Fedkiw R. A fast hybrid k-means level set algorithm for segmentation. Proceedings of 4th Annual Hawaii International Conference on Statistics and Mathematics (2005) 281-291 Also as Stanford Technical Report, 2002
17. Ito K., Kunisch K., and Li Z. Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225-1242
18. Lie J., Lysaker M., and Tai X.-C. Piecewise constant level set methods and image segmentation. In: Kimmel R., Sochen N., and Weickert J. (Eds). Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005. Lecture Notes in Computer Science vol. 3459 (2005), Springer, Heidelberg 573-584
19. J. Lie, M. Lysaker, X.-C. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. Image Process., in press. Also as UCLA, Applied Math., CAM-report-04-31, 2004
20. Lie J., Lysaker M., and Tai X.-C. A variant of the levelset method and applications to image segmentation. Math. Comp. 75 (2006) 1155-1174
21. Litman A. Reconstruction by level sets of n-ary scattering obstacles. Inverse Problems 21 6 (2005) S131-S152
22. Lu T., Neittaanmäki P., and Tai X.-C. A parallel splitting up method and its application to Navier-Stokes equations. Appl. Math. Lett. 4 2 (1991) 25-29
23. Lu T., Neittaanmäki P., and Tai X.-C. A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations. RAIRO Modél. Math. Anal. Numér. 26 6 (1992) 673-708
24. Nilssen T.K., and Tai X.C. Parameter estimation with the augmented Lagrangian method for a parabolic equation. J. Optim. Theory Appl. 124 2 (2005) 435-453
25. Osher S., and Sethian J.A. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49
26. Santosa F. A level-set approach for inverse problems involving obstacles. ESAIM: Contr. Optim. Calc. Var. 1 (1996) 17-33
27. J. Shen, Gamma-convergence approximation to piecewise constant Mumford-Shah segmentation, Tech. Rep. CAM-report-05-16, UCLA Dep. Math., 2005
28. B. Song, T. Chan, A fast algorithm for level set based optimization, CAM-UCLA, 68, 2002
29. Tai X.-C., and Chan T.F. A survey on multiple level set methods with some applications for identifying piecewise constant functions. Int. J. Numer. Anal. Modelling 1 1 (2004) 25-48
30. X.-C. Tai, C. Yao, Fast piecewise constant level set method with newton updating, Technical Report, UCLA, Applied Math. CAM-report-05-52, 2005
31. X.-C. Tai, O. Christiansen, P. Lin, I. Skjaelaaen, A remark on the mbo scheme and some piecewise constant level set methods, Technical Report, UCLA, Applied Mathematics, CAM-report-05-24, 2005