Level Set Methods for Structural Inversion and Image Reconstruction

  • Oliver Dorn
  • Dominique Lesselier
Reference work entry


In this chapter, an introduction is given into the use of level set techniques for inverse problems and image reconstruction. Several approaches are presented which have been developed and proposed in the literature since the publication of the original (and seminal) paper by F. Santosa in 1996 on this topic. The emphasis of this chapter, however, is not so much on providing an exhaustive overview of all ideas developed so far, but on the goal of outlining the general idea of structural inversion by level sets, which means the reconstruction of complicated images with interfaces from indirectly measured data. As case studies, recent results (in 2D) from microwave breast screening, history matching in reservoir engineering, and crack detection are presented in order to demonstrate the general ideas outlined in this chapter on practically relevant and instructive examples. Various references and suggestions for further research are given as well.


Descent Direction History Match Shape Evolution Topological Derivative Shape Derivative 
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.



OD thanks Diego Álvarez, Natalia Irishina, Miguel Moscoso and Rossmary Villegas for their collaboration on the exciting topic of level set methods in image reconstruction, and for providing figures which have been included in this chapter. He thanks the Spanish Ministerio de Educacion y Ciencia (Grants FIS2004-22546-E and FIS2007-62673), the European Union (Grant FP6-503259), the French CNRS and Univ. Paris Sud 11, and the Research Councils UK for their support of some of the work which has been presented in this chapter. DL thanks Jean Cea for having introduced him to the fascinating world of shape optimal design, Fadil Santosa for his contribution to his understanding of the linkage between shape optimal design and level set evolutions, and Jean-Paul Zolésio for his precious help on both topics, plus his many insights on topological derivatives.

References and Further Reading

  1. 1.
    Abascal JFPJ, Lambert M, Lesselier D, Dorn O (2009) 3-D eddy-current imaging of metal tubes by gradient-based, controlled evolution of level sets. IEEE Trans Magn 44:4721–4729CrossRefGoogle Scholar
  2. 2.
    Alexandrov O, Santosa F (2005) A topology preserving level set method for shape optimization. J Comput Phys 204:121–130MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194: 363–393MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alvarez D, Dorn O, Irishina N, Moscoso M (2009) Crack detection using a level set strategy. J Comput Phys 228:5710–57211MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ammari H, Calmon P, Iakovleva E (2008) Direct elastic imaging of a small inclusion. SIAM J Imaging Sci 1:169–187MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ammari H, Kang H (2004) Reconstruction of small inhomogeneities from boundary measurements. Lecture notes in mathematics, vol 1846. Springer, BerlinMATHCrossRefGoogle Scholar
  7. 7.
    Amstutz S, Andrä H (2005) A new algorithm for topology optimization using a level-set method. J Comput Phys 216:573–588CrossRefGoogle Scholar
  8. 8.
    Ascher UM, Huang H, van den Doel K (2007) Artificial time integration. BIT Numer Math 47:3–25MATHCrossRefGoogle Scholar
  9. 9.
    Bal G, Ren K (2006) Reconstruction of singular surfaces by shape sensitivity analysis and level set method. Math Model Meth Appl Sci 16:1347–1374MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ben Hadj Miled MK, Miller EL (2007) A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography. Inverse Prob 23:2375–2400MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ben Ameur H, Burger M, Hackl B (2004) Level set methods for geometric inverse problems in linear elasticity. Inverse Prob 20: 673–696MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Benedetti M, Lesselier D, Lambert M, Massa A (2010) Multiple-shape reconstruction by means of mutliregion level sets. IEEE Trans Geosci Remote Sens 48:2330–2342CrossRefGoogle Scholar
  13. 13.
    Berg JM, Holmstrom K (1999) On parameter estimation using level sets. SIAM J Control Optim 37:1372–1393MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Berre I, Lien M, Mannseth T (2007) A level set corrector to an adaptive multiscale permeability prediction. Comput Geosci 11: 27–42MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Bonnet M, Guzina BB (2003) Sounding of finite solid bodies by way of topological derivative. Int J Numer Methods Eng 61:2344–2373MathSciNetCrossRefGoogle Scholar
  16. 16.
    Burger M (2001) A level set method for inverse problems. Inverse Prob 17:1327–1355MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Burger M, Osher S (2005) A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16:263–301MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Burger M (2003) A framework for the construction of level set methods for shape optimization and reconstruction. Inter Free Bound 5: 301–329MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Burger M (2004) Levenberg-Marquardt level set methods for inverse obstacle problems. Inverse Prob 20:259–282MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194:344–362MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Carpio A, Rapún M-L (2008) Solving inhomogeneous inverse problems by topological derivative methods. Inverse Prob 24:045014CrossRefGoogle Scholar
  22. 22.
    Céa J, Gioan A, Michel J (1973) Quelques résultats sur l’identification de domains. Calcolo 10(3–4):207–232MathSciNetCrossRefGoogle Scholar
  23. 23.
    Céa J, Haug EJ (eds) 1981 Optimization of distributed parameter structures. Sijhoff & Noordhoff, Alphen aan den RijnMATHGoogle Scholar
  24. 24.
    Céa J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimizations connection. Comput Meth Appl Mech Eng 188:713–726MATHCrossRefGoogle Scholar
  25. 25.
    Chan TF, Vese LA (2001) Active contours without edges. IEEE Trans Image Process 10:266–277MATHCrossRefGoogle Scholar
  26. 26.
    Chan TF, Tai X-C (2003) Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J Comput Phys 193:40–66MathSciNetCrossRefGoogle Scholar
  27. 27.
    Chung ET, Chan TF, Tai XC (2005) Electrical impedance tomography using level set representation and total variational regularization. J Comput Phys 205:357–372MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    DeCezaro A, Leitão A, Tai X-C (2009) On multiple level-set regularization methods for inverse problems. Inverse Prob 25:035004CrossRefGoogle Scholar
  29. 29.
    Delfour MC, Zolésio J-P (1988) Shape sensitivity analysis via min max differentiability. SIAM J Control Optim 26:34–86CrossRefGoogle Scholar
  30. 30.
    Delfour MC, Zolésio J-P (2001) Shapes and geometries: analysis, differential calculus and optimization (SIAM advances in design and control). SIAM, PhiladelphiaMATHGoogle Scholar
  31. 31.
    Dorn O, Lesselier D (2006) Level set methods for inverse scattering. Inverse Prob 22:R67–R131. doi:10.1088/0266-5611/22/4/R01MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Dorn O, Lesselier D (2009) Level set methods for inverse scattering - some recent developments. Inverse Prob 25:125001. doi:10.1088/0266-5611/25/12/125001MathSciNetCrossRefGoogle Scholar
  33. 33.
    Dorn O, Lesselier D 2007 Level set techniques for structural inversion in medical imaging. In: Deformable models. Springer, New York, pp 61–90CrossRefGoogle Scholar
  34. 34.
    Dorn O, Villegas R (2008) History matching of petroleum reservoirs using a level set technique. Inverse Prob 24:035015MathSciNetCrossRefGoogle Scholar
  35. 35.
    Dorn O, Miller E, Rappaport C (2000) A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Prob 16:1119–1156MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Duflot M (2007) A study of the representation of cracks with level sets. Int J Numer Methods Eng 70:1261–1302MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems (mathematics and its applications), vol 375. Kluwer, DordrechtGoogle Scholar
  38. 38.
    Fang W (2007) Multi-phase permittivity reconstruction in electrical capacitance tomography by level set methods. Inverse Prob Sci Eng 15:213–247MATHCrossRefGoogle Scholar
  39. 39.
    Feijóo RA, Novotny AA, Taroco E, Padra C (2003) The topological derivative for the Poisson problem. Math Model Meth Appl Sci 13: 1–20CrossRefGoogle Scholar
  40. 40.
    Feijóo GR (2004) A new method in inverse scattering based on the topological derivative. Inverse Prob 20:1819–1840MATHCrossRefGoogle Scholar
  41. 41.
    Feng H, Karl WC, Castanon DA (2003) A curve evolution approach to object-based tomographic reconstruction. IEEE Trans Image Process 12:44–57MathSciNetCrossRefGoogle Scholar
  42. 42.
    Ferrayé R, Dauvignac JY, Pichot C (2003) An inverse scattering method based on contour deformations by means of a level set method using frequency hopping technique. IEEE Trans Antennas Propagat 51:1100–1113CrossRefGoogle Scholar
  43. 43.
    Frühauf F, Scherzer O, Leitao A (2005) Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators. SIAM J Numer Anal 43:767–786MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    González-Rodriguez P, Kindelan M, Moscoso M, Dorn O (2005) History matching problem in reservoir engineering using the propagation back-propagation method. Inverse Prob 21:565–590MATHCrossRefGoogle Scholar
  45. 45.
    Guzina BB, Bonnet M (2006) Small-inclusion asymptotic for inverse problems in acoustics. Inverse Prob 22:1761MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Haber E (2004) A multilevel level-set method for optimizing eigenvalues in shape design problems. J Comput Phys 198:518–534MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Hackl B (2007) Methods for reliable topology changes for perimeter-regularized geometric inverse problems. SIAM J Numer Anal 45: 2201–2227MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Harabetian E, Osher S (1998) Regularization of ill-posed problems via the level set approach. SIAM J Appl Math 58:1689–1706MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Hettlich F (1995) Fréchet derivatives in inverse obstacle scattering. Inverse Prob 11:371–382MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Hintermüller M, Ring W (2003) A second order shape optimization approach for image segmentation. SIAM J Appl Math 64:442–467MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Hou S, Solna K, Zhao H (2004) Imaging of location and geometry for extended targets using the response matrix. J Comput Phys 199:317–338MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Irishina N, Alvarez D, Dorn O, Moscoso M (2010) Structural level set inversion for microwave breast screening. Inverse Prob 26:035015MathSciNetCrossRefGoogle Scholar
  53. 53.
    Ito K, Kunisch K, Li Z (2001) Level-set approach to an inverse interface problem. Inverse Prob 17:1225–1242MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Ito K (2002) Level set methods for variational problems and application. In: Desch W, Kappel F, Kunisch K (eds) Control and estimation of distributed parameter systems. Birkhäuser, Basel, pp 203–217Google Scholar
  55. 55.
    Jacob M, Bresler Y, Toronov V, Zhang X, Webb A (2006) Level set algorithm for the reconstruction of functional activation in near-infrared spectroscopic imaging. J Biomed Opt 11:064029CrossRefGoogle Scholar
  56. 56.
    Kao CY, Osher S, Yablonovitch E (2005) Maximizing band gaps in two-dimentional photonic crystals by using level set methods. Appl Phys B 81:235–244CrossRefGoogle Scholar
  57. 57.
    Klann E, Ramlau R, Ring W (2008) A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. J Comput Phys 221:539–557Google Scholar
  58. 58.
    Kortschak B, Brandstätter B (2005) A FEM-BEM approach using level-sets in electrical capacitance tomography. COMPEL 24: 591–605MATHGoogle Scholar
  59. 59.
    Leitão A, Alves MM (2007) On level set type methods for elliptic Cauchy problems. Inverse Prob 23:2207–2222MATHCrossRefGoogle Scholar
  60. 60.
    Leitao A, Scherzer O (2003) On the relation between constraint regularization, level sets and shape optimization. Inverse Prob 19:L1–L11MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Lie J, Lysaker M, Tai X (2006) A variant of the level set method and applications to image segmentation. Math Comput 75:1155–1174MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Lie J, Lysaker M, Tai X (2006) A binary level set method and some applications for Mumford-Shah image segmentation. IEEE Trans Image Process 15:1171–1181CrossRefGoogle Scholar
  63. 63.
    Litman A, Lesselier D, Santosa D (1998) Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Prob 14:685–706MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Litman A (2005) Reconstruction by level sets of n-ary scattering obstacles. Inverse Prob 21:S131–S152MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Liu K, Yang X, Liu D et al (2010) Spectrally resolved three-dimensional bioluminescence tomography with a level-set strategy. J Opt Soc Am A 27:1413–1423CrossRefGoogle Scholar
  66. 66.
    Lu Z, Robinson BA (2006) Parameter identification using the level set method. Geophys Res Lett 33:L06404CrossRefGoogle Scholar
  67. 67.
    Luo Z, Tong LY, Luo JZ et al (2009) Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J Comput Phys 228:2643–2659MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Lysaker M, Chan TF, Li H, Tai X-C (2007) Level set method for positron emission tomography. Int J Biomed Imaging 2007:15. doi:10.1155/2007/26950Google Scholar
  69. 69.
    Masmoudi M, Pommier J, Samet B (2005) The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Prob 21:547–564MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Natterer F, Wübbeling F (2001) Mathematical methods in image reconstruction (monographs on mathematical modeling and computation), vol 5. SIAM, PhiladelphiaCrossRefGoogle Scholar
  72. 72.
    Nielsen LK, Li H, Tai XC, Aanonsen SI, Espedal M (2008) Reservoir description using a binary level set model. Comput Visual Sci 13(1):41–58MathSciNetCrossRefGoogle Scholar
  73. 73.
    Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Meth Appl Mech Eng 192:803–829MATHCrossRefGoogle Scholar
  74. 74.
    Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Osher S, Santosa F (2001) Level set methods for optimisation problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum. J Comput Phys 171: 272–288MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New YorkMATHGoogle Scholar
  77. 77.
    Park WK, Lesselier D (2009) Reconstruction of thin electromagnetic inclusions by a level set method. Inverse Prob 25:085010MathSciNetCrossRefGoogle Scholar
  78. 78.
    Ramananjaona C, Lambert M, Lesselier D, Zolésio J-P (2001) Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation. Inverse Prob 17:1087–1111MATHCrossRefGoogle Scholar
  79. 79.
    Ramananjaona C, Lambert M, Lesselier D, Zolésio J-P (2002) On novel developments of controlled evolution of level sets in the field of inverse shape problems. Radio Sci 37:8010CrossRefGoogle Scholar
  80. 80.
    Ramlau R, Ring W (2007) A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data. J Comput Phys 221:539–557MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Rocha de Faria J, Novotny AA, Feijóo RA, Taroco E (2009) First- and second-order topological sensitivity analysis for inclusions. Inverse Prob Sci Eng 17:665–679MATHCrossRefGoogle Scholar
  82. 82.
    Santosa F (1996) A level set approach for inverse problems involving obstacles. ESAIM Contr Optim Calc Var 1:17–33MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Schumacher A, Kobolev VV, Eschenauer HA (1994) Bubble method for topology and shape optimization of structures. J Struct Optim 8:42–51CrossRefGoogle Scholar
  84. 84.
    Schweiger M, Arridge SR, Dorn O, Zacharopoulos A, Kolehmainen V (2006) Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique. Opt Lett 31:471–473CrossRefGoogle Scholar
  85. 85.
    Sethian JA (1999) Level set methods and fast marching methods, 2nd edn. Cambridge University Press, CambridgeMATHGoogle Scholar
  86. 86.
    Sokolowski J, Zochowski A (1999) On topological derivative in shape optimization. SIAM J Control Optim 37:1251–1272MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Sokolowski J, Zolésio J-P (1992) Introduction to shape optimization: shape sensitivity analysis (springer series in computational mathematics), vol 16. Springer, BerlinMATHGoogle Scholar
  88. 88.
    Soleimani M (2007) Level-set method applied to magnetic induction tomography using experimental data. Res Nondestr Eval 18(1): 1–12MathSciNetCrossRefGoogle Scholar
  89. 89.
    Soleimani M, Lionheart WRB, Dorn O (2005) Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data. Inverse Prob Sci Eng 14:193–210CrossRefGoogle Scholar
  90. 90.
    Soleimani M, Dorn O, Lionheart WRB (2006) A narrowband level set method applied to EIT in brain for cryosurgery monitoring. IEEE Trans Biomed Eng 53:2257–2264CrossRefGoogle Scholar
  91. 91.
    Suri JS, Liu K, Singh S, Laxminarayan SN, Zeng X, Reden L (2002) Shape recovery algorithms using level sets in 2D/3D medical imagery: a state-of-the-art review. IEEE Trans Inf Technol Biomed 6:8–28CrossRefGoogle Scholar
  92. 92.
    Tai X-C, Chan TF (2004) A survey on multiple level set methods with applications for identifying piecewise constant functions. Int J Numer Anal Model 1:25–47MathSciNetMATHGoogle Scholar
  93. 93.
    van den Doel K et al (2007) Dynamic level set regularization for large distributed parameter estimation problems. Inverse Prob 23: 1271–1288MATHCrossRefGoogle Scholar
  94. 94.
    Van den Doel K, Ascher UM (2006) On level set regularization for highly ill-posed distributed parameter estimation problems. J Comput Phys 216:707–723MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Vese LA, Chan TF (2002) A multiphase level set framework for image segmentation using the Mumford-Shah model. Int J Comput Vision 50:271–293MATHCrossRefGoogle Scholar
  96. 96.
    Wang M, Wang X (2004) Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Meth Appl Mech Eng 193:469–496MATHCrossRefGoogle Scholar
  97. 97.
    Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78(4): 379–402MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Ye JC, Bresler Y, Moulin P (2002) A self-referencing level-set method for image reconstruction from sparse Fourier samples. Int J Comput Vision 50:253–270MATHCrossRefGoogle Scholar
  99. 99.
    Zhao H-K, Chan T, Merriman B, Osher S (1996) A variational level set approach to multiphase motion. J Comput Phys 127:179–195MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Oliver Dorn
  • Dominique Lesselier

There are no affiliations available

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