Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Linear Inverse Problems Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Level Set Methods for Structural Inversion and Image Reconstruction

Authors : Oliver Dorn, Dominique Lesselier

Published in: Handbook of Mathematical Methods in Imaging

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Iterative Solution Methods
next chapter Expansion Methods
Literature
1.
Abascal, J.F.P.J., Lambert, M., Lesselier, D., Dorn, O.: 3-D eddy-current imaging of metal tubes by gradient-based, controlled evolution of level sets. IEEE Trans. Magn. 44, 4721–4729 (2009)CrossRef
2.
3.
Allaire, G., Jouve, F., Toader, A.-M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)CrossRefMATHMathSciNet
4.
Alvarez, D., Dorn, O., Irishina, N., Moscoso, M.: Crack detection using a level set strategy. J. Comput. Phys. 228, 5710–57211 (2009)CrossRefMATHMathSciNet
5.
6.
Ammari, H., Kang, H.: Reconstruction of Small Inhomogeneities from Boundary Measurements. Lecture Notes in Mathematics, vol. 1846. Springer, Berlin (2004)
7.
Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216, 573–588 (2005)CrossRef
8.
Ascher, U.M., Huang, H., van den Doel, K.: Artificial time integration. BIT Numer. Math. 47, 3–25 (2007)CrossRefMATH
9.
Bal, G., Ren, K.: Reconstruction of singular surfaces by shape sensitivity analysis and level set method. Math. Models Methods Appl. Sci. 16, 1347–1374 (2006)CrossRefMATHMathSciNet
10.
Ben Ameur, H., Burger, M., Hackl, B.: Level set methods for geometric inverse problems in linear elasticity. Inverse Probl. 20, 673–696 (2004)CrossRefMATHMathSciNet
11.
Benedetti, M., Lesselier, D., Lambert, M., Massa, A.: Multiple-shape reconstruction by means of mutliregion level sets. IEEE Trans. Geosci. Remote Sens. 48, 2330–2342 (2010)CrossRef
12.
Ben Hadj Miled, M.K., Miller, E.L.: A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography. Inverse Probl. 23, 2375–2400 (2007)
13.
14.
Berre, I., Lien, M., Mannseth, T.: A level set corrector to an adaptive multiscale permeability prediction. Comput. Geosci. 11, 27–42 (2007)CrossRefMATHMathSciNet
15.
Bonnet, M., Guzina, B.B.: Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61, 2344–2373 (2003)CrossRefMathSciNet
16.
17.
Burger, M.: A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5, 301–329 (2003)CrossRefMATHMathSciNet
18.
Burger, M.: Levenberg-Marquardt level set methods for inverse obstacle problems. Inverse Probl. 20, 259–282 (2004)CrossRefMATH
19.
Burger, M., Hackl, B., Ring, W.: Incorporating topological derivatives into level set methods. J. Comput. Phys. 194, 344–362 (2004)CrossRefMATHMathSciNet
20.
Burger, M., Osher, S.: A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16, 263–301 (2005)CrossRefMATHMathSciNet
21.
Carpio, A., Rapún, M.-L.: Solving inhomogeneous inverse problems by topological derivative methods. Inverse Probl. 24, 045014 (2008)CrossRef
22.
Céa, J., Garreau, S., Guillaume, P., Masmoudi, M.: The shape and topological optimizations connection. Comput. Methods Appl. Mech. Eng. 188, 713–726 (2000)CrossRefMATH
23.
Céa, J., Gioan, A., Michel, J.: Quelques résultats sur l’identification de domains. Calcolo 10(3–4), 207–232 (1973)CrossRefMathSciNet
24.
Céa, J., Haug, E.J. (eds.): Optimization of Distributed Parameter Structures. Sijhoff & Noordhoff, Alphen aan den Rijn (1981)
25.
Chan, T.F., Tai, X.-C.: Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193, 40–66 (2003)CrossRefMathSciNet
26.
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001)CrossRefMATH
27.
Chung, E.T., Chan, T.F., Tai, X.C.: Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205, 357–372 (2005)CrossRefMATHMathSciNet
28.
DeCezaro, A., Leitão, A., Tai, X.-C.: On multiple level-set regularization methods for inverse problems. Inverse Probl. 25, 035004 (2009)CrossRef
29.
Delfour, M.C., Zolésio, J.-P.: Shape sensitivity analysis via min max differentiability. SIAM J. Control Optim. 26, 34–86 (1988)CrossRef
30.
Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM Advances in Design and Control. SIAM, Philadelphia (2001)
31.
Dorn, O., Lesselier, D.: Level set methods for inverse scattering. Inverse Probl. 22, R67–R131 (2006). doi:10.1088/0266-5611/22/4/R01CrossRefMATHMathSciNet
32.
Dorn, O., Lesselier, D.: Level set techniques for structural inversion in medical imaging. In: Suri, J.S., Farag, A.A. (eds.) Deformable Models, pp. 61–90. Springer, New York (2007)CrossRef
33.
Dorn, O., Lesselier, D.: Level set methods for inverse scattering – some recent developments. Inverse Probl. 25, 125001 (2009). doi:10.1088/0266-5611/25/12/125001CrossRefMathSciNet
34.
Dorn, O., Miller, E., Rappaport, C.: A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Probl. 16, 1119–1156 (2000)CrossRefMATHMathSciNet
35.
Dorn, O., Villegas, R.: History matching of petroleum reservoirs using a level set technique. Inverse Probl. 24, 035015 (2008)CrossRefMathSciNet
36.
37.
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Kluwer, Dordrecht (1996)
38.
Fang, W.: Multi-phase permittivity reconstruction in electrical capacitance tomography by level set methods. Inverse Probl. Sci. Eng. 15, 213–247 (2007)CrossRefMATHMathSciNet
39.
Feijóo, G.R.: A new method in inverse scattering based on the topological derivative. Inverse Probl. 20, 1819–1840 (2004)CrossRefMATH
40.
Feijóo, R.A., Novotny, A.A., Taroco, E., Padra, C.: The topological derivative for the Poisson problem. Math. Model Meth. Appl. Sci. 13, 1–20 (2003)CrossRef
41.
Feng, H., Karl, W.C., Castanon, D.A.: A curve evolution approach to object-based tomographic reconstruction. IEEE Trans. Image Process. 12, 44–57 (2003)CrossRefMathSciNet
42.
Ferrayé, R., Dauvignac, J.Y., Pichot, C.: An inverse scattering method based on contour deformations by means of a level set method using frequency hopping technique. IEEE Trans. Antennas Propag. 51, 1100–1113 (2003)CrossRef
43.
Frühauf, F., Scherzer, O., Leitao, A.: Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators. SIAM J. Numer. Anal. 43, 767–786 (2005)CrossRefMATHMathSciNet
44.
González-Rodriguez, P., Kindelan, M., Moscoso, M., Dorn, O.: History matching problem in reservoir engineering using the propagation back-propagation method. Inverse Probl. 21, 565–590 (2005)CrossRefMATH
45.
46.
Haber, E.: A multilevel level-set method for optimizing eigenvalues in shape design problems. J. Comput. Phys. 198, 518–534 (2004)CrossRefMATHMathSciNet
47.
Hackl, B.: Methods for reliable topology changes for perimeter-regularized geometric inverse problems. SIAM J. Numer. Anal. 45, 2201–2227 (2007)CrossRefMATHMathSciNet
48.
Harabetian, E., Osher, S.: Regularization of ill-posed problems via the level set approach. SIAM J. Appl. Math. 58, 1689–1706 (1998)CrossRefMATHMathSciNet
49.
50.
Hintermüller, M, Ring, W.: A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64, 442–467 (2003)CrossRefMATHMathSciNet
51.
Hou, S., Solna, K., Zhao, H.: Imaging of location and geometry for extended targets using the response matrix. J. Comput. Phys. 199, 317–338 (2004)CrossRefMATHMathSciNet
52.
Irishina, N., Alvarez, D., Dorn, O., Moscoso, M.: Structural level set inversion for microwave breast screening. Inverse Probl. 26, 035015 (2010)CrossRefMathSciNet
53.
Ito, K.: Level set methods for variational problems and application. In: Desch, W., Kappel, F., Kunisch, K. (eds.) Control and Estimation of Distributed Parameter Systems, pp. 203–217. Birkhäuser, Basel (2002)
54.
55.
Jacob, M., Bresler, Y., Toronov, V., Zhang, X., Webb, A.: Level set algorithm for the reconstruction of functional activation in near-infrared spectroscopic imaging. J. Biomed. Opt. 11, 064029 (2006)CrossRef
56.
Kao, C.Y., Osher, S., Yablonovitch, E.: Maximizing band gaps in two-dimentional photonic crystals by using level set methods. Appl. Phys. B 81, 235–244 (2005)CrossRef
57.
Klann, E., Ramlau, R., Ring, W.: A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. J. Comput. Phys. 221, 539–557 (2008)
58.
Kortschak, B., Brandstätter, B.: A FEM-BEM approach using level-sets in electrical capacitance tomography. COMPEL 24, 591–605 (2005)CrossRefMATH
59.
Leitão, A., Alves, M.M.: On level set type methods for elliptic Cauchy problems. Inverse Probl. 23, 2207–2222 (2007)CrossRefMATH
60.
Leitao, A., Scherzer, O.: On the relation between constraint regularization, level sets and shape optimization. Inverse Probl. 19, L1–L11 (2003)CrossRefMATHMathSciNet
61.
Lie, J., Lysaker, M., Tai, X.: A variant of the level set method and applications to image segmentation. Math. Comput. 75, 1155–1174 (2006)CrossRefMATHMathSciNet
62.
Lie, J., Lysaker, M., Tai, X.: A binary level set method and some applications for Mumford-Shah image segmentation. IEEE Trans. Image Process. 15, 1171–1181 (2006)CrossRefMATH
63.
64.
Litman, A., Lesselier, D., Santosa, D.: Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Probl. 14, 685–706 (1998)CrossRefMATHMathSciNet
65.
Liu, K., Yang, X., Liu, D., et al.: Spectrally resolved three-dimensional bioluminescence tomography with a level-set strategy. J. Opt. Soc. Am. A 27, 1413–1423 (2010)CrossRef
66.
Lu, Z., Robinson, B.A.: Parameter identification using the level set method. Geophys. Res. Lett. 33, L06404 (2006)
67.
Luo, Z., Tong, L.Y., Luo, J.Z., et al.: Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J. Comput. Phys. 228, 2643–2659 (2009)CrossRefMATHMathSciNet
68.
Lysaker, M., Chan, T.F., Li, H., Tai, X.-C.: Level set method for positron emission tomography. Int. J. Biomed. Imaging 2007, 15 (2007). doi:10.1155/2007/26950
69.
Masmoudi, M., Pommier, J., Samet, B.: The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Probl. 21, 547–564 (2005)CrossRefMATHMathSciNet
70.
Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)CrossRefMATHMathSciNet
71.
Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. Monographs on Mathematical Modeling and Computation, vol. 5. SIAM, Philadelphia (2001)
72.
Nielsen, L.K., Li, H., Tai, X.C., Aanonsen, S.I., Espedal, M.: Reservoir description using a binary level set model. Comput. Vis. Sci. 13(1), 41–58 (2008)CrossRefMathSciNet
73.
Novotny, A.A., Feijóo, R.A., Taroco, E., Padra, C.: Topological sensitivity analysis. Comput. Methods Appl. Mech. Eng. 192, 803–829 (2003)CrossRefMATH
74.
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)CrossRefMATH
75.
Osher, S., Santosa, F.: Level set methods for optimisation problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171, 272–288 (2001)MATHMathSciNet
76.
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)CrossRefMATHMathSciNet
77.
Park, W.K., Lesselier, D.: Reconstruction of thin electromagnetic inclusions by a level set method. Inverse Probl. 25, 085010 (2009)CrossRefMathSciNet
78.
Ramananjaona, C., Lambert, M., Lesselier, D., Zolésio, J.-P.: Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation. Inverse Probl. 17, 1087–1111 (2001)CrossRefMATH
79.
Ramananjaona, C., Lambert, M., Lesselier, D., Zolésio, J.-P.: On novel developments of controlled evolution of level sets in the field of inverse shape problems. Radio Sci. 37, 8010 (2002)CrossRef
80.
Ramlau, R., Ring, W.: A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data. J. Comput. Phys. 221, 539–557 (2007)CrossRefMATHMathSciNet
81.
Rocha de Faria, J., Novotny, A.A., Feijóo, R.A., Taroco, E.: First- and second-order topological sensitivity analysis for inclusions. Inverse Probl. Sci. Eng. 17, 665–679 (2009)
82.
83.
Schumacher, A., Kobolev, V.V., Eschenauer, H.A.: Bubble method for topology and shape optimization of structures. J. Struct. Optim. 8, 42–51 (1994)CrossRef
84.
Schweiger, M., Arridge, S.R., Dorn, O., Zacharopoulos, A., Kolehmainen, V.: Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique. Opt. Lett. 31, 471–473 (2006)CrossRef
85.
Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press, Cambridge (1999)MATH
86.
87.
Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992)
88.
Soleimani, M.: Level-set method applied to magnetic induction tomography using experimental data. Res. Nondestruct. Eval. 18(1), 1–12 (2007)CrossRefMathSciNet
89.
Soleimani, M., Dorn, O., Lionheart, W.R.B.: A narrowband level set method applied to EIT in brain for cryosurgery monitoring. IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006)CrossRef
90.
Soleimani, M., Lionheart, W.R.B., Dorn, O.: Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data. Inverse Probl. Sci. Eng. 14, 193–210 (2005)CrossRef
91.
Suri, J.S., Liu, K., Singh, S., Laxminarayan, S.N., Zeng, X., Reden, L.: Shape recovery algorithms using level sets in 2D/3D medical imagery: a state-of-the-art review. IEEE Trans. Inf. Technol. Biomed. 6, 8–28 (2002)CrossRef
92.
Tai, X.-C., Chan, T.F.: A survey on multiple level set methods with applications for identifying piecewise constant functions. Int. J. Numer. Anal. Model. 1, 25–47 (2004)MATHMathSciNet
93.
Van den Doel, K., Ascher, U.M.: On level set regularization for highly ill-posed distributed parameter estimation problems. J. Comput. Phys. 216, 707–723 (2006)CrossRefMATHMathSciNet
94.
van den Doel, K., et al.: Dynamic level set regularization for large distributed parameter estimation problems. Inverse Probl. 23, 1271–1288 (2007)CrossRefMATH
95.
Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford-Shah model. Int. J. Comput. Vis. 50, 271–293 (2002)CrossRefMATH
96.
Wang, M., Wang, X.: Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193, 469–496 (2004)CrossRefMATH
97.
Wei, P., Wang, M.Y.: Piecewise constant level set method for structural topology optimization. Int. J. Numer. Methods Eng. 78(4), 379–402 (2009)CrossRefMATH
98.
Ye, J.C., Bresler, Y., Moulin, P.: A self-referencing level-set method for image reconstruction from sparse Fourier samples. Int. J. Comput. Vis. 50, 253–270 (2002)CrossRefMATH
99.
Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)CrossRefMATHMathSciNet
Metadata
Title
Level Set Methods for Structural Inversion and Image Reconstruction
Authors
Oliver Dorn
Dominique Lesselier
Copyright Year
2015
Publisher
Springer New York
Book
Handbook of Mathematical Methods in Imaging

Print ISBN: 978-1-4939-0789-2

Electronic ISBN: 978-1-4939-0790-8

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-1-4939-0790-8_11

Premium Partner

    Image Credits