Level Set Techniques For Structural Inversion In Medical Imaging

  • Oliver Dorn
  • Dominique Lesselier
Part of the Topics in Biomedical Engineering. International Book Series book series (ITBE)

Most biological bodies are structured in the sense that they contain quite well-defined interfaces between regions of different types of tissue or anatomical material. Extracting structural information from medical or biological images has been an important research topic for a long time. Recently, much attention has been devoted to quite novel techniques for the direct recovery of structural information from physically measured data. These techniques differ from more traditional image processing and image segmentation techniques by the fact that they try to recover structured images not from already given pixel or voxelbased reconstructions (obtained, e.g., using traditional medical inversion techniques), but directly from the given raw data. This has the advantage that the final result is guaranteed to satisfy the imposed criteria of data fitness as well as those of the given structural prior information. The ‘level-set-technique’ [1–3] plays an important role in many of these novel structural inversion approaches, due to its capability of modeling topological changes during the typically iterative inversion process. In this text we will provide a brief introduction into some techniques that have been developed recently for solving structural inverse problems using a level set technique.


Inverse Problem Electrical Impedance Tomography Diffuse Optical Tomography Linear Inverse Problem Structural Inversion 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Osher S, Sethian JA. 1988. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12-49.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Osher S, Fedkiw R. 2003. Level set methods and dynamic implicit surfaces. New York: Springer.zbMATHGoogle Scholar
  3. 3.
    Sethian JA. 1999. Level set methods and fast marching methods, 2nd ed. Cambridge: Cambridge UP.zbMATHGoogle Scholar
  4. 4.
    Santosa F. 1996. A level set approach for inverse problems involving obstacles. ESAIM Control Optim Calculus Variations 1:17-33.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Litman A, Lesselier D, Santosa D. 1998. Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level set. Inverse Probl 14:685-706.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feng H, Karl WC, Castanon D. 2000. Tomographic reconstruction using curve evolution. Proc IEEE Int Conf Computer Vision and Pattern Recognition 1:361-366.Google Scholar
  7. 7.
    Dorn O, Miller E, Rappaport C. 2000. A shape reconstruction method for electromagnetic to- mography using adjoint fields and level sets. Inverse Probl 16:1119-1156.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Burger M. 2001. A level set method for inverse problems. Inverse Probl 17:1327-1355.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ito K, Kunisch K, Li Z. 2001. Level set approach to an inverse interface problem. Inverse Probl 17:1225-1242.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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 Probl 17: 1087-1111.zbMATHCrossRefGoogle Scholar
  11. 11.
    Ramananjaona C, Lambert M, Lesselier D. 2001. Shape inversion from TM and TE real data by controlled evolution of level sets. Inverse Probl 17:1585-1595.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Burger M, Osher S. 2005. A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16:263-301.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dorn O, Lesselier D. 2006. Level set methods for inverse scattering, Inverse Probl 22:R67-R131.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tai X-C, Chan TF. 2004. A survey on multiple level set methods with applications for identifying piecewise constant functions Int J Num Anal Model 1:25-47.zbMATHMathSciNetGoogle Scholar
  15. 15.
    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 6:8-28.CrossRefGoogle Scholar
  16. 16.
    Case K, Zweifel P. 1967. Linear transport theory. New York: Addison Wesley.zbMATHGoogle Scholar
  17. 17.
    Kak AC, Slaney M. 2001. Principles of computerized tomographic imaging. SIAM Classics in Applied Mathematics, Vol. 33. Philadelphia: SIAM.Google Scholar
  18. 18.
    Natterer F. 1986. The mathematics of computerized tomography. Stuttgart: Teubner.zbMATHGoogle Scholar
  19. 19.
    Natterer F, W übbeling F. 2001. Mathematical methods in image reconstruction. Monographs on Mathematical Modeling and Computation, Vol. 5. Philadelphia: SIAM.Google Scholar
  20. 20.
    Radon J. 1917. U¨ber die bestimmung von funktionen durch ihre integralwerte l ängs gewisser mannigfaltigkeiten. Ber S äch Akad der Wiss Leipzig, Math-Phys Kl 69:262-267.Google Scholar
  21. 21.
    Feng H, Karl WC, Castanon DA. 2003. A curve evolution approach to object-based tomographic reconstruction. IEEE Trans Image Process 12:44-57.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Delfour MC, Zolésio J-P. 2001. Shapes and geometries: analysis, differential calculus and opti-mization. SIAM Advances in Design and Control. Philadelphia: SIAM.Google Scholar
  23. 23.
    Sokolowski J, Zolésio J-P. 1992. Introduction to shape optimization: shape sensitivity analysis. Springer series in Computational Mathematics, Vol. 16. Berlin: Springer.Google Scholar
  24. 24.
    Mumford D, Shah J. 1989. Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577-685.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Chan TF, Vese LA. 2001. Active contours without edges. IEEE Trans Image Process 10:266-277.zbMATHCrossRefGoogle Scholar
  26. 26.
    Vese LA, Chan TF. 2002. A multiphase level set framework for image segmentation using the Mumford-Shah model. Int J Comput Vision 50 271-293.zbMATHCrossRefGoogle Scholar
  27. 27.
    Chan TF, Tai X-C. 2003. Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J Comput Phys 193:40-66.CrossRefMathSciNetGoogle Scholar
  28. 28.
    Lysaker M, Chan T, Tai X-C. 2004. Level set method for positron emission tomography. UCLA- CAM Preprint 04-30.Google Scholar
  29. 29.
    Whitaker RT, Elangovan V. 2002. A direct approach to estimating surfaces in tomographic data. Med Imaging Anal 6, 235-249.CrossRefGoogle Scholar
  30. 30.
    Ye JC, Bresler Y, Moulin P. 2002. A self-referencing level set method for image reconstruction from sparse Fourier samples. Int J Comput Vision 50:253-270.zbMATHCrossRefGoogle Scholar
  31. 31.
    Arridge SR. 1999. Optical tomography in medical imaging. Inverse Probl 15:R41-R93.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Okada E, Firbank M, Schweiger M, Arridge SR, Cope M, Delpy DT. 1997. Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head. Appl Opt 36(1):21-31.CrossRefGoogle Scholar
  33. 33.
    Dorn O. 1998. A transport-backtransport method for optical tomography. Inverse Probl 14:1107-1130.zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Heino J, Arridge S, Sikora J, Somersalo E. 2003. Anisotropic effects in highly scattering media. Phys Rev E, 68(3):031908-1-031908-8.Google Scholar
  35. 35.
    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(4):471-473.CrossRefGoogle Scholar
  36. 36.
    Dierkes T, Dorn O, Natterer F, Palamodov V, Sielschott H. 2002. Frechet derivatives for some bilinear inverse problems SIAM J Appl Math 62:2092-2113.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Dorn O. 2000. Scattering and absorption transport sensitivity functions for optical tomography. Opt Express 7:492-506.CrossRefGoogle Scholar
  38. 38.
    Dorn O, Miller E, Rappaport C. 2001. Shape reconstruction in 2D from limited-view multifre-quency electromagnetic data. Radon transform and tomography. AMS series on Contemporary Mathematics, Vol. 278, pp. 97-122.MathSciNetGoogle Scholar
  39. 39.
    Ferray é R, Dauvignac J-Y, Pichot C. 2003. An inverse scattering method based on contour defor-mations by means of a level set method using frequency hopping technique. IEEE Trans Antennas Propag 51:1100-1113.CrossRefGoogle Scholar
  40. 40.
    Ferray é R, Dauvignac J-Y, Pichot C. 2003. Reconstruction of complex and multiple shape object contours using a level set method J Electromagn Waves Appl 17:153-181.CrossRefMathSciNetGoogle Scholar
  41. 41.
    Irishina N, Moscoso M, Dorn O. 2006. Detection of small tumors in microwave medical imaging using level sets and MUSIC. Proceedings of the progress in electromagnetics research symposium, Cambridge, MA, March 26-29, 2006. To appear.
  42. 42.
    LitmanA. 2005. Reconstruction by level sets of n-ary scattering obstacles. Inverse Probl 21:S131-S152.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Ramananjaona C, Lambert M, Lesselier D, Zolesio J-P. 2003. On novel developments of controlled evolution of level sets in the field of inverse shape problems. Radio Sci 38:1-9.Google Scholar
  44. 44.
    Ascher UM, Van den Doel K. 2005. On level set regularization for highly ill-posed dis-tributed parameter estimation problems. J Comput Phys. To appear. kv-doel/publications/keesUri05.pdf
  45. 45.
    Chung ET, Chan TF, Tai XC. 2005. Electrical impedance tomography using level set representation and total variational regularization. J Comput Phys 205:357-372.zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Leitao A, Scherzer O. 2003. On the relation between constraint regularization, level sets and shape optimization. Inverse Probl 19:L1-L11.zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Soleimani M, Lionheart WRB, Dorn O. 2005. Level set reconstruction of conductivity and per-mittivity from boundary electrical measurements using experimental data. Inverse Probl Sci Eng 14:193-210.CrossRefGoogle Scholar
  48. 48.
    Soleimani M, Dorn O, Lionheart WRB. 2006. A narrowband level set method applied to EIT in brain for cryosurgery monitoring. IEEE Trans Biomed Eng. To appear.Google Scholar
  49. 49.
    Calderero F, Ghodrati A, Brooks DH, Tadmor G, MacLeod R. 2005. A method to reconstruct activation wavefronts without isotropy assumptions using a level set approach. In Lecture Notes in Computer Science, Vol. 3504: Functional imaging and modeling of the heart: third international workshop, FIMH 2005, Barcelona, Spain, June 2-4, 2005, pp. 195-204. Ed.AF Frangi, PI Radeva, A Santos, M Hernandez. Berlin: Springer.Google Scholar
  50. 50.
    Lysaker OM, Nielsen BF. 2006. Toward a level set framework for infarction modeling: an inverse problem. Int J Num Anal Model 3:377-394.zbMATHMathSciNetGoogle Scholar
  51. 51.
    Bal G, Ren K. 2005. Reconstruction of singular surfaces by shape sensitivity analysis and level set method. Preprint, Columbia University. gb2030/PAPERS/Sing LevelSet.pdf.
  52. 52.
    Dorn O. 2004. Shape reconstruction in scattering media with voids using a transport model and level sets. Can Appl Math Q 10:239-275.MathSciNetGoogle Scholar
  53. 53.
    Dorn O. 2006. Shape reconstruction for an inverse radiative transfer problem arising in medical imaging. In Numerical methods for multidimensional radiative transfer problems. Springer series Computational Science and Engineering. Ed. G Kanschat, E Meink öhn, R Rannacher, R Wehrse. Springer: Berlin. To appear.Google Scholar
  54. 54.
    Ishimaru A. 1978. Wave propagation and scattering in random media. New York: Academic Press.Google Scholar
  55. 55.
    Natterer F, W übbeling F. 1995. A propagation-backpropagation method for ultrasound tomogra- phy. Inverse Probl 11:1225-1232.zbMATHCrossRefGoogle Scholar
  56. 56.
    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 171:272-288.zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Osher S, Paragios N. 2003. Geometric level set methods in imaging, vision and graphics. Berlin: Springer.zbMATHGoogle Scholar
  58. 58.
    Sikora J, Zacharopoulos A, Douiri A, Schweiger M, Horesh L, Arridge S, Ripoll J. 2006. Diffuse photon propagation in multilayered geometries. Phys Med Biol 51:497-516.CrossRefGoogle Scholar
  59. 59.
    Zacharopoulos A, Arridge S, Dorn O, Kolehmainen V, Sikora J. 2006. 3D shape reconstruc-tion in optical tomography using spherical harmonics and BEM. Proceedings of the progress in electromagnetics research symposium, Cambridge, MA, March 26-29, 2006. To appear.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Oliver Dorn
    • 1
  • Dominique Lesselier
    • 2
  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésEspaña
  2. 2.Laboratoire des Signaux et Systèmes Gif sur YvetteFrance

Personalised recommendations