Skip to main content
Log in

Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

An inverse problem is considered where the structure of multiple sound-soft planar obstacles is to be determined given the direction of the incoming acoustic field and knowledge of the corresponding total field on a curve located outside the obstacles. A local uniqueness result is given for this inverse problem suggesting that the reconstruction can be achieved by a single incident wave. A numerical procedure based on the concept of the topological derivative of an associated cost functional is used to produce images of the obstacles. No a priori assumption about the number of obstacles present is needed. Numerical results are included showing that accurate reconstructions can be obtained and that the proposed method is capable of finding both the shapes and the number of obstacles with one or a few incident waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    MATH  Google Scholar 

  2. Ben Hassen, F., Liu, J., Potthast, R.: On source analysis by wave splitting with applications in inverse scattering of multiple obstacles. J. Comput. Math. 25(3), 266–281 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Carpio, A., Rapun, M.L.: Topological Derivatives for Shape Reconstruction. Lect. Not. Mat., vol. 1943, pp. 85–131 (2008)

  4. Carpio, A., Rapun, M.L.: Solving inverse inhomogeneous problems by topological derivative methods. Inverse Probl. 24, 045014 (2008)

    Article  MathSciNet  Google Scholar 

  5. Carpio, A., Rapun, M.L.: Domain reconstruction by thermal measurements. J. Comput. Phys. 227(17), 8083–8106 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carpio, A., Rapun, M.L.: An iterative method for parameter identification and shape reconstruction. Inverse Probl. Sci. Eng. 18, 35–50 (2010)

    Article  MATH  Google Scholar 

  7. Colton, D., Gieberman, K., Monk, P.: A regularized sampling method for solving three dimensional inverse scattering problems. SIAM J. Sci. Comput. 21, 2316–2330 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering. Springer, Berlin (1998)

    MATH  Google Scholar 

  9. Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)

    MATH  Google Scholar 

  10. Colton, D., Sleeman, B.D.: Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. 31, 253–259 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Feijoo, G.R.: A new method in inverse scattering based on the topological derivative. Inverse Probl. 20, 1819–1840 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Feijoo, G.R., Oberai, A.A., Pinsky, P.M.: An application of shape optimization in the solution of inverse acoustic scattering problems. Inverse Probl. 20, 199–228 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality. Inverse Probl. 21, 1195–1205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Guzina, B.B., Bonnet, M.: Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics. Inverse Probl. 22, 1761–1785 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Guzina, B.B., Chikichev, I.: From imaging to material identification: A generalized concept of topological sensitivity. J. Mech. Phys. Solids 55, 245–279 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ivanyshyn, O., Johansson, T.: Nonlinear integral equations methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equ. Appl. 19(3), 289–308 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Johansson, T., Sleeman, B.D.: Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern. IMA J. Appl. Math. 72, 96–112 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Keller, J.B., Givoli, D.: Exact non-reflecting boundary conditions. J. Comput. Phys. 82, 172–192 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kirsch, A.: The domain derivative and two applications in inverse scattering theory. Inverse Probl. 9, 81–96 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21, 1207–1223 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Linton, C.M., Martin, P.A.: Multiple scattering by random configuration of circular cylinders: Second-order corrections for the effective wavenumber. J. Acoust. Soc. Am. 117, 3413–3423 (2005)

    Article  Google Scholar 

  22. Litman, A., Lesselier, D., Santosa, F.: Reconstruction of a two dimensional binary obstacle by controlled evolution of a level set. Inverse Probl. 14, 685–706 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Martin, P.A.: Multiple Scattering, Interaction of Time-Harmonic Waves with N Obstacles. Cambridge Univ. Press, Cambridge (2006)

    MATH  Google Scholar 

  24. Potthast, R.: A survey on sampling and probe methods for inverse problems, Topical Review. Inverse Probl. 22, R1–R47 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Samet, B., Amstutz, S., Masmoudi, M.: The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42, 1523–1544 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Santosa, F.: A level set approach for inverse problems involving obstacles. ESAIM Control Optim. Calc. Var. 1, 17–33 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sleeman, B.D.: The inverse problem of acoustic scattering. Applied Mathematics Institute Technical Report No. 114 A, University of Delaware, Newark, 1981

  28. Twersky, V.: Multiple scattering of radiation by an arbitrary configuration of parallel cylinders. J. Acoust. Soc. Am. 24, 42–46 (1952)

    Article  Google Scholar 

  29. Twersky, V.: Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders. J. Appl. Phys. 23, 407–414 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  30. Young, J.W., Bertrand, J.C.: Multiple scattering by two cylinders. J. Acoust. Soc. Am. 58, 1190–1195 (1975)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M.-L. Rapún.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carpio, A., Johansson, B.T. & Rapún, ML. Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields. J Math Imaging Vis 36, 185–199 (2010). https://doi.org/10.1007/s10851-009-0182-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-009-0182-x

Navigation