Journal of Mathematical Imaging and Vision

, Volume 49, Issue 2, pp 273–288 | Cite as

Parameter Identification in Photothermal Imaging

  • A. Carpio
  • M.-L. Rapún


We propose a technique to reconstruct the geometry of inclusions and their material parameters in thermal scattering near surfaces. The imaging problem is reformulated as a constrained optimization problem with a finite number of stationary constraints. The unknown domains and their parameters are the design variables. A descent method combining topological derivative analysis to find improved guesses of the objects and gradient iterations to correct their material parameters provides reasonable reconstructions.


Shape reconstruction Parameter identification Nondestructive testing Topological derivative Laplace transform Heat equation Thermal waves 



The authors are partially supported by the Spanish Government research project TRA2010–18054 and the Spanish Ministerio de Economia y Competitividad Grants No. FIS2011-28838-C02-02, and No. FIS2010-22438-E.


  1. 1.
    Allaire, G., de Gournay, F., Jouve, F., Toader, A.M.: Structural optimization using topological and shape sensitivity via a level-set method. Control Cybern. 34, 59–80 (2005) MATHGoogle Scholar
  2. 2.
    Almond, D.P., Patel, P.M.: Photothermal Science and Techniques. Chapman and Hall, London (1996) Google Scholar
  3. 3.
    Banks, H.T., Kojima, F.: Boundary shape identification problems in two-dimensional domains related to thermal testing of materials. Q. Appl. Math. 47, 273–293 (1989) MATHMathSciNetGoogle Scholar
  4. 4.
    Banks, H.T., Kojima, F., Winfree, W.P.: Boundary estimation problems arising in thermal tomography. Inverse Probl. 6, 897–921 (1990) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cakoni, F., Colton, D., Monk, P.: The determination of the surface conductivity of a partially coated dielectric. SIAM J. Appl. Math. 65, 767–789 (2005) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Carpio, A., Rapún, M.-L.: Solving inverse inhomogeneous problems by topological derivative methods. Inverse Probl. 24, 045014 (2008) CrossRefGoogle Scholar
  7. 7.
    Carpio, A., Rapún, M.-L.: Domain reconstruction by photothermal techniques. J. Comput. Phys. 227, 8083–8106 (2008) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Carpio, A., Rapún, M.-L.: Hybrid topological derivative and gradient based methods for electrical impedance tomography. Inverse Probl. 28, 095010 (2012) CrossRefGoogle Scholar
  9. 9.
    Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106, 367–413 (1985) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Elden, L., Berntsson, F., Reginska, T.: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput. 21, 2187–2205 (2000) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Feijoo, G.R.: A new method in inverse scattering based on the topological derivative. Inverse Probl. 20, 1819–1840 (2004) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Garrido, F., Salazar, A.: Thermal wave scattering from spheres. J. Appl. Phys. 95, 140–149 (2004) CrossRefGoogle Scholar
  13. 13.
    Guzina, B.B., Bonnet, M.: Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics. Inverse Probl. 22, 1761–1785 (2006) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Guzina, B.B., Chikichev, I.: From imaging to material identification: a generalized concept of topological sensitivity. J. Mech. Phys. Solids 55, 245–279 (2007) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Heath, D.M., Welch, C.S., Winfree, W.P.: Quantitative thermal diffusivity measurements of composites. In: Review of Progress in Quantitative Non-Destructive Evaluation, vol. 5B, pp. 1125–1132. Plenum, New York (1986) Google Scholar
  16. 16.
    Hohage, T., Rapún, M.-L., Sayas, F.-J.: Detecting corrosion using thermal measurements. Inverse Probl. 23, 53–72 (2007) CrossRefMATHGoogle Scholar
  17. 17.
    Hohage, T., Sayas, F.-J.: Numerical approximation of a heat diffusion problem by boundary element methods using the Laplace transform. Numer. Math. 102, 67–92 (2005) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (1998) CrossRefMATHGoogle Scholar
  19. 19.
    López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Mandelis, A.: Diffusion-Wave Fields. Mathematical Methods and Green Functions. Springer, New York (2001) CrossRefMATHGoogle Scholar
  21. 21.
    Mandelis, A.: Diffusion waves and their uses. Phys. Today 53, 29–34 (2000) CrossRefGoogle Scholar
  22. 22.
    Nicolaides, L., Mandelis, A.: Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions. Inverse Probl. 13, 1393–1412 (1997) CrossRefMATHGoogle Scholar
  23. 23.
    Ocáriz, A., Sánchez-Lavega, A., Salazar, A.: Photothermal study of subsurface cylindrical structures II. Experimental results. J. Appl. Phys. 81, 7561–7566 (1997) CrossRefGoogle Scholar
  24. 24.
    Rapún, M.-L., Sayas, F.-J.: Boundary integral approximation of a heat diffusion problem in time-harmonic regime. Numer. Algorithms 41, 127–160 (2006) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Rapún, M.-L., Sayas, F.-J.: Boundary element simulation of thermal waves. Arch. Comput. Methods Eng. 14, 3–46 (2007) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Salazar, A., Sánchez-Lavega, A., Celorrio, R.: Scattering of cylindrical thermal waves in fiber composites: in-plane thermal diffusivity. J. Appl. Phys. 93, 4536–4542 (2003) CrossRefGoogle Scholar
  27. 27.
    Terrón, J.M., Salazar, A., Sánchez-Lavega, A.: General solution for the thermal wave scattering in fiber composites. J. Appl. Phys. 91, 1087–1098 (2002) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dept. Matemática AplicadaUniversidad Complutense de MadridMadridSpain
  2. 2.Dept. Fundamentos Matemáticos, ETSI AeronáuticosUniversidad Politécnica de MadridMadridSpain

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