Abstract

A wavelet domain, nonlinear inverse scattering approach is presented for imaging subsurface defects in a material sample, given observations of scattered thermal waves. Unlike methods using the Born linearization, our inversion scheme is based on the full wave-field model describing the propagation of thermal waves. Multiresolution techniques are employed to regularize and to lower the computational burden of this ill-posed imaging problem. We use newly developed wavelet-based regularization methods to resolve better the edge structures of defects relative to reconstructions obtained with smoothness-type regularizers. A nonlinear approximation to the exact forward-scattering model is introduced to simplify the inversion with little loss in accuracy. We demonstrate this approach on cross-section imaging problems by using synthetically generated scattering data from transmission and backprojection geometries.

© 1998 Optical Society of America

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  1. L. Nicolaides, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions,” Inverse Probl. 13, 1339–1412 (1997).
  2. L. Nicolaides, M. Munidasa, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with backpropagation and transmission reconstructions: experimental,” Inverse Probl. 13, 1413–1425 (1997).
    [CrossRef]
  3. O. Pade, A. Mandelis, “Computational thermal-wave slice tomography with backpropagation and transmission reconstructions,” Rev. Sci. Instrum. 64, 3548–3562 (1993).
    [CrossRef]
  4. A. Mandelis, “Theory of photothermal-wave diffraction and interference in condensed media,” J. Opt. Soc. Am. A 6, 298–308 (1989).
    [CrossRef]
  5. A. Mandelis, “Green’s functions in thermal-wave physics: Cartesian coordinate representations,” J. Appl. Phys. 78, 647–655 (1995).
    [CrossRef]
  6. C. Torres-Verdı́n, T. M. Habashy, “Rapid 2.5-D forward modeling and inversion via a new nonlinear scattering approximation,” Radio Sci. 29, 1051–1079 (1994).
    [CrossRef]
  7. T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
    [CrossRef]
  8. T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
    [CrossRef]
  9. O. Pade, A. Mandelis, “Thermal-wave slice tomography using wave-field reconstruction,” Inverse Probl. 10, 185–197 (1994).
    [CrossRef]
  10. T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
    [CrossRef]
  11. D. Lesselier, B. Duchene, “Wave-field inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science, W. R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996).
  12. T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
    [CrossRef]
  13. E. L. Miller, A. S. Willsky, “Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation,” Radio Sci. 31, 51–67 (1996).
    [CrossRef]
  14. G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
    [CrossRef]
  15. M. Bertero, “Linear inverse and Ill-posed problems,” in Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, Boston, 1989), Vol. 75, pp. 1–120.
  16. E. L. Miller, A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. Geosci. Remote Sens. 34, 346–357 (1996).
    [CrossRef]
  17. M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
    [CrossRef]
  18. Y. Meyer, Wavelets and Operators (Cambridge U. Press, Cambridge, 1995).
  19. R. F. Harrington, Field Computations by Moment Methods (Macmillan, New York, 1968).
  20. B. Wang, J. C. Moulder, J. P. Basart, “Wavelets in the solution of the volume integral equation: application to eddy current modeling,” J. Appl. Phys. 81, 6397–6406 (1997).
    [CrossRef]
  21. M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (A. K. Peters, Wellesley, Mass., 1994).
  22. A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
    [CrossRef]
  23. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
    [CrossRef]
  24. G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).
  25. M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
    [CrossRef]
  26. P. E. Gill, W. Murry, M. H. Wright, Practical Optimization (Academic, San Diego, 1981).
  27. A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
    [CrossRef]
  28. P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
    [CrossRef] [PubMed]
  29. E. L. Miller, “The application of multiscale and statistical techniques to the solution of inverse problems,” (MIT Laboratory for Information and Decision Systems, Cambridge, Mass., 1994).
  30. B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
    [CrossRef]
  31. E. L. Miller, A. S. Willsky, “Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems,” Multidimens. Syst. Signal Process. 8, 151–184 (1997).
    [CrossRef]

1997 (6)

L. Nicolaides, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions,” Inverse Probl. 13, 1339–1412 (1997).

L. Nicolaides, M. Munidasa, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with backpropagation and transmission reconstructions: experimental,” Inverse Probl. 13, 1413–1425 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

B. Wang, J. C. Moulder, J. P. Basart, “Wavelets in the solution of the volume integral equation: application to eddy current modeling,” J. Appl. Phys. 81, 6397–6406 (1997).
[CrossRef]

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

E. L. Miller, A. S. Willsky, “Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems,” Multidimens. Syst. Signal Process. 8, 151–184 (1997).
[CrossRef]

1996 (2)

E. L. Miller, A. S. Willsky, “Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation,” Radio Sci. 31, 51–67 (1996).
[CrossRef]

E. L. Miller, A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. Geosci. Remote Sens. 34, 346–357 (1996).
[CrossRef]

1995 (1)

A. Mandelis, “Green’s functions in thermal-wave physics: Cartesian coordinate representations,” J. Appl. Phys. 78, 647–655 (1995).
[CrossRef]

1994 (3)

C. Torres-Verdı́n, T. M. Habashy, “Rapid 2.5-D forward modeling and inversion via a new nonlinear scattering approximation,” Radio Sci. 29, 1051–1079 (1994).
[CrossRef]

T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
[CrossRef]

O. Pade, A. Mandelis, “Thermal-wave slice tomography using wave-field reconstruction,” Inverse Probl. 10, 185–197 (1994).
[CrossRef]

1993 (4)

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

O. Pade, A. Mandelis, “Computational thermal-wave slice tomography with backpropagation and transmission reconstructions,” Rev. Sci. Instrum. 64, 3548–3562 (1993).
[CrossRef]

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
[CrossRef]

1991 (1)

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

1989 (1)

1988 (2)

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

1986 (2)

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
[CrossRef]

1985 (1)

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Alpert, B.

B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
[CrossRef]

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Barlund, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Basart, J. P.

B. Wang, J. C. Moulder, J. P. Basart, “Wavelets in the solution of the volume integral equation: application to eddy current modeling,” J. Appl. Phys. 81, 6397–6406 (1997).
[CrossRef]

Bednar, J. B.

A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
[CrossRef]

Bertero, M.

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

M. Bertero, “Linear inverse and Ill-posed problems,” in Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, Boston, 1989), Vol. 75, pp. 1–120.

Beylkin, G.

B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
[CrossRef]

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Chew, W. C.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

Chow, E. Y.

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

Cohen, A.

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

Coifman, R.

B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
[CrossRef]

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Daubechies, I.

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

Duchene, B.

D. Lesselier, B. Duchene, “Wave-field inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science, W. R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996).

Gersztenkorn, A.

A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
[CrossRef]

Gill, P. E.

P. E. Gill, W. Murry, M. H. Wright, Practical Optimization (Academic, San Diego, 1981).

Groom, R. W.

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Habashy, T. M.

C. Torres-Verdı́n, T. M. Habashy, “Rapid 2.5-D forward modeling and inversion via a new nonlinear scattering approximation,” Radio Sci. 29, 1051–1079 (1994).
[CrossRef]

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

T. M. Habashy, W. C. Chew, E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Sci. 21, 635–645 (1986).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computations by Moment Methods (Macmillan, New York, 1968).

Hohmann, G.

T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
[CrossRef]

Isernia, T.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Jawerth, B.

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

Lesselier, D.

D. Lesselier, B. Duchene, “Wave-field inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science, W. R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996).

Lines, L. R.

A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
[CrossRef]

Mandelis, A.

L. Nicolaides, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions,” Inverse Probl. 13, 1339–1412 (1997).

L. Nicolaides, M. Munidasa, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with backpropagation and transmission reconstructions: experimental,” Inverse Probl. 13, 1413–1425 (1997).
[CrossRef]

A. Mandelis, “Green’s functions in thermal-wave physics: Cartesian coordinate representations,” J. Appl. Phys. 78, 647–655 (1995).
[CrossRef]

O. Pade, A. Mandelis, “Thermal-wave slice tomography using wave-field reconstruction,” Inverse Probl. 10, 185–197 (1994).
[CrossRef]

O. Pade, A. Mandelis, “Computational thermal-wave slice tomography with backpropagation and transmission reconstructions,” Rev. Sci. Instrum. 64, 3548–3562 (1993).
[CrossRef]

A. Mandelis, “Theory of photothermal-wave diffraction and interference in condensed media,” J. Opt. Soc. Am. A 6, 298–308 (1989).
[CrossRef]

Meyer, Y.

Y. Meyer, Wavelets and Operators (Cambridge U. Press, Cambridge, 1995).

Miller, E. L.

E. L. Miller, A. S. Willsky, “Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems,” Multidimens. Syst. Signal Process. 8, 151–184 (1997).
[CrossRef]

E. L. Miller, A. S. Willsky, “Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation,” Radio Sci. 31, 51–67 (1996).
[CrossRef]

E. L. Miller, A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. Geosci. Remote Sens. 34, 346–357 (1996).
[CrossRef]

E. L. Miller, “The application of multiscale and statistical techniques to the solution of inverse problems,” (MIT Laboratory for Information and Decision Systems, Cambridge, Mass., 1994).

Mol, C. D.

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Moulder, J. C.

B. Wang, J. C. Moulder, J. P. Basart, “Wavelets in the solution of the volume integral equation: application to eddy current modeling,” J. Appl. Phys. 81, 6397–6406 (1997).
[CrossRef]

Munidasa, M.

L. Nicolaides, M. Munidasa, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with backpropagation and transmission reconstructions: experimental,” Inverse Probl. 13, 1413–1425 (1997).
[CrossRef]

Murry, W.

P. E. Gill, W. Murry, M. H. Wright, Practical Optimization (Academic, San Diego, 1981).

Nguyen, T.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).

Nicolaides, L.

L. Nicolaides, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions,” Inverse Probl. 13, 1339–1412 (1997).

L. Nicolaides, M. Munidasa, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with backpropagation and transmission reconstructions: experimental,” Inverse Probl. 13, 1413–1425 (1997).
[CrossRef]

Oristaglio, M.

T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
[CrossRef]

Pade, O.

O. Pade, A. Mandelis, “Thermal-wave slice tomography using wave-field reconstruction,” Inverse Probl. 10, 185–197 (1994).
[CrossRef]

O. Pade, A. Mandelis, “Computational thermal-wave slice tomography with backpropagation and transmission reconstructions,” Rev. Sci. Instrum. 64, 3548–3562 (1993).
[CrossRef]

Pascazio, V.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Pierri, R.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Pike, E. R.

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Rokhlin, V.

B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 159–184 (1993).
[CrossRef]

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Spies, B. R.

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Strang, G.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).

Torres-Verdi´n, C.

C. Torres-Verdı́n, T. M. Habashy, “Rapid 2.5-D forward modeling and inversion via a new nonlinear scattering approximation,” Radio Sci. 29, 1051–1079 (1994).
[CrossRef]

Tripp, A.

T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
[CrossRef]

Vial, P.

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

Wang, B.

B. Wang, J. C. Moulder, J. P. Basart, “Wavelets in the solution of the volume integral equation: application to eddy current modeling,” J. Appl. Phys. 81, 6397–6406 (1997).
[CrossRef]

Wang, T.

T. Wang, M. Oristaglio, A. Tripp, G. Hohmann, “Inversion of diffusive transient electromagnetic data by a conjugate-gradient method,” Radio Sci. 29, 1143–1156 (1994).
[CrossRef]

Wickerhauser, M. V.

M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (A. K. Peters, Wellesley, Mass., 1994).

Willsky, A. S.

E. L. Miller, A. S. Willsky, “Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems,” Multidimens. Syst. Signal Process. 8, 151–184 (1997).
[CrossRef]

E. L. Miller, A. S. Willsky, “Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation,” Radio Sci. 31, 51–67 (1996).
[CrossRef]

E. L. Miller, A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. Geosci. Remote Sens. 34, 346–357 (1996).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murry, M. H. Wright, Practical Optimization (Academic, San Diego, 1981).

Appl. Comput. Harmon. Anal. (1)

A. Cohen, I. Daubechies, B. Jawerth, P. Vial, “Multiresolution analysis, wavelets and fast algorithms on an interval,” Appl. Comput. Harmon. Anal. 1, 54–81 (1993).
[CrossRef]

Commun. Pure Appl. Math. (2)

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical algorithms. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Geophysics (1)

A. Gersztenkorn, J. B. Bednar, L. R. Lines, “Robust iterative inversion for the one-dimensional acoustic wave equation,” Geophysics 51, 357–368 (1986).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

E. L. Miller, A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. Geosci. Remote Sens. 34, 346–357 (1996).
[CrossRef]

IEEE Trans. Image Process. (1)

P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlund, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Inverse Probl. (5)

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

M. Bertero, C. D. Mol, E. R. Pike, “Linear inverse problems with discrete data. II. Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

L. Nicolaides, A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions,” Inverse Probl. 13, 1339–1412 (1997).

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Figures (9)

Fig. 1
Fig. 1

Experimental setup for thermal-wave slice tomography. Incident thermal waves originating from a point on the top of the material sample interact with defects, giving rise to scattered fields, whose effects are measured by arrays located at the top and bottom of the sample. The objective of the inverse problem is to image the internal structure of the material on the basis of these measurements. The incidence point is generally scanned across the top.

Fig. 2
Fig. 2

Plots of Daubechies four-tap wavelet basis functions: ψ4,1(x) (solid curve), ψ4,3(x) (dashed curve), ψ5,13(x) (dotted–dashed curve), ψ5,17(x) (dotted curve).

Fig. 3
Fig. 3

Wavelet transform of a pulse function. The original function is given by the top trace, and the wavelet coefficients at a variety of scales are shown in the lower traces. Finer-scale information is conveyed in traces closer to the top. The wavelet coefficients characterize the local discontinuity structure of the function.

Fig. 4
Fig. 4

Plots of l(x) (dashed curve) and l¯(x) (solid curve) used to implement wavelet domain edge-preserving regularization. Both functions are identical for x0, but l¯(x) is better behaved near the origin, thereby aiding in numerical implementation.

Fig. 5
Fig. 5

True object function, inversion results, and error versus iteration for a single defect located near the top of the material. The laser source position is along the line x=0.

Fig. 6
Fig. 6

True object function, inversion results, and error versus iteration for a single defect located near the top of the material. Here we compare reconstructions obtained by using (b) only backpropagation data, (c) only transmission data, and (d) a combination of both in the nonlinear inversion algorithm. The laser source position is along the line x=0.

Fig. 7
Fig. 7

True object function, inversion results, and error versus iteration for a single defect located near the bottom of the material. The laser source position is along the line x=0.

Fig. 8
Fig. 8

True object function, inversion results, and error versus iteration for a single, small-defect problem. The laser source position is along the line x=0.

Fig. 9
Fig. 9

True object function, inversion results, and error versus iteration for a two-defect problem. The laser source position is along the line x=0.

Equations (35)

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yi(rk)=AG(rk, r)Ti(r)g(r)dr+ni(rk),
Ti(r)=T¯i(r)+AG(r, r)Ti(r)g(r)dr,
yi=LiD(Ti)g+ni,
Ti=T¯i+GD(g)Ti.
yi=hi(g)+ni,
hi(g)=LiD([I-GD(g)]-1T¯i)g.
a(x)=j=-k=-αj,kψj,k(x),
a(x)=k=-a0,kϕ0,k(x)+j=0Fak=-αj,nψj,n(x),
αˇ=Wa,y aˇWa,xT.
α=Waa.
Wiyi=[WiLiWgT][WgD(Ti)WgT](Wgg)+Wini,
WiTi=WiT¯i+[WiGWgT][WgD(g)WiT](WiTi),
ηi=ΛiΔ(θi)γ+νi,
θi=θ¯i+ΓΔ(γ)θi,
ηi=ΛiΔ([I-ΓΔ(γ)]-1θ¯i)γΥi(γ)+νi.
η=Υ(γ)+ν,
γˆ=arg minγ C(γ),
C (γ)=12η-Υ(γ)R-12+λ2ρT(γ)ρ(γ),
γˆn+1=γˆn+sn,
sn=arg minσ2 C (γˆn+s),
s=[IT(γˆn)R-1IT(γˆn)+σ2LT(γˆn)L(γˆn)]-1×IT(γˆn)R-1[η-Υ(γˆn)]-LT(γˆn)L(γˆn).
[D]iidi=2-(αxjx,i+αyjy,i).
ρT(γ)=[d11/2|γ1|p/2d21/2|γ2|p/2dN1/2|γN|p/2],
[L(γˆn)]i,i=di1/2 p2l([γˆn]i),
l(x)=sign(x)|x|p/2-1,
l¯(x)=x+|x|2-p/2,
Ii(γ)=ΛiΔ(θi)+Λi[I-ΓΔ(γ)]-1ΓΔ(θi),
Ξn+1[I-ΓΔ(γˆn+1)]-1=[I-ΓΔ(γˆn+sn)]-1,
Ξn+1=[I-ΓΔ(γˆn+sn)]-1
=[I-ΓΔ(γˆn)-ΓΔ(sn)]-1,
=Ξn[I-ΓΔ(sn)Ξn]-1,
Ξn+ΞnΓΔ(sn)Ξn,
Ξn+1=Ξn+ΞnΓΔ(sn)Ξn,
Ξn+1=truncδ(Ξn+truncδ(ΞnΓ)truncδ[Δ(γˆn)Ξn]),
SNRi=10 log10 h(g)22Niqi2,

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