Abstract

Two-dimensional target characterization using inverse profiling approaches with total-field phaseless data is discussed. Two different inversion schemes are compared. In the first one, the intensity-only data are exploited in a minimization scheme, thanks to a proper definition of the cost functional. Specific normalization and starting guess are introduced to avoid the need for global optimization methods. In the second scheme [J. Opt. Soc. Am. A 21, 622 (2004) ], one exploits the field properties and the theoretical results on the inversion of quadratic operators to derive a two-step solution strategy, wherein the (complex) scattered fields embedded in the available data are retrieved first and then a traditional inverse scattering problem is solved. In both cases, the analytical properties of the fields allow one to properly fix the measurement setup and identify the more convenient strategy to adopt. Also, indications on the number and types of sources and receivers to be used are given. Results from experimental data show the efficiency of these approaches and the tools introduced.

© 2008 Optical Society of America

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References

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  1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).
  2. P. C. Chaumet, K. Belkebir, and A. Sentenac, "Superresolution of three-dimensional optical imaging by use of evanescent waves," Opt. Lett. 29, 2740-2742 (2004).
    [CrossRef] [PubMed]
  3. P. C. Chaumet, K. Belkebir, and R. Lencrerot, "Three-dimensional optical imaging in layered media," Opt. Express 14, 3415-3426 (2006).
    [CrossRef] [PubMed]
  4. M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037-2046 (2006).
    [CrossRef] [PubMed]
  5. V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
    [CrossRef] [PubMed]
  6. N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
    [CrossRef]
  7. L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
    [CrossRef]
  8. M. H. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356-1363 (1992).
    [CrossRef]
  9. M. H. Maleki and A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithms from optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
    [CrossRef]
  10. T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
    [CrossRef]
  11. S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
    [CrossRef]
  12. G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
    [CrossRef]
  13. O. M. Bucci, L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
    [CrossRef]
  14. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
    [CrossRef]
  15. O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
    [CrossRef]
  16. O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
    [CrossRef]
  17. L. Crocco, M. D'Urso, and T. Isernia, "Faithful non-linear imaging from only-amplitude measurements of incident and total fields," Opt. Express 15, 3804-3815 (2007).
    [CrossRef] [PubMed]
  18. R. E. Kleinman and P. M. van den Berg, "A contrast source inversion method," Inverse Probl. 13, 1607-1620 (1997).
    [CrossRef]
  19. O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, "Degree of nonlinearity and a new solution procedure in scalar 2-D inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
    [CrossRef]
  20. T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE GRSS Geosci. Remote Sens. Lett. 1, 327-331 (2004).
    [CrossRef]
  21. L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source extended born method against real data: the TM case," Inverse Probl. 21, S33-S50 (2005).
    [CrossRef]
  22. T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
    [CrossRef]
  23. K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data," Inverse Probl. 17, 1565-2028(2001).
    [CrossRef]
  24. K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, S1-S165 (2005).
    [CrossRef]
  25. J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
    [CrossRef]

2007 (1)

2006 (4)

2005 (3)

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source extended born method against real data: the TM case," Inverse Probl. 21, S33-S50 (2005).
[CrossRef]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, S1-S165 (2005).
[CrossRef]

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
[CrossRef]

2004 (3)

2003 (1)

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

2002 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

2001 (4)

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, "Degree of nonlinearity and a new solution procedure in scalar 2-D inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[CrossRef]

T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
[CrossRef]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data," Inverse Probl. 17, 1565-2028(2001).
[CrossRef]

1999 (1)

1998 (2)

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

1997 (3)

O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
[CrossRef]

R. E. Kleinman and P. M. van den Berg, "A contrast source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

1993 (1)

1992 (1)

Azaro, R.

G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
[CrossRef]

Belkebir, K.

P. C. Chaumet, K. Belkebir, and R. Lencrerot, "Three-dimensional optical imaging in layered media," Opt. Express 14, 3415-3426 (2006).
[CrossRef] [PubMed]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, S1-S165 (2005).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Superresolution of three-dimensional optical imaging by use of evanescent waves," Opt. Lett. 29, 2740-2742 (2004).
[CrossRef] [PubMed]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data," Inverse Probl. 17, 1565-2028(2001).
[CrossRef]

Bucci, O. M.

O. M. Bucci, L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, "Degree of nonlinearity and a new solution procedure in scalar 2-D inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[CrossRef]

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
[CrossRef]

Caorsi, S.

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Cardace, N.

Chaumet, P. C.

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

Crocco, L.

Destouches, N.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Devaney, A. J.

Donelli, M.

G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
[CrossRef]

D'Urso, M.

Eyraud, C.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
[CrossRef]

Fiddy, M. A.

Franceschini, G.

G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
[CrossRef]

Geffrin, J.-M.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
[CrossRef]

Gennarelli, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

Giovannini, H.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Guérin, C. A.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Harada, H.

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Isernia, T.

L. Crocco, M. D'Urso, and T. Isernia, "Faithful non-linear imaging from only-amplitude measurements of incident and total fields," Opt. Express 15, 3804-3815 (2007).
[CrossRef] [PubMed]

O. M. Bucci, L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source extended born method against real data: the TM case," Inverse Probl. 21, S33-S50 (2005).
[CrossRef]

T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE GRSS Geosci. Remote Sens. Lett. 1, 327-331 (2004).
[CrossRef]

L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
[CrossRef]

O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, "Degree of nonlinearity and a new solution procedure in scalar 2-D inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[CrossRef]

T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
[CrossRef]

O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
[CrossRef]

Kleinman, R. E.

R. E. Kleinman and P. M. van den Berg, "A contrast source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

Lencrerot, R.

Leone, G.

Lequime, M.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Maleki, M. H.

Massa, A.

G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
[CrossRef]

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Pascazio, V.

T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
[CrossRef]

Pastorino, M.

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Pierri, R.

T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
[CrossRef]

Randazzo, A.

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Sabouroux, P.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
[CrossRef]

Saillard, M.

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, S1-S165 (2005).
[CrossRef]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data," Inverse Probl. 17, 1565-2028(2001).
[CrossRef]

Savarese, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

Schatzberg, A.

Sentenac, A.

Soldovieri, F.

Takenaka, T.

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Tanaka, M.

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Testorf, M.

van den Berg, P. M.

R. E. Kleinman and P. M. van den Berg, "A contrast source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

Wall, D. J. N.

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

IEEE GRSS Geosci. Remote Sens. Lett. (1)

T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE GRSS Geosci. Remote Sens. Lett. 1, 327-331 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetics fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (3)

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

G. Franceschini, M. Donelli, R. Azaro, and A. Massa, "Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach," IEEE Trans. Geosci. Remote Sens. 41, 3527-3539 (2006).
[CrossRef]

T. Isernia, V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens. 39, 1596-1607 (2001).
[CrossRef]

Inverse Probl. (5)

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data," Inverse Probl. 17, 1565-2028(2001).
[CrossRef]

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, S1-S165 (2005).
[CrossRef]

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: experimental set-up and measurement precision," Inverse Probl. 21, S117-S130 (2005).
[CrossRef]

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source extended born method against real data: the TM case," Inverse Probl. 21, S33-S50 (2005).
[CrossRef]

R. E. Kleinman and P. M. van den Berg, "A contrast source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

J. Microsc. (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (6)

Microwave Opt. Technol. Lett. (1)

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Opt. Commun. (1)

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Radio Sci. (1)

O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
[CrossRef]

Other (1)

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem. A two-dimensional target with cross section Ω. Γ (with radius b) is the circle where the sources ( T x ) and probes ( R x ) are located. a is the radius of the minimum circle enclosing the unknown targets.

Fig. 2
Fig. 2

TwinDielTM dataset: Intensity and phase patterns of (a), (b) the measured scattered fields and (c), d) those reconstructed by solving Eq. (16) of the two-step procedure in Section 4.

Fig. 3
Fig. 3

Real part of the reconstructed contrast function for the TWINDIELTM dataset when using Eq. (17) of the two-step procedure in Section 4. The maximum value of the estimated contrast function is 2.10.

Fig. 4
Fig. 4

Real part of the reconstructed contrast function for the TWINDIELTM dataset with the one-step approach of Section 3. The background solution has been used as starting guess. The maximum value of the estimated contrast function is 0.67.

Fig. 5
Fig. 5

Real part of the reconstructed contrast function for the TWINDIELTM dataset obtained with the one-step procedure of Section 3. The modified backpropagation solution has been used as starting guess in the inversion procedure. The maximum value of the estimated contrast function is 1.85.

Fig. 6
Fig. 6

Real part of the reconstructed contrast function for the FOAMDIELEXTTM dataset obtained when following the two-step approach of Section 4. The background solution has been used as starting guess. The maximum value of the estimated contrast function is 1.34.

Fig. 7
Fig. 7

Real part of the reconstructed contrast function for the FOAMDIELEXTTM dataset by using the one-step approach of Section 3. The modified backpropagation solution has been used as starting guess. The maximum value of the estimated contrast function is 1.75.

Fig. 8
Fig. 8

Real part of the reconstructed contrast function for the FOAMMETEXTTM dataset by using the one-step approach of Section 3. The modified backpropagation solution has been used as starting guess.

Fig. 9
Fig. 9

Imaginary part of the reconstructed contrast function for the FOAMMETEXTTM dataset by using the one-step approach of Section 3. The modified backpropagation solution has been used as starting guess.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

E i l ( r ) = E l i ( r ) u z = A ω μ 0 4 H 0 ( 2 ) ( k b r r l ) u z ,
E l ( r Γ ) = E l i ( r Γ ) + E l s ( r Γ ) = E l i ( r Γ ) + D G ( r , r ) J l ( r ) d r ,
J l ( r D ) = χ ( r D ) E l i ( r D ) + χ ( r D ) D G ( r , r ) J l ( r ) d r ,
J l ( r ) ξ ( r ) E l i ( r ) = ξ ( r ) D G ( r , r ) [ J l ( r ) J l ( r ) ] d r = ξ ( r ) G mod ( J l ) ,
ξ ( r ) = χ ( r ) 1 χ ( r ) f D ( r ) , f D ( r ) = D G ( r , r ) d r ,
G mod ( J l ) = D G ( r , r ) [ J l ( r ) J l ( r ) ] d r = D G ( r , r ) J l ( r ) d r J l ( r ) f D ( r ) .
E l s = K J l ; J l = ξ E l i + ξ G mod ( J l ) ,
ξ ( r ) = p = 1 P a p ψ p ( r ) ,
J l ( r ) = q = 1 Q c q l ϕ q ( r ) l = 1 , , L ,
J ( ξ ) = l = 1 L α l I l obs E l i + K J l ( ξ ) 2 W Γ 2 ,
w l ( r Γ ) = 1 I l obs ( r Γ ) + ε ,
L ( ξ , J ) = l = 1 L { α l I l obs E l i + K J l ( ξ ) 2 W Γ 2 + β l J l ξ E l i ξ G mod ( J l ) D 2 } ,
ξ L = 2 l = 1 L β l [ E l i + G mod ( J l ) ] * [ J l ξ E l i ξ G mod ( J l ) ] ,
J l L = 4 α l K [ ( E l i + K J l ) ( I obs E l i + K J l 2 ) w l ] + 2 β l [ I ξ G mod ] [ J l ξ E l i ξ G mod ( J l ) ] ,
J l , 0 ( r ) = γ E l i ( r ) ,
ξ 0 ( r ) = min a p J l , 0 ξ E l i ξ G mod ( J l , 0 ) D 2 ,
F ( f ) = l = 1 L I l obs E l i 2 E l s 2 2 R e ( E l i E l s * ) W Γ 2 .
L ( ξ , J ) = l = 1 L { α l ̂ K J l E l est Γ 2 + β l J l ξ E l i ξ G mod ( J l ) D 2 } ,
J ( r ) χ ( r ) E i ( r ) = χ ( r ) D G ( r , r ) [ J ( r ) J ( r ) ] d r + χ ( r ) J ( r ) D G ( r , r ) d r .
J ( r ) χ ( r ) J ( r ) f D ( r ) = χ ( r ) E i ( r ) + χ ( r ) D G ( r , r ) [ J ( r ) J ( r ) ] d r ,
f D ( r ) = D G ( r , r ) d r .
ξ ( r ) = χ ( r ) 1 χ ( r ) f D ( r ) ,
G mod ( J ) = D G ( r , r ) [ J ( r ) J ( r ) ] d r = D G ( r , r ) J ( r ) d r J ( r ) f D ( r ) ,

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