Abstract

A new solution approach to inverse scattering from aspect-limited phaseless measurements of the total field is introduced and discussed. In analogy with the case of measurements on closed curves [J. Opt. Soc. Am. A 21, 622 (2004) ], the procedure splits the problem into two different steps. In the first step, amplitude and phase of the scattered field are estimated from only amplitude information of the total field. By properly extending the concept of reduced radiated field to the case of scattered fields (as a function of both illumination and measurement variables) and taking advantage of the properties of the square amplitude distribution of the total field, criteria are given for an optimal choice of the measurement setup and a successful retrieval. Then the complex permittivity profile is reconstructed in the second step, starting from the scattered fields estimated in the previous step. Numerical examples are provided to assess the effectiveness of the whole chain in the presence of noise-corrupted data and the relevance of the representation introduced for the scattered fields.

© 2006 Optical Society of America

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  1. 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]
  2. T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
    [Crossref]
  3. R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
    [Crossref]
  4. Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).
  5. S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
    [Crossref]
  6. W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1999).
    [Crossref]
  7. M. Bertero, "Linear inverse and ill-posed problems," in Advances in Electronics and Electron Physics (Academic, 1989), Vol. 75, pp. 1-120.
  8. O. M. Bucci and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci. 32, 2123-2138 (1997).
    [Crossref]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. O. M. Bucci and G. Franceschetti, "On the spatial bandwidth of the scattered fields," IEEE Trans. Antennas Propag. 35, 1445-1455 (1987).
    [Crossref]
  15. O. M. Bucci and G. Franceschetti, "On the degrees of freedom of the scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1999).
    [Crossref]
  16. O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
    [Crossref]
  17. R. Snieder, "The role of nonlinearity in inverse problems," Inverse Probl. 14, 387-404 (1998).
    [Crossref]
  18. R. Potthast, "On a concept of uniqueness in inverse scattering for a finite number of incident waves," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 58, 666-682 (1998).
    [Crossref]
  19. I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
    [Crossref]
  20. D. Colton and R. Krees, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).
  21. Strictly speaking, see Ref. ; the Dirichlet polynomial DN(x)=sin[(2N+1)x/2]/[(2N+1)sin(x/2)] should be used in Eq. . However, this is really required only if the observation domain totally encircles the source.
  22. O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
    [Crossref]
  23. R. E. Kleinman and P. M. van den Berg, "An extended modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
    [Crossref]
  24. O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, "Degree of nonlinearity and a new solution procedure in scalar two-dimensional inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
    [Crossref]
  25. L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," Inverse Probl. 21, S33-S50 (2005).
    [Crossref]
  26. T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
    [Crossref]
  27. For an efficient exploitation of the fast-Fourier-transform (FFT) codes in the solution of Eq. , the first integer that is a power of 2 not smaller than the needed number of measures is considered in the numerical procedures. Note that this convenient number is achieved by interpolating the measured independent samples via 2D FFT techniques.
  28. W. C. Chew and J. L. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous bodies," IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
    [Crossref]
  29. P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
    [Crossref]
  30. L. Crocco and T. Isernia, "Inverse scattering with real data: detecting and imaging homogeneous dielectric objects," Inverse Probl. 17, 1573-1583 (2001).
    [Crossref]
  31. I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
    [Crossref]
  32. 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]
  33. 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]
  34. I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

2005 (1)

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," Inverse Probl. 21, S33-S50 (2005).
[Crossref]

2004 (4)

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]

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]

T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
[Crossref]

I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
[Crossref]

2003 (3)

O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
[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]

P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
[Crossref]

2002 (1)

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

2001 (5)

S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
[Crossref]

R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
[Crossref]

L. Crocco and T. Isernia, "Inverse scattering with real data: detecting and imaging homogeneous dielectric objects," Inverse Probl. 17, 1573-1583 (2001).
[Crossref]

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 two-dimensional inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[Crossref]

1999 (3)

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1999).
[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 G. Franceschetti, "On the degrees of freedom of the scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1999).
[Crossref]

1998 (3)

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

R. Snieder, "The role of nonlinearity in inverse problems," Inverse Probl. 14, 387-404 (1998).
[Crossref]

R. Potthast, "On a concept of uniqueness in inverse scattering for a finite number of incident waves," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 58, 666-682 (1998).
[Crossref]

1997 (3)

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

T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
[Crossref]

1995 (1)

W. C. Chew and J. L. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous bodies," IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[Crossref]

1993 (2)

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]

R. E. Kleinman and P. M. van den Berg, "An extended modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[Crossref]

1992 (1)

1987 (1)

O. M. Bucci and G. Franceschetti, "On the spatial bandwidth of the scattered fields," IEEE Trans. Antennas Propag. 35, 1445-1455 (1987).
[Crossref]

Abubakar, A.

P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
[Crossref]

R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
[Crossref]

Belkebir, K.

Bertero, M.

M. Bertero, "Linear inverse and ill-posed problems," in Advances in Electronics and Electron Physics (Academic, 1989), Vol. 75, pp. 1-120.

Bloemenkamp, R. F.

R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
[Crossref]

Bryan, J. A.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Bucci, O. M.

O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
[Crossref]

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

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of the scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1999).
[Crossref]

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic 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]

O. M. Bucci and G. Franceschetti, "On the spatial bandwidth of the scattered fields," IEEE Trans. Antennas Propag. 35, 1445-1455 (1987).
[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]

S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
[Crossref]

Capozzoli, A.

O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
[Crossref]

Cardace, N.

Catapano, I.

I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

Chaumet, P. C.

Chew, W. C.

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1999).
[Crossref]

W. C. Chew and J. L. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous bodies," IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[Crossref]

Colton, D.

D. Colton and R. Krees, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

Crocco, L.

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," Inverse Probl. 21, S33-S50 (2005).
[Crossref]

I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
[Crossref]

T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (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 two-dimensional inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[Crossref]

L. Crocco and T. Isernia, "Inverse scattering with real data: detecting and imaging homogeneous dielectric objects," Inverse Probl. 17, 1573-1583 (2001).
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
[Crossref]

D'Elia, G.

O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
[Crossref]

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.

S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
[Crossref]

D'Urso, M.

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," 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 Trans. Geosci. Remote Sens. Lett. 1, 331-337 (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]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

Fokkema, J. T.

P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
[Crossref]

Franceschetti, G.

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of the scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1999).
[Crossref]

O. M. Bucci and G. Franceschetti, "On the spatial bandwidth of the scattered fields," IEEE Trans. Antennas Propag. 35, 1445-1455 (1987).
[Crossref]

Gennarelli, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic 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, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," Inverse Probl. 21, S33-S50 (2005).
[Crossref]

I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
[Crossref]

T. Isernia, L. Crocco, and M. D'Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (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 two-dimensional inverse scattering problems," J. Opt. Soc. Am. A 18, 1832-1843 (2001).
[Crossref]

L. Crocco and T. Isernia, "Inverse scattering with real data: detecting and imaging homogeneous dielectric objects," Inverse Probl. 17, 1573-1583 (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]

T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
[Crossref]

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

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

Joines, W. T.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Kleinman, R. E.

R. E. Kleinman and P. M. van den Berg, "An extended modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[Crossref]

Krees, R.

D. Colton and R. Krees, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

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]

Lin, J. L.

W. C. Chew and J. L. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous bodies," IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[Crossref]

Liu, Q. O.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Maleki, M. H.

Massa, 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).
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S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
[Crossref]

Nolte, L. W.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Pascazio, V.

T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
[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, 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).
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T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
[Crossref]

Potthast, R.

R. Potthast, "On a concept of uniqueness in inverse scattering for a finite number of incident waves," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 58, 666-682 (1998).
[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]

Savarese, C.

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

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R. Snieder, "The role of nonlinearity in inverse problems," Inverse Probl. 14, 387-404 (1998).
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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]

van den Berg, P.

R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
[Crossref]

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P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
[Crossref]

R. E. Kleinman and P. M. van den Berg, "An extended modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[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]

Wang, T. T.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Wang, Y. M.

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1999).
[Crossref]

Ybarra, G. A.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

Zhang, Z. Q.

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

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W. C. Chew and J. L. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous bodies," IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[Crossref]

IEEE Trans. Antennas Propag. (5)

O. M. Bucci, A. Capozzoli, and G. D'Elia, "A novel effective approach to scatterers localization problems," IEEE Trans. Antennas Propag. 51, 2079-2090 (2003).
[Crossref]

Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, "Active microwave imaging. I. 2-D forward and inverse scattering methods," IEEE Trans. Antennas Propag. 50, 123-133 (2002).

O. M. Bucci and G. Franceschetti, "On the spatial bandwidth of the scattered fields," IEEE Trans. Antennas Propag. 35, 1445-1455 (1987).
[Crossref]

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of the scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1999).
[Crossref]

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic 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. (2)

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]

T. Isernia, V. Pascazio, and R. Pierri, "A non-linear estimation method in tomographic imaging," IEEE Trans. Geosci. Remote Sens. 35, 910-923 (1997).
[Crossref]

IEEE Trans. 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 Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
[Crossref]

IEEE Trans. Med. Imaging (1)

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1999).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

S. Caorsi, M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech. 52, 1217-1228 (2001).
[Crossref]

Inverse Probl. (4)

R. F. Bloemenkamp, A. Abubakar, and P. van den Berg, "Inversion of experimental multi-frequency data using the contrast source inversion method," in special section "Testing Inversion Algorithm Against Experimental Data," Inverse Probl. 17, 1611-1622 (2001).
[Crossref]

R. Snieder, "The role of nonlinearity in inverse problems," Inverse Probl. 14, 387-404 (1998).
[Crossref]

L. Crocco, M. D'Urso, and T. Isernia, "Testing the contrast source—extended Born inversion method against real data: the TM case," in special section, "Testing Inversion Algorithm against Experimental Data: Inhomogeneous Target DATA 2005," Inverse Probl. 21, S33-S50 (2005).
[Crossref]

L. Crocco and T. Isernia, "Inverse scattering with real data: detecting and imaging homogeneous dielectric objects," Inverse Probl. 17, 1573-1583 (2001).
[Crossref]

J. Opt. Soc. Am. A (5)

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. Lett. (1)

Radio Sci. (4)

P. M. van den Berg, A. Abubakar, and J. T. Fokkema, "Multiplicative regularization for contrast profile inversion," Radio Sci. 38, 23-1-23-10 (2003).
[Crossref]

R. E. Kleinman and P. M. van den Berg, "An extended modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[Crossref]

I. Catapano, L. Crocco, and T. Isernia, "A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media," Radio Sci. 39, RS1012, (2004).
[Crossref]

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

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

R. Potthast, "On a concept of uniqueness in inverse scattering for a finite number of incident waves," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 58, 666-682 (1998).
[Crossref]

Other (6)

M. Bertero, "Linear inverse and ill-posed problems," in Advances in Electronics and Electron Physics (Academic, 1989), Vol. 75, pp. 1-120.

D. Colton and R. Krees, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

Strictly speaking, see Ref. ; the Dirichlet polynomial DN(x)=sin[(2N+1)x/2]/[(2N+1)sin(x/2)] should be used in Eq. . However, this is really required only if the observation domain totally encircles the source.

For an efficient exploitation of the fast-Fourier-transform (FFT) codes in the solution of Eq. , the first integer that is a power of 2 not smaller than the needed number of measures is considered in the numerical procedures. Note that this convenient number is achieved by interpolating the measured independent samples via 2D FFT techniques.

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Advances in microwave tomography: phaseless measurements and layered background," in Proceedings of the 2nd International Workshop on Advanced Ground Penetrating Radar, A.Yarovoy, ed. (IEEE, 2003), pp. 183-188.
[Crossref]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "Faithful phaseless microwave tomography," in Progress in Electromagnetic Research (PIERS) 2006 Cambridge Abstracts (The Elecromagnetics Academy, 2006), p. 458.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Comparison between the real part of the (a) actual and (b) estimated scattered field for the case of a single object.

Fig. 3
Fig. 3

Comparison between the imaginary part of the (a) actual and (b) estimated scattered field for the case of a single object.

Fig. 4
Fig. 4

Comparison between the (a) real parts and (b) imaginary parts of the reconstructed field (dashed curves) and the actual field (solid curves) for a source located in y s = 7.3 λ as a function of the locations of the receivers.

Fig. 5
Fig. 5

Behavior of the (a) real parts and (b) imaginary parts of the reconstructed field (dashed curves) and the actual one (solid curves) for a source located in y s = 3.5 λ as a function of the locations of the receivers.

Fig. 6
Fig. 6

Comparison between the real part of the (a) actual and (b) estimated scattered field by using a uniform representation for the scattered field in terms of the observation variable.

Fig. 7
Fig. 7

Comparison between the imaginary part of the (a) actual and (b) estimated scattered field by using a uniform representation for the scattered field in terms of the observation variable.

Fig. 8
Fig. 8

Real part of the contrast for the case of a multiple object.

Fig. 9
Fig. 9

Comparison between the real part of the (a) actual and (b) estimated scattered field for the case of a multiple object.

Fig. 10
Fig. 10

Comparison between the imaginary part of the (a) actual and (b) estimated scattered field for the case of a multiple object.

Fig. 11
Fig. 11

Real part of the reconstructed contrast function.

Equations (26)

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χ ( r ) = [ ε ̃ ( r ) ε b 1 ] , r Ω .
J ( r s , r ) = χ ( r ) E inc , i ( r s , r ) + χ ( r ) Ω g ( r r ) J ( r s , r ) d r = χ ( r ) E inc , i ( r s , r ) + χ ( r ) A i [ J ] ( r s , r ) , r Ω ,
E tot ( r s , r o ) = E inc , e ( r s , r o ) + E s ( r s , r o ) = E inc , e ( r s , r o ) + Ω g ( r o r ) J ( r s , r ) d r = E inc , e ( r s , r o ) + A e [ J ] ( r s , r o ) , r s Γ s , r o Γ o ,
F w ( ξ ) = n = N N F w ( ξ n ) sinc [ γ W ξ ( ξ ξ n ) ] ,
F s ( ξ s , ξ o ) = E s ( ξ s , ξ o ) exp [ j ψ o ( ξ o ) ] exp [ j ψ s ( ξ s ) ] .
ξ s = ξ o = ϑ , ψ s ( r ) = ψ o ( r ) = k r 2 a 2 k a cos 1 ( a r ) ,
W E s = k a ,
ξ ̂ o ( y o ) = π ( R 1 R 2 ) 2 L s , ψ ̂ o ( y o ) = k 2 ( R 1 + R 2 ) k L s 2 ,
W E inc , e = k L s π ,
R 1 = d 2 + ( y o + L s 2 ) 2 , R 2 = d 2 + ( y o L s 2 ) 2 ,
F inc , e ( ξ ̂ s , ξ ̂ o ) = E inc , e ( ξ ̂ s , ξ ̂ o ) exp [ j ψ ̂ o ( ξ ̂ o ) ] exp [ j ψ ̂ s ( ξ ̂ s ) ] ,
B [ E s ( r s , r o ) ] = E s ( r s , r o ) 2 + 2 Re [ E s ( r s , r o ) E inc , e ( r s , r o ) * ] ,
B [ E s ( r s , r o ) ] = M ̃ 2 ( r s , r o ) E inc , e ( r s , r o ) 2 ,
w E s 2 ( s ) = 2 w E s ( s ) .
Re ( E s E inc , e * ) = 1 2 ( E s E inc , e * + E s * E inc , e ) .
E s ( s ) E inc , e ( s ) * = F s ( s ) F inc , e ( s ) * exp { j [ ψ ( s ) ψ ̂ ( s ) ] } ,
w Int ( s ) = w E s ( s ) + w E inc , e ( s ) + ( ψ ψ ̂ ) s .
w Tot ( s ) = max [ w Int ( s ) , w E s 2 ( s ) ] .
L d 2 ( d 2 a ) 2 1 .
N Tot = N + M + Int [ γ π ( ψ ψ ̂ ) 0 L 2 ] = N + M + Δ N .
R a = 4 N Tot 2 2 × 4 N 2 = 1 2 ( N Tot N ) 2 1 2 ( 1 + M N ) 2 .
W ξ = max ξ [ w ( ξ ) ] = max ξ [ max r D d ψ ( ξ ) d ξ k R ( ξ , r ) ξ ] ,
R = r ( ξ ) r ,
ψ ( ξ ) = k 2 0 s ( ξ ) [ max r D R s + min r D R s ] d s ,
ξ = ξ ( s ) = k 2 W ξ 0 s [ max r D R s min r D R s ] d s ,
N ( s ¯ ) = Int [ γ ξ ( s ¯ ) W ξ π ] = Int [ γ π k 2 0 s ¯ [ max r D R s min r D R s ] d s ] = Int [ γ π 0 s ¯ w ( s ) d s ] .

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