Abstract

A novel method of optical diffraction tomography (ODT) to image weakly scattering, electrically large, two-dimensional (2D) objects using the far-zone scattered field data is presented. The proposed technique is based on the expansion of the target object function in terms of Fourier–Bessel basis functions and an alternative approximation for the total electric field within the support of the investigated scatterer. Analytical (Mie) plane-wave scattering by a layered, circularly symmetric, lossy cylinder, and finite-difference time-domain simulations involving plane-wave scattering by a more general, lossless phantom are utilized to compare the performance of the proposed method with that of the standard ODT techniques, which are based on the Born approximation and the Fourier diffraction theorem. Quantitative and qualitative superiority of the presented method is demonstrated. The proposed 2D technique can be readily extended to more realistic three-dimensional cases. With proper (cylindrical–spherical) receiver configuration, the proposed method can be used without being confined to far-zone observations.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
  2. A. Kirsch, “The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
    [CrossRef]
  3. F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
    [CrossRef]
  4. S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
    [CrossRef]
  5. E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635, 2007.
    [CrossRef]
  6. E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
    [CrossRef] [PubMed]
  7. F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).
  8. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  9. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221-228 (1992).
    [CrossRef] [PubMed]
  10. 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]
  11. G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890-1892 (2002).
    [CrossRef]
  12. G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, “Spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 230-238 (2005).
    [CrossRef]
  13. T. C. Wedberg and J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39-54 (1995).
    [CrossRef]
  14. P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857-859 (2004).
    [CrossRef] [PubMed]
  15. A. J. Devaney and M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855-870 (2003).
    [CrossRef]
  16. P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338-2347 (2005).
    [CrossRef]
  17. J. Y. Cheng and A. J. Devaney, “Inverse scattering and diffraction tomography in cylindrical background media,” J. Opt. Soc. Am. A 23, 1038-1047 (2006).
    [CrossRef]
  18. M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
    [CrossRef] [PubMed]
  19. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).
  21. E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410-412 (1999).
    [CrossRef]
  22. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  23. T. C. Wedberg, J. J. Stamnes, and W. Singer, “Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575-6581 (1995).
    [CrossRef] [PubMed]
  24. H. E. Bussey and J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer: numerical values,” IEEE Trans. Antennas Propag. 23, 723-725 (1975).
    [CrossRef]
  25. M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.
  26. K. Yee, “Numerical solution of inital boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  27. K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC-Press, 1993).
  28. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

2007 (2)

E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635, 2007.
[CrossRef]

2006 (2)

J. Y. Cheng and A. J. Devaney, “Inverse scattering and diffraction tomography in cylindrical background media,” J. Opt. Soc. Am. A 23, 1038-1047 (2006).
[CrossRef]

S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

2005 (2)

2004 (2)

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857-859 (2004).
[CrossRef] [PubMed]

2003 (1)

A. J. Devaney and M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855-870 (2003).
[CrossRef]

2002 (2)

A. Kirsch, “The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890-1892 (2002).
[CrossRef]

2001 (1)

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

1999 (1)

E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410-412 (1999).
[CrossRef]

1995 (2)

1992 (2)

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]

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1975 (1)

H. E. Bussey and J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer: numerical values,” IEEE Trans. Antennas Propag. 23, 723-725 (1975).
[CrossRef]

1966 (1)

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Alviggi, C.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Anastasio, M. A.

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Broquetas, A.

M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.

Bussey, H. E.

H. E. Bussey and J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer: numerical values,” IEEE Trans. Antennas Propag. 23, 723-725 (1975).
[CrossRef]

Cakoni, F.

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).

Cheng, J. Y.

Colton, D.

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).

Dale, B.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Dennison, M.

A. J. Devaney and M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855-870 (2003).
[CrossRef]

Devaney, A. J.

J. Y. Cheng and A. J. Devaney, “Inverse scattering and diffraction tomography in cylindrical background media,” J. Opt. Soc. Am. A 23, 1038-1047 (2006).
[CrossRef]

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338-2347 (2005).
[CrossRef]

P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857-859 (2004).
[CrossRef] [PubMed]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

A. J. Devaney and M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855-870 (2003).
[CrossRef]

E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410-412 (1999).
[CrossRef]

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]

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

di Matteo, L.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Ferrando, M.

M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.

Gbur, G.

Gruber, F. K.

E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

Guo, P.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Hernandez, R. D.

Hou, S.

S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

Huang, Y.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Jofre, L.

M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Kirsch, A.

A. Kirsch, “The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC-Press, 1993).

Lev-Ari, H.

Lombardi, L.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC-Press, 1993).

Maleki, M. H.

Marengo, E. A.

E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635, 2007.
[CrossRef]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410-412 (1999).
[CrossRef]

Marino, M.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Pisaturo, M.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Placida, G. D.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Richmond, J. H.

H. E. Bussey and J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer: numerical values,” IEEE Trans. Antennas Propag. 23, 723-725 (1975).
[CrossRef]

Rius, J. M.

M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.

Schatzberg, A.

Shi, D.

Simonetti, F.

E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

Singer, W.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Solna, K.

S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

Stamnes, J. J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Wedberg, T. C.

Wilding, M.

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

Wolf, E.

Yee, K.

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zhao, H.

S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

Appl. Opt. (1)

Hum. Reprod. (1)

M. Wilding, B. Dale, M. Marino, L. di Matteo, C. Alviggi, M. Pisaturo, L. Lombardi, and G. D. Placida, “Mitochondrial aggregation patterns and activity in human oocytes and preimplantation embryos,” Hum. Reprod. 16, 909-917 (2001).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (3)

E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410-412 (1999).
[CrossRef]

H. E. Bussey and J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer: numerical values,” IEEE Trans. Antennas Propag. 23, 723-725 (1975).
[CrossRef]

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Image Process. (2)

E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-reversal MUSIC imaging of extended targets,” IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

Inverse Probl. (3)

A. Kirsch, “The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

S. Hou, K. Solna, and H. Zhao, “A direct imaging algorithm for extended targets,” Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

A. J. Devaney and M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855-870 (2003).
[CrossRef]

J. Acoust. Soc. Am. (1)

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

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

Opt. Lett. (2)

Pure Appl. Opt. (1)

T. C. Wedberg and J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (8)

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

M. Ferrando, A. Broquetas, L. Jofre, and J. M. Rius, “Microwave imaging of multilayer cylinders using optimization techniques,” in IEEE AP-S International Symposium (IEEE, 1989), pp. 1716-1719.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC-Press, 1993).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Investigated problem and the ODT setup.

Fig. 2
Fig. 2

Reconstructions of the (a) real and (b) imaginary parts of O ( ρ ) with various algorithms at λ = 1.5 μ m .

Fig. 3
Fig. 3

Reconstructions of the (a) real and (b) imaginary parts of O ( ρ ) with various algorithms at λ = 1.25 μ m .

Fig. 4
Fig. 4

Reconstructions of the (a) real and (b) imaginary parts of O ( ρ ) with various algorithms at λ = 1 μ m .

Fig. 5
Fig. 5

Reconstructions of the (a) real and (b) imaginary parts of O ( ρ ) with various algorithms at λ = 0.75 μ m .

Fig. 6
Fig. 6

Object function profile O ( ρ , ϕ ) for the lossless dielectric phantom.

Fig. 7
Fig. 7

Reconstruction of O ( ρ , ϕ ) with the (a) proposed, (b) FBP, and (c) FBDT techniques at λ = 1.5 μ m .

Fig. 8
Fig. 8

Reconstruction of O ( ρ , ϕ ) with the (a) proposed, (b) FBP, and (c) FBDT techniques at λ = 1.25 μ m .

Fig. 9
Fig. 9

Reconstruction of O ( ρ , ϕ ) with the (a) proposed, (b) FBP, and (c) FBDT techniques at λ = 1 μ m .

Fig. 10
Fig. 10

Reconstruction of O ( ρ , ϕ ) with the (a) proposed, (b) FBP, and (c) FBDT techniques at λ = 0.75 μ m .

Tables (2)

Tables Icon

Table 1 RMSE Values for the Circularly Symmetric Lossy Cylinder

Tables Icon

Table 2 RMSE Values for the Lossless General Phantom

Equations (59)

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

J eq = j ω ( ϵ ϵ 0 ) E ,
M eq = j ω ( μ μ 0 ) H ,
f ( ϕ ) = lim ρ { exp ( j k ρ ) ρ E s ( ρ , ϕ ) } .
E s ( ρ , ϕ ) = j k η 0 2 π 0 a J eq ( ρ , ϕ ) G ( ρ , ϕ ; ρ , ϕ ) ρ d ρ d ϕ ,
G ( ρ , ϕ ; ρ , ϕ ) = 1 4 j H 0 ( 2 ) ( k ρ ρ ) .
H 0 ( 2 ) ( k ρ ρ ) = n = J n ( k ρ ) H n ( 2 ) ( k ρ ) exp [ j n ( ϕ ϕ ) ] ,
lim ρ H n ( 2 ) ( k ρ ) = j n 2 j π k exp ( j k ρ ) ρ .
f ( ϕ ) = A n = α n exp ( j n ϕ ) ,
A = η j k 8 π ,
α n = j n 0 2 π 0 a J e q ( ρ , ϕ ) J n ( k ρ ) exp ( j n ϕ ) ρ d ρ d ϕ .
f ( ϕ ) A n = [ k a ] [ k a ] α n exp ( j n ϕ ) .
α n i = 1 2 π A 0 2 π f i ( ϕ ) exp ( j n ϕ ) d ϕ .
j n 0 2 π 0 a J e q i ( ρ , ϕ ) J n ( k ρ ) exp ( j n ϕ ) ρ d ρ d ϕ = α n i = 1 2 π A 0 2 π f i ( ϕ ) exp ( j n ϕ ) d ϕ .
J e q i ( ρ , ϕ ) = j ω ( ϵ ϵ 0 ) E t i ( ρ , ϕ ) = j k η O ( ρ , ϕ ) E t i ( ρ , ϕ ) ,
O ( ρ , ϕ ) = ϵ r ( ρ , ϕ ) 1 .
O ( ρ , ϕ ) = m = 1 M n = N N c m n g m n ( ρ , ϕ ) ,
g m n ( ρ , ϕ ) = J 0 ( χ m a ρ ) exp ( j n ϕ ) .
g m n ( ρ , ϕ ) , g m n ( ρ , ϕ ) 0 2 π 0 a g m n ( ρ , ϕ ) g m n * ( ρ , ϕ ) ρ d ρ d ϕ = [ π a 2 J 1 2 ( χ m ) ] δ ( m m ) δ ( n n ) .
E t i ( ρ , ϕ ) = m = F m i ( ρ ) exp ( j m ϕ ) ρ a .
m = 1 M n = N N c m n [ 2 π k η j u + 1 0 a F u n i ( ρ ) J 0 ( χ m a ρ ) J u ( k ρ ) ρ d ρ ] = α u i u k a , 1 i I .
E t i ( ρ , ϕ ) = E inc
b m i = exp ( j m ϕ i ) j m J m ( k a ) H m ( 2 ) ( k a ) J m ( k a ) H m ( 2 ) ( k a ) J m ( k a a ) H m ( 2 ) ( k a ) n a J m ( k a a ) H m ( 2 ) ( k a )
k a = n a k .
E s i ( ρ , ϕ ) = n = c n i H n ( 2 ) ( k ρ ) exp ( j n ϕ ) ρ a ,
c n i = exp ( j n ϕ i ) j n n e i J n ( k a ) J n ( k e i a ) J n ( k a ) J n ( k e i a ) J n ( k e i a ) H n ( 2 ) ( k a ) n e i J n ( k e i a ) H n ( 2 ) ( k a )
k e i = n e i k .
f e i ( ϕ ) A n = [ k a ] [ k a ] γ n i exp ( j n ϕ ) ,
C i ( z ) = n = [ k a ] [ k a ] γ n i ( z ) α n i 2
E s i ( ρ , ϕ ) = k η 4 m = j m α m i H m ( 2 ) ( k ρ ) exp ( j m ϕ ) ρ a .
E inc i ( ρ , ϕ ) = m = j m exp ( j m ϕ i ) J m ( k ρ ) exp ( j m ϕ ) .
H ϕ = 1 j ω μ E z ρ .
d m i J m ( n m i k a ) = exp ( j m ϕ i ) J m ( k a ) k η 4 α m i H m ( 2 ) ( k a ) ,
d m i n m i J m ( n m i k a ) = exp ( j m ϕ i ) J m ( k a ) k η 4 α m i H m ( 2 ) ( k a ) ,
C m ( x ) = C m 1 ( x ) m x C m ( x ) .
n m i J m 1 ( n m i k a ) J m ( n m i k a ) = β m i H m 1 ( 2 ) ( k a ) + J m 1 ( k a ) β m i H m ( 2 ) ( k a ) + J m ( k a ) ,
β m i = 1 4 k η α m i exp ( j m ϕ i ) .
α n i = 2 j n π k η H n ( 2 ) ( k ρ 0 ) 0 2 π E s i ( ρ 0 , ϕ ) exp ( j n ϕ ) d ϕ .
E inc i ( ρ , ϕ ) = m = j m γ m i J m ( k ρ ) exp ( j m ϕ ) .
β m i = 1 4 k η α m i γ m i .
O ( x , y ) = 1 2 π 0 2 π Π ϕ i * ( x sin ϕ i y cos ϕ i , x cos ϕ i + y sin ϕ i ) d ϕ i ,
Π ϕ i ( ξ , η ) = 1 2 π k k Π ̃ ϕ i ( κ , η ) exp ( j κ ξ ) d κ ,
Π ̃ ϕ i ( κ , η ) = Γ ̃ ϕ i ( κ ) κ exp [ j ( k γ ) ( l 0 η ) ] ,
γ = k 2 κ 2 ,
Γ ̃ ϕ i ( κ ) = Γ ϕ i ( ξ ) exp ( j κ ξ ) d ξ ,
Γ ϕ i ( ξ ) = j k [ E s ( ξ ) E inc ( ξ ) ] * .
O ( ρ ) = 1 π 0 k { Γ ̃ ( ξ ) exp [ j ( k γ ) l 0 ] } * J 0 [ 2 k ( k γ ) ρ ] κ d κ .
E s i ( ρ , ϕ ) = k 2 object O ( ρ , ϕ ) E t i ( ρ , ϕ ) G ( ρ , ϕ ; ρ , ϕ ) d s .
G ( ρ , ϕ ; x , y ) = 1 4 j 2 j π k exp ( j k ρ ) ρ exp [ j k ( x cos ϕ + y sin ϕ ) ] ,
E inc i ( x , y ) = exp [ j k ( x cos ϕ i + y sin ϕ i ) ] .
O ̃ ( u , v ) O ( x , y ) exp ( j u x ) exp ( j v y ) d x d y = 8 π j k 3 f i ( ϕ ) ,
u = k ( cos ϕ i cos ϕ ) ,
v = k ( sin ϕ i sin ϕ ) .
O ( x , y ) = ( 1 2 π ) 2 O ̃ ( u , v ) exp ( j u x ) exp ( j v y ) d u d v ,
O ( ρ ) = ( 1 2 π ) 2 0 2 k O ̃ ( κ ) κ [ 0 2 π exp ( j κ cos α ρ ) d α ] d κ .
0 2 π exp ( j κ cos α ρ ) d α = 2 π J 0 ( κ ρ ) ,
O ( ρ ) = 1 2 π 0 2 k O ̃ ( κ ) J 0 ( κ ρ ) κ d κ .
κ = k 2 ( 1 cos ϕ ) ,
O ̃ ( κ ) = 8 π j k 3 f ( ϕ ) .
RMSE = object O r O a 2 d s object O a 2 d s .

Metrics