Abstract

A nonlinear signal processing method based on cepstral filtering has been developed to provide an approximate solution to the inverse scattering problem in two dimensions. It has been used to recover images of strongly scattering objects from measured far-field scattering data and is applied here to synthesize structures with prescribed scattering characteristics. An example is shown to illustrate the synthesis method. The scattering properties of the resulting structures are verified using a finite difference time domain method. The inverse scattering method is straightforward to implement and requires reprocessing of the scattered field data in order to ensure that the function describing the secondary source (contrast source function) has the properties of being a minimum phase function. This is accomplished by a numerical preprocessing step involving an artificial reference wave.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
    [CrossRef]
  2. D. Colton and M. Piana, "The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves," Inverse Probl. 14, 597-614 (1998).
    [CrossRef]
  3. F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990).
    [CrossRef]
  4. M. A. Fiddy, M. Testorf and U. Shahid, "Minimum-phase -based inverse scattering method applied to IPS008," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy and R. P. Millane, eds., Proc. SPIE 5562, 188-195 (2004).
    [CrossRef]
  5. U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
    [CrossRef]
  6. M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
    [CrossRef]
  7. M. Testorf and M. A. Fiddy, "Imaging from real scattered field data using a linear spectral estimation technique," Inverse Probl. 17, 1645-1658 (2001).
    [CrossRef]
  8. M. A. Fiddy and U. Shahid, "Minimum phase and zero distributions in 2D," in Optical Information Systems, B. Javidi and D. Psaltis, eds., SPIE Proc 5202, 201-208 (2003).
    [CrossRef]
  9. M. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001).
    [CrossRef]
  10. D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985).
    [CrossRef]
  11. D. Dudegeon and R. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, NJ, 1978).
  12. R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
    [CrossRef]
  13. Y. Aizenberg and A. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, (American Math. Society, Providence RI, 1982).
  14. R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

2005 (1)

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
[CrossRef]

2001 (2)

M. Testorf and M. A. Fiddy, "Imaging from real scattered field data using a linear spectral estimation technique," Inverse Probl. 17, 1645-1658 (2001).
[CrossRef]

M. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001).
[CrossRef]

1999 (1)

R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

1998 (1)

D. Colton and M. Piana, "The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves," Inverse Probl. 14, 597-614 (1998).
[CrossRef]

1997 (1)

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

1990 (1)

F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990).
[CrossRef]

1985 (1)

D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
[CrossRef]

1974 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Barlaud, M.

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

Blanc-Feraud, L.

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Colton, D.

D. Colton and M. Piana, "The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves," Inverse Probl. 14, 597-614 (1998).
[CrossRef]

Fiddy, M. A.

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
[CrossRef]

M. Testorf and M. A. Fiddy, "Imaging from real scattered field data using a linear spectral estimation technique," Inverse Probl. 17, 1645-1658 (2001).
[CrossRef]

M. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001).
[CrossRef]

F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
[CrossRef]

Kleinman, R. E.

R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

Larsen, L. E.

M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
[CrossRef]

Lin, F. C.

F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990).
[CrossRef]

Lobel, P.

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

McGahan, R. V.

R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

Piana, M.

D. Colton and M. Piana, "The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves," Inverse Probl. 14, 597-614 (1998).
[CrossRef]

Pichot, Ch.

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

Raghuramireddy, D.

D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985).
[CrossRef]

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Shahid, U.

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
[CrossRef]

Testorf, M.

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
[CrossRef]

M. Testorf and M. A. Fiddy, "Imaging from real scattered field data using a linear spectral estimation technique," Inverse Probl. 17, 1645-1658 (2001).
[CrossRef]

M. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001).
[CrossRef]

Unbehauen, R.

D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985).
[CrossRef]

IEEE Magazine (1)

R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

IEEE Trans. Acoust. Speech and Signal Process. (1)

D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-869, (1984).
[CrossRef]

Int. J. Imaging Syst. Technol. (2)

P. Lobel, Ch. Pichot, L. Blanc-Feraud and M. Barlaud, "Microwave imaging: reconstructions from experimental data using conjugate gradient and enhancement by edge-preserving regularization," Int. J. Imaging Syst. Technol. 8, 337-342 (1997).
[CrossRef]

F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990).
[CrossRef]

Inverse Probl. (3)

D. Colton and M. Piana, "The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves," Inverse Probl. 14, 597-614 (1998).
[CrossRef]

M. Testorf and M. A. Fiddy, "Imaging from real scattered field data using a linear spectral estimation technique," Inverse Probl. 17, 1645-1658 (2001).
[CrossRef]

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data," Inverse Probl. 21, S153-S164 (2005).
[CrossRef]

J. Phys. D (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway and G. Ross, "The application of dispersion relations (Hilbert transforms) to phase retrieval," J. Phys. D 7, L65-68 (1974).
[CrossRef]

Waves Random Media (1)

M. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001).
[CrossRef]

Other (4)

M. A. Fiddy and U. Shahid, "Minimum phase and zero distributions in 2D," in Optical Information Systems, B. Javidi and D. Psaltis, eds., SPIE Proc 5202, 201-208 (2003).
[CrossRef]

M. A. Fiddy, M. Testorf and U. Shahid, "Minimum-phase -based inverse scattering method applied to IPS008," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy and R. P. Millane, eds., Proc. SPIE 5562, 188-195 (2004).
[CrossRef]

Y. Aizenberg and A. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, (American Math. Society, Providence RI, 1982).

D. Dudegeon and R. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, NJ, 1978).

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 (8)

Fig. 1.
Fig. 1.

Scattering geometry.

Fig. 2.
Fig. 2.

Plane wave scattering in the Fourier space (k-space) of the object. The direction of the incident field r̂0 and the direction r̂ S of a particular plane wave component of the scattered field define a point at the Ewald circle. Changing the wavelength of the incident radiation as well as the directions of incident and measured scattered field components provides sufficient information about k-space to compute the image of the object via an inverse Fourier transform.

Fig. 3.
Fig. 3.

The reconstruction of V〈Ψ〉 (left) and V using cepstral filtering.

Fig. 4.
Fig. 4.

The top figure shows the modification to the estimated structure for IPS010 which, when illuminated in the -y direction (see above) would have very little if any scattering in the forward direction. The lower figure shows a different structure which would exhibit little forward scatter when illuminated in the +x direction.

Fig. 5.
Fig. 5.

Scattering amplitude in the forward direction over a range of angles for illumination in the +x direction. The green curve illustrates the scattered field from (our reconstruction of) IPS010, the red and blue curves from the two synthesized structures.

Fig. 6.
Fig. 6.

Forward scattering amplitudes for both synthesized objects when illuminated from the -y direction. Here both scattered fields show some reduction in amplitude in the forward direction compared to IPS010. In addition, the fields exhibit strong sidelobes.

Fig. 7.
Fig. 7.

The k-space data removal imposed for each of the two new synthesized objects is illustrated. The axes indicate the k-space origin and the central image shows the distribution of energy in the original IPS010 measured data.

Fig. 8.
Fig. 8.

Series of images of V〈Ψ〉 following application of a low-pass filter in k-space with increasing radius (compare with Fig. 3); we emphasize that only filtering in the cepstral domain, after enforcing the minimum phase condition, provides an estimates of V(r).

Equations (4)

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

Ψ s ( r , k r ̂ 0 ) = exp ( ikr ) 8 πkr D d 2 r ' V ( r ' ) Ψ ( r ' , k r ̂ 0 ) exp ( ik r ̂ r ' ) ,
V B ( r , k r ̂ 0 ) V ( r ) Ψ ( r , k r ̂ 0 ) Ψ 0 ( r , k r ̂ 0 )
log ( V Ψ ) = log V + log Ψ + i [ arg ( V ) + arg ( Ψ ) ]
F ( u ) = + f ( x ) exp ( i 2 π ux ) dx

Metrics