## Inverse scattering method applied to the synthesis of strongly scattering structures

Optics Express, Vol. 14, Issue 5, pp. 2037-2046 (2006)

http://dx.doi.org/10.1364/OE.14.002037

Acrobat PDF (8716 KB)

### 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

## 1. Introduction

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]

3. F. C. Lin and M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. **2**, 76–95 (1990). [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]

*physical*premise upon which they are based is rarely acceptable in practice [6

6. M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. **MTT-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]

## 2. Nonlinear inverse scattering algorithm

*V*(

_{B}**r**,

*k*

**r**̂

_{0})i.e.

**r**

_{0}. In diffraction tomography, the scattered field data for all incident field directions is combined in k-space and a Fourier inversion of that data one expects to provide an estimate for V(

**r**). When the first Born approximation is valid, then Ψ(

**r**,

*k*

**r**̂

_{0})~Ψ

_{0}(

**r**,

*k*

**r**̂

_{0}) and

*V*(

_{B}**r**)~

*V*(

**r**) at least to within low-pass spatial filtering effects resulting from the available k-space coverage. When the first Born approximation is not valid then

*V*(

_{B}**r**)~

*V*(

**r**)⟨Ψ(

**r**)⟩ where ⟨Ψ(

**r**)⟩ is a complex and noise-like term with a characteristic range of spatial frequencies dominated by the bandwidth of the source.

*V*〈Ψ〉 can be separated using homomorphic filtering. This product exhibits spatial fluctuations characteristic of the wavelengths being employed as well as the spatial fluctuations of the permittivity. For incremental illumination wavelength changes, the 〈Ψ〉 term will change quite considerably but

*V*(

**r**) need not. The first step in homomorphic filtering is to take the logarithm of the product of

*V*〈Ψ〉 and then employ standard spatial filtering in the associated Fourier domain to remove the field component. However, since

*V*〈Ψ〉) and taking its Fourier transform. When the magnitude of the term

*V*〈Ψ〉 is close to zero, its logarithm becomes singular, and when the phase of

*V*〈Ψ〉 has a range that exceeds 2π, the resulting wrapped phase introduces spurious spatial frequencies in the cepstrum. Phase unwrapping is exceedingly difficult, especially in two and higher dimensional problems, because of zeros in the field, which are associated with wavefront dislocations [8

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]

*V*〈Ψ〉) generates the cepstrum of the function [9

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. , **ASSP-33**, 1335–1337 (1985). [CrossRef]

## 3. Minimum-phase based homomorphic filtering

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]

*Optical Information Systems*,
B. Javidi and D. Psaltis, eds., SPIE Proc **5202**, 201–208 (2003). [CrossRef]

*only*for minimum phase functions. An important feature of a minimum phase function is that the phase is a continuous function bounded between -π and π and “minimum” in this sense indicates that the phase is already unwrapped. It is possible to enforce the minimum phase condition on a function by applying Rouche’s theorem [12

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]

*V*〈Ψ〉 by introducing a reference point in the data or k- space domain to make it minimum phase prior to taking its logarithm. The first step is to make the data in k-space causal by moving it to one quadrant of k-space and then place a reference point at the origin of k-space. This is equivalent to adding a reference wave to

*V*〈Ψ〉 which needs an amplitude just large enough to ensure that phase of

*V*〈Ψ〉 is continuous and lies within the bounds of -

*π*and +

*π*. This is readily determined by inspection of the phase of the reconstructed image of

*V*〈Ψ〉, i.e. the image resulting from the direct Fourier inversion of the data in k-space. The implementation of the homomorphic filtering algorithm then requires that a low pass filter be applied in the cepstral domain until the wavelike features associated with 〈Ψ〉 are removed from the resulting image. This spatial filtering is successful to the extent to which the field internal to the scattering structure has spatial frequencies that are distinct from those of log(

*V*). Using different source frequencies can resolve this problem.

## 4. Reconstructions and structure synthesis

*V*〈Ψ〉 is shown computed with the PDFT algorithm [7

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]

*V*is shown. This is obtained by adding the reference as described in the previous section. Then the logarithm of the resulting Fourier inverse is computed and Fourier transformed into the cepstral domain of the signal. After applying a linear filter (low-pass) the Fourier inverse of the resulting cepstrum is computed and the final image is obtained after exponentiation. For a detailed description of this procedure we refer to references [4

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]

## 5. Conclusions

*V*(

**r**) from the image of

*V*〈Ψ〉 obtained using conventional diffraction tomography. The quality of the reconstruction depends on a number of factors, including the quantity and quality of the data made available. It is obvious that simple low-pass filtering of the spectrum of

*V*〈Ψ〉 as opposed to the cepstrum, will do nothing. This is illustrated in the sequence of images shown in Fig. 8.

## References and links

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. |

2. | D. Colton and M. Piana, “The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves,” Inverse Probl. |

3. | F. C. Lin and M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. |

4. | M. A. Fiddy, M. Testorf, and U. Shahid, “Minimum-phase -based inverse scattering method applied to IPS008,” in |

5. | U. Shahid, M. Testorf, and M. A. Fiddy, “Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data,” Inverse Probl. |

6. | M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. |

7. | M. Testorf and M. A. Fiddy, “Imaging from real scattered field data using a linear spectral estimation technique,” Inverse Probl. |

8. | M. A. Fiddy and U. Shahid, “Minimum phase and zero distributions in 2D,” in |

9. | M. Testorf and M. A. Fiddy, “Algorithms for data evaluation applied to the detection of buried objects,” Waves Random Media |

10. | D. Raghuramireddy and R. Unbehauen, ”The two dimensional differential cepstrum,” IEEE Trans. Acoust. Speech and Signal Process. , |

11. | D. Dudegeon and R. Mersereau, |

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 |

13. | Y. Aizenberg and A. Yuzhakov, |

14. | R. V. McGahan and R. E. Kleinman, “Special session on image reconstruction using real data,” IEEE Magazine |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(220.4000) Optical design and fabrication : Microstructure fabrication

(290.0290) Scattering : Scattering

(290.3200) Scattering : Inverse scattering

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: January 3, 2006

Revised Manuscript: February 24, 2006

Manuscript Accepted: February 28, 2006

Published: March 6, 2006

**Virtual Issues**

Vol. 1, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express **14**, 2037-2046 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-5-2037

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### References

- 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]
- 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]
- F. C. Lin and M. A. Fiddy, "Image estimation from scattered field data," Int. J. Imaging Syst. Technol. 2, 76-95 (1990). [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]
- 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. 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]
- 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. 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. Testorf and M. A. Fiddy, "Algorithms for data evaluation applied to the detection of buried objects," Waves Random Media 11, 535-547 (2001). [CrossRef]
- D. Raghuramireddy and R. Unbehauen, "The two dimensional differential cepstrum," IEEE Trans. Acoust. Speech and Signal Process. 33, 1335-1337 (1985). [CrossRef]
- D. Dudegeon and R. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, NJ, 1978).
- 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]
- Y. Aizenberg and A. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, (American Math. Society, Providence RI, 1982).
- R. V. McGahan and R. E. Kleinman, "Special session on image reconstruction using real data," IEEE Magazine 41, 34-51 (1999).

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