## Reconstructing the contour of metallic planar objects from only intensity scattered field data over a single plane

Optics Express, Vol. 16, Issue 13, pp. 9468-9479 (2008)

http://dx.doi.org/10.1364/OE.16.009468

Acrobat PDF (149 KB)

### Abstract

The inverse scattering problem of recovering the contour of planar metallic scattering objects from only the amplitude of the scattered field is considered. A two step reconstruction procedure is proposed: first the phase of the scattered field is retrieved by solving a phase retrieval problem; then the objects’ supports are reconstructed from the retrieved scattered field. Differently form previous approaches, (see [

© 2008 Optical Society of America

## 1. Introduction

1. R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. **54**, 655–665 (2006). [CrossRef]

2. R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. **29**, 1157–1169 (1994). [CrossRef]

3. E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-Reversal MUSIC Imaging of Extended Targets,” IEEE Trans. Imag. Process. **16**, 1967–1984 (2007). [CrossRef]

^{1}.

4. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. **62**, 2385–2388 (1989). [CrossRef] [PubMed]

5. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Imag. Process. **1**, 221–228 (1991). [CrossRef]

4. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. **62**, 2385–2388 (1989). [CrossRef] [PubMed]

5. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Imag. Process. **1**, 221–228 (1991). [CrossRef]

6. 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]

7. T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, “Reconstruction algorithm of the refracrive index of a cylindrical object from the intensity measurement of the total field,” Microw. Opt. Techn. Lett. **14**, 182–188 (1997). [CrossRef]

8. 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]

9. T. Isernia, G. Leone, and R. Pierri, “Phase retreival of radiated fields,” Inv. Probl. **11**, 183–203 (1995). [CrossRef]

10. 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]

11. F. Soldovieri and R. Pierri, “Shape reconstruction of metallic objects from intensity scattered field data only,” Opt. Lett. **33**, 246–248 (2008). [CrossRef] [PubMed]

12. G. Hislop, G. J. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antenn. Propag. **55**, 2332–2341 (2007). [CrossRef]

11. F. Soldovieri and R. Pierri, “Shape reconstruction of metallic objects from intensity scattered field data only,” Opt. Lett. **33**, 246–248 (2008). [CrossRef] [PubMed]

12. G. Hislop, G. J. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antenn. Propag. **55**, 2332–2341 (2007). [CrossRef]

8. 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]

13. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retreival using multiple illumination wavelengths,” Opt. Lett. **33**, 309–311 (2008). [CrossRef] [PubMed]

9. T. Isernia, G. Leone, and R. Pierri, “Phase retreival of radiated fields,” Inv. Probl. **11**, 183–203 (1995). [CrossRef]

14. G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A **13**, 1546–1556 (1996). [CrossRef]

## 2. Formulation of the problem

*object-aperture*𝒪=[-

*X*,

_{O}*X*]×[-

_{O}*Y*,

_{O}*Y*] which is a subset of the

_{O}*x*-

*y*plane located at

*z*=0. Whereas the amplitude of the scattered field is collected over a single planar

*measurement-domain*𝓓=[-

*X*,

_{M}*X*]×[-

_{M}*Y*,

_{M}*Y*] in the

_{M}*x*-

*y*plane at

*z*=

*z*

_{1}. As incident field we consider a normally impinging plane wave (i.e., along -

*z*direction) at two different frequencies, say

*f*

_{1}and

*f*

_{2}, with

*f*

_{2}>

*f*

_{1}. The background medium is assumed to be the free-space whose dielectric permittivity and magnetic permeability are denoted by

*ε*

_{0}and

*μ*

_{0}, respectively.

*U*

_{Γ}the corresponding characteristic function, that is

*retrieve U*

_{Γ}

*from the amplitude of the scattered field M*

_{1}

*and M*

_{2}

*collected over the measurement aperture*𝓓

*at two different frequencies f*

_{1}

*and*

*f*

_{2}.

*U*

_{Γ}and

*M*, with

_{i}*i*∊(1,2). Hence, we start by considering the expression of the scattered field which arises from the interaction between the incident plane wave and the metallic planar scatterers.

*E̲*=

_{inc}*E*exp(

_{inc}*jk*)

_{i}z*î*, with

_{x}*f*. Note that the time dependence exp(

_{i}*j*2

*π f*) has been assumed and suppressed.

_{i}t*r̲*is the field point within the observation aperture 𝓓,

*G̳*(

*r̲*-

*r̲*′,

*f*) is the dyadic Green’s function and

_{i}*J̲*(

_{PO}*r̲*′,

*f*) is the current induced over the scattering objects under the PO approximation. In particular,

_{i}*r̲*′ ranges within 𝒪 and

*î*is the unitary vector normal to the scatterers’ surface, which for the case at hand coincides with

_{n}*î*.

_{z}*x*-component of the scattered field. Hence, the problem turns to be scalar. For such a case, by exploiting Eq. (3), Eq. (2) simplifies as

*x*-component of the scattered field still as

*E*but discarding the vectorial notation.

_{S}*G*(

_{xx}*r̲*-

*r̲*′,

*f*) [16] the scattered field over the measurement aperture 𝓓 at

_{i}*z*=

*z*

_{1}can be rewritten as

*u*,

*ν*and

*w*are the spectral variables conjugated of those spatial

_{i}*x*,

*y*and

*z*, respectively, with

*k*=2

_{i}*π*/

*λ*and

_{i}*λ*are the free-space wavenumber and the corresponding wavelength, respectively, at the frequency

_{i}*f*,

_{i}*Û*

_{Γ}is the Fourier transform of the induced current support

*f*

_{1}and

*f*

_{2}we aim at reconstructing the function

*U*

_{Γ}. To this end, a reconstruction procedure in two steps is adopted. In the first step we retrieve

*Û*

_{Γ}by the data

*M*

_{1}and

*M*

_{2}. Then,

*U*

_{Γ}is retrieved from

*Û*

_{Γ}.

11. F. Soldovieri and R. Pierri, “Shape reconstruction of metallic objects from intensity scattered field data only,” Opt. Lett. **33**, 246–248 (2008). [CrossRef] [PubMed]

*Û*

_{Γ}of the following non-linear operator

*E*has been assumed equal to one and the square amplitude of the scattered field

_{inc}*M*

^{2}

_{1}and

*M*

^{2}

_{2}have been considered as data instead of

*M*

_{1}and

*M*

_{2}.

9. T. Isernia, G. Leone, and R. Pierri, “Phase retreival of radiated fields,” Inv. Probl. **11**, 183–203 (1995). [CrossRef]

*adequate*ratio between independent data and unknowns is available. This circumstance can be meet by assuming the knowledge of the scattered field amplitude in more than one measurement condition. In the paper by Soldovieri and Pierri [11

**33**, 246–248 (2008). [CrossRef] [PubMed]

*U*

_{Γ}which does not depend on the frequency whereas the way the induced surface current depends on frequency is a priori known due to the adopted PO approximation. In particular, for the configuration considered in this paper, the induced surface current does not depend on the frequency (as is evident from Eq. (3)). Accordingly, observations (of the scattered field intensity) at different frequencies can be exploited for conveying an increased amount of

*information*on the unknown, and simultaneously adopted in the reconstruction procedure. From a mathematical point of view, this leads to the the inversion of an operator similar to the one in [11

**33**, 246–248 (2008). [CrossRef] [PubMed]

*ℒ*

_{1}and

*ℒ*

_{2}, that is from the exponential terms exp[-

*jz*

_{1}(

*w*

_{2}-

*w*

_{1})], whereas in the approach presented in [11

**33**, 246–248 (2008). [CrossRef] [PubMed]

*jw*(

*z*

_{2}-

*z*

_{1})], with

*w*defined as above but at the single employed frequency, and

*z*

_{2}and

*z*

_{1}were the quotas of the two planar measurement domains.

*ambiguities*of the phase retrieval problem [17

17. I. Sabba Sefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. **26**, 2141–2160 (1985). [CrossRef]

*Û*

_{Γ}has been retrieved, the current support is obtained by passing

*Û*

_{Γ}through the adjoint of the Fourier operator 𝓕, that is

*Ũ*

_{Γ}being the reconstructed version of

*U*

_{Γ}.

## 3. The solution algorithm

Û _

_{Γ},

_{1}and

_{2}are the propagator operators at the two different frequencies as in (5) discretized according to the representation adopted for the data and the unknown. Note, that

*M*

^{2}

_{1}and

*M*

^{2}

_{2}.

19. O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by finite and nonredundant number of samples,” IEEE Trans. Ant. Prop. **46**, 351–359 (1998). [CrossRef]

*Û*

_{Γ}. In this case the a priori knowledge we have says us that

*Û*

_{Γ}, being the Fourier transform of a function compactly supported over 𝒪, is an entire band-limited function [20

20. A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. of IEEE **68**, 308–328 (1980). [CrossRef]

*Û*

_{Γ}can be expanded in a sampling series as

*Û*

_{Γnm}are the samples of

*Û*

_{Γ}taken at the Nyquist rate, the truncation indexes

*N*and

*M*are chosen as explained below and

*sinc*(

*x*), as usual, is defined as

*ℒ*

_{1}and

*ℒ*

_{2}can be limited to only the

*visible*part of the plane wave spectrum as evanescent waves give rise to an exponentially decaying (with

*z*) contribution. Therefore, the truncation indexes are chosen so that

*Û*

_{Γ},

*U*

_{Γ}is reconstructed by means of (9).

*N*and

*M*, the best way (in terms of the norm of the representation error) to represent a band-limited function projected over a compact support (in our case the circle of radius

*k*

_{2}in the

*u*-

*v*plane) is by means of a finite series of the singular functions of the operator

*P*𝓕

_{VC}*P*

_{𝒪}, where 𝓕 is defined as above and

*P*

_{𝒪}and

*P*are projector operators over the investigation domain 𝒪 and the circular visible domain, respectively. However, the representation adopted in Eq. (11) is useful to achieve a very efficient implementation of the solution algorithm as such a choice enables most of the operations to be performed thanks to the FFT technique.

_{VC}*x*and

*y*[19

19. O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by finite and nonredundant number of samples,” IEEE Trans. Ant. Prop. **46**, 351–359 (1998). [CrossRef]

21. R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over buonded domains,” Inv. Prob. **14**, 321–337 (1998). [CrossRef]

*Û*

_{Γnm}to evaluate the updating direction in the minimizing procedure.

*L*minimization steps where the accuracy of the solution is improved by gradually increasing the number of unknowns [23

23. R. Pierri, G. D’Elia, and F. Soldovieri, “A two probes scanning phaseless near-field far-field transformation technique,” IEEE Trans. Ant. Prop. **47**, 792–802 (1999). [CrossRef]

*l*-th step we consider

*N*=2

*l*+1 and

*M*=2

*l*+1. The obtained solution is then used as starting point in the next step by considering

*N*=2(

*l*+1)+1 and

*M*=2(

*l*+1)+1. This procedure continues until all the samples falling within the visible domain at the higher frequency, as dictated by Eq. (15), are considered. Accordingly, a resolution of

*λ*

_{2}/2, along both

*x*and

*y*, is expected while reconstructing

*U*

_{Γ}.

14. G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A **13**, 1546–1556 (1996). [CrossRef]

*c*=

*a*./

*b*means a vector whose components are given as

*c*=

_{n}*a*/

_{n}*b*and

_{n}*η*is factor avoiding division by zero.

*dB*below the maximum of the reconstruction which corresponds to the sidelobe level of the expected point spread function.

## 4. Numerical analysis

*f*

_{2}=2

*f*

_{1}, a measurement aperture 𝓓=[-16

*λ*

_{2},16

*λ*

_{2}]×[-16

*λ*

_{2},16

*λ*

_{2}] located at the quota

*z*

_{1}=5

*λ*

_{2}and an object-aperture 𝒪=[-15

*λ*

_{2},15

*λ*

_{2}]×[-15

*λ*

_{2},15

*λ*

_{2}]. Accordingly, the scattered field data at the lower frequency

*f*

_{1}consist of 64×64 measurements taken

*λ*

_{2}/2 apart each other both along

*x*and

*y*, while for the frequency

*f*

_{2}we have 128×128 measurements spaced by

*λ*

_{2}/4.

**c**of the same figure the reconstruction have been obtained by enlarging the number of unknowns so as to consider all the visible part dictated by

*f*

_{2}. As can be seen, even though the reconstruction error quantified as ∥

*U*

_{Γ}-

*Û*

_{Γ}∥

^{2}/∥

*U*

_{Γ}∥

^{2}is not particularly low (for example in the case of panel

**c**it is equal to 0.12) the reconstruction results allow us to correctly determine the location and the geometry of the metallic plate.

**c**and

**d**of Fig. 1. In fact, the reconstruction shown in panel

**d**have been obtained by a single minimization of (10) by simultaneously searching for all the 2

*N*×2

*M*unknowns. In particular, in order to improve the reconstruction, the procedure have been followed by a weighted minimization according to the functional in Eq. (14) (instead for the case of panel

**c**such further step did not give a relevant improvement and hence is not reported here). In this case, the error in the data space evaluated as for previous case, when the procedure arrests returns -34.7 dB which is much higher than the one obtained for the case reported in panel

**c**(see blue line of Fig. 2). Indeed, due to the reduced ration between the number of data and unknowns the procedurewas not able to achieve the true solution.

*λ*

_{2}(see Fig. 3, panel

**a**).

**c**). As expected in this case the reconstruction error increases (now ∥

*U*

_{Γ}-

*Û*

_{Γ}∥

^{2}/∥

*U*

_{Γ}∥

^{2}returns 0.24 whereas in the previous example it was 0.12). Nevertheless, the scatterer’s contour is retrieved and the reconstruction is very similar to corresponding reconstruction reported in Fig. 1.

**33**, 246–248 (2008). [CrossRef] [PubMed]

**33**, 246–248 (2008). [CrossRef] [PubMed]

## 5. Conclusion

*U*

_{Γ}of metallic planar scatterers which are known to reside within the object aperture from the amplitude of their scattered field has been addressed. A two step inversion procedure has been presented. First, a phase retrieval problem has solved to retrieve (both in amplitude and phase) the Fourier transform

*Û*

_{Γ}of the support function

*U*

_{Γ}. Then,

*U*

_{Γ}has been determined through 𝓕

^{†}

*Û*

_{Γ}. The phase retrieval stage has taken advantage from considering as data the square amplitude of the scattered field which has allowed to cast the problem as the inversion of quadratic operator whose features concerning the occurrence of local minima has been well understood [9

**11**, 183–203 (1995). [CrossRef]

14. G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A **13**, 1546–1556 (1996). [CrossRef]

12. G. Hislop, G. J. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antenn. Propag. **55**, 2332–2341 (2007). [CrossRef]

## Footnotes

1 | Note that references [1 1. R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 2. R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 3. E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-Reversal MUSIC Imaging of Extended Targets,” IEEE Trans. Imag. Process. |

## References and links

1. | R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. |

2. | R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. |

3. | E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-Reversal MUSIC Imaging of Extended Targets,” IEEE Trans. Imag. Process. |

4. | A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. |

5. | A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Imag. Process. |

6. | M. H. Maleki, A. J. Devaney, and A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A |

7. | T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, “Reconstruction algorithm of the refracrive index of a cylindrical object from the intensity measurement of the total field,” Microw. Opt. Techn. Lett. |

8. | 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 |

9. | T. Isernia, G. Leone, and R. Pierri, “Phase retreival of radiated fields,” Inv. Probl. |

10. | E. A. Marengo, R.D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A |

11. | F. Soldovieri and R. Pierri, “Shape reconstruction of metallic objects from intensity scattered field data only,” Opt. Lett. |

12. | G. Hislop, G. J. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antenn. Propag. |

13. | P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retreival using multiple illumination wavelengths,” Opt. Lett. |

14. | G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A |

15. | C. A. Balanis, |

16. | P. C. Clemmow, |

17. | I. Sabba Sefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. |

18. | T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,” J. Elettromagn. Waves Appl. |

19. | O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by finite and nonredundant number of samples,” IEEE Trans. Ant. Prop. |

20. | A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. of IEEE |

21. | R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over buonded domains,” Inv. Prob. |

22. | D. Luenberger, Linear and Nonlinear Programming. Reading, (Addison-Wesley, 1987). |

23. | R. Pierri, G. D’Elia, and F. Soldovieri, “A two probes scanning phaseless near-field far-field transformation technique,” IEEE Trans. Ant. Prop. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(100.3200) Image processing : Inverse scattering

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 17, 2008

Revised Manuscript: April 27, 2008

Manuscript Accepted: May 14, 2008

Published: June 12, 2008

**Citation**

F. Soldovieri, R. Solimene, and R. Pierri, "Reconstructing the contour of metallic planar objects from only intensity scattered field data over a single plane," Opt. Express **16**, 9468-9479 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9468

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### References

- R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, "Beyond physical optics SVD shape reconstruction of metallic cylinders," IEEE Trans. Antennas Propag. 54, 655-665 (2006). [CrossRef]
- R. E. Kleinman and P. M. van den Berg, "Two-dimensional location and shape reconstruction, Radio Sci. 29, 1157-1169 (1994). [CrossRef]
- E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-Reversal MUSIC Imaging of Extended Targets, IEEE Trans. Imag. Process. 16, 1967-1984 (2007). [CrossRef]
- A. J. Devaney, "Structure determination from intensity measurements in scattering experiments," Phys. Rev. Lett. 62, 2385-2388 (1989). [CrossRef] [PubMed]
- A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Imag. Process. 1, 221-228 (1991). [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]
- T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refracrive index of a cylindrical object from the intensity measurement of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997). [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]
- T. Isernia, G. Leone, and R. Pierri, "Phase retreival of radiated fields," Inverse Probl. 11, 183-203 (1995). [CrossRef]
- 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. Soldovieri and R. Pierri, "Shape reconstruction of metallic objects from intensity scattered field data only," Opt. Lett. 33, 246-248 (2008). [CrossRef] [PubMed]
- G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007). [CrossRef]
- P. Bao, F. Zhang, G. Pedrini, and W. Osten, "Phase retreival using multiple illumination wavelengths," Opt. Lett. 33, 309-311 (2008). [CrossRef] [PubMed]
- G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996). [CrossRef]
- C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, Hoboken, N J, 1989).
- P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Wiley-IEEE Press, Hoboken, N J, 1996).
- I. Sabba Sefanescu, "On the phase retrieval problem in two dimensions," J. Math. Phys. 26, 2141-2160 (1985). [CrossRef]
- T. Isernia, G. Leone, and R. Pierri, "Phaseless near field techniques: uniqueness condition and attainment of the solution," J. Electron. Waves Appl. 8, 889-908 (1994).
- O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by finite and nonredundant number of samples," IEEE Trans. Ant. Prop. 46, 351-359 (1998). [CrossRef]
- A. Requicha, "The zeros of entire functions: theory and engineering applications," Proc. of IEEE 68, 308-328 (1980). [CrossRef]
- R. Pierri and F. Soldovieri, "On the information content of the radiated fields in the near zone over buonded domains," InverseProbl. 14, 321-337 (1998). [CrossRef]
- D. Luenberger, Linear and Nonlinear Programming. Reading, (Addison-Wesley, 1987).
- R. Pierri, G. D???Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Ant. Prop. 47, 792-802 (1999). [CrossRef]

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