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 [11] for example), here the amplitude of the scattered field is assumed known over a single plane in near zone but at two different frequencies. In this way, while the frequency diversity increases the number of independent data, relevant for ensuring the reliability of the phase retrieval stage, to perform measurements on a single plane allows some practical advantages. Numerical results show the performances achievable by the proposed reconstruction scheme with respect to the local minima problem and the stability against the noise on data.

© 2008 Optical Society of America

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References

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  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]
  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," Microwave Opt. Technol. 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," Inverse 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]
  13. P. Bao, F. Zhang, G. Pedrini, and W. Osten, "Phase retreival using multiple illumination wavelengths," Opt. Lett. 33, 309-311 (2008).
    [CrossRef] [PubMed]
  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]
  15. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, Hoboken, N J, 1989).
  16. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Wiley-IEEE Press, Hoboken, N J, 1996).
  17. I. Sabba Sefanescu, "On the phase retrieval problem in two dimensions," J. Math. Phys. 26, 2141-2160 (1985).
    [CrossRef]
  18. 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).
  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]
  20. A. Requicha, "The zeros of entire functions: theory and engineering applications," Proc. of IEEE 68, 308-328 (1980).
    [CrossRef]
  21. 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]
  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. 47, 792-802 (1999).
    [CrossRef]

2008

2007

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]

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]

G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007).
[CrossRef]

2006

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]

2004

1999

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]

1998

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]

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]

1997

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]

1996

1995

T. Isernia, G. Leone, and R. Pierri, "Phase retreival of radiated fields," Inverse Probl.  11, 183-203 (1995).
[CrossRef]

1994

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).

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

1992

1991

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Imag. Process. 1, 221-228 (1991).
[CrossRef]

1989

A. J. Devaney, "Structure determination from intensity measurements in scattering experiments," Phys. Rev. Lett. 62, 2385-2388 (1989).
[CrossRef] [PubMed]

1985

I. Sabba Sefanescu, "On the phase retrieval problem in two dimensions," J. Math. Phys. 26, 2141-2160 (1985).
[CrossRef]

1980

A. Requicha, "The zeros of entire functions: theory and engineering applications," Proc. of IEEE 68, 308-328 (1980).
[CrossRef]

Bao, P.

Bucci, O. M.

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]

Crocco, L.

D???Elia, G.

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]

D???Urso, M.

Devaney, A. J.

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. Imag. Process. 1, 221-228 (1991).
[CrossRef]

A. J. Devaney, "Structure determination from intensity measurements in scattering experiments," Phys. Rev. Lett. 62, 2385-2388 (1989).
[CrossRef] [PubMed]

Gennarelli, C.

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]

Gruber, F. K.

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]

Harada, H.

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]

Hellicar, A.

G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007).
[CrossRef]

Hernandez, R.D.

Hislop, G.

G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007).
[CrossRef]

Isernia, T.

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]

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).

James, G. J.

G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007).
[CrossRef]

Kleinman, R. E.

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

Leone, G.

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]

T. Isernia, G. Leone, and R. Pierri, "Phase retreival of radiated fields," Inverse Probl.  11, 183-203 (1995).
[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).

Lev-Ari, H.

Liseno, A.

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]

Maleki, M. H.

Marengo, E. A.

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]

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]

Osten, W.

Pedrini, G.

Pierri, R.

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

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. 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]

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]

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]

T. Isernia, G. Leone, and R. Pierri, "Phase retreival of radiated fields," Inverse Probl.  11, 183-203 (1995).
[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).

Requicha, A.

A. Requicha, "The zeros of entire functions: theory and engineering applications," Proc. of IEEE 68, 308-328 (1980).
[CrossRef]

Sabba Sefanescu, I.

I. Sabba Sefanescu, "On the phase retrieval problem in two dimensions," J. Math. Phys. 26, 2141-2160 (1985).
[CrossRef]

Savarese, C.

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]

Schatzberg, A.

Simonetti, F.

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]

Soldovieri, F.

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

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. 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]

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]

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]

Solimene, R.

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]

Takenaka, T.

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]

Tanaka, M.

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]

van den Berg, P. M.

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

Wall, D. J. N.

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]

Zhang, F.

IEEE Trans. Ant. Prop.

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]

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]

IEEE Trans. Antenn. Propag.

G. Hislop, G. J. James, and A. Hellicar, "Phase retrieval of scattered fields," IEEE Trans. Antenn. Propag. 55, 2332-2341 (2007).
[CrossRef]

IEEE Trans. Antennas Propag.

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]

IEEE Trans. Imag. Process.

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, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Imag. Process. 1, 221-228 (1991).
[CrossRef]

Inverse Probl.

T. Isernia, G. Leone, and R. Pierri, "Phase retreival of radiated fields," Inverse Probl.  11, 183-203 (1995).
[CrossRef]

J. Electron. Waves Appl.

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).

J. Math. Phys.

I. Sabba Sefanescu, "On the phase retrieval problem in two dimensions," J. Math. Phys. 26, 2141-2160 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

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]

Opt. Lett.

Phys. Rev. Lett.

A. J. Devaney, "Structure determination from intensity measurements in scattering experiments," Phys. Rev. Lett. 62, 2385-2388 (1989).
[CrossRef] [PubMed]

Probl.

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]

Proc. of IEEE

A. Requicha, "The zeros of entire functions: theory and engineering applications," Proc. of IEEE 68, 308-328 (1980).
[CrossRef]

Radio Sci.

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

Other

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).

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

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Figures (4)

Fig. 1.
Fig. 1.

a: actual scatterer. b: reconstruction obtained by considering up to (2N 1+1)(2M 1+ 1) unknown coefficients for the Û Γ representation in the minimization of the cost functional (10). c: reconstruction obtained by enlarging the number of unknown coefficients up to (2N+1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N+1)(2M+1) unknowns by first minimizing the cost functional (10) and then adopting the results as starting point for minimizing the weighted functional (14). Results are shown in dB scale.

Fig. 2.
Fig. 2.

Behavior of the error in the data space defined as Φ( Γ)/∥( 2 1, 2 2)∥2. Blue line refers to the case reported in Fig. 1 whereas red line refers to the case reported in Fig. 3. Markers * denotes the transition by two different minimization steps accordingly to the progressive increasing of the searched for number of unknown coefficients.

Fig. 3.
Fig. 3.

a: actual scatterer. b: reconstruction obtained by considering up to (2N 1+1)(2M 1+1) unknown coefficients for the Û Γ representation in the minimization of the cost functional (10). c: reconstruction obtained by enlarging the number of unknown coefficients up to (2N+1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N+1)(2M+1) unknowns by first minimizing the cost functional (10) and then adopting the results as starting point for minimizing the weighted functional (14). Results are shown in dB scale.

Fig. 4.
Fig. 4.

Reconstruction for the same scatterer and for the same situation as in Fig. 1 panel c but data have been corrupted by an additive 10% uniformly distributed noise. Result is shown in dB scale.

Equations (16)

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

U Γ ( r ) = { 1 for r Γ 0 otherwise .
E S ¯ ( r ¯ , f i ) = j 2 π f i μ 0 𝒪 G = ( r ¯ r ¯ , f i ) J ¯ P O ( r ¯ , f i ) d r ¯ ,
J ¯ P O ( r ¯ , f i ) = 2 i n ̂ × H ¯ inc ( r ¯ , f i ) U Γ ( r ¯ ) = 2 ζ i z ̂ × E inc U Γ ( r ¯ ) i y ̂ = 2 ζ E inc U Γ ( r ¯ ) i x , ̂
E S ( r ¯ , f i ) = j 4 π f i μ 0 ζ 𝒪 G x x ( r ¯ r ¯ , f i ) E inc U Γ ( r ¯ ) d r ¯ ,
E S ( x , y , z 1 , f i ) = ( i J ̂ ) ( x , y , z 1 , f i ) = F i v 2 + w i 2 w i ×
exp ( jux jvy j w i z 1 ) U ̂ Γ ( u , v ) dudv ,
U ̂ ( u , v ) = 𝓕 U Γ = E inc U Γ ( x , y ) exp ( j u x + j v y ) d x d y ,
𝒜 : U ̂ Γ ( M 1 2 , M 2 2 ) = ( 1 U ̂ Γ 2 , 2 U ̂ Γ 2 ) ,
Φ ( U ̂ Γ ) = ( M 1 2 , M 2 2 ) ( 1 U ̂ Γ 2 , 2 U ̂ Γ 2 ) 2 = 1 U ̂ Γ 2 M 1 2 2 + 2 U ̂ Γ 2 M 2 2 2 .
U ˜ Γ ( x , y ) = U ̂ Γ .
Φ ( U ̂ ¯ Γ ) = ( M ¯ ˜ 1 2 , M ¯ ˜ 2 2 ) ( = 1 U ¯ Γ ̂ 2 , = 2 U ¯ Γ ̂ 2 ) 2 = = 1 U ¯ Γ ̂ 2 M ¯ ˜ 1 2 2 + = 2 U Γ ¯ ̂ 2 M ¯ ˜ 2 2 2 .
U ̂ Γ ( u , ν ) = N N M M U ̂ Γ n m sinc ( u X O n π ) sinc ( ν Y O m π ) ,
sin c ( x ) = sin x x .
( n π X O ) 2 + ( m π Y O ) 2 k 2 2 ( n , m ) : n ( N , N ) , m ( M , M ) .
Ψ ( U ¯ ̂ Γ ) = ( 1 = U ¯ ̂ Γ 2 M ¯ ˜ 1 2 ) . ( M ¯ ˜ 1 2 + η ) 2 + ( 2 = U ¯ ̂ Γ 2 M ¯ ˜ 2 2 ) . ( M ¯ ˜ 2 2 + η ) 2 ,
( n π X O ) 2 + ( m π Y O ) 2 k 1 2 ( n , m ) : n ( N 1 , N 1 ) , m ( M 1 , M 1 ) .

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