Abstract

In weak scattering diffraction tomography, there is a well-known Fourier relationship between the field scattered by an unknown object and the scattering potential that characterizes the object. For these weakly scattering objects, those that satisfy the Born or Rytov approximation, the scattered field generated from a sequence of different illumination directions complements Fourier data on the unknown scattering potential. A low-pass reconstruction is found by Fourier inversion. A simulation is presented in which the scattered far field from several strongly scattering penetrable two-dimensional cylinders is backpropagated into the object domain. For these more strongly scattering objects, a single wavelength and a single illumination direction are used to provide limited Fourier data on the product of the scattering potential and the total field. A nonlinear filtering technique, known as differential cepstral filtering, is used to isolate the scattering potential and to suppress artifacts introduced by the perturbing field component. Reconstructed images calculated by this technique from exact scattered-field data from cylindrically symmetric objects are shown.

© 1996 Optical Society of America

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  1. J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
    [Crossref]
  2. F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
    [Crossref]
  3. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [Crossref]
  4. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [PubMed]
  5. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [Crossref]
  6. L. Zapalowski, M. A. Fiddy, S. Leeman, “On inverse scattering in the first Born approximation,” in Ultrasonics Symposium IEEE (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 827–830.
  7. A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
    [Crossref]
  8. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  9. J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
    [Crossref]
  10. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [Crossref]
  11. A. Polydoros, A. T. Fam, “The differential cepstrum: definitions and properties,” in IEEE International Symposium on Circuits and Systems Proceedings (Institute of Electrical and Electronics Engineers, New York, 1981), Vol. 1, pp. 77–80.
  12. D. Raghuramireddy, R. Unbehauen, “The two-dimensional differential cepstrum,”IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 1335–1337 (1985).
    [Crossref]
  13. D. Raghuramireddy, R. Unbehauen, “Multidimensional homomorphic deconvolution systems,” in Proceedings of the Twenty-Fourth Conference on Decision and Control, G. F. Franklin, ed. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 1606–1612.
  14. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [Crossref]
  15. M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
    [Crossref]
  16. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,”J. Math. Phys. 19, 1526–1531 (1978).
    [Crossref]
  17. C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
    [Crossref]
  18. A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, non-iterative object reconstruction from incomplete data using a prioriknowledge,”J. Opt. Soc. Am. 73, 1466–1469 (1983).
    [Crossref]

1995 (1)

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

1990 (1)

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

1986 (1)

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

1985 (2)

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[Crossref]

D. Raghuramireddy, R. Unbehauen, “The two-dimensional differential cepstrum,”IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 1335–1337 (1985).
[Crossref]

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[Crossref]

1983 (2)

1982 (1)

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

1978 (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,”J. Math. Phys. 19, 1526–1531 (1978).
[Crossref]

1977 (1)

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[Crossref]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1968 (1)

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[Crossref]

1965 (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

Byrne, C. L.

Darling, A. M.

Devaney, A. J.

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

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,”J. Math. Phys. 19, 1526–1531 (1978).
[Crossref]

Fam, A. T.

A. Polydoros, A. T. Fam, “The differential cepstrum: definitions and properties,” in IEEE International Symposium on Circuits and Systems Proceedings (Institute of Electrical and Electronics Engineers, New York, 1981), Vol. 1, pp. 77–80.

Fiddy, M. A.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

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

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[Crossref]

A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, non-iterative object reconstruction from incomplete data using a prioriknowledge,”J. Opt. Soc. Am. 73, 1466–1469 (1983).
[Crossref]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
[Crossref]

L. Zapalowski, M. A. Fiddy, S. Leeman, “On inverse scattering in the first Born approximation,” in Ultrasonics Symposium IEEE (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 827–830.

Fitzgerald, R. M.

Hall, T. J.

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[Crossref]

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[Crossref]

Leeman, S.

L. Zapalowski, M. A. Fiddy, S. Leeman, “On inverse scattering in the first Born approximation,” in Ultrasonics Symposium IEEE (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 827–830.

Lin, F. C.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

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

McGahan, R. V.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

Morris, J. B.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[Crossref]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Polydoros, A.

A. Polydoros, A. T. Fam, “The differential cepstrum: definitions and properties,” in IEEE International Symposium on Circuits and Systems Proceedings (Institute of Electrical and Electronics Engineers, New York, 1981), Vol. 1, pp. 77–80.

Pommet, D. A.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

Raghuramireddy, D.

D. Raghuramireddy, R. Unbehauen, “The two-dimensional differential cepstrum,”IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 1335–1337 (1985).
[Crossref]

D. Raghuramireddy, R. Unbehauen, “Multidimensional homomorphic deconvolution systems,” in Proceedings of the Twenty-Fourth Conference on Decision and Control, G. F. Franklin, ed. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 1606–1612.

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[Crossref]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Scivier, M. S.

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[Crossref]

Stockham, T. G.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[Crossref]

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[Crossref]

Unbehauen, R.

D. Raghuramireddy, R. Unbehauen, “The two-dimensional differential cepstrum,”IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 1335–1337 (1985).
[Crossref]

D. Raghuramireddy, R. Unbehauen, “Multidimensional homomorphic deconvolution systems,” in Proceedings of the Twenty-Fourth Conference on Decision and Control, G. F. Franklin, ed. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 1606–1612.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Zapalowski, L.

L. Zapalowski, M. A. Fiddy, S. Leeman, “On inverse scattering in the first Born approximation,” in Ultrasonics Symposium IEEE (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 827–830.

IEEE Trans. Acoust. Speech Signal Process. (2)

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[Crossref]

D. Raghuramireddy, R. Unbehauen, “The two-dimensional differential cepstrum,”IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 1335–1337 (1985).
[Crossref]

IEEE Trans. Antennas Propag. (2)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering objects,”IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[Crossref]

Int. J. Imaging Syst. Technol. (1)

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

J. Math. Phys. (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,”J. Math. Phys. 19, 1526–1531 (1978).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Proc. IEEE (1)

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[Crossref]

Ultrason. Imaging (1)

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

Other (4)

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

L. Zapalowski, M. A. Fiddy, S. Leeman, “On inverse scattering in the first Born approximation,” in Ultrasonics Symposium IEEE (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 827–830.

A. Polydoros, A. T. Fam, “The differential cepstrum: definitions and properties,” in IEEE International Symposium on Circuits and Systems Proceedings (Institute of Electrical and Electronics Engineers, New York, 1981), Vol. 1, pp. 77–80.

D. Raghuramireddy, R. Unbehauen, “Multidimensional homomorphic deconvolution systems,” in Proceedings of the Twenty-Fourth Conference on Decision and Control, G. F. Franklin, ed. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 1606–1612.

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Figures (16)

Fig. 1
Fig. 1

2D inverse scattering experiment.

Fig. 2
Fig. 2

Differential cepstral filtering method.

Fig. 3
Fig. 3

(a) Magnitude of VΨ/Ψi for object 1.1/1.03. (b) Magnitude of VΨ/Ψi for object 1.03/1.1.

Fig. 4
Fig. 4

(a) Magnitude of differential cepstrum for object 1.1/1.03. (b) Magnitude of differential cepstrum for object 1.03/1.1.

Fig. 5
Fig. 5

(a) Reconstruction after filtering in differential cepstral domain for object 1.1/1.03. (b) Reconstruction after filtering in differential cepstral domain for object 1.03/1.1.

Fig. 6
Fig. 6

(a) Magnitude of backpropagated VΨ/Ψi for object 1.1/1.03. (b) Magnitude of backpropagated VΨ/Ψi for object 1.03/1.1.

Fig. 7
Fig. 7

(a) Figure 6(a), zeroed beyond object support. (b) Figure 6(b), zeroed beyond object support.

Fig. 8
Fig. 8

(a) Magnitude of differential cepstrum of Fig. 7(a). (b) Magnitude of differential cepstrum of Fig. 7(b).

Fig. 9
Fig. 9

(a) Reconstruction after filtering in differential cepstral domain for backpropagated object 1.1/1.03. (b) Reconstruction after filtering in differential cepstral domain for backpropagated object 1.1/1.03.

Fig. 10
Fig. 10

(a) Magnitude of VΨ/Ψi for object 4/2. (b) Magnitude of VΨ/Ψi for object 2/4.

Fig. 11
Fig. 11

(a) Magnitude of differential cepstrum for object 4/2. (b) Magnitude of differential cepstrum for object 2/4.

Fig. 12
Fig. 12

Reconstruction after filtering in differential cepstral domain for object 4/2. (b) Reconstruction after filtering in differential cepstral domain for object 2/4.

Fig. 13
Fig. 13

(a) Magnitude of backpropagated VΨ/Ψi for object 4/2. (b) Magnitude of backpropagated VΨ/Ψi for object 2/4.

Fig. 14
Fig. 14

(a) Figure 13(a), zeroed beyond object support, (b) Figure 13(b), zeroed beyond object support.

Fig. 15
Fig. 15

(a) Magnitude of differential cepstram of Fig. 13(a). (b) Magnitude of differential cepstram of Fig. 13(b).

Fig. 16
Fig. 16

(a) Reconstruction after filter in differential cepstral domain for backpropagated object 4/2. (b) Reconstruction after filtering in differential cepstral domain for backpropagated object 2/4.

Equations (21)

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[ 2 + k 0 2 r ( r ) ] Ψ ( r ) = 0 ,
( 2 + k 0 2 ) Ψ ( r ) = - k 0 2 V ( r ) Ψ ( r ) .
( 2 + k 0 2 ) Ψ i ( r ) = 0.
Ψ ( r , r ^ i ) = exp ( i k 0 r ^ i · r ) - k 0 2 D V ( r ) Ψ ( r , r ^ i ) G ( r , r ) d r ,
( 2 + k 0 2 ) G ( r , r ) = - δ ( r - r ) .
Ψ s ( r , r ^ i ) = i k 0 2 4 D V ( r ) Ψ ( r , r ^ i ) H 0 ( 1 ) ( k 0 r - r ) d r .
Ψ s ff ( r , r ^ i ) = 2 k 0 2 π 3 r exp [ i ( k 0 r + π / 4 ) ] F ( r , r ^ i ) ,
F ( r , r ^ i ) 1 ( 2 π ) 2 D V ( r ) Ψ ( r , r ^ i ) exp ( - i k 0 r ^ · r ) d r .
F BA ( r , r ^ i ) = 1 ( 2 π ) 2 D V ( r ) exp [ - i k 0 ( r ^ - r ^ i ) · r ] d r .
V ˜ ( k ) 1 ( 2 π ) 2 - V ( r ) exp ( - i k · r ) d r ,
V l p ( r ) = k 0 2 2 ( 2 π ) 2 - π π - π π V ˜ [ k 0 ( r ^ - r ^ i ) ] × exp [ i k 0 ( r ^ - r ^ i ) · r ] 1 - ( r ^ · r ^ i ) 2 d ϕ s d ϕ i .
F ( r , r ^ i ) = 1 ( 2 π ) 2 D V eff ( r , r ^ i ) exp [ - i k 0 ( r ^ - r ^ i ) · r ] d r ,
V eff ( r , r ^ i ) V ( r ) Ψ ( r , r ^ i ) Ψ i ( r , r ^ i ) .
V ˜ eff ( k ) 1 ( 2 π ) 2 - V eff ( r , r ^ i ) exp ( - i k · r ) d r ,             r ^ i is constant ,
S ( r , r ^ i ) = k 0 2 2 - π π V ˜ eff [ k 0 ( r ^ - r ^ i ) ] × exp [ i k 0 ( r ^ - r ^ i ) · r ] 1 - ( r ^ · r ^ i ) 2 d ϕ s             for r ^ i is constant ,
S ( r , r ^ i ) = [ V ( r ) Ψ ( r , r ^ i ) Ψ i ( r , r ^ i ) ] * * h ( r , r ^ i )             for r i is constant
log [ V ( r ) Ψ ( r , r ^ i ) Ψ i ( r , r ^ i ) ] = log V ( r ) + log | Ψ ( r , r ^ i ) Ψ i ( r , r ^ i ) | + i { arg [ V ( r ) ] + arg [ Ψ ( r , r ^ i ) Ψ i ( r , r ^ i ) ] + 2 π k } ,             k = 0 , 1 , 2 ,
δ δ x [ V ( x , y ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) ] [ V ( x , y ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) ] = δ δ x V ( x , y ) V ( x , y ) + δ δ x Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i )
δ δ x [ V ( x , y ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) ] [ V ( x , y ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) ] * | V ( x , y ) Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) | 2 + δ ,
δ δ x Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) / Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i )
δ δ x V ( x , y ) / V ( x , y ) , δ δ x Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i ) / Ψ ( x , y , ϕ i ) Ψ i ( x , y , ϕ i )

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