Abstract

A new algorithm to estimate the two-dimensional local frequencies of phase interferometric data is described. With a complex sine-wave model, demonstration is given that a conventional multiple-signal classification (MUSIC) algorithm can be used in spite of multiplicative noise perturbations. A faster algorithm dedicated to the processing of interferograms is developed and a measure of confidence in the estimate is proposed. We studied numerical performances using synthetic fringes. As a result of the frequency estimation, knowledge of the fringe local width and orientation can be applied to restore noisy phase data. Results of a complex phase filter are presented for real interferograms obtained from synthetic aperture radar images.

© 1996 Optical Society of America

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References

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  1. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–854 (1986).
    [CrossRef]
  2. H. A. Vrooman, A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  3. D. Massonnet, T. Rabaute, “Radar interferometry: limits and potentials,” IEEE Trans. Geosci. Remote Sensing 31, 445–464 (1993).
    [CrossRef]
  4. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  5. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  6. C. Prati, M. Giani, N. Leuratti, “SAR interferometry: a 2D phase unwrapping technique based on phase and absolute values informations,” in Proceedings of the 1990 International Geoscience and Remote Sensing Society Conference, R. Mills, ed. (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 3, pp. 2043–2046.
  7. H. Takajo, T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  8. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
    [CrossRef]
  9. P. Stoica, A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989).
    [CrossRef]
  10. E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
    [CrossRef]
  11. J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
    [CrossRef]
  12. W. K. Pratt, “Vector space formulation of two dimensional signal processing operations,” Comput. Graphics Image Process. 4, 1–24 (1975).
    [CrossRef]
  13. A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).
  14. I. Bloch, “Information combination operators for data fusion: a comparative review with classification,” IEEE Trans. Syst. Man Cybernetics. 26, 52–67 (1996).
    [CrossRef]
  15. E. Trouvé, H. Maître, “Wrapped phase restoration and unwrapping obstacles classification in SAR interferometry,” paper presented at the Symposium on Retrieval of Bio- and Geophysical Parameters from SAR Data for Land Applications, 1995Toulouse, France (Centre National d'Etudes Spatiales–IEEE Geoscience and Remote Sensing Society).
  16. H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echos,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
    [CrossRef]
  17. J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
    [CrossRef]
  18. E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Electr. Eng. Part F 139, 147–159 (1992).
  19. D. Massonnet, F. Perlant, T. Rabaute, “Précision et niveau d'artéfacts dans les calculs de modèles numériques de terrain par interférométrie,” in From Optics to Radar, SPOT and ERS Applications (Cepadues Editions, Paris,1993), pp. 511–523, in English.

1996 (1)

I. Bloch, “Information combination operators for data fusion: a comparative review with classification,” IEEE Trans. Syst. Man Cybernetics. 26, 52–67 (1996).
[CrossRef]

1995 (1)

E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
[CrossRef]

1994 (2)

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

1993 (1)

D. Massonnet, T. Rabaute, “Radar interferometry: limits and potentials,” IEEE Trans. Geosci. Remote Sensing 31, 445–464 (1993).
[CrossRef]

1992 (2)

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Electr. Eng. Part F 139, 147–159 (1992).

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echos,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
[CrossRef]

1991 (2)

1989 (1)

P. Stoica, A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989).
[CrossRef]

1988 (2)

1986 (1)

1975 (1)

W. K. Pratt, “Vector space formulation of two dimensional signal processing operations,” Comput. Graphics Image Process. 4, 1–24 (1975).
[CrossRef]

Ainsworth, T. L.

J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
[CrossRef]

Bloch, I.

I. Bloch, “Information combination operators for data fusion: a comparative review with classification,” IEEE Trans. Syst. Man Cybernetics. 26, 52–67 (1996).
[CrossRef]

Bone, D. J.

Cha, S. S.

E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

Giani, M.

C. Prati, M. Giani, N. Leuratti, “SAR interferometry: a 2D phase unwrapping technique based on phase and absolute values informations,” in Proceedings of the 1990 International Geoscience and Remote Sensing Society Conference, R. Mills, ed. (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 3, pp. 2043–2046.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
[CrossRef]

Grunes, M. R.

J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
[CrossRef]

Hoppel, K. W.

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

Joo, W.

E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
[CrossRef]

Kreis, T.

Lee, J. S.

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
[CrossRef]

Leuratti, N.

C. Prati, M. Giani, N. Leuratti, “SAR interferometry: a 2D phase unwrapping technique based on phase and absolute values informations,” in Proceedings of the 1990 International Geoscience and Remote Sensing Society Conference, R. Mills, ed. (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 3, pp. 2043–2046.

Maas, A. M.

Maître, H.

E. Trouvé, H. Maître, “Wrapped phase restoration and unwrapping obstacles classification in SAR interferometry,” paper presented at the Symposium on Retrieval of Bio- and Geophysical Parameters from SAR Data for Land Applications, 1995Toulouse, France (Centre National d'Etudes Spatiales–IEEE Geoscience and Remote Sensing Society).

Mango, S. A.

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

Martin, J. M.

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Electr. Eng. Part F 139, 147–159 (1992).

Massonnet, D.

D. Massonnet, T. Rabaute, “Radar interferometry: limits and potentials,” IEEE Trans. Geosci. Remote Sensing 31, 445–464 (1993).
[CrossRef]

D. Massonnet, F. Perlant, T. Rabaute, “Précision et niveau d'artéfacts dans les calculs de modèles numériques de terrain par interférométrie,” in From Optics to Radar, SPOT and ERS Applications (Cepadues Editions, Paris,1993), pp. 511–523, in English.

Miller, A. R.

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

Nehorai, A.

P. Stoica, A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989).
[CrossRef]

Perlant, F.

D. Massonnet, F. Perlant, T. Rabaute, “Précision et niveau d'artéfacts dans les calculs de modèles numériques de terrain par interférométrie,” in From Optics to Radar, SPOT and ERS Applications (Cepadues Editions, Paris,1993), pp. 511–523, in English.

Prati, C.

C. Prati, M. Giani, N. Leuratti, “SAR interferometry: a 2D phase unwrapping technique based on phase and absolute values informations,” in Proceedings of the 1990 International Geoscience and Remote Sensing Society Conference, R. Mills, ed. (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 3, pp. 2043–2046.

Pratt, W. K.

W. K. Pratt, “Vector space formulation of two dimensional signal processing operations,” Comput. Graphics Image Process. 4, 1–24 (1975).
[CrossRef]

Rabaute, T.

D. Massonnet, T. Rabaute, “Radar interferometry: limits and potentials,” IEEE Trans. Geosci. Remote Sensing 31, 445–464 (1993).
[CrossRef]

D. Massonnet, F. Perlant, T. Rabaute, “Précision et niveau d'artéfacts dans les calculs de modèles numériques de terrain par interférométrie,” in From Optics to Radar, SPOT and ERS Applications (Cepadues Editions, Paris,1993), pp. 511–523, in English.

Rabinowitz, P.

A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).

Ralston, A.

A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).

Rodriguez, E.

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Electr. Eng. Part F 139, 147–159 (1992).

Romero, L. A.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

Stoica, P.

P. Stoica, A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989).
[CrossRef]

Takahashi, T.

Takajo, H.

Trouvé, E.

E. Trouvé, H. Maître, “Wrapped phase restoration and unwrapping obstacles classification in SAR interferometry,” paper presented at the Symposium on Retrieval of Bio- and Geophysical Parameters from SAR Data for Land Applications, 1995Toulouse, France (Centre National d'Etudes Spatiales–IEEE Geoscience and Remote Sensing Society).

Villasenor, J.

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echos,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
[CrossRef]

Vrooman, H. A.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Yu, E.

E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
[CrossRef]

Zebker, H. A.

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echos,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
[CrossRef]

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Appl. Opt. (2)

Comput. Graphics Image Process. (1)

W. K. Pratt, “Vector space formulation of two dimensional signal processing operations,” Comput. Graphics Image Process. 4, 1–24 (1975).
[CrossRef]

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

P. Stoica, A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust. Speech Signal Process. 37, 720–741 (1989).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (3)

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echos,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
[CrossRef]

J. S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sensing 32, 1017–1028 (1994).
[CrossRef]

D. Massonnet, T. Rabaute, “Radar interferometry: limits and potentials,” IEEE Trans. Geosci. Remote Sensing 31, 445–464 (1993).
[CrossRef]

IEEE Trans. Syst. Man Cybernetics. (1)

I. Bloch, “Information combination operators for data fusion: a comparative review with classification,” IEEE Trans. Syst. Man Cybernetics. 26, 52–67 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

Opt. Eng. (1)

E. Yu, S. S. Cha, W. Joo, “Use of interferometric directionality for noise reduction,” Opt. Eng. 34, 173–182 (1995).
[CrossRef]

Proc. Inst. Electr. Eng. Part F (1)

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Electr. Eng. Part F 139, 147–159 (1992).

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Other (5)

C. Prati, M. Giani, N. Leuratti, “SAR interferometry: a 2D phase unwrapping technique based on phase and absolute values informations,” in Proceedings of the 1990 International Geoscience and Remote Sensing Society Conference, R. Mills, ed. (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 3, pp. 2043–2046.

E. Trouvé, H. Maître, “Wrapped phase restoration and unwrapping obstacles classification in SAR interferometry,” paper presented at the Symposium on Retrieval of Bio- and Geophysical Parameters from SAR Data for Land Applications, 1995Toulouse, France (Centre National d'Etudes Spatiales–IEEE Geoscience and Remote Sensing Society).

D. Massonnet, F. Perlant, T. Rabaute, “Précision et niveau d'artéfacts dans les calculs de modèles numériques de terrain par interférométrie,” in From Optics to Radar, SPOT and ERS Applications (Cepadues Editions, Paris,1993), pp. 511–523, in English.

J. S. Lee, T. L. Ainsworth, M. R. Grunes, R. M. Goldstein, “Noise filtering of interferometric S.A.R. images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 735–742 (1994).
[CrossRef]

A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).

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Figures (4)

Fig. 1
Fig. 1

Synthetic fringes: a 320 × 320 image extracted from a circular pattern with noise level br = π/2. Samples of the frequencies measured by the proposed method for Ds = 3 and De = 9 are plotted as short lines with orientation equal to θ ̂ and length proportional to the fringe radial width L ̂ = 1 / f ̂ x 2 + f ̂ y 2 .

Fig. 2
Fig. 2

Relative width error | L L ̂ | / L versus fringe width. Even small window sizes (Ds = 3, De = 9) give good results for a noise level that increases to 5π/8. Larger windows (Ds = 6, De = 15) permit a correct estimation for a very high noise level (br = 3π/4).

Fig. 3
Fig. 3

Confidence measurements versus frequency estimation error for window sizes (Ds = 3, De = 9): (1a), (1b), and 1(c) represent, respectively, confidences (1 − Ud ), (1 − Uf ), and C measured on synthetic fringes with a noise level of br = 3π/4; (2) confidence C for lower noise (br = 5π/8).

Fig. 4
Fig. 4

SAR interferogram taken over the Etna volcano in Sicilia (240 × 180 subimages): (a) Original fringes processed from the ERS-1 satellite SAR images 127 September and 1 November 19922, with 2 × 4 complex multilooking, a copyright of the Centre National d'Etudes Spatiales. (b) Restored wrapped phase obtained from 5 × 5 complex filtering with compensation of fringe patterns given by 2-D local frequencies. (c) Level of confidence associated with the frequency estimation. Low confidence, in dark, points out the areas where correct estimation could not be performed.

Equations (35)

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

ϕ ̂ = ϕ + η ( mod 2 π ) .
s n ( k , l ) = s ( k , l ) b ( k , l ) = exp [ i ϕ ( k , l ) ] exp [ i η ( k , l ) ] ,
s ( k , l ) = exp [ i 2 π ( k f x + l f y ) ] .
γ b ( m , n ) = exp [ i η ( k , l ) ] exp [ i η ( k , l ) ] ,
γ b ( m , n ) = { 1 if ( m , n ) = ( 0 , 0 ) K elsewhere ,
γ s n ( m , n ) = s n ( k , l ) s n ( k , l ) ¯ = exp [ i 2 π ( m f x + n f y ) ] × exp [ i η ( m , n ) ] exp [ i η ( 0 , 0 ) ] = γ s ( m , n ) γ b ( m , n ) = K γ s ( m , n ) + ( 1 K ) δ m , n ,
Γ s b = K Γ s + ( 1 K ) I ,
x n ( k , l ) = x ( k , l ) + b ( k , l ) = j = 1 P a j exp [ i 2 π ( k f x j + l f y j ) ] + b ( k , l ) .
e j = [ 1 exp ( i 2 π f x j ) exp [ i 2 π ( D s 1 ) f x j ] exp ( i 2 π f y j ) exp [ i 2 π ( f x j + f y j ) ] exp { i 2 π [ ( D s 1 ) f x j + f y j ] } exp [ i 2 π ( D s 1 ) f y j ] exp { i 2 π [ ( D s 1 ) f x j + ( D s 1 ) f y j ] } ] .
Γ x = V ,
Γ x n = V + σ b 2 I .
Γ x n = K e 1 e 1 + ( 1 K ) I .
Γ s = e ( f x , f y ) e ( f x , f y ) ,
λ 1 = D K + 1 K .
L j + 1 = a x L j for j = j 0 + k D s , j 0 [ 1 , D s 1 ] , k [ 0 , D s 1 ] .
L j + D s = a y L j for j [ 1 , D s ( D s 1 ) ] .
v 2 a x v 1 = 0 ,
w 2 a y w 1 = 0 ,
v 1 ( v 2 a x v 1 ) = 0 ,
a x = exp ( i 2 π f x ) = v 1 v 2 v 1 2 , a y = exp ( i 2 π f y ) = w 1 w 2 w 1 2 ,
Γ s n e ( f ̂ x , f ̂ y ) λ 1 e ( f ̂ x , f ̂ y ) = 0 ,
Γ s n λ 1 I = K [ Γ s D I ] .
[ Γ s n λ 1 I ] e ( f ̂ x , f ̂ y ) 2 K 2 D 2 ( D 1 ) .
U d = Γ s n e ( f ̂ x , f ̂ y ) λ 1 e ( f ̂ x , f ̂ y ) KD D 1 ,
U f x = v 2 a x v 1 2 v 2 2 = v 2 v 2 a x v 2 v 1 v 2 2 = 1 | v 1 v 2 | 2 v 1 2 v 2 2 , U f y = 1 | w 1 w 2 | 2 w 1 2 w 2 2 ,
d f = | f x | d f x + | f y | d f y f x 2 + f y 2 .
U f = | f ̂ x | U f x + | f ̂ y | U f y 2 f ̂ x 2 + f ̂ y 2 ,
C = 2 ( 1 U d ) ( 1 U f ) ( 1 U d ) + ( 1 U f ) .
δ f = 1 2 | f x | δ f x + | f y | δ f y f x 2 f y 2 ,
ϕ ̂ ( k , l ) = arg [ ρ 1 exp ( i ϕ 1 ) ρ 2 exp ( i ϕ 2 ) ρ 1 2 ρ 2 2 ] ,
ϕ υ ( k , l ) = 2 π ( k f x + l f y ) .
ϕ ( P ) = arg ( ( k , l ) W exp { i [ ϕ ̂ ( k , l ) ϕ υ ( k , l ) ] } ) .
ϕ ( P ) = arg ( ( k , l ) W exp [ i ϕ υ ( k , l ) ] × { ρ 1 ρ 2 exp [ i ( ϕ 1 ϕ 2 ) ] } ) .
γ b ( m , n ) = exp [ i η ( m , n ) ] exp [ i η ( 0 , 0 ) ] = exp [ i η ( m , n ) ] exp [ i η ( 0 , 0 ) ] = exp [ i η ( 0 , 0 ) ] exp [ i η ( 0 , 0 ) ] = | exp [ i η ( 0 , 0 ) ] | 2 = | + exp ( i η ) p ( η ) d η | 2 = | p ˙ ( 1 2 π ) | 2 ,
( f ) = sin ( 2 π b r f ) 2 π b r f , K = ( sin b r b r ) 2 .

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