Abstract

We derive hydrodynamic equations for the point-spread function of an imaging system looking through atmospheric turbulence at an incoherent object. These are derived from the hydrodynamics of the index of refraction of the air. We use the path integral representation of the paraxial approximation for wave propagation through turbulence. We then study the case of a frozen turbulent refractive index field being advected past the imaging system with a constant wind and discuss the implications for optical flow estimation. We conclude by discussing possible directions for future work.

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  1. D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
    [CrossRef]
  2. I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.
  3. D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.
  4. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1985), Vol. 22, pp. 341–398.
  5. G. Potvin, J. L. Forand, and D. Dion, “A parametric model for simulating turbulence effects on imaging systems,” Tech. Rep. TR 2006-787 (DRDC Valcartier, 2007).
  6. V. I. Klyatskin and V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–339 (1970).
  7. G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. 61, 161–168 (1971).
    [CrossRef]
  8. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed. (Dover, 2010).
  9. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  10. M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
    [CrossRef]
  11. M. I. Charnotskii, “Turbulence effects on the image of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996).
    [CrossRef]
  12. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.
  13. L. S. Schulman, Techniques and Applications of Path Integration (Dover, 2005).
  14. B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
    [CrossRef]
  15. D. Sun, S. Roth, and M. J. Black, “Secrets of optical flow estimation and their principles,” in Proceedings of the IEEE International Conference on Computer Vision & Pattern Recognition (IEEE, 2010), pp. 2432–2439.
  16. E. L. Andreas, “Estimating Cn2 over snow and sea ice from meteorological data,” J. Opt. Soc. Am. A 5, 481–495 (1988).
    [CrossRef]
  17. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, 1988).

1999 (1)

1996 (1)

1993 (1)

1988 (1)

1981 (1)

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

1979 (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

1971 (1)

1970 (1)

V. I. Klyatskin and V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–339 (1970).

Andreas, E. L.

Black, M. J.

D. Sun, S. Roth, and M. J. Black, “Secrets of optical flow estimation and their principles,” in Proceedings of the IEEE International Conference on Computer Vision & Pattern Recognition (IEEE, 2010), pp. 2432–2439.

Charnotskii, M. I.

M. I. Charnotskii, “Turbulence effects on the image of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996).
[CrossRef]

M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.

Clyde, D.

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Dion, D.

G. Potvin, J. L. Forand, and D. Dion, “A parametric model for simulating turbulence effects on imaging systems,” Tech. Rep. TR 2006-787 (DRDC Valcartier, 2007).

Eichmann, G.

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1985), Vol. 22, pp. 341–398.

Feynman, R. P.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed. (Dover, 2010).

Forand, J. L.

G. Potvin, J. L. Forand, and D. Dion, “A parametric model for simulating turbulence effects on imaging systems,” Tech. Rep. TR 2006-787 (DRDC Valcartier, 2007).

Fraser, D.

D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
[CrossRef]

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.

Hege, K.

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed. (Dover, 2010).

Horn, B. K. P.

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Klyatskin, V. I.

V. I. Klyatskin and V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–339 (1970).

Lambert, A.

D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
[CrossRef]

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.

Potvin, G.

G. Potvin, J. L. Forand, and D. Dion, “A parametric model for simulating turbulence effects on imaging systems,” Tech. Rep. TR 2006-787 (DRDC Valcartier, 2007).

Roth, S.

D. Sun, S. Roth, and M. J. Black, “Secrets of optical flow estimation and their principles,” in Proceedings of the IEEE International Conference on Computer Vision & Pattern Recognition (IEEE, 2010), pp. 2432–2439.

Schulman, L. S.

L. S. Schulman, Techniques and Applications of Path Integration (Dover, 2005).

Schunck, B. G.

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Scott-Fleming, I.

D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

Stull, R. B.

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, 1988).

Sun, D.

D. Sun, S. Roth, and M. J. Black, “Secrets of optical flow estimation and their principles,” in Proceedings of the IEEE International Conference on Computer Vision & Pattern Recognition (IEEE, 2010), pp. 2432–2439.

Tatarskii, V. I.

V. I. Klyatskin and V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–339 (1970).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.

Thorpe, G.

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.

Artif. Intell. (1)

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Sov. Phys. JETP (1)

V. I. Klyatskin and V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–339 (1970).

Other (9)

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, 1988).

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed. (Dover, 2010).

D. Sun, S. Roth, and M. J. Black, “Secrets of optical flow estimation and their principles,” in Proceedings of the IEEE International Conference on Computer Vision & Pattern Recognition (IEEE, 2010), pp. 2432–2439.

I. Scott-Fleming, K. Hege, D. Clyde, D. Fraser, and A. Lambert, “Gradient based optical flow techniques for tracking image motion due to atmospheric turbulence,” in Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America, 2001), paper STuB3.

D. Clyde, I. Scott-Fleming, D. Fraser, and A. Lambert, “Application of optical flow techniques in the restoration of non-uniformly warped images,” in Proceedings of the 6th Digital Image Computing: Techniques and Applications (DICTA) (Australian Pattern Recognition Society, 2002), pp. 195–200.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1985), Vol. 22, pp. 341–398.

G. Potvin, J. L. Forand, and D. Dion, “A parametric model for simulating turbulence effects on imaging systems,” Tech. Rep. TR 2006-787 (DRDC Valcartier, 2007).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1993), Vol. 32, pp. 203–266.

L. S. Schulman, Techniques and Applications of Path Integration (Dover, 2005).

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Figures (2)

Fig. 1.
Fig. 1.

Generalized Huygens–Fresnel principle starts with the electromagnetic field (represented by arrows) at the object plane (A), each point emitting a spherical wave (B) that gets distorted by the atmospheric turbulence. The waves go through a thin lens (C) and are focused on the image plane (D).

Fig. 2.
Fig. 2.

Sample path where the y coordinate between the object and lens planes is evenly divided into discrete steps a distance ε apart.

Equations (82)

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E l ( ρ l ) = d 2 ρ o E o ( ρ o ) U ( ρ o , 0 , ρ l , L ) .
U ( ρ o , 0 , ρ l , L ) = A 1 D ρ ( y ) exp [ i k S ( ρ o , 0 , ρ l , L ) ] ,
S ( ρ o , 0 , ρ l , L ) = 0 L d y L ( x , z , x ˙ , z ˙ ; y ) ,
L ( x , z , x ˙ , z ˙ ; y ) = ( 1 + n ( x , y , z ) ) ( 1 + x ˙ 2 + z ˙ 2 ) 1 / 2 ,
L ( x , z , x ˙ , z ˙ ; y ) = 1 2 ( x ˙ 2 + z ˙ 2 ) + n ( x , y , z ) .
U ( ρ o , 0 , ρ l , L ) e i k L A D ρ ( y ) exp [ i k S ( ρ o , 0 , ρ l , L ) ] ,
u ( ρ o , 0 , ρ l , L ) = A 1 D ρ ( y ) exp [ i k S ( ρ o , 0 , ρ l , L ) ] .
t u = i k A D ρ ( y ) exp [ i k S ] ( 0 L d y t n ) ,
F ( ρ o , 0 , ρ l , L ) = A 1 D ρ ( y ) F { ρ ( y ) } exp [ i k S ( ρ o , 0 , ρ l , L ) ] ,
t n + · j + y j y = s ,
t u = i k 0 L d y ( · j + y j y ) + i k 0 L d y s .
u ( x o , 0 , x l , L ) = lim N A N 1 d x 1 d x N 1 exp [ i k i = 0 N 1 S ( i , i + 1 ) ] ,
S ( i , i + 1 ) = L ( x i + x i + 1 2 , x i + 1 x i ε ; y i + y i + 1 2 ) ε ,
· j + y j y ( x i + x i + 1 ) j x ( x i + x i + 1 2 ; y i + y i + 1 2 ) + ( y i + y i + 1 ) j y ( x i + x i + 1 2 ; y i + y i + 1 2 ) .
· j + y j y ( x i + x i + 1 ) [ j x ( x i + 1 x i ε ) j y ] + ( y i + y i + 1 ) j y + ( x i + 1 x i ε ) ( x i + x i + 1 ) j y ,
1 ε x i [ S ( i 1 , i ) + S ( i , i + 1 ) ] = x i + 1 + x i 1 2 x i ε 2 + x i [ n ( x i 1 + x i 2 ; y i 1 + y i 2 ) + n ( x i + x i + 1 2 ; y i + y i + 1 2 ) ] ,
1 ε x i [ S ( i 1 , i ) + S ( i , i + 1 ) ] = n x x ¨ .
x 0 S ( 0 , 1 ) + x N S ( N 1 , N ) = x N x N 1 ε x 1 x 0 ε + ε x 0 n ( x 0 + x 1 2 ; y 0 + y 1 2 ) + ε x N n ( x N 1 + x N 2 ; y N 1 + y N 2 ) ,
x 0 S ( 0 , 1 ) + x N S ( N 1 , N ) x ˙ ( L ) x ˙ ( 0 ) = 0 L d y x ¨ .
0 L d y ( · j + y j y ) = ( o + l ) · 0 L d y ( j ρ ˙ j y ) + 0 L d y ( y j y + ρ ˙ · j y ) i k ( 0 L d y ( j ρ ˙ j y ) ) · ( 0 L d y n ) ,
0 L d y f x = ( 0 L d y δ δ x ( y ) ) ( 0 L d y f ) ,
0 L d y δ δ x ( y ) = lim N i = 0 N x i .
( 0 L d y δ δ x ( y ) ) ( 0 L d y f ) = ( x ( 0 ) + x ( L ) ) ( 0 L d y f ) i k ( 0 L d y f ) ( 0 L d y n x ) .
0 L d y ( y j y + ρ ˙ · j y ) = j y ( ρ l , L ) j y ( ρ o , 0 ) ,
0 L d y ( y j y + ρ ˙ · j y ) = ( j y ( ρ l , L ) j y ( ρ o , 0 ) ) u .
t u + ( o + l ) · f + i ω u = σ ,
f ( ρ l , L , ρ o , 0 , t ) = i k 0 L d y ( j ρ ˙ j y ) ,
ω ( ρ l , L , ρ o , 0 , t ) = k [ j y ( ρ l , L , t ) j y ( ρ o , 0 , t ) ] ,
σ ( ρ l , L , ρ o , 0 , t ) = i k 0 L d y s k 2 ( 0 L d y ( j ρ ˙ j y ) ) · ( 0 L d y n ) ,
j ρ ˙ j y = n ( u ρ ˙ v ) .
t j i + j m i j = g i ,
t f i = i k 0 L d y t ( j i ρ ˙ i j y ) k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y t n ) .
I = i k 0 L d y t ( j i ρ ˙ i j y ) = i k 0 L d y ( j m i j + y m i y ρ ˙ i j m y j ρ ˙ i y m y y ) + i k 0 L d y ( g i ρ ˙ i g y ) .
I = ( 0 L d y δ j ) τ ^ i j ( 1 ) i a i u + i b i + i k 0 L d y ( g i ρ ˙ i g y ρ ¨ i m y y ) ,
τ ^ i j ( 1 ) = i k 0 L d y ( m i j ρ ˙ i m y j ρ ˙ j m y i + ρ ˙ i ρ ˙ j m y y )
a i ( ρ l , L , ρ o , 0 , t ) = k [ m i y ( ρ l , L , t ) m i y ( ρ o , 0 , t ) ]
b i ( ρ l , L , ρ o , 0 , t ) = k [ ρ ˙ i ( L ) m y y ( ρ l , L , t ) ρ ˙ i ( 0 ) m y y ( ρ o , 0 , t ) ]
I = ( o , j + l , j ) τ i j ( 1 ) + i k τ ^ i j ( 1 ) ( 0 L d y j n ) i a i u + i b i + i k 0 L d y ( g i ρ ˙ i g y ρ ¨ i m y y ) ,
II = k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y t n ) = k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y ( j j j + y j y ) ) k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y s ) ,
II = k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y j ( j j ρ ˙ j j y ) ) i ω f i k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y s ) .
II = ( o , j + l , j ) τ i j ( 2 ) + i k τ ^ i j ( 2 ) ( 0 L d y j n ) + h i i ω f i k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y s ) ,
τ ^ i j ( 2 ) = k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y ( j j ρ ˙ j j y ) ) + k 2 2 δ i j ( 0 L d y ( j k ρ ˙ k j y ) ) ( 0 L d y ( j k ρ ˙ k j y ) )
h ( ρ l , L , ρ o , 0 , t ) = k 2 ( 0 L d y ( j ρ ˙ j y ) ) × ( 0 L d y × ( j ρ ˙ j y ) )
t f i + ( o , j + l , j ) τ i j + i ω f i + i a i u = γ i + h i + i b i ,
γ i ( ρ l , L , ρ o , 0 , t ) = i k 0 L d y ( g i ρ ˙ i g y ρ ¨ i m y y ) + i k τ ^ i j ( 0 L d y j n ) k 2 ( 0 L d y ( j i ρ ˙ i j y ) ) ( 0 L d y s ) .
[ t + c · ( o + l ) ] c i = 1 u j τ i j + γ i u + h i u + i ( b i u a i ) ,
E i ( α i , t ) = K exp [ i k ( L + f ) α i 2 2 ] d 2 α o E o ( L α o ) T ( α o , α i , t ) ,
T ( α o , α i , t ) = d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] u ( L α o , 0 , ρ l , L , t ) ,
W ( ρ l ) = 1 for | ρ l | D / 2 , W ( ρ l ) = 0 for | ρ l | > D / 2 .
t T + ( o + i ) · F + i O + B = Σ .
F ( α o , α i , t ) = 1 L d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] f ( L α o , 0 , ρ l , L , t ) ,
O ( α o , α i , t ) = d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] × ω ( L α o , 0 , ρ l , L , t ) u ( L α o , 0 , ρ l , L , t ) ,
B ( α o , α i , t ) = D 2 0 2 π d φ exp [ i π μ 2 ( L α i ρ l ) 2 ] f ( L α o , 0 , ρ l , L , t ) · ρ ^ l ,
Σ ( α o , α i , t ) = d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] σ ( L α o , 0 , ρ l , L , t ) .
I i ( α i , t ) = | K | 2 d 2 α o d 2 α o Γ ( α o , α o ) T ( α o , α i , t ) T * ( α o , α i , t ) ,
Γ ( α o , α o ) = E o ( L α o ) E o * ( L α o ) o .
Γ ( α o , α o ) = λ 2 π L 2 I o ( α o + α o 2 ) δ ( α o α o ) ,
I i ( α i , t ) = d 2 α o I o ( α o ) P ( α o , α i , t ) ,
P ( α o , α i , t ) = λ 2 | K | 2 π L 2 | T ( α o , α i , t ) | 2 .
t P + ( o + i ) · Φ + i O + T * B + T B * = T * Σ + T Σ * + F · ( o + i ) T * + F * · ( o + i ) T ,
Φ ( α o , α i , t ) = T * ( α o , α i , t ) F ( α o , α i , t ) + T ( α o , α i , t ) F * ( α o , α i , t )
O ( α o , α i , t ) = T * ( α o , α i , t ) O ( α o , α i , t ) T ( α o , α i , t ) O * ( α o , α i , t )
( o + i ) T = d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] ( o + L l ) u D L 2 0 2 π d φ exp [ i π μ 2 ( L α i ρ l ) 2 ] u ρ ^ l ,
( x o + x l ) u ( x o , 0 , x l , L , t ) = lim ε 0 u ( x o + ϵ , 0 , x l + ϵ , L , t ) u ( x o , 0 , x l , L , t ) ϵ .
u ( x o + ϵ , 0 , x l + ϵ , L , t ) u ( x o , 0 , x l , L , t ) + i k ϵ 0 L d y x n ,
( x o + x l ) u ( x o , 0 , x l , L , t ) = i k 0 L d y x n .
X ( α o , α i , t ) = i k L d 2 ρ l W ( ρ l ) exp [ i π μ 2 ( L α i ρ l ) 2 ] 0 L d y n .
Z ( α o , α i , t ) = D L 2 0 2 π d φ exp [ i π μ 2 ( L α i ρ l ) 2 ] u ρ ^ l ,
t P + ( o + i ) · Φ + i O + B = S ,
B = T * B + T B * + F · Z * + F * · Z ,
S = T * Σ + T Σ * + F · X * + F * · X .
n ( x , y , t ) = n ( x U t , y ) .
t u + U · ( o + l ) u = 0 ,
t T + U L · ( o + i ) T + U L · β = 0 ,
β = L D 2 0 2 π d φ exp [ i π μ 2 ( L α i ρ l ) 2 ] u ρ ^ l
t P + U L · ( o + i ) P + U L · V = 0 ,
V = T * β + T β * .
t I i + U L · i I i + U L · V = U L · G ,
V ( α i , t ) = d 2 α o I o ( α o ) V ( α o , α i , t ) ,
G ( α i , t ) = d 2 α o P ( α o , α i , t ) o I o ( α o )
t I i + U · i I i = 0 .
U = U L + U L · A ˜ U L · B ˜ ,

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