Abstract

I present a simple theoretical model for dewarped imaging through a turbulent medium and calculate the degree of superresolution that can be attained by dewarping the distorted instantaneous images registered through a turbulent atmosphere. The estimates show that on 1km near the ground propagation path, spatial frequencies of the dewarped image can exceed the diffraction limit three times with a probability up to 10%.

© 2008 Optical Society of America

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  1. M. I. Charnotskii, V. A. Myakinin, and V. U. Zavorotny, “Observation of superresolution in nonisoplanatic imaging through turbulence,” J. Opt. Soc. Am. A 7, 1345-1350 (1990).
    [CrossRef]
  2. M. I. Charnotskii, “Imaging in turbulence beyond diffraction limit,” Proc. SPIE 2534, 289-297 (1995).
    [CrossRef]
  3. 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]
  4. A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
    [CrossRef]
  5. A. J. Lambert and D. Fraser, “Superresolution in imagery arising from observation through anisoplanatic distortion,” Proc. SPIE 5562, 65-75 (2004).
    [CrossRef]
  6. D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.
  7. Z. Zalevsky, S. Rozental, and M. Meller, “Usage of turbulence for superresolved imaging,” Opt. Lett. 32, 1072-1074 (2007).
    [CrossRef] [PubMed]
  8. S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation through Random Media (Springer-Verlag, 1989).
  9. M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492-501 (1993).
    [CrossRef]

2007 (1)

2004 (1)

A. J. Lambert and D. Fraser, “Superresolution in imagery arising from observation through anisoplanatic distortion,” Proc. SPIE 5562, 65-75 (2004).
[CrossRef]

2002 (1)

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

1999 (1)

1995 (1)

M. I. Charnotskii, “Imaging in turbulence beyond diffraction limit,” Proc. SPIE 2534, 289-297 (1995).
[CrossRef]

1993 (1)

1990 (1)

Charnotskii, M. I.

Clyde, D.

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

Fraser, D.

A. J. Lambert and D. Fraser, “Superresolution in imagery arising from observation through anisoplanatic distortion,” Proc. SPIE 5562, 65-75 (2004).
[CrossRef]

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

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]

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

Hunt, B. R.

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

Jahromi, M. R. S.

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation through Random Media (Springer-Verlag, 1989).

Lambert, A.

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

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]

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

Lambert, A. J.

A. J. Lambert and D. Fraser, “Superresolution in imagery arising from observation through anisoplanatic distortion,” Proc. SPIE 5562, 65-75 (2004).
[CrossRef]

Meller, M.

Myakinin, V. A.

Rozental, S.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation through Random Media (Springer-Verlag, 1989).

Tahtali, M.

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation through Random Media (Springer-Verlag, 1989).

Thorpe, G.

Zalevsky, Z.

Zavorotny, V. U.

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Proc. SPIE (3)

A. Lambert, D. Fraser, M. R. S. Jahromi, and B. R. Hunt, “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence,” Proc. SPIE 4792, 35-43 (2002).
[CrossRef]

A. J. Lambert and D. Fraser, “Superresolution in imagery arising from observation through anisoplanatic distortion,” Proc. SPIE 5562, 65-75 (2004).
[CrossRef]

M. I. Charnotskii, “Imaging in turbulence beyond diffraction limit,” Proc. SPIE 2534, 289-297 (1995).
[CrossRef]

Other (2)

S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation through Random Media (Springer-Verlag, 1989).

D. Fraser, A. Lambert, M. R. S. Jahromi, M. Tahtali, and D. Clyde, “Anisoplanatic image restoration at ADFA,” in Proceedings of Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, eds. (CSIRO, 2003), pp. 19-28.

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Figures (4)

Fig. 1
Fig. 1

Problem geometry.

Fig. 2
Fig. 2

Average highest and lowest resolutions.

Fig. 3
Fig. 3

Probability distribution for resolution along the major axis.

Fig. 4
Fig. 4

Probability distribution for resolution along the minor axis.

Tables (1)

Tables Icon

Table 1 Estimates of Parameters γ 2 and s 1 for Horizontal Propagation Paths

Equations (44)

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u ( r , z 1 ) = k 2 π i | z 1 z 0 | exp ( i k | z 1 z 0 | ) d 2 ρ u ( ρ , z 0 ) exp [ i k 2 | z 1 z 0 | ( r ρ ) 2 ] g ( ρ , z 0 , r , z 1 ) ,
u D ( r D ) = k 2 4 π 2 L L 1 exp [ i k ( L + L 1 ) ] d 2 r OB u ( r OB ) d 2 r AP A ( r AP ) exp [ i k 2 L ( r AP r OB ) 2 + i k 2 L 1 ( r D r AP ) 2 i k 2 F r AP 2 ] g ( r AP , L , r OB , 0 ) .
I IM ( R IM ) = k 2 4 π 2 L 2 k 2 2 π 2 L 1 2 d 2 R AP d 2 ρ AP A ( R AP + 1 2 ρ AP ) A ( R AP 1 2 ρ AP ) d 2 R OB O ( R OB ) exp [ i k ( R OB L + R IM L 1 ) · ρ AP ] g ( R OB , L , R AP + 1 2 ρ AP , 0 ) g * ( R OB , L , R AP 1 2 ρ AP , 0 ) ,
I ^ IM ( κ ) 1 4 π 2 d 2 R I IM ( R ) exp ( i κ · R ) = 1 4 π 2 k 2 2 π 2 L 2 d 2 R OB O ( R OB ) d 2 R AP A ( R AP κ L 1 2 k ) A ( R AP + κ L 1 2 k ) × exp ( i L 1 L R OB · κ ) g ( R OB , L , R AP κ L 1 2 k , 0 ) g * ( R OB , L , R AP + κ L 1 2 k , 0 ) .
I IM ( R IM ) = d 2 R OB O ( R OB ) P ( R OB , R IM + R OB L 1 L ) .
P ( R OB , r ) = k 2 4 π 2 L 2 k 2 2 π 2 L 1 2 d 2 R AP d 2 ρ AP A ( R AP + 1 2 ρ AP ) A ( R AP 1 2 ρ AP ) exp ( i k L 1 r · ρ AP ) g ( R OB , L , R AP + 1 2 ρ AP , 0 ) g * ( R OB , L , R AP 1 2 ρ AP , 0 ) .
P LE ( r ) = k 2 4 π 2 L 2 k 2 2 π 2 L 1 2 d 2 R AP d 2 ρ AP A ( R AP + 1 2 ρ AP ) A ( R AP 1 2 ρ AP ) exp ( i k L 1 r · ρ AP ) exp [ 1 2 D ( ρ AP ) ] ,
exp [ 1 2 D ( ρ AP ) ] g ( R OB , L , R AP + 1 2 ρ AP , 0 ) g * ( R OB , L , R AP 1 2 ρ AP , 0 ) .
P ( R OB , r ) = Q ( R OB , r γ ( R OB ) L 1 ) ,
Q ( R OB , r ) = k 2 4 π 2 L 2 k 2 2 π 2 L 1 2 d 2 R AP d 2 ρ AP A ( R AP + 1 2 ρ AP ) A ( R AP 1 2 ρ AP ) exp ( i k γ ( R OB ) · ρ AP i k L 1 r · ρ AP ) g ( R OB , L , R AP + 1 2 ρ AP , 0 ) g * ( R OB , L , R AP 1 2 ρ AP , 0 ) .
g ( R OB , L , R AP , 0 ) = exp ( i k γ ( R OB ) · R AP ) ,
I SE ( R IM ) = d 2 R OB O ( R OB ) Q ( R OB , R IM + R OB L 1 L ) ,
P SE ( r ) = Q ( R OB , r ) = k 2 4 π 2 L 2 k 2 2 π 2 L 1 2 d 2 R AP d 2 ρ AP A ( R AP + 1 2 ρ AP ) A ( R AP 1 2 ρ AP ) exp ( i k L 1 r · ρ AP ) Ψ ( R AP , ρ AP ) ,
I IM ( R IM ) = d 2 R OB O ( R OB ) Q ( R OB , R IM + R OB L 1 L L 1 γ ) .
I IM ( R IM ) O ( R IM L L 1 + L γ ) .
I DW ( R ) = I IM ( R L 1 γ ( R ) ) .
γ ( R 0 + Δ R ) γ ( R 0 ) + γ ( R 0 ) · Δ R = γ i + γ i R j Δ R j ,
I DW ( R 0 + Δ R ) = I IM ( R 0 L 1 γ ( R 0 ) + M · Δ R ) ,
M = E L 1 γ ( R 0 ) = ( 1 γ x x γ x y γ y x 1 γ y y ) ,
I ^ DW ( κ ) 1 4 π 2 d 2 Δ R I DW ( R 0 + Δ R ) exp ( i κ · R ) = 1 4 π 2 d 2 K I ^ IM ( K ) exp [ i K · ( R 0 L 1 γ ( R 0 ) ) ] d 2 Δ R exp [ i ( K · M κ ) · Δ R ] .
K = κ · M 1 = K ( κ ) ,
I ^ DW ( κ ) 1 M exp [ i κ · M 1 ( R 0 L 1 γ ( R 0 ) ) ] I ^ IM ( K ( κ ) ) .
γ x x 2 = γ y y 2 = s 1 , γ x y 2 = γ x x γ y y = s 2 , γ x x γ x y = γ x y γ y y = 0 .
P ( γ x x , γ x y , γ y y ) = 1 2 π 3 s 2 ( s 1 2 s 2 2 ) exp [ s 1 γ x x 2 2 s 2 γ x x γ y y + s 1 γ y y 2 2 ( s 1 2 s 2 2 ) γ x y 2 s 2 ] .
κ ( K ) = K M = ( K x K x γ x x K y γ y x , K y K x γ x y K y γ y y ) .
M = ( 1 + γ x x + γ y y + ( γ x x + γ y y ) 2 + 4 γ x y 2 2 0 0 1 + γ x x + γ y y ( γ x x + γ y y ) 2 + 4 γ x y 2 2 ) .
κ 1 = K DIF ( 1 + γ x x + γ y y + ( γ x x γ y y ) 2 + 4 γ x y 2 2 ) , κ 2 = K DIF ( 1 + γ x x + γ y y ( γ x x γ y y ) 2 + 4 γ x y 2 2 ) .
κ 1 , 2 K DIF = 1 ± 1 2 ( γ x x γ y y ) 2 + 4 γ x y 2 = 1 ± 1 2 d γ x x d γ y y d γ x y ( γ x x γ y y ) 2 + 4 γ x y 2 P ( γ x x , γ y y , γ x y ) = 1 ± 1 π 2 s 2 ( s 1 s 2 ) d y d z y 2 + z 2 exp ( y 2 ( s 1 s 2 ) z 2 2 s 2 ) ,
1 P ( κ 1 N K DIF ) = P ( κ 1 N K DIF ) = Ω ( N ) d γ x x d γ y y d γ x y P ( γ x x , γ y y , γ x y ) ,
Ω ( N ) = { γ x x + γ y y + ( γ x x γ y y ) 2 + 4 γ x y 2 2 ( N 1 ) } .
P ( κ 1 N K DIF ) = 1 g ( Z ) , g ( Z ) 1 2 [ 1 + erf ( 3 2 Z ) ] 1 2 3 erf ( 1 2 Z 2 ) [ 1 + erf ( 1 2 Z ) ] , Z N 1 s 1 .
P ( κ 2 N K DIF ) = 1 g ( Y ) , Y 1 N s 1 .
γ ( R OB ) = k 4 π 0 L d z z d 2 R AP a 2 ( R AP ) d 2 r ε ( z , r + R OB z L + R AP L z L ) sin ( k r 2 2 z ( L z ) ) .
γ ( R OB ) = 1 2 0 L d z ( L z ) d 2 R AP a 2 ( R AP ) ε ( z , R OB z L + R AP L z L ) .
ε ( z , r ) ε ( z , r ) = δ ( z z ) A ( z , r r ) .
ε ( z , r ) ε ( z , r ) = d p d 2 κ Φ ε ( z , p , κ ) exp [ i p ( z z ) + i κ · ( r r ) ] ,
A ( z , r r ) = 2 π d 2 κ Φ ε ( z , 0 , κ ) exp [ i κ · ( r r ) ] .
γ 2 = π 2 0 L d z ( 1 z L ) 2 d 2 κ a ^ 2 ( κ ) κ 2 Φ ε ( z , 0 , κ ) ,
s 1 = π 2 0 L d z ( 1 z L ) 2 z 2 d 2 κ a ^ 2 ( κ ) κ x 4 Φ ε ( z , 0 , κ ) , s 2 = π 2 0 L d z ( 1 z L ) 2 z 2 d 2 κ a ^ 2 ( κ ) κ x 2 κ y 2 Φ ε ( z , 0 , κ ) .
Φ ε ( z , κ ) = 0.033 C ε 2 κ 11 / 3 ,
γ 2 = c γ a 1 / 3 0 L d z ( 1 z L ) 5 / 3 C ε 2 ( z ) ,
s 1 = c s a 7 / 3 0 L d z z 2 ( 1 z L ) 1 / 3 C ε 2 ( z ) ,
c γ = 0.033 π 2 Γ ( 1 6 ) 2 5 / 6 1.017 , c s = 0.033 π 2 3 Γ ( 1 6 ) 2 17 / 6 0.127 .
γ 2 = 0.38 C ε 2 L a 1 / 3 , s 1 = 0.086 a 7 / 3 C ε 2 L 3 .

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