Abstract

The problem of imaging through turbulent media has been studied frequently in connection with astronomical imaging and airborne radars. Therefore most image restoration methods encountered in the literature assume a stationary object, e.g., a star or a piece of land. In this paper the problem of interferometric measurements of slowly moving or deforming objects in the presence of air disturbances and vibrations is discussed. Measurement noise is reduced by postprocessing the data with a digital noise suppression filter that uses a reference noise signal measured on a small stationary plate inserted in the field of view. The method has proven successful in reducing noise in the vicinity of the reference point where the size of the usable area depends on the degree of spatial correlation in the noise, which in turn depends on the spatial scales present in the air turbulence. Vibrations among the optical components in the setup tend to produce noise that is highly correlated across the field of view and is thus efficiently reduced by the filter.

© 2008 Optical Society of America

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References

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    [CrossRef]
  4. L. M. H. Ullander and H. Hellsten, “Low-frequency ultra-wideband arrayantenna SAR for stationary and moving target imaging,” Proc. SPIE 3704, 35-45 (1999).
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  8. J. J. M Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Engineer. 26, 131-1501997.
    [CrossRef]
  9. J. J. M Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  10. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627-3632 (1991).
  11. S. T. Alexander, Adaptive Signal Processing: Theory and Appplications (Springer-Verlag, 1986).

2001 (1)

1999 (1)

L. M. H. Ullander and H. Hellsten, “Low-frequency ultra-wideband arrayantenna SAR for stationary and moving target imaging,” Proc. SPIE 3704, 35-45 (1999).

1998 (1)

1997 (1)

J. J. M Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Engineer. 26, 131-1501997.
[CrossRef]

1994 (1)

K. F. M. Tateiba, “Restoration of holographic image degraded by atmospheric turbulence and through spatial filtering,” J. Electromagn. Waves Appl. 8, 315-327 (1994).
[CrossRef]

1991 (1)

1982 (1)

1973 (1)

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319-320 (1973).

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

Alexander, S. T.

S. T. Alexander, Adaptive Signal Processing: Theory and Appplications (Springer-Verlag, 1986).

Bates, R. H. T.

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319-320 (1973).

Bone, D. J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gough, P. T.

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319-320 (1973).

Gren, P. S. S.

Hellsten, H.

L. M. H. Ullander and H. Hellsten, “Low-frequency ultra-wideband arrayantenna SAR for stationary and moving target imaging,” Proc. SPIE 3704, 35-45 (1999).

Huntley, J. J. M

J. J. M Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

J. J. M Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Engineer. 26, 131-1501997.
[CrossRef]

Kobayashi, S.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

Li, X.

Takeda, M. H. I.

Tateiba, K. F. M.

K. F. M. Tateiba, “Restoration of holographic image degraded by atmospheric turbulence and through spatial filtering,” J. Electromagn. Waves Appl. 8, 315-327 (1994).
[CrossRef]

Ullander, L. M. H.

L. M. H. Ullander and H. Hellsten, “Low-frequency ultra-wideband arrayantenna SAR for stationary and moving target imaging,” Proc. SPIE 3704, 35-45 (1999).

Appl. Opt. (3)

Astron. Astrophys. (2)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319-320 (1973).

J. Electromagn. Waves Appl. (1)

K. F. M. Tateiba, “Restoration of holographic image degraded by atmospheric turbulence and through spatial filtering,” J. Electromagn. Waves Appl. 8, 315-327 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lasers Engineer. (1)

J. J. M Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Engineer. 26, 131-1501997.
[CrossRef]

Proc. SPIE (1)

L. M. H. Ullander and H. Hellsten, “Low-frequency ultra-wideband arrayantenna SAR for stationary and moving target imaging,” Proc. SPIE 3704, 35-45 (1999).

Other (2)

J. W. Goodman, Statistical Optics (Wiley, 2000).

S. T. Alexander, Adaptive Signal Processing: Theory and Appplications (Springer-Verlag, 1986).

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Figures (11)

Fig. 1
Fig. 1

Schematic of the problem. An interferometric setup measures the complex amplitude of the light from the object in an image plane as a function of time. The measurement is disturbed by a region of turbulent air between the object and imaging optics.

Fig. 2
Fig. 2

Connection between the phase changes at an image point and the refractive index distribution of the turbulent air.

Fig. 3
Fig. 3

Block diagram of the noise reduction filter.

Fig. 4
Fig. 4

Schematic of the experimental setup.

Fig. 5
Fig. 5

Deformation of the object after 10 s .

Fig. 6
Fig. 6

Unfiltered deformation at point A as a function of time (see Fig. 5) together with the true deformation.

Fig. 7
Fig. 7

MSE between the filtered measurement and true deformation as a function of the forgetting factor of the filter with and without prefiltering.

Fig. 8
Fig. 8

MSE between the filtered measurement and true deformation as a function of the impulse response length of the filter.

Fig. 9
Fig. 9

Filtered deformation at A as a function of time together with the true deformation.

Fig. 10
Fig. 10

Filtered deformation at B as a function of time together with the true deformation.

Fig. 11
Fig. 11

Noise energy before and after filtering as a function of the distance from R. Energies are normalized so the maximum values are one. Also shown is the ratio of the noise energy after filtering to that before filtering (also normalized).

Equations (16)

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U tot ( t ) A tot ( t ) · e i · ϕ tot ( t ) = j A j · e i · ϕ j ( t ) .
ϕ j ( t ) = k · path j n ( x , y , z , t ) d s ,
ϕ tot ( t ) = U W { tan 1 [ Im ( U tot ( t ) ) Re ( U tot ( t ) ) ] } g [ ϕ 1 ( t ) , ϕ 2 ( t ) , , ϕ N ( t ) ] .
E { ϕ tot ( t ) } = g ( ϕ 1 , , ϕ N ) · f ϕ ( ϕ 1 , , ϕ N ; t ) d ϕ 1 d ϕ N ,
E { ϕ tot ( t 1 ) · ϕ tot ( t 2 ) } = g ( ϕ 1 1 , , ϕ N 1 ) · g ( ϕ 1 2 , , ϕ N 2 ) · f ϕ ( ϕ 1 1 , , ϕ N 1 , ϕ 1 2 , , ϕ N 2 ; t 1 , t 2 ) d ϕ 1 1 d ϕ N 1 · d ϕ 1 2 d ϕ N 2 .
E { ϕ j ( t ) } = k · path j E { n ( x , y , z , t ) } d s , E { ϕ m ( t 1 ) · ϕ j ( t 2 ) } = k 2 · path m path j E { n ( x 1 , y 1 , z 1 , t 1 ) · n ( x 2 , y 2 , z 2 , t 2 ) } d s 1 d s 2 ( m j ) , E { ϕ j ( t 1 ) · ϕ j ( t 2 ) } = k 2 · path j path j E { n ( x 1 , y 1 , z 1 , t 1 ) · n ( x 2 , y 2 , z 2 , t 2 ) } d s 1 d s 2 .
E { e 2 [ k ] } = E { S 2 [ k τ ] } + E { ( n [ k τ ] y [ k ] ) 2 } + 2 · E { S [ k τ ] · ( n [ k ] τ y [ k ] ) } .
X ¯ L [ k ] = [ x [ k ] , x [ k 1 ] , , x [ k L ] ] T ,
W ¯ L [ k ] = [ W 0 [ k ] , W 1 [ k ] , , W L [ k ] ] .
E { X ¯ L [ k ] · X ¯ L T [ k ] } · W ¯ L * [ k ] R X X [ k ] · W ¯ L * [ k ] = E { d [ k τ ] · X ¯ L [ k ] } P D X [ k ] .
R X X [ k ] i = 1 N λ | k i | · X ¯ L [ i ] · X ¯ L T [ i ] ,
P D X [ k ] i = 1 N λ | k i | · d [ i τ ] · X ¯ L [ i ] .
R X X [ k ] = i = 1 k λ k i · X ¯ L [ i ] · X ¯ L T [ i ] + i = k + 1 N λ i k · X ¯ L [ i ] · X ¯ L T [ i ] = R X X 1 [ k ] + R X X 2 [ k ] ,
P ¯ D X [ k ] = i = 1 k λ k i · d [ i τ ] · X ¯ L [ i ] + i = k + 1 N λ i k · d [ i τ ] · X ¯ L [ i ] = P ¯ D X 1 [ k ] + P ¯ D X 2 [ k ] ,
R X X 1 [ k ] = λ · R X X 1 [ k 1 ] + X ¯ L [ k ] · X ¯ L T [ k ] , R X X 2 [ k ] = 1 λ · R X X 2 [ k 1 ] X ¯ L [ k ] · X ¯ L T [ k ] ,
P ¯ D X 1 [ k ] = λ · P D X 1 [ k 1 ] + d [ k τ ] · X ¯ L [ k ] , P ¯ D X 2 [ k ] = 1 λ · P ¯ D X 2 [ k 1 ] d [ k τ ] · X ¯ L [ k ] .

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