Abstract

We present a two-step deconvolution method for restoring images degraded by atmospheric turbulence. The first step is linear space-invariant filtering, and the second step is a nonhomogeneous Markov process. This nonhomogeneous method preserves the discontinuities of the original image better than the homogeneous method does.

© 1996 Optical Society of America

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References

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  1. J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.
  2. R. K. Ward, IEEE Trans. Image Process. 2, 369 (1993).
    [CrossRef]
  3. J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
    [CrossRef]
  4. G. Demoment, IEEE Trans. Acoust. Speech Signal Process. 37, 2024 (1989).
    [CrossRef]
  5. S. Geman, D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6, 721 (1984).
    [CrossRef]
  6. M. Barlaud, L. Blanc-Féraud, Trait. Signal 10, 171 (1986).
  7. H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
    [CrossRef]
  8. J. Besag, J. R. Stat. Soc. B 48, 259 (1986).
  9. T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

1993 (1)

R. K. Ward, IEEE Trans. Image Process. 2, 369 (1993).
[CrossRef]

1991 (1)

H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
[CrossRef]

1990 (1)

J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
[CrossRef]

1989 (1)

G. Demoment, IEEE Trans. Acoust. Speech Signal Process. 37, 2024 (1989).
[CrossRef]

1988 (1)

T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

1986 (2)

M. Barlaud, L. Blanc-Féraud, Trait. Signal 10, 171 (1986).

J. Besag, J. R. Stat. Soc. B 48, 259 (1986).

1984 (1)

S. Geman, D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6, 721 (1984).
[CrossRef]

Barlaud, M.

M. Barlaud, L. Blanc-Féraud, Trait. Signal 10, 171 (1986).

Besag, J.

J. Besag, J. R. Stat. Soc. B 48, 259 (1986).

Blanc-Féraud, L.

M. Barlaud, L. Blanc-Féraud, Trait. Signal 10, 171 (1986).

Carlson, L.

T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

Dainty, J. C.

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.

Delp, E. J.

H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
[CrossRef]

Demoment, G.

G. Demoment, IEEE Trans. Acoust. Speech Signal Process. 37, 2024 (1989).
[CrossRef]

Fienup, J. R.

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
[CrossRef]

Gelfand, S. B.

H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
[CrossRef]

Geman, D.

S. Geman, D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6, 721 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6, 721 (1984).
[CrossRef]

Goldring, T.

T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
[CrossRef]

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
[CrossRef]

Tan, H. L.

H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
[CrossRef]

Ulrich, P.

T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

Ward, R. K.

R. K. Ward, IEEE Trans. Image Process. 2, 369 (1993).
[CrossRef]

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

G. Demoment, IEEE Trans. Acoust. Speech Signal Process. 37, 2024 (1989).
[CrossRef]

IEEE Trans. Image Process. (1)

R. K. Ward, IEEE Trans. Image Process. 2, 369 (1993).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

S. Geman, D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6, 721 (1984).
[CrossRef]

H. L. Tan, S. B. Gelfand, E. J. Delp, IEEE Trans. Pattern Anal. Mach. Intell. 14, 3 (1991).
[CrossRef]

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

J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 9, 1598 (1990).
[CrossRef]

J. R. Stat. Soc. B (1)

J. Besag, J. R. Stat. Soc. B 48, 259 (1986).

Proc. SPIE (1)

T. Goldring, L. Carlson, P. Ulrich, Proc. SPIE 874, 307 (1988).

Trait. Signal (1)

M. Barlaud, L. Blanc-Féraud, Trait. Signal 10, 171 (1986).

Other (1)

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.

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Figures (5)

Fig. 1
Fig. 1

Original scene.

Fig. 2
Fig. 2

Observed image v, SNRv = 30 dB.

Fig. 3
Fig. 3

u e ( 1 ), image filtered by a least-squares filter. This is the initial image of the iterated conditional mode. λ1 = 0.21.

Fig. 4
Fig. 4

Image filtered by a least-squares filter with the noise removed. λ = 5.81.

Fig. 5
Fig. 5

u e ( 2 ), image after the Markovian processing of u e ( 1 ).

Equations (10)

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v = d * u + n ,
h ^ ( i , j ) = d ^ * ( i , j ) d ^ ( i , j ) 2 + λ c ^ ( i , j ) ,
h ^ ( i , j ) = d ^ * ( i , j ) d ^ ( i , j ) 2 + λ 1 c ^ ( i , j ) d ^ ( i , j ) 2 + λ 1 c ^ ( i , j ) d ^ ( i , j ) 2 + λ c ^ ( i , j ) λ 1 < λ ,
u ^ e ( 1 ) ( i , j ) = d ^ * ( i , j ) v ^ ( i , j ) d ^ ( i , j ) 2 + λ 1 c ^ ( i , j ) .
u e ( 2 ) = argmin x , l [ E ( x , l ) ] ,
E ( x , l ) = E 1 ( x ) + δ E 2 ( x , 1 ) + μ E 3 ( l ) ,
E 1 ( x ) = ( i , j ) ( N × N ) u e ( 1 ) ( i , j ) - x ( i , j ) 2 ,
E 2 ( x , l ) = ( i , j ) ( N × N ) [ s , t l ( s , t , i , j ) x ( s , t ) - x ( i , j ) ] 2 ,
E 3 ( l ) = ( i , j ) ( N × N ) C l ( V l i j ) ,
g ^ ( i , j ) = N 11 / 3 ( i 2 + j 2 ) 11 / 6             ( i , j < N / 2 ) .

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