Abstract

The presence of speckle in radar images reduces the radiometric resolution and renders less efficient the procedures for texture class discrimination. We present an algorithm devoted to speckle reduction in syntheticaperture radar images based on a homomorphic filter coupled with a Wiener filter. To construct the Wiener filter we analytically evaluate the autocorrelation function of the noise, starting from the first two orders of statistics of the noise, before performing the homomorphic transformation (a logarithmic one, in the case of multiplicative noise), and the autocorrelation of the noise-free image is evaluated by an iterative procedure. The algorithm, tested on both simulated and actual synthetic-aperture radar images, provides very promising results and shows the usefulness of the proposed method.

© 1995 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]
  2. L. J. Porcello, N. G. Massey, R. B. Innes, J. M. Marks, “Speckle reduction in synthetic-aperture radars,” J. Opt. Soc. Am. 66, 1305–1311 (1976).
    [CrossRef]
  3. F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
    [CrossRef]
  4. A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
    [CrossRef]
  5. V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
    [CrossRef]
  6. D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
    [CrossRef]
  7. H. H. Arsenault, M. Levesque, “Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise,” Appl. Opt. 23,845–850 (1984).
    [CrossRef] [PubMed]
  8. A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
    [CrossRef]
  9. A. D. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892–1899 (1991).
    [CrossRef]
  10. F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
    [CrossRef]
  11. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, New York, 1990).
  12. J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).
  13. B. Levine, Fondements Theoriques de la Radiotechnique Statistique (Mir, Moscow, 1965).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

1991 (1)

A. D. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892–1899 (1991).
[CrossRef]

1990 (1)

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
[CrossRef]

1986 (1)

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

1985 (1)

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

1984 (1)

1983 (1)

F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
[CrossRef]

1981 (1)

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

1976 (1)

1968 (1)

A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Arsenault, H. H.

Blackledge, J. M.

J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).

Brisco, B.

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

Chavel, P.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

Chin, R. T.

A. D. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892–1899 (1991).
[CrossRef]

Croft, C.

F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
[CrossRef]

Frost, V. S.

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

Held, D. N.

F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
[CrossRef]

Hillery, A. D.

A. D. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892–1899 (1991).
[CrossRef]

Holtzman, J. C.

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Innes, R. B.

Kouyate, F.

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

Kuan, D. T.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

Lee Williams, T. H.

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

Levesque, M.

Levine, B.

B. Levine, Fondements Theoriques de la Radiotechnique Statistique (Mir, Moscow, 1965).

Li, F. K.

F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
[CrossRef]

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, New York, 1990).

Lopes, A.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
[CrossRef]

Marks, J. M.

Massey, N. G.

Nezry, E.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Porcello, L. J.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

Sawchuk, A. A.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

Shafer, R. W.

A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Shanmugam, K. S.

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Smith, S. A.

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Stiles, J. A.

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Stockam, T. G.

A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Strand, T. C.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

Touzi, R.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
[CrossRef]

Ulaby, F. T.

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Geosci. Remote Sensing (3)

F. K. Li, C. Croft, D. N. Held, “Comparison of several techniques to obtain multiple-look SAR imagery,” IEEE Trans. Geosci. Remote Sensing GE-21, 370–375 (1983).
[CrossRef]

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene hetereogeneity,”IEEE Trans. Geosci. Remote Sensing 28, 992–1000 (1990).
[CrossRef]

F. T. Ulaby, F. Kouyate, B. Brisco, T. H. Lee Williams, “Textural information in SAR images,” IEEE Trans. Geosci. Remote Sensing GE-24, 235–245 (1986).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 165–177 (1985).
[CrossRef]

IEEE Trans. Signal Process. (1)

A. D. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892–1899 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

A. V. Oppenheim, R. W. Shafer, T. G. Stockam, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

V. S. Frost, J. A. Stiles, K. S. Shanmugam, J. C. Holtzman, S. A. Smith, “An adaptive filter for smoothing noisy radar images,” Proc. IEEE 69, 133–135 (1981).
[CrossRef]

Other (5)

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, New York, 1990).

J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).

B. Levine, Fondements Theoriques de la Radiotechnique Statistique (Mir, Moscow, 1965).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Canonical representation of homomorphic systems.

Fig. 2
Fig. 2

Theoretical (solid curve) and experimental (dashed curve) pdf’s of noise n′ after the log process.

Fig. 3
Fig. 3

Theoretical (solid curve) and experimental (dashed curve) acf s of noise n′ after the log process.

Fig. 4
Fig. 4

(a) Simulated ideal (noise-free) image, (b) simulated speckled image, (c) one-shot Wiener-filtered image, (d) iterative Wiener-filtered image after 30 iterations.

Fig. 5
Fig. 5

Behavior of the NMSE in the iterative method versus the number of iterations.

Fig. 6
Fig. 6

Behavior of the ENL in the iterative method versus the number of iterations.

Fig. 7
Fig. 7

Gray-level histograms: simulated ideal (noise-free) image shown in Fig. 4(a) (solid curve), simulated speckled image shown in Fig. 4(b) (dashed curve), one-shot Wienerfiltered image shown in Fig. 4(c) (dotted curve), iterative Wiener-filtered image shown in Fig. 4(d) (dashed–dotted curve).

Fig. 8
Fig. 8

(a)–(c): Vertical cut of the ideal noise-free image (solid curve) and of (a) the original speckled signal before processing (dashed curve), (b) the speckled image after ideal one-shot Wiener filtering (dotted curve), (c) the speckled signal after iterative Wiener filtering (dashed–dotted curve). (d) Detail of (c), highlighting the transition between two different reflectivity levels.

Fig. 9
Fig. 9

(a) Actual single-look airborne SAR image relative to the E-SAR mission (X band), (b) iterative Wiener-filtered airborne SAR image relative to the E-SAR mission.

Fig. 10
Fig. 10

(a) Actual single-look airborne SAR image relative to the E-SAR mission (C band), (b) iterative Wiener-filtered airborne SAR image relative to the E-SAR mission.

Fig. 11
Fig. 11

(a) Actual single-look spaceborne SAR image relative to the ERS-1 mission (C band), (b) iterative Wiener-filtered spaceborne SAR image relative to the ERS-1 mission.

Fig. 12
Fig. 12

Logarithm operation performed by block H1.

Equations (27)

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z ( x , r ) = y ( x , r ) n ( x , r ) + n a ( x , r ) ,
f N ( n ) = M M Γ [ M ] n M 1 exp ( M n ) , n 0 ,
f Z | Y ( z | y ) = M M Γ [ M ] y M z M 1 exp ( M z y ) , z , y 0 .
f N ( n ) = exp ( n ) , n 0 ,
f Z | Y ( z | y ) = 1 y exp ( z y ) , z , y 0 .
R n n ( x ¯ , r ¯ ) = E [ n ( x , r ) n ( x + x ¯ , r + r ¯ ) ] = [ 1 + sinc 2 ( x ¯ / ρ x ) sinc 2 ( r ¯ / ρ r ) ] ,
E [ z | y ] = 0 z f Z | Y ( z | y ) d z = y .
H [ z 1 ( ) z 2 ( ) ] = H [ z 1 ( ) ] H [ z 2 ( ) ] , H [ c : z 1 ( ) ] = c ¬ H [ z 1 ( ) ] .
z = log ( z ) = log ( y n ) = log ( y ) + log ( n ) = y + n .
W ( ξ , η ) = S y ( ξ , η ) P * ( ξ , η ) S y ( ξ , η ) | P ( ξ , η ) | 2 + S n ( ξ , η ) ,
s ( x , r ) = y ̂ ( x , r ) = z ( x , r ) w ( x , r ) ,
s ( x , r ) = y ̂ ( x , r ) = exp [ y ̂ ( x , r ) ] ,
f N ( n ) = exp ( n e n ) ,
R n n ( x ¯ , r ¯ ) = ( 1 | μ n | 2 ) k = 1 | μ n | 2 k × [ i = 1 k 1 i γ + log ( 1 | μ n | 2 ) ] 2 ,
R n n ( x ¯ , r ¯ ) = E [ n ] 2 [ 1 + | μ n ( x ¯ , r ¯ ) | 2 ] .
S y ̂ ( 0 ) = S z .
W ( 1 ) = S y ̂ ( 0 ) S y ̂ ( 0 ) + S n = S z S z + S n ,
S y ̂ ( 1 ) = S z | W ( 1 ) | 2 .
W ( k ) = S y ̂ ( k 1 ) S y ̂ ( k 1 ) + S n , S y ̂ ( k ) = S z | W ( k ) | 2
ENL ( p ) = E [ p ] 2 E [ ( p E [ p ] 2 ) ] = E [ p ] 2 Var [ p ] ,
f N ( n ) = exp ( n ) , n 0 .
f N 1 N 2 ( n 1 , n 2 ) = exp [ n 1 + n 2 E [ n ] ( 1 | μ n | 2 ) ] E [ n ] 2 ( 1 | μ n | 2 ) × I 0 [ 2 n 1 n 2 | μ n | E [ n ] ( 1 | μ n | 2 ) ] , n 1 , n 2 0 ,
f N ( n ) = exp ( n e n ) .
E [ n ] = n f N ( n ) d n = 0 log ( n ) exp ( n ) d n = γ ,
R n n ( x ¯ , r ¯ ) = E [ n ( x , r ) n ( x + x ¯ , r + r ¯ ) ] = 0 0 log ( n 1 ) log ( n 2 ) f N 1 N 2 ( n 1 , n 2 ) d n 1 d n 2 = 0 0 log ( n 1 ) log ( n 2 ) 1 E [ n ] 2 ( 1 | μ n | 2 ) × exp [ n 1 + n 2 E [ n ] ( 1 | μ n | 2 ) ] × I 0 [ 2 n 1 n 2 | μ n | E [ n ] ( 1 | μ n | 2 ) ] d n 1 d n 2 .
I 0 ( ξ ) = k = ( ξ / 2 ) 2 k ( k ! ) 2
R n n ( x ¯ , r ¯ ) = 1 a 0 0 log ( n 1 ) log ( n 2 ) exp ( n 1 + n 2 a ) × k = 0 1 ( k ! ) 2 ( n 1 n 2 | μ n | 2 a 2 ) k d n 1 d n 2 = 1 a k = 0 | μ n | 2 k ( k ! ) 2 a 2 k 0 log ( n 1 ) exp ( n 1 a ) n 1 k d n 1 × 0 log ( n 2 ) exp ( n 2 a ) n 2 k d n 2 = a k = 0 | μ n | 2 k [ i = 1 k 1 i γ + log ( a ) ] 2 ,

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