Abstract

A novel Markov-random-field model for speckled synthetic aperture radar (SAR) imagery is derived according to the physical, spatial statistical properties of speckle noise in coherent imaging. A convex Gibbs energy function for speckled images is derived and utilized to perform speckle-compensating image estimation. The image estimation is formed by computing the conditional expectation of the noisy image at each pixel given its neighbors, which is further expressed in terms of the derived Gibbs energy function. The efficacy of the proposed technique, in terms of reducing speckle noise while preserving spatial resolution, is studied by using both real and simulated SAR imagery. Using a number of commonly used metrics, the performance of the proposed technique is shown to surpass that of existing speckle-noise-filtering methods such as the Gamma MAP, the modified Lee, and the enhanced Frost.

© 2006 Optical Society of America

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  2. A. Doerry and F. Dickey, "Synthetic aperture radar," Opt. Photon. News, November 2004, pp. 28-33.
  3. F. T. Ulaby, R. K. Moore, and A. K. Fung, Radar Remote Sensing and Surface Scattering and Emission Theory, Vol. 2 of Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1982).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering of multiplicative noise," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157-165 (1982).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  13. H. Xie, L. Pierce, and F. Ulaby, "SAR speckle reduction using wavelet denoising and Markov random field modeling," IEEE Trans. Geosci. Remote Sens. 40, 2196-2212 (2002).
    [CrossRef]
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  16. O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle modeling and reduction in synthetic aperture radar imagery," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2005), Vol. 3, pp. 317-320.
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    [CrossRef]
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    [CrossRef]
  22. J. Besag, "Spatial interaction and the statistical analysis of lattice systems," J. R. Stat. Soc. Ser. B. Methodol. 6, 192-236 (1974).
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  32. F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
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  34. S. Gupta, R. C. Chauhan, and S. C. Sexana, "Wavelet-based statistical approach for speckle reduction in medical ultrasound images," Med. Biol. Eng. Comput. 42, 189-192 (2004).
    [CrossRef] [PubMed]
  35. L. Gagnon and A. Jouan, "Speckle filtering of SAR images—a comparative study between complex-wavelet-based and standard filters," in Wavelet Applications in Signal and Image Processing V, A.Aldroubi, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 3169, 80-91 (1997).
  36. J. R. Sveinsson and J. A. Benediktsson, "Almost translation invariant wavelet transformations for speckle reduction of SAR images," IEEE Trans. Geosci. Remote Sens. 41, 2404-2408 (2003).
    [CrossRef]
  37. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of state calculations by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
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    [CrossRef]

2005

M. Tawarmalani and N. V. Sahinidis, "A polyhedral branch-and-cut approach to global optimization," Math. Program. 103, 225-249 (2005).
[CrossRef]

2004

S. Gupta, R. C. Chauhan, and S. C. Sexana, "Wavelet-based statistical approach for speckle reduction in medical ultrasound images," Med. Biol. Eng. Comput. 42, 189-192 (2004).
[CrossRef] [PubMed]

2003

J. R. Sveinsson and J. A. Benediktsson, "Almost translation invariant wavelet transformations for speckle reduction of SAR images," IEEE Trans. Geosci. Remote Sens. 41, 2404-2408 (2003).
[CrossRef]

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy tailed modeling," IEEE Trans. Geosci. Remote Sens. 41, 1773-1784 (2003).
[CrossRef]

2002

F. Argenti and L. Alparone, "Speckle removal from SAR images in the undecimated wavelet domain," IEEE Trans. Geosci. Remote Sens. 40, 2363-2374 (2002).
[CrossRef]

H. Xie, L. Pierce, and F. Ulaby, "SAR speckle reduction using wavelet denoising and Markov random field modeling," IEEE Trans. Geosci. Remote Sens. 40, 2196-2212 (2002).
[CrossRef]

R. Touzi, "A review of speckle filtering in the context of estimation theory," IEEE Trans. Geosci. Remote Sens. 40, 2392-2404 (2002).
[CrossRef]

1997

F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
[CrossRef] [PubMed]

1990

S. Quegan, "Interpolation and sampling in SAR images," IEEE Trans. Geosci. Remote Sens. 28, 641-646 (1990).
[CrossRef]

A. Lopes, R. Touzi, and E. Nezry, "Adaptive speckle filters and scene heterogeneity," IEEE Trans. Geosci. Remote Sens. 28, 992-1000 (1990).
[CrossRef]

1986

F. Ulaby, F. Kouyate, B. Brisco, and T. Williams, "Texture information in SAR images," IEEE Trans. Geosci. Remote Sens. 24, 235-245 (1986).
[CrossRef]

1985

T. R. Crimmins, "Geometric filter for speckle reduction," Appl. Opt. 24, 1438-1443 (1985).
[CrossRef] [PubMed]

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

1982

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering of multiplicative noise," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157-165 (1982).
[CrossRef]

1981

R. Keys, "Cubic convolution interpolation for digital image processing," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1153-1160 (1981).
[CrossRef]

J. Lee, "Speckle analysis and smoothing of synthetic aperture radar images," Comput. Graph. Image Process. 17, 24-32 (1981).
[CrossRef]

1974

J. Besag, "Spatial interaction and the statistical analysis of lattice systems," J. R. Stat. Soc. Ser. B. Methodol. 6, 192-236 (1974).

1953

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of state calculations by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Achim, A.

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy tailed modeling," IEEE Trans. Geosci. Remote Sens. 41, 1773-1784 (2003).
[CrossRef]

A. Achim, A. Bezerianos, and P. Tsakalides, "Wavelet-based ultrasound image denoising using an alpha-stable prior probability model," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), Vol. 2, pp. 221-224.

Alparone, L.

F. Argenti and L. Alparone, "Speckle removal from SAR images in the undecimated wavelet domain," IEEE Trans. Geosci. Remote Sens. 40, 2363-2374 (2002).
[CrossRef]

Argenti, F.

F. Argenti and L. Alparone, "Speckle removal from SAR images in the undecimated wavelet domain," IEEE Trans. Geosci. Remote Sens. 40, 2363-2374 (2002).
[CrossRef]

Benediktsson, J. A.

J. R. Sveinsson and J. A. Benediktsson, "Almost translation invariant wavelet transformations for speckle reduction of SAR images," IEEE Trans. Geosci. Remote Sens. 41, 2404-2408 (2003).
[CrossRef]

Besag, J.

J. Besag, "Spatial interaction and the statistical analysis of lattice systems," J. R. Stat. Soc. Ser. B. Methodol. 6, 192-236 (1974).

Bezerianos, A.

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy tailed modeling," IEEE Trans. Geosci. Remote Sens. 41, 1773-1784 (2003).
[CrossRef]

A. Achim, A. Bezerianos, and P. Tsakalides, "Wavelet-based ultrasound image denoising using an alpha-stable prior probability model," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), Vol. 2, pp. 221-224.

Brisco, B.

F. Ulaby, F. Kouyate, B. Brisco, and T. Williams, "Texture information in SAR images," IEEE Trans. Geosci. Remote Sens. 24, 235-245 (1986).
[CrossRef]

Chauhan, R. C.

S. Gupta, R. C. Chauhan, and S. C. Sexana, "Wavelet-based statistical approach for speckle reduction in medical ultrasound images," Med. Biol. Eng. Comput. 42, 189-192 (2004).
[CrossRef] [PubMed]

Chavel, P.

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

Chow, Y. S.

Y. S. Chow and H. Teicher, Probability Theory (Springer-Verlag, 1997).
[CrossRef]

Crimmins, T. R.

Cumming, I.

B. Scheuchl, D. Flett, G. Staples, G. Davidson, and I. Cumming, "Preliminary classification results of simulated RADARSAT-2 polarimetric sea ice data," presented at the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, ESA/ESRIN, Frascati, Italy (January 14-16, 2003).

Dainty, J.

J. Dainty, Topics in Applied Physics: Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Davidson, G.

B. Scheuchl, D. Flett, G. Staples, G. Davidson, and I. Cumming, "Preliminary classification results of simulated RADARSAT-2 polarimetric sea ice data," presented at the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, ESA/ESRIN, Frascati, Italy (January 14-16, 2003).

Dickey, F.

A. Doerry and F. Dickey, "Synthetic aperture radar," Opt. Photon. News, November 2004, pp. 28-33.

Doerry, A.

A. Doerry and F. Dickey, "Synthetic aperture radar," Opt. Photon. News, November 2004, pp. 28-33.

S. Tsunoda, F. Pace, J. Stence, M. Woodring, W. H. Hensley, A. Doerry, and B. Walker, "Lynx: a high-resolution synthetic aperture radar," in Radar Sensor Technology IV, R.Trebits and J.L.Kurtz, eds., Proc. SPIE 3704, 20-27 (1999).

Flett, D.

B. Scheuchl, D. Flett, G. Staples, G. Davidson, and I. Cumming, "Preliminary classification results of simulated RADARSAT-2 polarimetric sea ice data," presented at the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, ESA/ESRIN, Frascati, Italy (January 14-16, 2003).

Floreby, L.

F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
[CrossRef] [PubMed]

Franceschetti, G.

G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, 1999).

Frost, V.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering of multiplicative noise," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157-165 (1982).
[CrossRef]

Fung, A. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Radar Remote Sensing and Surface Scattering and Emission Theory, Vol. 2 of Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1982).

Gagnon, L.

L. Gagnon and A. Jouan, "Speckle filtering of SAR images—a comparative study between complex-wavelet-based and standard filters," in Wavelet Applications in Signal and Image Processing V, A.Aldroubi, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 3169, 80-91 (1997).

Goodman, J.

J. Goodman, Statistical Optics (Wiley-Interscience, 1985).

Gupta, S.

S. Gupta, R. C. Chauhan, and S. C. Sexana, "Wavelet-based statistical approach for speckle reduction in medical ultrasound images," Med. Biol. Eng. Comput. 42, 189-192 (2004).
[CrossRef] [PubMed]

Hayat, M. M.

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle reduction of SAR images using a physically based Markov random field model and simulated annealing," in Algorithms for Synthetic Aperture Radar Imagery XII, F.G.Zelnio and F.D.Garber, eds., Proc. SPIE 5808, 210-221 (2005).

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle modeling and reduction in synthetic aperture radar imagery," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2005), Vol. 3, pp. 317-320.

O. Lankoande, M. M. Hayat, and B. Santhanam, "Segmentation of SAR images based on Markov random field model," in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (IEEE, 2005), pp. 2956-2961.
[CrossRef]

Hensley, W. H.

S. Tsunoda, F. Pace, J. Stence, M. Woodring, W. H. Hensley, A. Doerry, and B. Walker, "Lynx: a high-resolution synthetic aperture radar," in Radar Sensor Technology IV, R.Trebits and J.L.Kurtz, eds., Proc. SPIE 3704, 20-27 (1999).

Holtzman, J.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering of multiplicative noise," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157-165 (1982).
[CrossRef]

Jouan, A.

L. Gagnon and A. Jouan, "Speckle filtering of SAR images—a comparative study between complex-wavelet-based and standard filters," in Wavelet Applications in Signal and Image Processing V, A.Aldroubi, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 3169, 80-91 (1997).

Keys, R.

R. Keys, "Cubic convolution interpolation for digital image processing," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1153-1160 (1981).
[CrossRef]

Kinderman, R.

R. Kinderman and J. Snell, Markov Random Fields and Their Applications (American Mathematical Society,1980).
[CrossRef]

Kouyate, F.

F. Ulaby, F. Kouyate, B. Brisco, and T. Williams, "Texture information in SAR images," IEEE Trans. Geosci. Remote Sens. 24, 235-245 (1986).
[CrossRef]

Kuan, D. T.

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

Lanari, R.

G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing (CRC Press, 1999).

Lankoande, O.

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle modeling and reduction in synthetic aperture radar imagery," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2005), Vol. 3, pp. 317-320.

O. Lankoande, M. M. Hayat, and B. Santhanam, "Segmentation of SAR images based on Markov random field model," in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (IEEE, 2005), pp. 2956-2961.
[CrossRef]

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle reduction of SAR images using a physically based Markov random field model and simulated annealing," in Algorithms for Synthetic Aperture Radar Imagery XII, F.G.Zelnio and F.D.Garber, eds., Proc. SPIE 5808, 210-221 (2005).

Lee, J.

J. Lee, "Speckle analysis and smoothing of synthetic aperture radar images," Comput. Graph. Image Process. 17, 24-32 (1981).
[CrossRef]

Lopes, A.

A. Lopes, R. Touzi, and E. Nezry, "Adaptive speckle filters and scene heterogeneity," IEEE Trans. Geosci. Remote Sens. 28, 992-1000 (1990).
[CrossRef]

Lovstrom, B.

F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
[CrossRef] [PubMed]

Maheshwari, C.

C. Maheshwari, A. Neumaier, and H. Schichl, "Convexity and concavity detection," Tech. Rep., Universitat Wien, A-1010 Vienna, Austria (July 2003), http://www.mat.univie.ac.at/~herman/papers.html.

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of state calculations by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Moore, R. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Radar Remote Sensing and Surface Scattering and Emission Theory, Vol. 2 of Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1982).

Neumaier, A.

C. Maheshwari, A. Neumaier, and H. Schichl, "Convexity and concavity detection," Tech. Rep., Universitat Wien, A-1010 Vienna, Austria (July 2003), http://www.mat.univie.ac.at/~herman/papers.html.

Nezry, E.

A. Lopes, R. Touzi, and E. Nezry, "Adaptive speckle filters and scene heterogeneity," IEEE Trans. Geosci. Remote Sens. 28, 992-1000 (1990).
[CrossRef]

Oliver, C.

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images (SciTech, 2004).

Pace, F.

S. Tsunoda, F. Pace, J. Stence, M. Woodring, W. H. Hensley, A. Doerry, and B. Walker, "Lynx: a high-resolution synthetic aperture radar," in Radar Sensor Technology IV, R.Trebits and J.L.Kurtz, eds., Proc. SPIE 3704, 20-27 (1999).

Pierce, L.

H. Xie, L. Pierce, and F. Ulaby, "SAR speckle reduction using wavelet denoising and Markov random field modeling," IEEE Trans. Geosci. Remote Sens. 40, 2196-2212 (2002).
[CrossRef]

Quegan, S.

S. Quegan, "Interpolation and sampling in SAR images," IEEE Trans. Geosci. Remote Sens. 28, 641-646 (1990).
[CrossRef]

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images (SciTech, 2004).

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of state calculations by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of state calculations by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Sahinidis, N. V.

M. Tawarmalani and N. V. Sahinidis, "A polyhedral branch-and-cut approach to global optimization," Math. Program. 103, 225-249 (2005).
[CrossRef]

Salomonsson, G.

F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
[CrossRef] [PubMed]

Santhanam, B.

O. Lankoande, M. M. Hayat, and B. Santhanam, "Segmentation of SAR images based on Markov random field model," in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (IEEE, 2005), pp. 2956-2961.
[CrossRef]

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle reduction of SAR images using a physically based Markov random field model and simulated annealing," in Algorithms for Synthetic Aperture Radar Imagery XII, F.G.Zelnio and F.D.Garber, eds., Proc. SPIE 5808, 210-221 (2005).

O. Lankoande, M. M. Hayat, and B. Santhanam, "Speckle modeling and reduction in synthetic aperture radar imagery," in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2005), Vol. 3, pp. 317-320.

Sattar, F.

F. Sattar, L. Floreby, G. Salomonsson, and B. Lovstrom, "Image enhancement based on a nonlinear multiscale method," IEEE Trans. Image Process. 6, 888-895 (1997).
[CrossRef] [PubMed]

Sawchuk, A. A.

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

Scheuchl, B.

B. Scheuchl, D. Flett, G. Staples, G. Davidson, and I. Cumming, "Preliminary classification results of simulated RADARSAT-2 polarimetric sea ice data," presented at the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, ESA/ESRIN, Frascati, Italy (January 14-16, 2003).

Schichl, H.

C. Maheshwari, A. Neumaier, and H. Schichl, "Convexity and concavity detection," Tech. Rep., Universitat Wien, A-1010 Vienna, Austria (July 2003), http://www.mat.univie.ac.at/~herman/papers.html.

Sexana, S. C.

S. Gupta, R. C. Chauhan, and S. C. Sexana, "Wavelet-based statistical approach for speckle reduction in medical ultrasound images," Med. Biol. Eng. Comput. 42, 189-192 (2004).
[CrossRef] [PubMed]

Shanmugan, K.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering of multiplicative noise," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157-165 (1982).
[CrossRef]

Snell, J.

R. Kinderman and J. Snell, Markov Random Fields and Their Applications (American Mathematical Society,1980).
[CrossRef]

Staples, G.

B. Scheuchl, D. Flett, G. Staples, G. Davidson, and I. Cumming, "Preliminary classification results of simulated RADARSAT-2 polarimetric sea ice data," presented at the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, ESA/ESRIN, Frascati, Italy (January 14-16, 2003).

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Figures (9)

Fig. 1
Fig. 1

(a) Segment of the graph representing the first-order MRF. (b) Illustration of two types of cliques. (c) Neighborhood of the pixel k according to the first-order MRF. I k i corresponds to the k i th pixel value.

Fig. 2
Fig. 2

Flow chart of the SA method undertaken to simulate speckled images.

Fig. 3
Fig. 3

(a) True scene. (b)–(f) Speckled versions of true scene with (b) T 0 0 and SMSE = 27.79 dB , (c) T 0 = 15 and SMSE = 12.51 dB , (d) T 0 = 50 and SMSE = 8.89 dB , (e) T 0 = 100 and SMSE = 7.96 dB , (f) T 0 = 500 and SMSE = 7.01 dB .

Fig. 4
Fig. 4

(a) Flow chart of the proposed speckle-reduction approach, (b) window W k centered on the k th pixel, (c) corresponding window after subtraction of i k .

Fig. 5
Fig. 5

(a) True scene, (b) speckled version of true scene with T 0 = 15 and SMSE = 12.51 dB , (c) modified Lee filtered version, (d) Gamma filtered version, (e) enhanced Frost filtered version, (f) proposed approach.

Fig. 6
Fig. 6

(a) Image 1. (b)–(e) Processed images using (b) modified Lee, (c) Gamma MAP, (d) enhanced Frost, and (e) proposed approaches.

Fig. 7
Fig. 7

(a) Image 2. (b)–(e) Processed images using (b) modified Lee, (c) Gamma MAP, (d) enhanced Frost, and (e) proposed approaches.

Fig. 8
Fig. 8

(a) Zooming around the “cross” of image 1. (b)–(e) Processed images using (b) modified Lee, (c) Gamma MAP, (d) enhanced Frost, and (e) proposed approaches.

Fig. 9
Fig. 9

Image 1: point-spread-function-like curve before (+) and after filtering using the proposed (▵), enhanced Frost (◻), and modified Lee (엯) approaches.

Tables (4)

Tables Icon

Table 1 Variation of the SMSE Parameter as a Function of T 0

Tables Icon

Table 2 Results of Speckle Reduction Using the Simulated Speckled Image

Tables Icon

Table 3 Speckle Reduction Using Image 1 a

Tables Icon

Table 4 Speckle Reduction Using Image 2 a

Equations (18)

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p I k j I k i ( i k j i k i ) = exp { μ ( r k i k j ) 2 i k i + i k j [ I μ ( r k i k j ) 2 ] O k j } [ 1 μ ( r k i k j ) 2 ] O k j I 0 ( 2 i k i i k j μ ( r k i k j ) [ 1 μ ( r k i k j ) 2 ] O k j ) ,
μ ( r k i k j ) = { α r k i k j [ 0 , 1 ) r k i k j 1 0 otherwise } .
p I k I k 1 , , I k 4 ( i k i k 1 , , i k 4 ) = p I k I k 1 ( i k i k 1 ) p I k I k 2 ( i k i k 2 ) p I k I k 3 ( i k i k 3 ) p I k I k 4 ( i k i k 4 ) [ p I k ( i k ) ] 3 .
p I k I k 1 , , I k 4 ( i k i k 1 , , i k 4 ) = exp { j = 1 4 ln [ B ( i k , i k j ) ] A ( i k , i k j ) B ( i k , i k j ) + ln [ I 0 ( C ( i k , i k j ) B ( i k , i k j ) ) ] 3 ln [ p I k ( i k ) ] } ,
p I k I k 1 , , I k 4 ( i k i k 1 , , i k 4 ) = exp [ U ( i k , i k 1 , , i k 4 ) ] ,
U ( i k , i k 1 , , i k 4 ) = V C 1 ( i k ) + V C 2 ( i k , i k 1 , , i k 4 ) ,
V C 1 ( i k ) = 3 ln [ p I k ( i k ) ] ,
V C 2 ( i k , i k 1 , , i k 4 ) = j = 1 4 { A ( i k , i k j ) B ( i k , i k j ) ln [ I 0 ( C ( i k , i k j ) B ( i k , i k j ) ) ] + ln [ B ( i k , i k j ) ] } .
1 R N < 2 .
MSE = K 1 i = 1 K ( I i I i ̂ ) 2 ,
β = Γ ( I H I H ¯ , I H ̂ I H ̂ ¯ ) Γ ( I H I H ¯ , I H I H ¯ ) Γ ( I H ̂ I H ̂ ¯ , I H ̂ I H ̂ ¯ ) ,
SMSE = 10 log 10 [ j = 1 K I j 2 j = 1 K ( I i I i ̂ ) 2 ] .
ENL = ( K 0 1 i A I i ̂ ) 2 K 0 1 i A I ̂ i 2 ( K 0 1 i A I i ̂ ) 2 ,
O k ̂ = E [ I k I \ { I k } ] ,
O k ̂ = E [ I k N k ] ,
var [ I k ] var [ O k ̂ ] = E [ I k 2 ] E 2 [ I k ] ( E [ E 2 [ I k N k ] ] E 2 [ E [ I k N k ] ] ) = E [ I k 2 ] E 2 [ I k ] ( E [ E 2 [ I k N k ] ] E 2 [ I k ] ) = E [ I k 2 ] E [ E 2 [ I k N k ] ] = E [ E [ I k 2 N k ] ] E [ E 2 [ I k N k ] ] = E [ E [ I k 2 N k ] E 2 [ I k N k ] ] 0 ,
O k ̂ = 0 i k p I k I k 1 , , I k 4 ( i k i k 1 , , i k 4 ) d i k = 0 i k exp [ U ( i k , i k 1 , , i k 4 ) ] d i k ,
O k ̂ = i k = W k 0 W k 8 i k p I k I k 1 , , I k 4 ( i k i k 1 , , i k 4 ) .

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