Abstract

Multiplicative noise is one common type of noise in imaging science. For coherent image-acquisition systems, such as synthetic aperture radar, the observed images are often contaminated by multiplicative noise. Total variation (TV) regularization has been widely researched for multiplicative noise removal in the literature due to its edge-preserving feature. However, the TV-based solutions sometimes have an undesirable staircase artifact. In this paper, we propose a model to take advantage of the good nature of the TV norm and high-order TV norm to balance the edge and smoothness region. Besides, we adopt a spatially regularization parameter updating scheme. Numerical results illustrate the efficiency of our method in terms of the signal-to-noise ratio and structure similarity index.

© 2013 Optical Society of America

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  1. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. Rev. D 60, 259–268 (1992).
  2. D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
    [CrossRef]
  3. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 89–97 (2004).
    [CrossRef]
  4. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model Simul. 4, 490–530 (2005).
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  5. A. Wong, A. Mishra, K. Bizheva, and D. A. Clausi, “General Bayesian estimation for speckle noise reduction in optical coherence tomography retinal imagery,” Opt. Express 18, 8338–8352 (2010).
    [CrossRef]
  6. C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images (SciTech Publishing, Inc., 2004).
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    [CrossRef]
  8. S. Yun and H. Woo, “A new multiplicative denoising variational model based on m-th root transformation,” IEEE Trans. Image Process. 21, 2523–2533 (2012).
    [CrossRef]
  9. J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).
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    [CrossRef]
  11. S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
    [CrossRef]
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  13. G. Aubert and J. F. Aujol, “A variational approach to remove multiplicative noise,” SIAM J. Appl. Math. 68, 925–946 (2008).
    [CrossRef]
  14. J. Shi and S. Osher, “A nonlinear inverse scale space method for a convex multiplicative noise model,” SIAM J. Imag. Sci. 1, 294–321 (2008).
    [CrossRef]
  15. Y. Huang, M. Ng, and Y. Wen, “A new total variation method for multiplicative noise removal,” SIAM J. Imag. Sci. 2, 20–40 (2009).
    [CrossRef]
  16. J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 1720–1730 (2010).
    [CrossRef]
  17. G. Steidl and T. Teuber, “Removing multiplicative noise by Douglas–Rachford splitting methods,” J. Math. Imag. Vis. 36, 168–184 (2010).
    [CrossRef]
  18. M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems 25, 123006 (2009).
    [CrossRef]
  19. M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” J. Sci. Comput. 19, 95–122 (2003).
    [CrossRef]
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    [CrossRef]
  21. Y. Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale variation models for image restoration,” J. Math. Imaging Vision 40, 82–104 (2011).
    [CrossRef]
  22. G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Variational denoising of partly textured images by spatially varying constraints,” IEEE Trans. Image Process. 15, 2281–2289 (2006).
    [CrossRef]
  23. F. Li, M. Ng, and C. Shen, “Multiplicative noise removal with spatial-varying regularization parameters,” SIAM J. Imag. Sci. 3, 1–20 (2010).
    [CrossRef]
  24. D. Q. Chen and L. Z. Cheng, “Spatially adapted total variation model to remove multiplicative noise,” IEEE Trans. Image Process. 21, 1650–1662 (2012).
    [CrossRef]
  25. T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
    [CrossRef]
  26. M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 121579–1590 (2003).
    [CrossRef]
  27. S. Lefkimmiatis, A. Bourquard, and M. Unser, “Hessian-based norm regularization for image restoration with biomedical applications,” IEEE Trans. Image Process. 21, 983–995 (2012).
    [CrossRef]
  28. H. Z. Chen, J. P. Song, and X. C. Tai, “A dual algorithm for minimization of the LLT model,” Adv. Comput. Math. 31, 115–130 (2009).
    [CrossRef]
  29. F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
    [CrossRef]
  30. M. Lysaker and X. C. Tai, “Iterative image restoration combining total variation minimization and a second-order functional,” Int. J. Comput. Vis. 66, 5–18 (2006).
    [CrossRef]
  31. K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. (2013). doi 10.1007/s10851-013-0445-4.
    [CrossRef]
  32. K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imag. Sci. 3, 492–526 (2010).
    [CrossRef]
  33. C. L. Wu and X. C. Tai, “Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,” SIAM J. Imag. Sci. 3, 300–339 (2010).
    [CrossRef]
  34. E. Esser, “Applications of Lagrangian-based alternating direction methods and connections to split Bregman,” (UCLA, 2009).
  35. T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” , (UCLA, 2012).
  36. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
    [CrossRef]

2012 (3)

D. Q. Chen and L. Z. Cheng, “Spatially adapted total variation model to remove multiplicative noise,” IEEE Trans. Image Process. 21, 1650–1662 (2012).
[CrossRef]

S. Lefkimmiatis, A. Bourquard, and M. Unser, “Hessian-based norm regularization for image restoration with biomedical applications,” IEEE Trans. Image Process. 21, 983–995 (2012).
[CrossRef]

S. Yun and H. Woo, “A new multiplicative denoising variational model based on m-th root transformation,” IEEE Trans. Image Process. 21, 2523–2533 (2012).
[CrossRef]

2011 (1)

Y. Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale variation models for image restoration,” J. Math. Imaging Vision 40, 82–104 (2011).
[CrossRef]

2010 (6)

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 1720–1730 (2010).
[CrossRef]

G. Steidl and T. Teuber, “Removing multiplicative noise by Douglas–Rachford splitting methods,” J. Math. Imag. Vis. 36, 168–184 (2010).
[CrossRef]

A. Wong, A. Mishra, K. Bizheva, and D. A. Clausi, “General Bayesian estimation for speckle noise reduction in optical coherence tomography retinal imagery,” Opt. Express 18, 8338–8352 (2010).
[CrossRef]

K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imag. Sci. 3, 492–526 (2010).
[CrossRef]

C. L. Wu and X. C. Tai, “Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,” SIAM J. Imag. Sci. 3, 300–339 (2010).
[CrossRef]

F. Li, M. Ng, and C. Shen, “Multiplicative noise removal with spatial-varying regularization parameters,” SIAM J. Imag. Sci. 3, 1–20 (2010).
[CrossRef]

2009 (4)

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems 25, 123006 (2009).
[CrossRef]

S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
[CrossRef]

Y. Huang, M. Ng, and Y. Wen, “A new total variation method for multiplicative noise removal,” SIAM J. Imag. Sci. 2, 20–40 (2009).
[CrossRef]

H. Z. Chen, J. P. Song, and X. C. Tai, “A dual algorithm for minimization of the LLT model,” Adv. Comput. Math. 31, 115–130 (2009).
[CrossRef]

2008 (3)

A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” J. Sci. Comput. 34, 209–236 (2008).
[CrossRef]

G. Aubert and J. F. Aujol, “A variational approach to remove multiplicative noise,” SIAM J. Appl. Math. 68, 925–946 (2008).
[CrossRef]

J. Shi and S. Osher, “A nonlinear inverse scale space method for a convex multiplicative noise model,” SIAM J. Imag. Sci. 1, 294–321 (2008).
[CrossRef]

2007 (1)

F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
[CrossRef]

2006 (2)

M. Lysaker and X. C. Tai, “Iterative image restoration combining total variation minimization and a second-order functional,” Int. J. Comput. Vis. 66, 5–18 (2006).
[CrossRef]

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Variational denoising of partly textured images by spatially varying constraints,” IEEE Trans. Image Process. 15, 2281–2289 (2006).
[CrossRef]

2005 (1)

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model Simul. 4, 490–530 (2005).
[CrossRef]

2004 (2)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 89–97 (2004).
[CrossRef]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

2003 (2)

M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 121579–1590 (2003).
[CrossRef]

M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” J. Sci. Comput. 19, 95–122 (2003).
[CrossRef]

2000 (1)

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

1995 (2)

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

P. J. Green, “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination,” Biometrika 82, 711–732 (1995).
[CrossRef]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. Rev. D 60, 259–268 (1992).

1976 (1)

Almansa, A.

A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” J. Sci. Comput. 34, 209–236 (2008).
[CrossRef]

Aubert, G.

G. Aubert and J. F. Aujol, “A variational approach to remove multiplicative noise,” SIAM J. Appl. Math. 68, 925–946 (2008).
[CrossRef]

Aujol, J. F.

G. Aubert and J. F. Aujol, “A variational approach to remove multiplicative noise,” SIAM J. Appl. Math. 68, 925–946 (2008).
[CrossRef]

Ballester, C.

A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” J. Sci. Comput. 34, 209–236 (2008).
[CrossRef]

Bertalmio, M.

M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” J. Sci. Comput. 19, 95–122 (2003).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems 25, 123006 (2009).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 1720–1730 (2010).
[CrossRef]

Bizheva, K.

Boccacci, P.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems 25, 123006 (2009).
[CrossRef]

Bourquard, A.

S. Lefkimmiatis, A. Bourquard, and M. Unser, “Hessian-based norm regularization for image restoration with biomedical applications,” IEEE Trans. Image Process. 21, 983–995 (2012).
[CrossRef]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Bredies, K.

K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imag. Sci. 3, 492–526 (2010).
[CrossRef]

Buades, A.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model Simul. 4, 490–530 (2005).
[CrossRef]

Caselles, V.

A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” J. Sci. Comput. 34, 209–236 (2008).
[CrossRef]

M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” J. Sci. Comput. 19, 95–122 (2003).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 89–97 (2004).
[CrossRef]

Chan, T.

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Chen, D. Q.

D. Q. Chen and L. Z. Cheng, “Spatially adapted total variation model to remove multiplicative noise,” IEEE Trans. Image Process. 21, 1650–1662 (2012).
[CrossRef]

Chen, H. Z.

H. Z. Chen, J. P. Song, and X. C. Tai, “A dual algorithm for minimization of the LLT model,” Adv. Comput. Math. 31, 115–130 (2009).
[CrossRef]

Cheng, L. Z.

D. Q. Chen and L. Z. Cheng, “Spatially adapted total variation model to remove multiplicative noise,” IEEE Trans. Image Process. 21, 1650–1662 (2012).
[CrossRef]

Clausi, D. A.

Coll, B.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model Simul. 4, 490–530 (2005).
[CrossRef]

Desiderà, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems 25, 123006 (2009).
[CrossRef]

Dong, Y. Q.

Y. Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale variation models for image restoration,” J. Math. Imaging Vision 40, 82–104 (2011).
[CrossRef]

Donoho, D.

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

Esser, E.

E. Esser, “Applications of Lagrangian-based alternating direction methods and connections to split Bregman,” (UCLA, 2009).

Fan, J. S.

F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. Rev. D 60, 259–268 (1992).

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 1720–1730 (2010).
[CrossRef]

Gao, G. Q.

S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
[CrossRef]

Gilboa, G.

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Variational denoising of partly textured images by spatially varying constraints,” IEEE Trans. Image Process. 15, 2281–2289 (2006).
[CrossRef]

Goldstein, T.

T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” , (UCLA, 2012).

Goodman, J.

J. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[CrossRef]

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).

Green, P. J.

P. J. Green, “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination,” Biometrika 82, 711–732 (1995).
[CrossRef]

Guo, X. J.

S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
[CrossRef]

Haro, G.

A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” J. Sci. Comput. 34, 209–236 (2008).
[CrossRef]

Hintermüller, M.

Y. Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale variation models for image restoration,” J. Math. Imaging Vision 40, 82–104 (2011).
[CrossRef]

Huang, S. Q.

S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
[CrossRef]

Huang, Y.

Y. Huang, M. Ng, and Y. Wen, “A new total variation method for multiplicative noise removal,” SIAM J. Imag. Sci. 2, 20–40 (2009).
[CrossRef]

Kunisch, K.

K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imag. Sci. 3, 492–526 (2010).
[CrossRef]

Lefkimmiatis, S.

S. Lefkimmiatis, A. Bourquard, and M. Unser, “Hessian-based norm regularization for image restoration with biomedical applications,” IEEE Trans. Image Process. 21, 983–995 (2012).
[CrossRef]

Li, F.

F. Li, M. Ng, and C. Shen, “Multiplicative noise removal with spatial-varying regularization parameters,” SIAM J. Imag. Sci. 3, 1–20 (2010).
[CrossRef]

F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
[CrossRef]

Lions, P.

L. Rudin, P. Lions, and S. Osher, “Multiplicative denoising and deblurring: theory and algorithms,” in Geometric Level Sets in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds. (Springer, 2003), pp. 103–119.

Liu, D. Z.

S. Q. Huang, D. Z. Liu, G. Q. Gao, and X. J. Guo, “A novel method for speckle noise reduction and ship target detection in SAR images,” Pattern Recogn. 42, 1533–1542 (2009).
[CrossRef]

Lundervold, A.

M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 121579–1590 (2003).
[CrossRef]

Lysaker, M.

M. Lysaker and X. C. Tai, “Iterative image restoration combining total variation minimization and a second-order functional,” Int. J. Comput. Vis. 66, 5–18 (2006).
[CrossRef]

M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 121579–1590 (2003).
[CrossRef]

Marquina, A.

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Mishra, A.

Morel, J. M.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model Simul. 4, 490–530 (2005).
[CrossRef]

Mulet, P.

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Ng, M.

F. Li, M. Ng, and C. Shen, “Multiplicative noise removal with spatial-varying regularization parameters,” SIAM J. Imag. Sci. 3, 1–20 (2010).
[CrossRef]

Y. Huang, M. Ng, and Y. Wen, “A new total variation method for multiplicative noise removal,” SIAM J. Imag. Sci. 2, 20–40 (2009).
[CrossRef]

O’Donoghue, B.

T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” , (UCLA, 2012).

Oliver, C.

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

Osher, S.

J. Shi and S. Osher, “A nonlinear inverse scale space method for a convex multiplicative noise model,” SIAM J. Imag. Sci. 1, 294–321 (2008).
[CrossRef]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. Rev. D 60, 259–268 (1992).

L. Rudin, P. Lions, and S. Osher, “Multiplicative denoising and deblurring: theory and algorithms,” in Geometric Level Sets in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds. (Springer, 2003), pp. 103–119.

Papafitsoros, K.

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. (2013). doi 10.1007/s10851-013-0445-4.
[CrossRef]

Pock, T.

K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imag. Sci. 3, 492–526 (2010).
[CrossRef]

Quegan, S.

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

Rincon-Camacho, M. M.

Y. Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale variation models for image restoration,” J. Math. Imaging Vision 40, 82–104 (2011).
[CrossRef]

Rougé, B.

M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” J. Sci. Comput. 19, 95–122 (2003).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. Rev. D 60, 259–268 (1992).

L. Rudin, P. Lions, and S. Osher, “Multiplicative denoising and deblurring: theory and algorithms,” in Geometric Level Sets in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds. (Springer, 2003), pp. 103–119.

Schönlieb, C. B.

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. (2013). doi 10.1007/s10851-013-0445-4.
[CrossRef]

Setzer, S.

T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” , (UCLA, 2012).

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Shen, C.

F. Li, M. Ng, and C. Shen, “Multiplicative noise removal with spatial-varying regularization parameters,” SIAM J. Imag. Sci. 3, 1–20 (2010).
[CrossRef]

Shen, C. L.

F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
[CrossRef]

Shen, C. M.

F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” J. Visual Commun. Image Rep. 18, 322–330 (2007).
[CrossRef]

Shi, J.

J. Shi and S. Osher, “A nonlinear inverse scale space method for a convex multiplicative noise model,” SIAM J. Imag. Sci. 1, 294–321 (2008).
[CrossRef]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Sochen, N.

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Figures (7)

Fig. 1.
Fig. 1.

Original images. (a) Barbara, (b) Elaine, (c) Ground, and (d) Lena.

Fig. 2.
Fig. 2.

Results of the Barbara image with L=25 restored by CC’s algorithm and the proposed SAHTV. (a) Noisy image with L=25. (b) Images denoised by CC’s algorithm. (c) Images denoised by SAHTV. (d) Weight function value α at convergence of SAHTV. (e) Final value of λ in CC’s algorithm. (f) Final value of λ in SAHTV.

Fig. 3.
Fig. 3.

Zoomed parts of the denoised images in Fig. 2 and the corresponding contour plots. (a) Zoomed part of the original Barbara image. (b) Zoomed part by CC’s algorithm. (c) Zoomed part by SAHTV. (d) Contour plot of the original zoomed part. (e) Contour plot of the zoomed part of the restored Barbara by CC’s algorithm. (f) Contour plot of the zoomed part of the restored Barbara by SAHTV.

Fig. 4.
Fig. 4.

Denoised images of Elaine image with L=10 and Ground image with L=5. (a) Restored Elaine by CC’ algorithm. (b) Restored Elaine by SAHTV. (c) Restored Ground by CC’s algorithm. (d) Restored Ground by SAHTV. (e) Final values of λ for restored Elaine in CC’s algorithm. (f) Final values of λ for restored Elaine in SAHTV. (g) Final values of λ for restored Ground in CC’s algorithm. (h) Final values of λ for restored Ground in SAHTV.

Fig. 5.
Fig. 5.

40th and 100th lines of the original, noisy, and restored images of the Lena image with L=10. In the second column, the dashed blue line is the original line, the solid red line is the restored line by our method, and the dashed dot black line is the restored line by CC’s algorithm.

Fig. 6.
Fig. 6.

30th and 200th lines of the original, noisy, and restored images of the Lena image with L=25. In the second column, the dashed blue line is the original line, the solid red line is the restored line by our method, and the dashed dot black line is the restored line by CC’s algorithm.

Fig. 7.
Fig. 7.

Despeckled images of real SAR images by SAHTV. For better visualization, we take a square root on the despeckled images. (a) SAR image (328×328). (b) Despeckled image. (c) SAR image (386×386). (d) Despeckled image. (e) SAR image (358×358). (f) Despeckled image. (g) SAR image (410×410). (h) Despeckled image.

Tables (4)

Tables Icon

Algorithm 3.1 SAHTV-based algorithm for multiplicative noise removal

Tables Icon

Algorithm 3.2 Alternating direction minimization method for solving the subproblem in Algorithm 3.1

Tables Icon

Algorithm 3.3 CC’s algorithm for multiplicative noise removal

Tables Icon

Table 1. Comparison of the Performance of the Proposed Method SAHTV with CC’s Algorithm

Equations (34)

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f=uη,
p(η)=LLηL1Γ(L)eLη,forη0,
minuBV(Ω)λΩ(logu+f/u)dx+TV(u),
TV(u)=sup{Ωudivξdx|ξC0(Ω;R2),ξL(Ω,R2)1}
minzBV(Ω)λΩ(z+fez)dx+TV(z).
minuBV(Ω)λΩ(uflogu)dx+TV(u),
u=argmaxuPU|F(u|f).
u=argmaxuPF|U(f|u)PU(u)PF(f).
PF|U(f|u)=LLfL1uLΓ(L)eLfu.
PU(u)=exp(γ(αϕ(u)+(1α)φ(u))),
log(PU|F(u|f))=log(PF|U(f|u))+log(PU(u))log(PF(f)).
minuΩαϕ(u)+(1α)φ(u)+λ(logu+f/u)dx,
minuΩα|u|+(1α)|2u|+λ(logu+f/u)dx,
ω(x,y)={1|Ωxr|,ifyxr/2,0,else,
F(u)(x)=Ωω(x,y)(f/ulogf/u)(y)dy.
minuS(Ω)Ωα|logu|+(1α)|2logu|dxs.t.F(u)1+εa.e.inΩ,
minzS(Ω)Ωα|z|+(1α)|2z|dxs.t.ζ(z)1+εa.e.inΩ,
ζ(z)(x)=F(ez)(x)=Ωω(x,y)(fez+zlogf)(y)dy.
minzS(Ω)Ωα|z|+(1α)|2z|+Ωλ(fez+z)dx.
λ˜k+1=λ˜k+δmax(ζ(zk)1ϵ,0),
λk+1=Ωω(x,y)λ˜k+1(x)dx,
minzS(Ω)f.*ez+z,λ+i,j=1nαi,j|zi,j|+(1αi,j)|2zi,j|.
|vi,j|=(Dx+vi,j)2+(Dy+vi,j)2,
|2vi,j|=(Dxx+v)i,j2+(Dxy++v)i,j2+(Dyx++v)i,j2+(Dyy+v)i,j2,
minuS(Ω)Ωα|u|+(1α)|2u|+Ωλ(uflogu)dx.
uλ=ezλ.
minuS(Ω)uf.*logu,λ+i,j=1nαi,j|ui,j|+(1αi,j)|2ui,j|.
α=γ+k|Gδ*z0|1+γ+k|Gδ*z0|,
αi,jk={1,if|ui,jk|max(|uk|)c,cis a constant;0.5cos(2π|ui,jk|cmax(|uk|))+0.5,else,
minv,p,qλ,vf.*logv+i,j=1nαi,j|pi,j|+(1αi,j)|qi,j|s.t.u=v,u=p,2u=q,
Lτ=λ,vf.*logv+b0,uv+b1,upi,j=1nαi,j|pi,j|+(1αi,j)|qi,j|+b2,2uq+τ2(uv22+up22+2uq22),
RelErr=uu˜2u2.
SNR=20log10uu¯2u˜u2,
SSIM=(2μuμu˜+C1)(2σuu˜+C2)(μu2+μu˜2+C1)(σu2+σu˜2+C2),

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