Abstract

Multidetector imaging systems often suffer from the problem of stripe noise and random noise, which greatly degrade the imaging quality. In this paper, we propose a variational destriping method that combines unidirectional total variation and framelet regularization. Total-variation-based regularizations are considered effective in removing different kinds of stripe noise, and framelet regularization can efficiently preserve the detail information. In essence, these two regularizations are complementary to each other. Moreover, the proposed method can also efficiently suppress random noise. The split Bregman iteration method is employed to solve the resulting minimization problem. Comparative results demonstrate that the proposed method significantly outperforms state-of-the-art destriping methods on both qualitative and quantitative assessments.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol.11(1), 7–16 (2011).
    [CrossRef] [PubMed]
  2. A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
    [CrossRef]
  3. A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).
  4. P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
    [CrossRef]
  5. J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001).
    [CrossRef]
  6. J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).
  7. B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express17(10), 8567–8591 (2009).
    [CrossRef] [PubMed]
  8. P. Mather, Computer Processing of Remotely-Sensed Images: An Introduction (Wiley, 2004).
  9. R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
    [CrossRef]
  10. J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).
  11. F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
    [CrossRef]
  12. H. F. Shen and L. P. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Rem. Sens.47(5), 1492–1502 (2009).
    [CrossRef]
  13. H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens.48(4), 1860–1871 (2010).
    [CrossRef]
  14. J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
    [CrossRef] [PubMed]
  15. M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens.49(8), 2924–2935 (2011).
    [CrossRef]
  16. N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
    [CrossRef]
  17. X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
    [CrossRef]
  18. B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
    [CrossRef]
  19. X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).
  20. L. X. Yan, H. Z. Fang, and S. Zhong, “Blind image deconvolution with spatially adaptive total variation regularization,” Opt. Lett.37(14), 2778–2780 (2012).
    [CrossRef] [PubMed]
  21. H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 2: total variation and l1 data fidelity,” Opt. Express18(3), 2894–2912 (2010).
    [CrossRef] [PubMed]
  22. E. Vera, P. Meza, and S. Torres, “Total variation approach for adaptive nonuniformity correction in focal-plane arrays,” Opt. Lett.36(2), 172–174 (2011).
    [CrossRef] [PubMed]
  23. M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt.49(19), 3741–3747 (2010).
    [CrossRef] [PubMed]
  24. J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
    [CrossRef]
  25. J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
    [CrossRef]
  26. J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
    [CrossRef] [PubMed]
  27. J. F. Cai, Framelet toolbox version 2.02, http://www.math.uiowa.edu/ jiancai/code/SplitBreg_Deblur.zip .
  28. T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009).
    [CrossRef]
  29. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
    [CrossRef]
  30. X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process.19(12), 3116–3132 (2010).
    [CrossRef] [PubMed]
  31. D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994).
    [CrossRef]
  32. L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
    [CrossRef] [PubMed]
  33. L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
    [CrossRef] [PubMed]
  34. A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
    [CrossRef]
  35. H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process.20(3), 670–680 (2011).
    [CrossRef] [PubMed]

2013 (1)

X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
[CrossRef]

2012 (4)

J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
[CrossRef] [PubMed]

L. X. Yan, H. Z. Fang, and S. Zhong, “Blind image deconvolution with spatially adaptive total variation regularization,” Opt. Lett.37(14), 2778–2780 (2012).
[CrossRef] [PubMed]

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

2011 (5)

E. Vera, P. Meza, and S. Torres, “Total variation approach for adaptive nonuniformity correction in focal-plane arrays,” Opt. Lett.36(2), 172–174 (2011).
[CrossRef] [PubMed]

H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process.20(3), 670–680 (2011).
[CrossRef] [PubMed]

M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens.49(8), 2924–2935 (2011).
[CrossRef]

N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
[CrossRef]

S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol.11(1), 7–16 (2011).
[CrossRef] [PubMed]

2010 (4)

H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens.48(4), 1860–1871 (2010).
[CrossRef]

X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process.19(12), 3116–3132 (2010).
[CrossRef] [PubMed]

M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt.49(19), 3741–3747 (2010).
[CrossRef] [PubMed]

H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 2: total variation and l1 data fidelity,” Opt. Express18(3), 2894–2912 (2010).
[CrossRef] [PubMed]

2009 (3)

T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009).
[CrossRef]

H. F. Shen and L. P. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Rem. Sens.47(5), 1492–1502 (2009).
[CrossRef]

B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express17(10), 8567–8591 (2009).
[CrossRef] [PubMed]

2008 (2)

J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
[CrossRef]

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

2007 (3)

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
[CrossRef]

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

2004 (1)

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

2003 (3)

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

2001 (2)

J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001).
[CrossRef]

A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
[CrossRef]

2000 (1)

F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
[CrossRef]

1995 (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

1994 (1)

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994).
[CrossRef]

1992 (1)

J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).

1988 (1)

R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
[CrossRef]

Acito, N.

N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
[CrossRef]

Appleby, R.

A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).

Barnes, W.

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

Becker, S.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

Bouali, M.

M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens.49(8), 2924–2935 (2011).
[CrossRef]

Cai, J. F.

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
[CrossRef]

Cannon, M.

R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
[CrossRef]

Carfantan, H.

H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens.48(4), 1860–1871 (2010).
[CrossRef]

Chan, R. H.

J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
[CrossRef]

Chang, C. I.

J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).

Chen, J. S.

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

Chen, S.-W.

S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol.11(1), 7–16 (2011).
[CrossRef] [PubMed]

Clason, C.

Corsini, G.

N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
[CrossRef]

Csillag, F.

F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
[CrossRef]

Datt, B.

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

Diani, M.

N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
[CrossRef]

Dong, B.

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

Donoho, D. L.

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994).
[CrossRef]

Fang, H. Z.

Fehrenbach, J.

J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
[CrossRef] [PubMed]

Freiberger, M.

Gadallah, F. L.

F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
[CrossRef]

Gao, H.

Gasser, P. H.

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

Gauckler, L.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

Goldstein, T.

T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009).
[CrossRef]

Guo, H. D.

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

Harvey, A. R.

A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).

Holzer, L.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

Idier, J.

H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens.48(4), 1860–1871 (2010).
[CrossRef]

Indutnyi, F.

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

Infante, S. O.

J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001).
[CrossRef]

Ji, H.

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994).
[CrossRef]

Jupp, D. L. B.

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

Kaech, A.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

Ladjal, S.

M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens.49(8), 2924–2935 (2011).
[CrossRef]

Lettington, A. H.

A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
[CrossRef]

Liao, H.

H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process.20(3), 670–680 (2011).
[CrossRef] [PubMed]

Liu, C.

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

Liu, X. Q.

X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
[CrossRef]

Lorenzo, C.

J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
[CrossRef] [PubMed]

Marone, F.

McVicar, T. R.

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

Meza, P.

Milanfar, P.

X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process.19(12), 3116–3132 (2010).
[CrossRef] [PubMed]

Muench, B.

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

Münch, B.

B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express17(10), 8567–8591 (2009).
[CrossRef] [PubMed]

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

Ng, M. K.

H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process.20(3), 670–680 (2011).
[CrossRef] [PubMed]

Osher, S.

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009).
[CrossRef]

Pakusch, J.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

Pan, J. J.

J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).

Pearlman, J. S.

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

Pellequer, J. L.

S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol.11(1), 7–16 (2011).
[CrossRef] [PubMed]

Rakwatin, P.

P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
[CrossRef]

Rollason, M. P.

A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
[CrossRef]

Salomonson, V.

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

Scharfetter, H.

Shao, Y.

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

Shen, H. F.

H. F. Shen and L. P. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Rem. Sens.47(5), 1492–1502 (2009).
[CrossRef]

Shen, Z. W.

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
[CrossRef]

Smith, E. J. M.

F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
[CrossRef]

Srinivasan, R.

R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
[CrossRef]

Stampanoni, M.

Sun, J. Q.

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

Takeuchi, W.

P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
[CrossRef]

Torres, J.

J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001).
[CrossRef]

Torres, S.

Trtik, P.

Tzimopoulou, S.

A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
[CrossRef]

Van Niel, T. G.

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

Vera, E.

Wang, W. M.

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

Wang, Y. L.

X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
[CrossRef]

Wegmann, M.

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

Weiss, P.

J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
[CrossRef] [PubMed]

Wepf, R.

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

White, J.

R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
[CrossRef]

Winnefeld, F.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

Xiong, X. X.

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

Yan, L. X.

Yasuoka, Y.

P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
[CrossRef]

Yuan, Y.

X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
[CrossRef]

Zhang, L. P.

H. F. Shen and L. P. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Rem. Sens.47(5), 1492–1502 (2009).
[CrossRef]

Zhao, H. K.

Zhong, S.

Zhu, B. Q.

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

Zhu, X.

X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process.19(12), 3116–3132 (2010).
[CrossRef] [PubMed]

Zingg, A.

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

Appl. Comput. Harmon. Anal. (1)

J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal.24(2), 131–149 (2008).
[CrossRef]

Appl. Opt. (1)

Biometrika (1)

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994).
[CrossRef]

BMC Struct. Biol. (1)

S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol.11(1), 7–16 (2011).
[CrossRef] [PubMed]

Cement Concr. Res. (1)

A. Zingg, L. Holzer, A. Kaech, F. Winnefeld, J. Pakusch, S. Becker, and L. Gauckler, “The microstructure of dispersed and non-dispersed fresh cement pastes-new in-sight by cryo-microscopy,” Cement Concr. Res.38(4), 522–529 (2008).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (9)

P. Rakwatin, W. Takeuchi, and Y. Yasuoka, “Stripe noise reduction in MODIS data by combining histogram matching with facet filter,” IEEE Trans. Geosci. Rem. Sens.45(6), 1844–1856 (2007).
[CrossRef]

J. S. Chen, Y. Shao, H. D. Guo, W. M. Wang, and B. Q. Zhu, “Destriping CMODIS data by power filtering,” IEEE Trans. Geosci. Rem. Sens.41(9), 2119–2124 (2003).

H. F. Shen and L. P. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Rem. Sens.47(5), 1492–1502 (2009).
[CrossRef]

H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens.48(4), 1860–1871 (2010).
[CrossRef]

M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens.49(8), 2924–2935 (2011).
[CrossRef]

N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens.49(4), 1325–1342 (2011).
[CrossRef]

X. Q. Liu, Y. L. Wang, and Y. Yuan, “Grahp-regularized low-rank representation for destriping of hyperspectral imges,” IEEE Trans. Geosci. Rem. Sens.51(7), 4009–4018 (2013).
[CrossRef]

B. Datt, T. R. McVicar, T. G. Van Niel, D. L. B. Jupp, and J. S. Pearlman, “Preprocessing EO-1 Hyperion hyperspectral data to support the application of agricultural indexes,” IEEE Trans. Geosci. Rem. Sens.41(6), 1246–1259 (2003).
[CrossRef]

X. X. Xiong, J. Q. Sun, W. Barnes, and V. Salomonson, “Multiyear on-orbit calibration and performance of Terra MODIS reflective solar bands,” IEEE Trans. Geosci. Rem. Sens.45, 879–889 (2007).

IEEE Trans. Image Process. (4)

J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process.21(10), 4420–4430 (2012).
[CrossRef] [PubMed]

H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process.20(3), 670–680 (2011).
[CrossRef] [PubMed]

X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process.19(12), 3116–3132 (2010).
[CrossRef] [PubMed]

J. F. Cai, H. Ji, C. Liu, and Z. W. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process.21(2), 562–572 (2012).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

Int. J. Remote Sens. (1)

F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens.21(12), 2505–2511 (2000).
[CrossRef]

J. Aeronautical (1)

A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).

J. Am. Math. Soc. (1)

J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc.25(4), 1033–1089 (2012).
[CrossRef]

J. Microsc. (2)

L. Holzer, F. Indutnyi, P. H. Gasser, B. Münch, and M. Wegmann, “Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography,” J. Microsc.216(1), 84–95 (2004).
[CrossRef] [PubMed]

L. Holzer, P. H. Gasser, A. Kaech, M. Wegmann, A. Zingg, R. Wepf, and B. Muench, “Cryo-FIB-nanotomography for quantitative analysis of particle structures in cement suspensions,” J. Microsc.227(3), 216–228 (2007).
[CrossRef] [PubMed]

Opt. Eng. (3)

R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988).
[CrossRef]

J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001).
[CrossRef]

A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng.40(2), 268–274 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Photogramm. Eng. Remote Sensing (1)

J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).

SIAM J. Imag. Sci. (1)

T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009).
[CrossRef]

Other (2)

J. F. Cai, Framelet toolbox version 2.02, http://www.math.uiowa.edu/ jiancai/code/SplitBreg_Deblur.zip .

P. Mather, Computer Processing of Remotely-Sensed Images: An Introduction (Wiley, 2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

(a) Regular stripes in moderate resolution imaging spectroradiometer (MODIS) image. (b) Irregular stripes in focused ion beam nanotomography (FIB-nt).

Fig. 2
Fig. 2

(a) Striped image. (b) Horizontal derivative. (c) Vertical derivative.

Fig. 3
Fig. 3

Simulated experiment with image containing severe stripes. (a) Original image. (b) Simulated striped image with severe additive stripes. Destriping results obtained using (c) wavelet-FFT, (d) Uni-TV, (e) proposed model without framelet regularization (FR) term, and (f) proposed algorithm.

Fig. 4
Fig. 4

Detail regions cropped from Fig. 3.

Fig. 5
Fig. 5

Simulated experiment using image with random noise. (a) Striped image with Gaussian noise. Destriping results obtained using (b) wavelet-FFT, (c) Uni-TV, (d) VSNR, (e) proposed model without framelet regularization (FR) term, (f) Uni-TV + wavelet shrinkage (WS), (g) VSNR + WS, (h) proposed model without framelet regularization (FR) term + WS, and (i) complete proposed method.

Fig. 6
Fig. 6

(a) Original FIB-nt striped image. Destriping results obtained using (b) wavelet-FFT, (c) Huber-Reg, (d) Uni-TV, (e) VSNR, and (f) proposed algorithm.

Fig. 7
Fig. 7

Mean cross-track profiles for images shown in Fig. 6.

Fig. 8
Fig. 8

(a) Original AFM striped image. Destriping results obtained using (b) wavelet-FFT, (c) Huber-Reg, (d) Uni-TV, (e) VSNR, and (f) proposed algorithm.

Fig. 9
Fig. 9

Mean cross-track profiles for images shown in Fig. 8.

Fig. 10
Fig. 10

(a) Original PMMW striped image. Destriping results obtained using (b) wavelet-FFT, (c) Huber-Reg, (d) Uni-TV, (e) VSNR, and (f) proposed algorithm.

Fig. 11
Fig. 11

(a) Original MODIS striped image. Destriping results obtained using (b) wavelet-FFT, (c) Huber-Reg, (d) Uni-TV, (e) VSNR, and (f) proposed algorithm.

Fig. 12
Fig. 12

Evolution of functional energy versus number of iterations for the proposed method.

Fig. 13
Fig. 13

(a) Image with weak stripe noise. (b) Image with severe stripe noise. (c)-(e) Curve of PSNR values versus parameters λ1, λ2, and λ3, respectively.

Tables (2)

Tables Icon

Table 1 PSNR (dB) values for Lena image with severe stripes.

Tables Icon

Table 2 PSNR (dB) and Q-metric values for Lena image with stripes and random noise.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

I s = u + n,
min u Φ(u I s ) + λR(u),
R(u) = R TV (u) + ωR FR (u).
R TV (u)= || y u| | 1 + τ || x (u I s )| | 1 ,
R FR (u)= Wu 1 ,
min u 1 2 u I s 2 2 + λ 1 Wu 1 + λ 2 y u 1 + λ 3 x (u I s ) 1 .
min u 1 2 u I s 2 2 + λ 1 d 1 + λ 2 d y 1 + λ 3 d x 1 s.t. d = Wu, d x = x (u I s ), d y = y u.
min u,d, d x , d y 1 2 u I s 2 2 + λ 1 d 1 + λ 2 d y 1 + λ 3 d x 1 + α 2 dWub 2 2 + β 2 d y y u b y 2 2 + γ 2 d x x (u I s ) b x 2 2 ,
min u 1 2 u I s 2 2 + α 2 d k Wu b k 2 2 + β 2 d y k y u b y k 2 2 + γ 2 d x k x (u I s ) b x k 2 2 ,
(α W T W+β y T y +γ x T x +I) u k+1 =α W T ( d k b k )+β y T ( d y T b y T )+γ x T ( d x k + x I s b x k )+ I s .
min d λ 1 d 1 + α 2 dW u k+1 b k 2 2 .
d k+1 =shrink(W u k+1 + b k , λ 1 α ),
shrink(r,ξ)= r | r | max(rξ,0).
{ d k+1 =shrink(W u k+1 + b k , λ 1 α ), d y k+1 =shrink( y u k+1 + b y k , λ 2 β ), d x k+1 =shrink( x ( u k+1 I s )+ b x k , λ 3 γ ).
{ b k+1 = b k +(W u k+1 d k+1 ), b y k+1 = b y k +( y u k+1 d y k+1 ), b x k+1 = b x k +( x ( u k+1 I s ) d x k+1 ).
PSNR=10 log 10 ( N u I s 2 ),

Metrics