## Robust destriping method with unidirectional total variation and framelet regularization |

Optics Express, Vol. 21, Issue 20, pp. 23307-23323 (2013)

http://dx.doi.org/10.1364/OE.21.023307

Acrobat PDF (9568 KB)

### 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

## 1. Introduction

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]

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]

7. B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express **17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

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]

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]

7. B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express **17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

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]

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]

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]

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]

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]

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]

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]

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]

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]

**I**represents the striped image, and

_{s}**u**is the latent unstriped image. The stripe noise

**n**includes regular stripes [Fig. 1(a)] with strictly horizontal or vertical direction characteristics and irregular stripes [Fig. 1(b)] (for e.g., a stripe that is not perfectly straight, or with an offset value that is not constant over the whole length).

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]

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]

- i) Unidirectional TV and framelet regularization are combined to address the destriping issue. The proposed method can effectively remove regular stripe noise with unidirectional TV regularization and preserve detail information via framelet regularization.
- ii) The proposed method works well not only on regular stripes, but it also has the ability to remove irregular stripes from different kinds of images. Moreover, it can suppress random noise as well.
- iii) The split Bregman algorithm is introduced to optimize the proposed model. The resulting optimization problem is split into several subproblems, which are very easy to implement.

## 2. Formulation and algorithm

### 2.1 Problem formulation

**u**) to remove the stripe noise as well as preserve the edges and detail information:

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]

7. B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express **17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

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]

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]

**W**represents the framelet transform using the filters of the framelet system. This framelet regularization term penalizes the

**u**. In this work, we use the B-splines framelet. Further details on the theory and implementation on the framelet transform can be found in [26

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.

**u**is:To simplify the notations used, we set the regularization parameters as

**u**are that the

### 2.2 Numerical algorithm

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

**u**in (6) into a constrained one by introducing three auxiliary variables

Algorithm Image destriping with sparsity regularizations |
---|

Input {Initialize |

While (do |

Update |

Solve (14) for |

Update |

end While |

Output: Destriping image = |

## 3. Experiments and discussion

**17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

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]

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]

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]

**17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

*N*denotes the total number of image pixels. Moreover, the non-referenced image quality Q-metric is used to evaluate the denoising performance [30

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]

### 3.1 Simulation experiments

31. D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika **81**(3), 425–455 (1994). [CrossRef]

### 3.2 Actual experiments

#### 3.2.1 Elimination of the waterfall effect in focused ion beam nanotomography imaging

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]

**17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

**17**(10), 8567–8591 (2009). [CrossRef] [PubMed]

#### 3.2.2 Removal of non-uniform stripes and random noise in atomic force microscope

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]

**11**(1), 7–16 (2011). [CrossRef] [PubMed]

**11**(1), 7–16 (2011). [CrossRef] [PubMed]

#### 3.2.3 Removal of periodic and severe stripes in passive millimeter-wave images

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]

#### 3.2.4 Removal of incomplete stripes in moderate resolution imaging spectroradiometer

### 3.3 Convergence rate of the algorithm and parameter determination

_{2}and λ

_{3}. We carried our testing using different noise levels for the parameter λ

_{1}, and we empiricallydetermined that the best PSNR values are achieved when λ

_{1}∈ [3, 5

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]

_{1}lies between 3 and 5. Experimental results for other images show that these parameter ranges are universal.

_{1}∈ [3, 5

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]

_{2}∈[0.1, 1], and λ

_{3}∈ [5

**40**(7), 1309–1314 (2001). [CrossRef]

_{2}depends on the degree of the image stripe. Images with severe stripes require a larger value of λ

_{2}. Images with Gaussian noise require a larger value λ

_{1}.

### 3.4. Limitation

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]

## 4. Conclusion

**u**

^{k + 1}is obtained via (10), the three equations given by (14) can be computed parallelly. Subsequently, the three equations given by (15) can be also calculated parallelly, and this ease of the calculation makes the proposed method more preferable for practical application. The qualitative and quantitative assessment results demonstrate that the proposed method consistently outperforms the other methods for all test images.

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]

## Acknowledgment

## References and links

1. | S.-W. Chen and J. L. Pellequer, “DeStripe: frequency-based algorithm for removing stripe noises from AFM images,” BMC Struct. Biol. |

2. | A. H. Lettington, S. Tzimopoulou, and M. P. Rollason, “Nonuniformity correction and restoration of passive millimeter-wave images,” Opt. Eng. |

3. | A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical |

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. |

5. | J. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng. |

6. | J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing |

7. | B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express |

8. | P. Mather, |

9. | R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng. |

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. |

11. | F. L. Gadallah, F. Csillag, and E. J. M. Smith, “Destriping multisensor imagery with moment matching,” Int. J. Remote Sens. |

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. |

13. | H. Carfantan and J. Idier, “Statistical linear destriping of satellite-based pushbroom-type images,” IEEE Trans. Geosci. Rem. Sens. |

14. | J. Fehrenbach, P. Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Trans. Image Process. |

15. | M. Bouali and S. Ladjal, “Toward optimal destriping of MODIS data using a unidirectional variational model,” IEEE Trans. Geosci. Rem. Sens. |

16. | N. Acito, M. Diani, and G. Corsini, “Subspace-based striping noise reduction in hyperspectral images,” IEEE Trans. Geosci. Rem. Sens. |

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. |

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. |

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. |

20. | L. X. Yan, H. Z. Fang, and S. Zhong, “Blind image deconvolution with spatially adaptive total variation regularization,” Opt. Lett. |

21. | H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 2: total variation and l1 data fidelity,” Opt. Express |

22. | E. Vera, P. Meza, and S. Torres, “Total variation approach for adaptive nonuniformity correction in focal-plane arrays,” Opt. Lett. |

23. | M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. |

24. | J. F. Cai, R. H. Chan, and Z. W. Shen, “A framelet-based image inpaiting algorithm,” Appl. Comput. Harmon. Anal. |

25. | J. F. Cai, B. Dong, S. Osher, and Z. W. Shen, “Image restoration: total variation; wavelet frames; and beyond,” J. Am. Math. Soc. |

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. |

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. |

29. | D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory |

30. | X. Zhu and P. Milanfar, “Automatic parameter selection for denoising algorithms using a no-reference measure of image content,” IEEE Trans. Image Process. |

31. | D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika |

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. |

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. |

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. |

35. | H. Liao and M. K. Ng, “Blind deconvolution using generalized cross-validation approach to regularization parameter estimation,” IEEE Trans. Image Process. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.3020) Image processing : Image reconstruction-restoration

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 28, 2013

Revised Manuscript: September 11, 2013

Manuscript Accepted: September 11, 2013

Published: September 24, 2013

**Citation**

Yi Chang, Houzhang Fang, Luxin Yan, and Hai Liu, "Robust destriping method with unidirectional total variation and framelet regularization," Opt. Express **21**, 23307-23323 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23307

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### References

- 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]
- 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]
- A. R. Harvey and R. Appleby, “Passive mm-wave imaging from UAVs using aperture synthesis,” J. Aeronautical107, 87–98 (2003).
- 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. Torres and S. O. Infante, “Wavelet analysis for the elimination of striping noise in satellite images,” Opt. Eng.40(7), 1309–1314 (2001). [CrossRef]
- J. J. Pan and C. I. Chang, “Destriping of Landsat MSS images by filtering techniques,” Photogramm. Eng. Remote Sensing58, 1417–1423 (1992).
- 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]
- P. Mather, Computer Processing of Remotely-Sensed Images: An Introduction (Wiley, 2004).
- R. Srinivasan, M. Cannon, and J. White, “Landsat data destriping using power filtering,” Opt. Eng.27, 939–943 (1988). [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).
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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, Framelet toolbox version 2.02, http://www.math.uiowa.edu/ jiancai/code/SplitBreg_Deblur.zip .
- T. Goldstein and S. Osher, “The split bregman method for L1 regularized problems,” SIAM J. Imag. Sci.2(2), 323–343 (2009). [CrossRef]
- D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995). [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]
- D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika81(3), 425–455 (1994). [CrossRef]
- 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]
- 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]
- 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]

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