## Construction model for total variation regularization parameter |

Optics Express, Vol. 22, Issue 9, pp. 10500-10508 (2014)

http://dx.doi.org/10.1364/OE.22.010500

Acrobat PDF (841 KB)

### Abstract

Image denoising is important for high-quality imaging in adaptive optics. Richardson-Lucy deconvolution with total variation(TV) regularization is commonly used in image denoising. The selection of TV regularization parameter is an essential issue, yet no systematic approach has been proposed. A construction model for TV regularization parameter is proposed in this paper. It consists of four fundamental elements, the properties of which are analyzed in details. The proposed model bears generality, making it apply to different image recovery scenarios. It can achieve effective spatially adaptive image recovery, which is reflected in both noise suppression and edge preservation. Simulations are provided as validation of recovery and demonstration of convergence speed and relative mean-square error.

© 2014 Optical Society of America

## 1. Introduction

1. A. V. Oppenheim, R. W. Schafer, and T. G. Stockham Jr., “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on , **16**(3), 437–466 (1968). [CrossRef]

2. G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. **13**(7), 547–549 (1988). [CrossRef] [PubMed]

3. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. **62**(1), 55–59 (1972). [CrossRef]

4. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. **79**, 745 (1974). [CrossRef]

5. G. Krishnamurthi, C. Y. Wang, G. Steyer, and D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. **18**(21), 22,324–22,338 (2010). [CrossRef]

6. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**(1), 58–65 (1995). [CrossRef]

7. D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. **36**(8), 1766–1775 (1997). [CrossRef] [PubMed]

8. D. A. Hope and S. M. Jefferies, “Compact multiframe blind deconvolution,” Opt. Lett. **36**(6), 867–869 (2011). [CrossRef] [PubMed]

9. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D **60**(1), 259–268 (1992). [CrossRef]

*et al.*propose the total variation(TV) regularization. Because of the differential properties of gradient function, TV regularization exhibits outstanding edge preservation results [10

10. 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]

11. 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]

*et al.*has proved in [12

12. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory **52**(2), 489–509 (2006). [CrossRef]

13. A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, and L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. **21**(1), 710–723 (2013). [CrossRef]

14. E. Y. Sidky, M. A. Anastasio, and X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. **18**(10), 10,404–10,422 (2010). [CrossRef]

23. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. **53**(17), 4777–4807 (2008). [CrossRef] [PubMed]

## 2. Mathematical description of the model

### 2.1. Introduction of RL deconvolution and TV regularization

**,**

*i***,**

*h***and**

*o***are the observed image, the point spread function, the original image and noise, respectively. The symbol * refers to convolution. In Eq. (1),**

*n***∈**

*x***denotes pixels**

*X***in the image space**

*x***, generally represented as (**

*X**x*,

*y*) in a two-dimensional Cartesian coordinate system. Moreover, the point spread function

**has a normalized imposition, i.e. ∑**

*h*

_{x}**(**

*h***) = 1.**

*x***. Thus, the conditional probability is represented as follows:**

*n**λ*is added and the functional

**(**

*J***) becomes**

*o***and**

*o***and search for zero points respectively, then apply D’Alembert convergence condition**

*h**n*+ 1) and (

*n*) are numbers of iterations, ‖·‖ represents

*ℓ*

_{2}-norm, div(·) refers to divergence,

*o*^{*}and

*h*^{*}are conjugates of

**and**

*o***, respectively, and Eq. (7) is for preservation of the normalization imposition of**

*h***.**

*h*### 2.2. Approximation in weak or moderate noise cases

**is negligible or relatively small, i.e. Eq. (1) is approximate as**

*n***, Eq. (5) is rewritten as where the divergence of**

*h**λ*, which is the only artificially introduced parameter, is of essence in TV regularization. In flat areas, noise contamination is the major problem, and the expression in Eq. (10) is small. Therefore,

*λ*is required to be large to achieve noise suppression. On the other hand, in higher gradient areas, there exists detailed information we want to preserve, and the divergence in Eq. (10) is large. As a result,

*λ*has to be small to preserve the details in higher gradient area, and the TV regularization algorithm basically degenerates to the unregularized RL deconvolution. In this way,

*λ*should be a bounded function with respect to

**, and the ideal selection of**

*x**λ*(

**) will achieve spatially adaptive regularization:**

*x**λ*(

**) is large in flat areas and small in higher gradient areas, and we define it as the**

*x**spatially adaptive property*of

*λ*(

**).**

*x*### 2.3. Proposition of our model and properties of λ(**x**) and fundamental elements

*λ*(

**) as follows: where**

*x**f*is

*constructor function*,

*EI*(

**) is**

*x**edge indicator*,

*λ*

_{0}is

*intensity factor*and

*β*is

*shaping f actor*. The two factors of

*λ*

_{0}and

*β*are positive constants, and they get their names from their vertical and horizontal effects on the graph of

*λ*(

**), respectively.**

*x**f*,

*EI*(

**),**

*x**λ*

_{0}and

*β*are

*fundamental elements*of the regularization parameter construction model, and the combination of the four fundamental elements makes the construction process systematic and convenient.

*λ*(

**) and the four fundamental elements. As discussed before,**

*x**λ*(

**) is a bounded function, therefore, let**

*x**m*be the minimum (or infimum) and let

*M*be the maximum (or supremum). In this way,

*λ*(

**) ∈ (**

*x**m*,

*M*]. The edge indicator

*EI*(

**) :**

*x***→ ℝ**

*X*^{+}∪ {0} is a non-negative function defined in the image space

**. It is used to distinguish the image “edgy degree”. Its value increases with the increase of image “edgy degree”, i.e.**

*X**EI*(

**) is small in flat areas and large in higher gradient areas. In order to adapt all edge indicators**

*x**EI*(

**) satisfying the aforementioned conditions to the proposed model, the domain of**

*x**f*is set as [0, +∞). If we consider the spatially adaptive property of

*λ*(

**) and the property of**

*x**EI*(

**) comprehensively, it is obvious that the constructor function**

*x**f*is a monotonic decreasing function. Its domain is [0, +∞) and its range is (

*m*,

*M*]. Furthermore,

*f*(

*x*) should be convex when

*x*is sufficiently large, since it is a monotonic decreasing function with the domain of [0, +∞) and the bounded range of (

*m*,

*M*].

*λ*

_{0}and the shaping factor

*β*. Since the regularization parameter

*λ*(

**) being too small will lead to insufficient noise suppression and it being too large will cause blur of image detailed information,**

*x**λ*

_{0}being too small and/or

*β*being too large will fail to suffice noise suppression, and

*λ*

_{0}being too large and/or

*β*being too small will blur the image details. As their names suggest,

*λ*

_{0}changes the overall intensity of

*λ*(

**) and**

*x**β*shapes

*λ*(

**) horizontally. Therefore, the values of**

*x**λ*

_{0}and

*β*should be chosen comprehensively.

- Constructor function
*f*(Property 1): The constructor function*f*is a monotonic decreasing function, and it should be convex when its argument is sufficiently large. Its domain is [0, +∞) and it ranges from (*m*,*M*], where*m*is near zero and*M*is an empirical value large enough to suppress noise in flat areas. - Edge indicator
*EI*() (Property 2): The edge indicator*x**EI*() is a non-negative function defined in the image space*x*. It increases with the increase of image “edgy degree”, i.e. it is small in flat areas and large in higher gradient areas.*X* - Intensity factor
*λ*_{0}and shaping factor*β*: The parameters of*λ*_{0}and*β*are positive scalar constants.

## 3. Simulation

*λ*(

**) with the four fundamental elements as in Eq. (11), and simulation results of image recovery with the regularization parameter are provided.**

*x**m*and

*M*as

*m*= 0 and

*M*= 1, and we choose some constructor functions according to Property 1. We choose

*f*in the category of elementary function, and some typical choices are

*f*

_{1}(

*x*) = 1/(1 +

*x*),

*f*

_{2}(

*x*) = 2/

*π*· arc cot(x),

*f*

_{3}(

*x*) = 1/(1 +

*x*

^{2}) and

*f*

_{4}(

*x*) = exp(−

*x*). The typicality of

*f*

_{1}∼

*f*

_{4}lies in the fact that there are six types of fundamental elementary functions, i.e. constants, exponentials, logarithms,

*n*-th roots, trigonometric functions and inverse trigonometric functions, and the choices of

*f*

_{1}∼

*f*

_{4}cover all possible types. Graphs of

*f*

_{1}∼

*f*

_{4}are shown in Fig. 1.

*λ*for comparison with our proposed model, therefore we choose

*f*

_{0}(

*x*) = 1, since

*f*

_{0}indicates a fixed value of

*λ*(

**) =**

*x**λ*

_{0}. The images in the following Fig. 2 and Fig. 4 are all greyscale images in the range of 0 to 255.

### 3.1. Recovery of a real noisy image

*EI*(

**), we choose a very intuitive one, the**

*x**ℓ*

_{2}-norm of gradient (first-order derivative) of the image, i.e.

*λ*

_{0}and

*β*are constants set as

*β*= 0.05 and

*λ*

_{0}= 0.02, respectively. The recovery process goes through 400 iterations. In Fig. 2, the blurred moon image and its five recovered versions corresponding to constructor functions

*f*

_{0}∼

*f*

_{4}are demonstrated. Figure 2(a) is the blurred moon image and Figs. 2(b)–2(f) are images recovered by constructor functions

*f*

_{0}∼

*f*

_{4}, respectively. It is clearly seen that Fig. 2(b) suffers from serious staircase effect, which is typical in fixed TV regularization parameter RL deconvolution. Moreover, the recovery is more and more effective from Fig. 2(c) to Fig. 2(f) and we can see more delicate details in the image.

*f*

_{0}does not converge at all. All constructor functions of

*f*

_{1}∼

*f*

_{4}realize fast convergence, and the iterations converge faster from

*f*

_{1}to

*f*

_{4}: the relative error of successive iterations becomes smaller for the same number of iterations. The convergence results of

*f*

_{4}stand out since the error is the smallest among the four constructor functions and it generally decreases when the number of iterations increases.

### 3.2. Recovery of a simulated noisy image

*D*(

**) from [26**

*x*26. H. Tian, H. Cai, J. Lai, and X. Xu, “Effective image noise removal based on difference eigenvalue,” in *Proceedings of ICIP 2011 International Conference on Image Processing* (Brussels, 2011). [CrossRef]

*EI*(

**). The modification lies in the Hessian matrix of the image, where we add a Gaussian filter**

*x**G*and makes convolution of

_{σ}*G*and each element of the Hessian matrix. (The size of Gaussian filter is 3 × 3 and the deviation

_{σ}*σ*is 0.5.) The functions of

*f*

_{0}∼

*f*

_{4}shown in Fig. 1 are still employed as constructor functions. For the intensity factor and the shaping factor, we maintain

*λ*

_{0}= 0.0105 for all four constructor functions, and set

*β*= 2.0 for

*f*

_{1}∼

*f*

_{3}and

*β*= 0.85 for

*f*

_{4}. The fourth

*β*is smaller because

*f*

_{4}decreases faster with

*x*increasing for the same

*β*as

*f*

_{1}∼

*f*

_{3}. The recovery process goes through 300 iterations. We can see Figs. 4(d)–4(g) exhibit visually the same recovery, which are all better than Fig. 4(c) since Fig. 4(c) suffers from insufficient luminance and obvious noise blur.

**and**

*o***respectively represent the original image and the recovered image. Relative mean-square error results are shown in Fig. 5(b). It is obvious that**

*ô**f*

_{0}fails to converge and the relative mean-square error of

*f*

_{0}is not well suppressed. As we can see, the relative mean-square error results of

*f*

_{1}∼

*f*

_{4}are almost the same, while the convergence results of

*f*

_{1}∼

*f*

_{4}differ. If we take both convergence and relative mean-square error into account,

*f*

_{2}and

*f*

_{3}perform better, which show both high converge speed and low relative mean-square error.

*f*(

*x*) and the edge indicator

*EI*(

**) satisfying Property 1 and Property 2 (including but not limited to examples provided in the simulations) with appropriate values of**

*x**λ*

_{0}and

*β*will achieve proper recovery results.

## 4. Conclusion

*λ*(

**) =**

*x**λ*

_{0}·

*f*(

*β*·

*EI*(

**)). It consists of the constructor function**

*x**f*, the edge indicator

*EI*(

**), the intensity factor**

*x**λ*

_{0}and the shaping factor

*β*as the four fundamental elements. The properties of the fundamental elements are provided in Property 1 and Property 2. Image recovery by means of the construction model becomes spatially adaptive, which is reflected as noise suppression in flat areas and detail preservation in higher gradient areas. Simulation curves of convergence and relative mean-square error illustrate high convergence speed and low recovery error of this model. Moreover, the proposed model bears generality, which offers a systematic way to construct TV regularization parameter, making it applicable in biomedical science, astronautics, remote sensing and other applications.

## Acknowledgments

## References and links

1. | A. V. Oppenheim, R. W. Schafer, and T. G. Stockham Jr., “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on , |

2. | G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. |

3. | W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. |

4. | L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. |

5. | G. Krishnamurthi, C. Y. Wang, G. Steyer, and D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. |

6. | D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A |

7. | D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. |

8. | D. A. Hope and S. M. Jefferies, “Compact multiframe blind deconvolution,” Opt. Lett. |

9. | L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D |

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

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

12. | E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory |

13. | A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, and L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. |

14. | E. Y. Sidky, M. A. Anastasio, and X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. |

15. | A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, and L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. |

16. | J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, and C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. |

17. | A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision |

18. | J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. |

19. | M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in |

20. | D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Expr. |

21. | M. Marim, M. Atlan, E. Angelini, and J. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. |

22. | A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, and M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. |

23. | E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. |

24. | L. M. Mugnier, J.-M. Conan, T. Fusco, and V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in |

25. | N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. |

26. | H. Tian, H. Cai, J. Lai, and X. Xu, “Effective image noise removal based on difference eigenvalue,” in |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(100.2000) Image processing : Digital image processing

(100.2980) Image processing : Image enhancement

(100.1455) Image processing : Blind deconvolution

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: March 20, 2014

Revised Manuscript: April 15, 2014

Manuscript Accepted: April 15, 2014

Published: April 23, 2014

**Citation**

Guanghua Gong, Hongming Zhang, and Minyu Yao, "Construction model for total variation regularization parameter," Opt. Express **22**, 10500-10508 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10500

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### References

- A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Audio and Electroacoustics, IEEE Transactions on, 16(3), 437–466 (1968). [CrossRef]
- G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13(7), 547–549 (1988). [CrossRef] [PubMed]
- W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62(1), 55–59 (1972). [CrossRef]
- L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745 (1974). [CrossRef]
- G. Krishnamurthi, C. Y. Wang, G. Steyer, D. L. Wilson, “Removal of subsurface fluorescence in cryo-imaging using deconvolution,” Opt. Expr. 18(21), 22,324–22,338 (2010). [CrossRef]
- D. A. Fish, A. M. Brinicombe, E. R. Pike, J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995). [CrossRef]
- D. S. C. Biggs, M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. 36(8), 1766–1775 (1997). [CrossRef] [PubMed]
- D. A. Hope, S. M. Jefferies, “Compact multiframe blind deconvolution,” Opt. Lett. 36(6), 867–869 (2011). [CrossRef] [PubMed]
- L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992). [CrossRef]
- E. Vera, P. Meza, 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, H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. 49(19), 3741–3747 (2010). [CrossRef] [PubMed]
- E. Candes, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006). [CrossRef]
- A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, L. J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. 21(1), 710–723 (2013). [CrossRef]
- E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Expr. 18(10), 10,404–10,422 (2010). [CrossRef]
- A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Expr. 21(10), 12,185–12,196 (2013). [CrossRef]
- J. I. Sperl, D. Bequé, G. P. Kudielka, K. Mahdi, P. M. Edic, C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Expr. 22(1), 450–462 (2014). [CrossRef]
- A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20(1–2), 89–97 (2004). [CrossRef]
- J. Dahl, P. C. Hansen, S. H. Jensen, T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. 53(1), 67–92 (2010). [CrossRef]
- M. Cetin, W. Karl, A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, 2002). [CrossRef]
- D. Brady, K. Choi, D. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Expr. 17(15), 13,040–13,049 (2009). [CrossRef]
- M. Marim, M. Atlan, E. Angelini, J. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35(6), 871–873 (2010). [CrossRef] [PubMed]
- A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Expr. 21(3), 3417–3433 (2013). [CrossRef]
- E. Y. Sidky, X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53(17), 4777–4807 (2008). [CrossRef] [PubMed]
- L. M. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence-degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998). [CrossRef]
- N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, P. Roux, J. C. Olivo-Marin, J. Zerubia, “Richardson-Lucy Algorithm with Total Variation Regularization for 3D Confocal Microscope Deconvolution,” Microsc. Res. Tech. 69(4), 260–266 (2006). [CrossRef] [PubMed]
- H. Tian, H. Cai, J. Lai, X. Xu, “Effective image noise removal based on difference eigenvalue,” in Proceedings of ICIP 2011 International Conference on Image Processing (Brussels, 2011). [CrossRef]

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