## Depth-variant deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method |

Optics Express, Vol. 21, Issue 23, pp. 27668-27681 (2013)

http://dx.doi.org/10.1364/OE.21.027668

Acrobat PDF (802 KB)

### Abstract

We investigated the deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method and the depth-variant point spread function (DV-PSF). We build the DV-PSF by fitting a parameterized theoretical PSF model to an experimental microbead image. On the basis of the constructed DV-PSF, we restore the 3D widefield microscopy by minimizing an objective function consisting of a negative Poisson likelihood function and a total variation regularization function. In simulations and experiments, the proposed method showed better performance than existing methods.

© 2013 OSA

## 1. Introduction

1. J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

1. J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

2. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag. **23**, 32–45 (2006). [CrossRef]

*true*3D image and the 3D depth-variant point spread function (DV-PSF) [3

3. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **21**, 1593–1601 (2004). [CrossRef]

5. J. G. McNally, C. Preza, J. A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A **11**, 1056–1067 (1994). [CrossRef]

1. J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

12. P. J. Verveer and T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A **14**, 1696–1706 (1997). [CrossRef]

*unknown*as well as

*depth variant*[13

13. S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**, 1601–1613 (1991). [CrossRef]

13. S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**, 1601–1613 (1991). [CrossRef]

7. J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A **16**, 2377–2391 (1999). [CrossRef]

*unknown*, in some methods, one first estimates the PSF by using a microbead image and uses the estimated PSF for deconvolution. Other methods attempt to estimate both the

*unknown*PSF and the 3D image simultaneously by using blind deconvolution methods [7

7. J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A **16**, 2377–2391 (1999). [CrossRef]

13. S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**, 1601–1613 (1991). [CrossRef]

*true*object using an analytically computed [3

3. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **21**, 1593–1601 (2004). [CrossRef]

4. J. Shaevitz and D. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A **24**, 2622–2627 (2007). [CrossRef]

**8**, 1601–1613 (1991). [CrossRef]

4. J. Shaevitz and D. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A **24**, 2622–2627 (2007). [CrossRef]

16. S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl. **25**, 015002 (2009). [CrossRef]

17. R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to
confocal and STED microscopy,” Sci. Rep. **3**, 2523(2013). [CrossRef] [PubMed]

3. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **21**, 1593–1601 (2004). [CrossRef]

8. J. Markham and J. A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **18**, 1062–1071 (2001). [CrossRef]

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

**21**, 1593–1601 (2004). [CrossRef]

2. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag. **23**, 32–45 (2006). [CrossRef]

*a priori*information on the true image to yield a more enhanced image [8

8. J. Markham and J. A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **18**, 1062–1071 (2001). [CrossRef]

10. J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE **2655**, 199–208 (1996) [CrossRef]

*a posteriori*estimation) have been seldom studied except for recent two investigations for confocal microscopy [20, 21]. On the contrary, there exist several investigations on regularized depth-invariant deconvolution methods [6

6. S. Joshi and M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A **10**, 1078–1085 (1993). [CrossRef] [PubMed]

8. J. Markham and J. A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **18**, 1062–1071 (2001). [CrossRef]

10. J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE **2655**, 199–208 (1996) [CrossRef]

12. P. J. Verveer and T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A **14**, 1696–1706 (1997). [CrossRef]

*a priori*information is that of a true object with small intensity variations in the spatial domain, which requires a coupled regularization function defined by intensity differences in adjacent pixels. However, probably for simplicity, existing methods are based either on the intensity of each pixel (not the difference between pixels) or on approximations for computing differences between neighboring pixels [20,22,23]. We suppose that the scarcity of coupled regularization functions can be attributed to the belief that performing an RL-type pixelwise update is impossible for a coupled regularization function [10

10. J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE **2655**, 199–208 (1996) [CrossRef]

6. S. Joshi and M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A **10**, 1078–1085 (1993). [CrossRef] [PubMed]

24. A. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imag. **14**, 132–137 (1995). [CrossRef]

26. J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag. **23**, 1165–1175 (2004). [CrossRef]

## 2. Problem formulation

### 2.1. Image and PSF model

### 2.2. Image restoration

**s**= [

*s*(

**x**

_{0}),

*s*(

**x**

_{1}),...,

*s*(

**x**

_{N}_{−1})] using

**g**= [

*g*(

**x**

_{0}),

*g*(

**x**

_{1}),...,

*g*(

**x**

_{N}_{−1})]. One of the most natural approaches for the restoration entails the determination of the image that maximizes the Poisson likelihood function using a constructed PSF (either depth invariant or depth variant). The maximization using a depth-invariant PSF can be solved by applying an EM algorithm, which leads to an INV-RL iteration [18

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

27. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. **1**, 113–122 (1982). [CrossRef]

**8**, 1601–1613 (1991). [CrossRef]

**21**, 1593–1601 (2004). [CrossRef]

*L*(

**s**) to the minimization problems of the surrogate functions Φ(

**s;s**

*), where*

^{k}**s**

*is the estimated value of*

^{k}**s**at the

*k*-th iteration. It is proved that the sequence of the minimizers of the surrogate functions converges to the minimizer of the original objective function, provided that the surrogate function satisfies the following two conditions [19]: To apply the optimization transfer principle, we first note that the negative log-likelihood function using the DV-PSF is defined as follows: where we denote

*h̃*(

_{o}**x**

*−*

_{i}**x**

*) =*

_{o}*h*(

**x**

*−*

_{i}**x**

_{0};

*z*,

_{o}*θ*) for notional simplicity. To design a surrogate function for the depth-variant Poisson likelihood function given the

*k*-th estimated image

**s**

*= [*

^{k}*s*(

^{k}**x**

_{0}),

*s*(

^{k}**x**

_{1}),...,

*s*(

^{k}**x**

_{N}_{−1})], we note the following identity [19]: Applying the convex inequality after taking the logarithm of Eq. (7) yields

*s*(

**x**

*) yields Equating the derivative with zero yields the following: where the PSF is often normalized such that ∑*

_{o}*=*

_{i}*h̃*(

_{o}**x**

*−*

_{i}**x**

*) = 1. Note that the iteration formula derived in Eq. (11) is the same as the usual RL iteration except that the convolution sum is based on the DV-PSF*

_{o}*h*(

_{o}**x**

*−*

_{i}**x**

*), which varies depending on*

_{o}*z*. Although the DV-RL method yields the MLE of the true image, it usually suffers from a noise amplification problem since the image restoration problem is ill-posed. To remedy the problem, one must incorporate a penalty function that penalizes intensity variations in the spatial domain.

_{o}## 3. Proposed method

### 3.1. Construction of a depth-variant PSF

*true*location of the microbead, the MLE of the unknown parameters are the joint estimation of the parameters and unknown location obtained by maximizing the Poisson likelihood function defined as follows: where It is to be noted that we assume

*s*(

**x**

*) =*

_{o}*δ*(

**x**−

**x**

*) for computing the likelihood function. One can find the optimal parameters and the location of the microbead using a nonlinear optimization method.*

_{o}### 3.2. Depth-variant penalized maximum likelihood estimation

*R*(

**s**) is augmented to the surrogate function in Eq. (9), the next estimate of an image intensity should be determined as follows: where

*γ*is the regularization parameter. Many regularization functions in image restoration can be represented by where

*D*= [

*d*] is a matrix used for determining the difference between adjacent pixels and

_{ij}*ψ*(·) is a potential function that determines the weight of the difference. One of the most well known potential functions is the total variation (TV) function, which is defined as where a small positive number

*ε*is used for making the function differentiable. The smoothed TV function defined in Eq. (16) was used previously and reported to show better performance than the pure TV function for some applications [29

29. J. F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vis. **34**, 307–327 (2009). [CrossRef]

30. S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl. **27**, 095001 (2011). [CrossRef]

**2655**, 199–208 (1996) [CrossRef]

*k*+1)-th estimation. This can be done by designing a separable quadratic surrogate function of the regularization function in Eq. (15) as follows [19, 25

25. J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. **4**, 1417–1429 (1995). [CrossRef] [PubMed]

26. J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag. **23**, 1165–1175 (2004). [CrossRef]

**s**

^{(}

^{k,m}^{)}is the estimated image at the

*m*-th subiteration for determining the (

*k*+ 1)-th estimated image,

*α*= [

*D*

**s**

^{(}

^{k,m}^{)}], and the quadratic function

*č*that guarantees the first inequality. One may determine the optimal curvature

*č*by following the Huber’s method [31]. The last inequality is due to a convex inequality, in which equality holds if

**s**=

**s**

^{(}

^{k,m}^{)}. On the basis of the separable quadratic surrogate function, the surrogate function of the regularization function

*R*(

_{s}**s; s**

^{(}

^{k,m}^{)}) satisfies the following condition [19]: It is to be noted that the surrogate function of the regularization function is additively separable. In addition, it is straightforward to show that the derivative of the surrogate function with respect to

*s*(

**x**

*) can be computed as follows [19]: By using the surrogate function, the minimizer*

_{o}*s*

^{(}

^{k,m}^{+1)}can be determined as follows: Taking the derivative of the objective function in Eq. (20) and equating it to zero yields the following quadratic equation [19]:

## 4. Results

### 4.1. PSF estimation

*μm*-diameter Fluoresbrite Yellow Green Fluorescent Microspheres (Polyscience Inc., PA, USA) and dried the solution onto a cover slip. Then, the sample was imaged using a 40×, 0.95 NA objective lens with an Olympus IX-71 inverted microscope. Using the 65 × 65 × 59 acquired image with spacings of 0.11 × 0.11 × 0.64

*μ*m, we estimated the PSF parameters and the location of the microbead using the method defined in Eq. (12). To estimate the parameters that minimize the objective function defined in Eq. (12), we used the simplex optimization method [28]. Figures 1(a) and 1(b) show the

*x*–

*z*sections of the acquired microbead image and the estimated PSF image (the estimated center location of the microbead on the

*z*axis is 26.9

*μ*m), respectively, whereas Figs. 1(c) and 1(d) show the

*x*–

*y*sections of the acquired bead and the estimated PSF image (a slice at

*z*= 36.7

*μ*m), respectively. The estimated

*n*and

_{i}*n*values are 1.53 and 1.13, respectively. The normalized correlation coefficient (COR) value of the acquired image and the estimated PSF image was 0.903.

_{s}### 4.2. Simulations

*μ*m-diameter microsphere object with its center location on the

*z*axis being at a depth of 10

*μ*m. Then, we convolved the synthetic object using the constructed PSF model under Poisson noise. We generated such synthetic noisy and blurred observed images (65 × 65 × 111, with spacings of 0.5

*μ*m for the

*x*,

*y*, and

*z*axes) for five different signal to noise ratios (SNRs) and attempted to restore the

*true*object using the INV-RL, INV-GEM, DV-RL, and DV-GEM methods. We repeated iterations of the four methods for 200 times, which ensures the relative change of objective function value is less than 1 ×

*e*

^{−5}for all the methods. We also tested the performance of the INV-RL and the DV-RL method with a stopping strategy [32

32. J. Llacer and E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imag. **8**, 186–193 (1989). [CrossRef]

*x*–

*z*profile of the

*true*microsphere-shape synthetic object and that of the noisy blurred image used for simulation. Figures 2(c)–2(h) show the restored images using the INV-RL, INV-RL with stopping strategy, INV-GEM, DV-RL, DV-RL with stopping strategy, and DV-GEM methods. For depth-invariant methods such as INV-RL and INV-GEM, we used the PSF model at

*z*= 10

*μ*m for deconvolution. Figures 3(a) and 3(b) show the

*x*–

*y*section (at

*z*= 10

*μ*m) of the true object and the noisy blurred image. Figures 3(c)–3(h) show the restored images using the INV-RL, INV-RL with stopping strategy, INV-GEM, DV-RL, DV-RL with stopping strategy, and DV-GEM methods, respectively. As one can see in the figures, invariant methods such as INV-RL and INV-GEM were unable to effectively restore the

*x*–

*z*section of the true object. Although the DV-RL method was able to restore the spherical shape in the

*x*–

*z*profile, the image was noisy because the method does not incorporate a regularization function. Although the stopping strategy was able to reduce noise a little bit more than the INV-RL and DV-RL methods without stopping strategy, the resulting images (Figs. 2(d), 2(g), 3(d) and 3(g)) are still noisy. Compared with other methods, the DV-GEM method restored the

*true*object more effectively even in the presence of noise.

*true*object and the restored images for the 10 restorations obtained by using the four methods. As shown in the table, the proposed DV-GEM method outperformed the other methods. Note that the performance of the depth-variant methods was better than that of the depth-invariant methods. In addition, the DV-GEM method outperformed the DV-RL method for all SNR cases. In particular, if the noise is severe, the performance of the DV-GEM method was much better than that of the DV-RL method. When the SNR was very good (e.g., 30 dB case), the performance of the DV-RL was almost the same as that of the DV-GEM method. For the INV-GEM method and the DV-GEM method, we selected the regularization parameter manually in a way such that restored images are smooth enough. One may attempt to automatically tune the regularization parameter using methods such as generalized cross validation [33

33. N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process. **10**, 1299–1308 (2001). [CrossRef]

### 4.3. Experiments

*μ*m-diameter Fluoresbrite Yellow Green Fluorescent Microspheres (Polyscience Inc., PA, USA) under the same condition as in the acquisition of the 0.2-

*μ*m-diameter microbeads for the PSF estimation. Then, we attempted to restore the

*true*object using the four methods. Figure 4(a) shows the

*x*–

*z*section of the acquired image while Figs. 4(b)–4(e) show the results using INV-RL, INV-GEM, DV-RL, and DV-GEM methods, respectively. For the depth-invariant methods such as INV-RL and INV-GEM, we used a PSF at

*z*= 17.97

*μ*m, which is around the center of the slices of the acquired microbead image, for deconvolution. As one can see in Figs. 4(b) and 4(c), the restored images based on the depth-invariant PSF have longer support size than the true image because of the incorrect PSF, especially along the

*z*axis. This result is consistent with the results from the simulations in the preceding subsection and results from previous investigations [34

34. F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express **13**, 10503–10522 (2005). [CrossRef] [PubMed]

*z*axis is closer to the truth. As expected, the DV-RL method generated a noisier image than the DV-GEM method. It is to be noted that unlike in the results from the simulation studies, the restored image along the

*z*axis is not perfectly spherical. We suspect that this is due to the use of a simplified version of the OPD parameterization and that this may be improved by using a more complete OPD model [13

**8**, 1601–1613 (1991). [CrossRef]

*x*–

*y*section of the acquired image at

*z*= 17.97

*μ*m and Figs. 5(b)–5(e) show the results using INV-RL, INV-GEM, DV-RL, and DV-GEM methods, respectively. As one can see in the figures, the TV penalty function was able to reduce noise in the restored images without blurring edges very much.

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods |

2. | P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag. |

3. | C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A |

4. | J. Shaevitz and D. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A |

5. | J. G. McNally, C. Preza, J. A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A |

6. | S. Joshi and M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A |

7. | J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A |

8. | J. Markham and J. A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A |

9. | C. Preza, M. I. Miller, J. Lewis, J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A |

10. | J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE |

11. | P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in |

12. | P. J. Verveer and T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A |

13. | S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A |

14. | F. Aguet, D. Van De Ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in |

15. | S. Ben Hadj, G. Blanc-Feraud, G. Aubert, and Engler, “Blind restoration of confocal microscopy images in presence of a depth-variant blur and poisson noise,” in |

16. | S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl. |

17. | R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to
confocal and STED microscopy,” Sci. Rep. |

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

19. | J. Fessler, “Image reconstruction: Algorithms and analysis,” Online preprint of book in preparation. |

20. | S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in |

21. | S. Ben Hadj and L. Blanc-Feraud, “Modeling and removing depth variant blur in 3D fluorescence microscopy,” in |

22. | N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in |

23. | P. J. Green, “On use of the EM for penalized likelihood estimation,” J. R. Stat. Soc. B |

24. | A. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imag. |

25. | J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. |

26. | J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag. |

27. | L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. |

28. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

29. | J. F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vis. |

30. | S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl. |

31. | P. Huber, |

32. | J. Llacer and E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imag. |

33. | N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process. |

34. | F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express |

**OCIS Codes**

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 6, 2013

Revised Manuscript: October 22, 2013

Manuscript Accepted: October 27, 2013

Published: November 4, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Jeongtae Kim, Suhyeon An, Sohyun Ahn, and Boyoung Kim, "Depth-variant deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method," Opt. Express **21**, 27668-27681 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27668

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### References

- J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999). [CrossRef] [PubMed]
- P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag.23, 32–45 (2006). [CrossRef]
- C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A21, 1593–1601 (2004). [CrossRef]
- J. Shaevitz and D. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A24, 2622–2627 (2007). [CrossRef]
- J. G. McNally, C. Preza, J. A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A11, 1056–1067 (1994). [CrossRef]
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