## Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals |

Optics Express, Vol. 21, Issue 17, pp. 20240-20253 (2013)

http://dx.doi.org/10.1364/OE.21.020240

Acrobat PDF (840 KB)

### Abstract

We investigated the penalized maximum likelihood estimation of lifetime and amplitude images for fluorescence lifetime imaging microscopy. The proposed method penalizes large variations in the lifetimes and amplitudes in the spatial domain to reduces noise in the images, which is a serious problem in the conventional maximum likelihood estimation method. For an effective optimization of the objective function, we applied an optimization transfer method that is based on a separable surrogate function. Simulations show that the proposed method outperforms the conventional MLE method in terms of the estimation accuracy, and the proposed method yielded less noisy images in real experiments.

© 2013 OSA

## 1. Introduction

5. P. J. Verveer, A. Squire, and P. I. Bastiaens, “Global analysis of fluorescence lifetime imaging microscopy data,” Biophys. J. **78**, 2127–2137 (2000). [CrossRef] [PubMed]

13. H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting fret-flim data,” Opt. Express **17**, 6493–6508 (2009). [CrossRef] [PubMed]

5. P. J. Verveer, A. Squire, and P. I. Bastiaens, “Global analysis of fluorescence lifetime imaging microscopy data,” Biophys. J. **78**, 2127–2137 (2000). [CrossRef] [PubMed]

13. H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting fret-flim data,” Opt. Express **17**, 6493–6508 (2009). [CrossRef] [PubMed]

5. P. J. Verveer, A. Squire, and P. I. Bastiaens, “Global analysis of fluorescence lifetime imaging microscopy data,” Biophys. J. **78**, 2127–2137 (2000). [CrossRef] [PubMed]

13. H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting fret-flim data,” Opt. Express **17**, 6493–6508 (2009). [CrossRef] [PubMed]

14. S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J. **87**, 2807–2817 (2004). [CrossRef] [PubMed]

3. W. Becker, “Fluorescence lifetime imaging techniques and applications,” J. Microsc. **247**, 119–136 (2012). [CrossRef] [PubMed]

15. K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum ph gradient,” Biophys. J. **83**, 1682–1690 (2002). [CrossRef] [PubMed]

2. C. W. Chang and M.-A. Mycek, “Enhancing precision in time-domain fluorescence lifetime imaging,” J. Biomed. Opt. **15**, 056013 (2010). [CrossRef] [PubMed]

16. A. Squire and P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. **193**, 36–49 (1999). [CrossRef] [PubMed]

17. D. Sud and M.-A. Mycek, “Image restoration for fluorescence lifetime imaging microscopy (FLIM),” Opt. Express **16**, 19192–19200 (2008). [CrossRef]

16. A. Squire and P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. **193**, 36–49 (1999). [CrossRef] [PubMed]

2. C. W. Chang and M.-A. Mycek, “Enhancing precision in time-domain fluorescence lifetime imaging,” J. Biomed. Opt. **15**, 056013 (2010). [CrossRef] [PubMed]

17. D. Sud and M.-A. Mycek, “Image restoration for fluorescence lifetime imaging microscopy (FLIM),” Opt. Express **16**, 19192–19200 (2008). [CrossRef]

18. M. Heilemann, D. P. Herten, R. Heintzmann, C. Cremer, C. Mller, P. Tinnefeld, K. D. Weston, J. Wolfrum, and M. Sauer, “High-resolution colocalization of single dye molecules by fluorescence lifetime imaging microscopy,” Anal. Chem. **74**, 3511–3517 (2002). [CrossRef] [PubMed]

3. W. Becker, “Fluorescence lifetime imaging techniques and applications,” J. Microsc. **247**, 119–136 (2012). [CrossRef] [PubMed]

19. B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem. **84**, 4725–4731 (2012). [CrossRef] [PubMed]

20. E. Gratton, S. Breusegem, J. Sutin, Q. Ruan, and N. Barry, “Fluorescence lifetime imaging for the two-photon microscope: time-domain and frequency-domain methods,” J. Biomed. Opt. **8**, 381–390 (2003). [CrossRef] [PubMed]

21. 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]

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]

21. 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]

21. 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]

## 2. Theory

### 2.1. Problem formulation

*x*,

_{i}*y*) at a time instance

_{j}*t*by the Poisson random variable,

_{k}*g*(

*t*,

_{k}*x*,

_{i}*y*), as follows [1

_{j}1. J. R. Lakowicz, *Principles of Fluorescence Spectroscopy* (Kluwer Academic/Plenum, 1999). [CrossRef]

*h*(

*t*), and

*f*(

*t*) is the multi-exponential decay function at (

*x*,

_{i}*y*) defined by where

_{j}*τ*(

_{p}*x*,

_{i}*y*) is the

_{j}*p*-th decay constant at (

*x*,

_{i}*y*), and

_{j}*A*(

_{p}*x*,

_{i}*y*) is the associated amplitude. We define the parameter vector at (

_{j}*x*,

_{i}*y*) as Using the parameter vector defined in Eq. (4), the

_{j}*p*-th decay time constant can be represented by We use this parameterization to determine each decay time constant uniquely. The goal of FLIM imaging is to obtain an accurate estimation of the decay and amplitude constants

*θ*at every location using the collected number of detected photon counts for an

_{i,j}*M*×

*N*image,

**g**= {

**g**

*|*

_{i,j}*i*= 0,,

*N*− 1,

*j*= 1,,

*M*− 1}, where

**g**

*= [*

_{i,j}*g*(

*t*

_{0},

*x*,

_{i}*y*),.,

_{j}*g*(

*t*

_{K−1},

*x*,

_{i}*y*)]. Because it is not possible to determine the lifetime accurately when the total number of detected photon counts is too small, we do not attempt to estimate lifetimes at such locations. In other words, we attempt to estimate the following parameter vector, where

_{j}*C*is a set that defines the indices of the pixel locations where the number of detected photon counts is larger than some threshold

*γ*. The log-likelihood function of the entire parameter vector Θ from the entire measurement data

**g**is defined as where the log-likelihood function of the parameter vector

*θ*from the measurement

_{i,j}**g**

*is The MLE of Θ from the measured data*

_{ij}**g**is given by Note that the size of the parameter vector is 2

*P*×

*𝒞*, where

*𝒞*s the cardinality of the set

*C*. Since

*L*(Θ;

**g**) is additively separable, the MLE of the parameter vector

*θ*for each pixel location can be determined independently from other parameter vectors: One problem of using such an MLE is the noise sensitivity, since the method does not consider spatial correlations between neighboring decay and amplitude constants.

_{i,j}### 2.2. Proposed method

*a priori*knowledge: the amplitudes and lifetimes do not change rapidly in the spatial domain. The objective function of the proposed method Φ(Θ;

**g**) consists of the weighted sum of the Poisson likelihood function and a penalty function

*R*(Θ) that discourages large variations in the spatial domain: where

*β*is a regularization parameter that determines the weight between the first data fidelity term and second penalty term. The penalty function measures the total roughness of the lifetime and associated amplitude maps and is defined as where the change of the

*p*-th component at the (

*i*,

*j*)-th pixel location in the horizontal direction is measured by and that in the vertical direction is measured by where the 2

*P*× (2

*P*×

*𝒞*) matrices

*i*,

*j*)-th pixel, respectively. A natural choice for

*φ*(·) is a quadratic function; however, it is well known that quadratic penalty functions over-smooth edges in restored images [23]. We instead adopt an edge preserving penalty function such as the TV penalty function, which is defined as follows [23]: where

*ε*> 0 is a small positive constant to ensure that the penalty function is differentiable at

*x*= 0. Since the functions defined in Eq. (13) and (14) are not additively separable functions of

*θ*, minimization of the objective function defined in Eq. (11) requires the simultaneous optimization of the 2

_{i,j}*P*×

*𝒞*-size parameter vector Θ, which is often intractable. To overcome this difficulty, we apply an optimization transfer approach to determine the minimizer of the objective function. For the minimization of an objective function Φ(

*θ*), the iterative minimization of a surrogate function Φ

*(*

^{s}*θ*;

*θ*) (

^{k}*θ*denotes the estimated value of

^{k}*θ*at the

*k*-th iteration) monotonically converges to the minimizer of the objective function Φ(

*θ*) if the surrogate function satisfies the following two conditions [23]: That is, the following sequence

*θ*monotonically converges to the minimizer of the objective function Φ(

^{k}*θ*): We design additively separable quadratic surrogate functions for the penalty functions defined in Eq. (13) and (14) so that we can estimate the parameters at each pixel independently. To do this, we first apply Huber’s method [25] to design the quadratic surrogate functions of the penalty functions and then apply De Pierre’s approach [24

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]

**4**, 1417–1429 (1995). [CrossRef] [PubMed]

*φ̇*is the derivative of

*φ*[23]. The curvature

*c*that guarantees the first inequality of a quadratic or TV penalty function can be determined using Huber’s method [25]. The last inequality in Eq. (19) is due to the convex inequality [24

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]

*p*-th component, the following inequality holds: where,

*R*([

_{v}*θ*]

_{i,j}*; Θ*

_{p}*) for*

^{k}*R*(Θ), satisfies the following inequality: where Note that

_{s}*R*(Θ

*) =*

^{k}*R*(Θ

_{s}*;Θ*

^{k}*). Since both the likelihood function and the surrogate function of the penalty function are additively separable, the minimizer of the weighted sum of the log-likelihood function and the additively separable surrogate function at the*

^{k}*k*-th iteration can be determined at each pixel location independently as follows: The minimization problem defined in Eq. (23) can be solved numerically. The minimizer sequence defined in Eq. (23) monotonically converges to the minimizer of the objective function defined in Eq. (11).

## 3. Results

*fmincon*function in the Optimization toolbox of the MATLAB. We run the methods on a workstation equipped with two Intel Xeon X5650 processors (2.67 GHz) and 96 GB memory.

### 3.1. Simulations

*τ*

_{1}and

*τ*

_{2}) and amplitudes (

*A*

_{1}and

*A*

_{2}) in different regions of the image and convolved these synthesized signals with an IRF function of [0.014, 0.042, 0.121, 0.273, 0.357, 0.194]. We then generated 261 Poisson random variables at each pixel, whose means were uniformly sampled values of the convolved decaying fluorescence signal. The true values of the lifetime, amplitude and the average photon counts in each region are summarized in Table 1.

*τ*

_{1},

*τ*

_{2},

*A*

_{1},

*A*

_{2}) were (250, 100, 0.5, 1.5) in PMLEQ method and (8, 6, 0.4, 0.5) in PMLETV method. We also applied the MLE method with binning of the measured photon counts (i.e., summing the photon counts of neighboring pixels) from 3 × 3 neighboring pixels, which is used to reduce the effect of noise in the commercial software SPCImage (Becker & Hickl GmbH). The parameters Fig. 1 shows the estimated lifetime and amplitude images using the MLE method, which as expected yielded noisy lifetime and amplitude images due to Poisson noise. To reduce the effect of Poisson noise, we applied the PMLEQ and PMLETV methods. Figure 2 shows the estimated images obtained from the PMLEQ method, and as can be seen, the PMLEQ method was able to reduce the effect of noise. However, the quadratic penalty function blurred the edges in the estimated images. The blurring can be reduced by applying the PMLETV method, as shown in Fig. 3. Figure 4 shows estimated images using the MLE method with binning. As shown in the figure, although the binning strategy was able to reduce the effect of noise, it yielded images with artifacts, especially in the region where lifetimes in neighboring pixels are different. This is due to that photon counts data from different lifetimes are summed together.

26. J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs,” IEEE Trans. Image Process. , **5**, 1346–1358 (1996). [CrossRef] [PubMed]

*τ*

_{1},

*τ*

_{2},

*A*

_{1},

*A*

_{2}). Table 3 summarizes the results and reveals that the PMLE methods yield images that are more similar to the true images than those images obtained with the MLE method and the MLE method with binning. Among the two PMLE methods, the PMLETV method yielded better results than the PMLEQ method.

### 3.2. Real data

#### 3.2.2. cell data

*τ*

_{1}and

*τ*

_{2}) and associated amplitudes (

*A*

_{1}and

*A*

_{2}) obtained using the MLE method. The MLE method yielded very noisy

*τ*

_{1}and

*τ*

_{2}maps, thereby making it difficult to identify meaningful pH states. Figure 7 shows the estimated lifetime and amplitudes obtained using the PMLEQ method, and those obtained using the PMLETV method are shown in Fig. 8. We can see that the effects of noise have been reduced in the estimated images obtained using these two methods. Compared with the images obtained with the PMLEQ method, the lifetime maps estimated using the PMLETV method show more distinct features. Figure 8(a) and Fig. 8(b) show areas where the lifetimes are larger than the surrounding area, and we suspect that the pH states in this area are different based on the lifetime images. The variation of the fluorescence lifetime of the fluorocein as a function of pH is well known phenomenon [27

27. H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved ph imaging of living cells,” Cytometry Part A **52A**, 77–89 (2003). [CrossRef]

28. C. Hille, M. Berg, L. Bressel, D. Munzke, P. Primus, H.-G. Lahmannsraben, and C. Dosche, “Time-domain fluorescence lifetime imaging for intracellular ph sensing in living tissues,” Anal. Bioanal. Chem. **391**, 1871–1879 (2008). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. R. Lakowicz, |

2. | C. W. Chang and M.-A. Mycek, “Enhancing precision in time-domain fluorescence lifetime imaging,” J. Biomed. Opt. |

3. | W. Becker, “Fluorescence lifetime imaging techniques and applications,” J. Microsc. |

4. | M. Kneen, J. Farinas, Y. Li, and A. Verkman, “Green fluorescent protein as a noninvasive intracellular ph indicator,” Biophys. J. |

5. | P. J. Verveer, A. Squire, and P. I. Bastiaens, “Global analysis of fluorescence lifetime imaging microscopy data,” Biophys. J. |

6. | Z. Bajzer, T. M. Therneau, J. C. Sharp, and F. G. Prendergast, “Maximum likelihood method for the analysis of time-resolved fluorescence decay curves,” Eur. Biophys. J. |

7. | M. Kollner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. |

8. | J. Kim and J. Seok, “Statistical properties of amplitude and decay parameter estimators for fluorescence lifetime imaging,” Opt. Express |

9. | H. Cramer, |

10. | H. C. Gerritsen, M. A. H. Asselbergs, A. V. Agronskaia, and W. G. J. H. M. Van Sark, “Fluorescence lifetime imaging in scanning microscopes: acquisition speed, photon economy and lifetime resolution,” J. Microsc. |

11. | L. P. Watkins and H. Yang, “Information bounds and optimal analysis of dynamic single molecule measurements,” Biophys. J. |

12. | J. Philip and K. Carlsson, “Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging,” J. Opt. Soc. Am. A |

13. | H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting fret-flim data,” Opt. Express |

14. | S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J. |

15. | K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum ph gradient,” Biophys. J. |

16. | A. Squire and P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. |

17. | D. Sud and M.-A. Mycek, “Image restoration for fluorescence lifetime imaging microscopy (FLIM),” Opt. Express |

18. | M. Heilemann, D. P. Herten, R. Heintzmann, C. Cremer, C. Mller, P. Tinnefeld, K. D. Weston, J. Wolfrum, and M. Sauer, “High-resolution colocalization of single dye molecules by fluorescence lifetime imaging microscopy,” Anal. Chem. |

19. | B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem. |

20. | E. Gratton, S. Breusegem, J. Sutin, Q. Ruan, and N. Barry, “Fluorescence lifetime imaging for the two-photon microscope: time-domain and frequency-domain methods,” J. Biomed. Opt. |

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

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

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

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

25. | P. Huber, |

26. | J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs,” IEEE Trans. Image Process. , |

27. | H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved ph imaging of living cells,” Cytometry Part A |

28. | C. Hille, M. Berg, L. Bressel, D. Munzke, P. Primus, H.-G. Lahmannsraben, and C. Dosche, “Time-domain fluorescence lifetime imaging for intracellular ph sensing in living tissues,” Anal. Bioanal. Chem. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(180.2520) Microscopy : Fluorescence microscopy

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Microscopy

**History**

Original Manuscript: June 26, 2013

Revised Manuscript: August 12, 2013

Manuscript Accepted: August 13, 2013

Published: August 21, 2013

**Virtual Issues**

Vol. 8, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Jeongtae Kim, Jiyeong Seok, Hwiin Lee, and Minyung Lee, "Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals," Opt. Express **21**, 20240-20253 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-17-20240

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### References

- J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic/Plenum, 1999). [CrossRef]
- C. W. Chang and M.-A. Mycek, “Enhancing precision in time-domain fluorescence lifetime imaging,” J. Biomed. Opt.15, 056013 (2010). [CrossRef] [PubMed]
- W. Becker, “Fluorescence lifetime imaging techniques and applications,” J. Microsc.247, 119–136 (2012). [CrossRef] [PubMed]
- M. Kneen, J. Farinas, Y. Li, and A. Verkman, “Green fluorescent protein as a noninvasive intracellular ph indicator,” Biophys. J.74, 1591–1599 (1998). [CrossRef] [PubMed]
- P. J. Verveer, A. Squire, and P. I. Bastiaens, “Global analysis of fluorescence lifetime imaging microscopy data,” Biophys. J.78, 2127–2137 (2000). [CrossRef] [PubMed]
- Z. Bajzer, T. M. Therneau, J. C. Sharp, and F. G. Prendergast, “Maximum likelihood method for the analysis of time-resolved fluorescence decay curves,” Eur. Biophys. J.20, 247–262 (1991). [CrossRef]
- M. Kollner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett.200, 199 –204 (1992). [CrossRef]
- J. Kim and J. Seok, “Statistical properties of amplitude and decay parameter estimators for fluorescence lifetime imaging,” Opt. Express21, 6061–6075 (2013). [CrossRef] [PubMed]
- H. Cramer, Mathematical Methods of Statistics (PMS-9), Princeton Landmarks in Mathematics and Physics (Princeton University Press, 1999).
- H. C. Gerritsen, M. A. H. Asselbergs, A. V. Agronskaia, and W. G. J. H. M. Van Sark, “Fluorescence lifetime imaging in scanning microscopes: acquisition speed, photon economy and lifetime resolution,” J. Microsc.206, 218–224 (2002). [CrossRef] [PubMed]
- L. P. Watkins and H. Yang, “Information bounds and optimal analysis of dynamic single molecule measurements,” Biophys. J.86, 4015 – 4029 (2004). [CrossRef] [PubMed]
- J. Philip and K. Carlsson, “Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging,” J. Opt. Soc. Am. A20, 368–379 (2003). [CrossRef]
- H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting fret-flim data,” Opt. Express17, 6493–6508 (2009). [CrossRef] [PubMed]
- S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–2817 (2004). [CrossRef] [PubMed]
- K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum ph gradient,” Biophys. J.83, 1682–1690 (2002). [CrossRef] [PubMed]
- A. Squire and P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc.193, 36–49 (1999). [CrossRef] [PubMed]
- D. Sud and M.-A. Mycek, “Image restoration for fluorescence lifetime imaging microscopy (FLIM),” Opt. Express16, 19192–19200 (2008). [CrossRef]
- M. Heilemann, D. P. Herten, R. Heintzmann, C. Cremer, C. Mller, P. Tinnefeld, K. D. Weston, J. Wolfrum, and M. Sauer, “High-resolution colocalization of single dye molecules by fluorescence lifetime imaging microscopy,” Anal. Chem.74, 3511–3517 (2002). [CrossRef] [PubMed]
- B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012). [CrossRef] [PubMed]
- E. Gratton, S. Breusegem, J. Sutin, Q. Ruan, and N. Barry, “Fluorescence lifetime imaging for the two-photon microscope: time-domain and frequency-domain methods,” J. Biomed. Opt.8, 381–390 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- J. Fessler, “Image reconstruction: Algorithms and analysis,” Online preprint of book in preparation.
- A. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imag., 14, 132–137 (1995). [CrossRef]
- P. Huber, Robust Statistics (Wiley, 1974).
- J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs,” IEEE Trans. Image Process., 5, 1346–1358 (1996). [CrossRef] [PubMed]
- H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved ph imaging of living cells,” Cytometry Part A52A, 77–89 (2003). [CrossRef]
- C. Hille, M. Berg, L. Bressel, D. Munzke, P. Primus, H.-G. Lahmannsraben, and C. Dosche, “Time-domain fluorescence lifetime imaging for intracellular ph sensing in living tissues,” Anal. Bioanal. Chem.391, 1871–1879 (2008). [CrossRef] [PubMed]

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