## High-density localization of active molecules using Structured Sparse Model and Bayesian Information Criterion |

Optics Express, Vol. 19, Issue 18, pp. 16963-16974 (2011)

http://dx.doi.org/10.1364/OE.19.016963

Acrobat PDF (3306 KB)

### Abstract

Localization-based super-resolution microscopy (or called localization microscopy) rely on repeated imaging and localization of active molecules, and the spatial resolution enhancement of localization microscopy is built upon the sacrifice of its temporal resolution. Developing algorithms for high-density localization of active molecules is a promising approach to increase the speed of localization microscopy. Here we present a new algorithm called SSM_BIC for such purpose. The SSM_BIC combines the advantages of the Structured Sparse Model (SSM) and the Bayesian Information Criterion (BIC). Through simulation and experimental studies, we evaluate systematically the performance between the SSM_BIC and the conventional Sparse algorithm in high-density localization of active molecules. We show that the SSM_BIC is superior in processing single molecule images with weak signal embedded in strong background.

© 2011 OSA

## 1. Introduction

1. S. W. Hell, “Far-field optical nanoscopy,” Science **316**(5828), 1153–1158 (2007). [CrossRef] [PubMed]

2. B. Huang, H. Babcock, and X. Zhuang, “Breaking the diffraction barrier: super-resolution imaging of cells,” Cell **143**(7), 1047–1058 (2010). [CrossRef] [PubMed]

3. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**(5793), 1642–1645 (2006). [CrossRef] [PubMed]

4. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. **91**(11), 4258–4272 (2006). [CrossRef] [PubMed]

5. M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**(10), 793–796 (2006). [CrossRef] [PubMed]

6. G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. **61**(1), 345–367 (2010). [CrossRef] [PubMed]

7. S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods **8**(6), 499–505 (2011). [CrossRef] [PubMed]

8. T. J. Gould, V. V. Verkhusha, and S. T. Hess, “Imaging biological structures with fluorescence photoactivation localization microscopy,” Nat. Protoc. **4**(3), 291–308 (2009). [CrossRef] [PubMed]

9. S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods **8**(4), 279–280 (2011). [CrossRef] [PubMed]

10. F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express **2**(5), 1377–1393 (2011). [CrossRef] [PubMed]

## 2 Simulation and experimental methods

### 2.1 The imaging model of multiple active molecules

*L*(

*O*) denotes the function that returns the fixed values in the positions (

_{1},O_{2},..., O_{n}*O*) of

_{1},O_{2},..., O_{n}*n*active molecules and zeros otherwise. Considered that PSF can be approached by two dimensional Gaussian function, the theoretical signal (

*F*) of pixel (

_{i,j}*i,j*) in detector plane can be written as [14

14. B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. **46**(10), 1819–1829 (2007). [CrossRef] [PubMed]

*and O*

_{lx}*represent the x- and y- coordinates of the molecule in position*

_{ly}*O, A*is the amplitude, and

_{l}*ω*is the width of Gaussian kernel. Taking the influence of noise (including mainly shot noise and background noise) into account, the finally observed signal (S

*) in pixel (*

_{i,j}*i,j*) is [15

15. M. Elbaum and P. Diament, “SNR in photocounting images of rough objects in partially coherent light,” Appl. Opt. **15**(9), 2268–2275 (1976). [CrossRef] [PubMed]

*N*is the intensity of background noise and

_{b}*Pois*(

*x*) is a Poisson random number with mean value of

*x*. The task of high-density localization of active molecules is thus to find the positions of multiple active molecules (

*O*) from the observed signal

_{1},O_{2},..., O_{n}*S*(

_{i,j}*i,j*= 1,2,…,

*m*).

### 2.2 Pre-estimation of the positions of active molecules using SSM

*i,j*), the corresponding observed signal

*S*should be larger than a threshold value, and that the distribution of active molecules is still sparse in sub-pixel scale. Then an SSM [11] can be obtained

_{i,j}*e*

_{i,j}is an indicator function whose value is 1 if there is one molecule in pixel (

*i,j*) and 0 if there is no molecule.

*L*is determined by the observed signal

_{ij}*S*with the following procedures.

_{i,j}*S*is firstly subtracted by the associated background (the average intensity in the edge of the extracted sub-region), then normalized by dividing by its maximum value (after background subtraction) to give

_{i,j}*L*. If

_{i,j}*L*is larger than a given threshold value, it keeps unchanged and zeros otherwise. In optimization problem (4), the inequality constraints describe the sparse distribution characteristic of active molecules. Specifically, the first inequality constraint indicates that the total number of active molecules does not exceed

_{i,j}*n*, while the following inequality constraints show that there is only one active molecule at most in a pixel and its neighborhood. Furthermore, the inequality constraints in optimization problem (4) guarantee that, in most cases, the optimal solver

*e** could be obtained using common optimization algorithm [16

16. Z. Gao, Y. Lai, and Z. Hu, “A generalized gradient projection method for optimization problems with equality and inequality constraints about arbitrary initial point,” Acta Appl. Math. Sin. **12**(1), 40–49 (1996). [CrossRef]

17. D. Mayne and E. Polak, “Feasible directions algorithms for optimization problems with equality and inequality constraints,” Math. Program. **11**(1), 67–80 (1976). [CrossRef]

*e*, pre-estimated positions of active molecules are given by the optimal solver

_{i,j}*e**.

*n*can effectively control the sparsity of pre-estimated molecules in a sub-region, and thus a reasonable pre-estimation of the positions of active molecules is beneficial for the procedures in the following sections. Considering the size of Airy disk, the density of molecules, and pixel size, the threshold value and

*n*were set to be 0.4 and 6, respectively, unless otherwise specified.

### 2.3 The selection of optimal model using BIC

*S*(

_{i,j}*i,j*= 1,2,…,

*m*). A detailed description is presented as follows.

*s*, we can obtain these sub-pixel positions using the maximum likelihood estimation [18

18. R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. **86**(2), 1185–1200 (2004). [CrossRef] [PubMed]

*ω*in Eq. (2) is unchanged. The symbols in Eq. (5) have the same meanings as those in Eqs. (2) and (3). Equation (5) can be solved using the optimization algorithm reported in [16

16. Z. Gao, Y. Lai, and Z. Hu, “A generalized gradient projection method for optimization problems with equality and inequality constraints about arbitrary initial point,” Acta Appl. Math. Sin. **12**(1), 40–49 (1996). [CrossRef]

17. D. Mayne and E. Polak, “Feasible directions algorithms for optimization problems with equality and inequality constraints,” Math. Program. **11**(1), 67–80 (1976). [CrossRef]

*F*. Then, we can approximately calculate the corresponding BIC statistics [12]

_{i,j}+ N_{b}*s*molecules, the molecule with minimum fitting amplitude

*A*

_{i}is removed and a new model describing the remaining

*s-*1 molecules is generated. This new model is used to fit the observed signal

*S*(

_{i,j}*i,j*= 1,2,…,

*m*), and the statistics BIC(

*s*-1) is calculated again. In this way, a series of statistics BIC(

*s*-2)

*,…,*BIC(1) can be obtained by repeating the above procedures. The model with minimum value of BIC

*(s*)*is defined to be the optimal model.

*S*is fitted to give positions of the active molecules, which are regarded as their real positions. Because the value of amplitude

_{i,j}*A*

_{i}is typically one or two orders of magnitude larger than the position parameters of active molecules in a sub-region, it is difficult to search for the optimal solver through iteration method. Therefore in the fitting the position parameters of active molecules were enlarged ten times and then recovered afterwards.

### 2.4 The extraction of sub-region

19. T. W. Quan, P. C. Li, F. Long, S. Q. Zeng, Q. M. Luo, P. N. Hedde, G. U. Nienhaus, and Z. L. Huang, “Ultra-fast, high-precision image analysis for localization-based super resolution microscopy,” Opt. Express **18**(11), 11867–11876 (2010). [CrossRef] [PubMed]

20. P. N. Hedde, J. Fuchs, F. Oswald, J. Wiedenmann, and G. U. Nienhaus, “Online image analysis software for photoactivation localization microscopy,” Nat. Methods **6**(10), 689–690 (2009). [CrossRef] [PubMed]

21. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**(5), 2775–2783 (2002). [CrossRef] [PubMed]

22. Y. Cheng, “Mean shift, mode seeking, and clustering,” IEEE Trans. Pattern Anal. Mach. Intell. **17**(8), 790–799 (1995). [CrossRef]

- Step 1. De-noising and sub-region extraction.
- Step 2. Pre-estimation of the positions of active molecules using SSM.
- Step 3. Select the optimal model by BIC and search for the positions of active molecules.
- Step 4. Cluster the positions of active molecules with mean-shift algorithm for another round of sub-region extraction.
- Step 5. Repeat
*Step 3*with the refined sub-regions found in*Step 4*.

### 2.5 Localization microscopy imaging

#### 2.5.1 Cell culture and plasmid transfection

_{2}and grown in Dulbecco’s modified Eagle’s medium (DMEM, Invitrogen). Transfection of the eukaryotic expression vector for actin bundles labeling, pcDNA3-lifeact-d2EosFP, was carried out using Lipofectamine LTX (Invitrogen) according to the manufacturer’s instructions. Cells were maintained for 24 h after transfection in culture medium and then cleaned and fixed for further single molecule imaging.

#### 2.5.2 Single molecule imaging

## 3. Results

19. T. W. Quan, P. C. Li, F. Long, S. Q. Zeng, Q. M. Luo, P. N. Hedde, G. U. Nienhaus, and Z. L. Huang, “Ultra-fast, high-precision image analysis for localization-based super resolution microscopy,” Opt. Express **18**(11), 11867–11876 (2010). [CrossRef] [PubMed]

20. P. N. Hedde, J. Fuchs, F. Oswald, J. Wiedenmann, and G. U. Nienhaus, “Online image analysis software for photoactivation localization microscopy,” Nat. Methods **6**(10), 689–690 (2009). [CrossRef] [PubMed]

23. S. Wolter, M. Schüttpelz, M. Tscherepanow, S. Van De Linde, M. Heilemann, and M. Sauer, “Real-time computation of subdiffraction-resolution fluorescence images,” J. Microsc. **237**(1), 12–22 (2010). [CrossRef] [PubMed]

19. T. W. Quan, P. C. Li, F. Long, S. Q. Zeng, Q. M. Luo, P. N. Hedde, G. U. Nienhaus, and Z. L. Huang, “Ultra-fast, high-precision image analysis for localization-based super resolution microscopy,” Opt. Express **18**(11), 11867–11876 (2010). [CrossRef] [PubMed]

### 3.1 Analysis of simulated images

### 3.2 Resolving active molecule pairs

### 3.3 Analysis of experimental data

## 4 Discussions

### 4.1 The capability of detecting weak signals with SSM_BIC

9. S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods **8**(4), 279–280 (2011). [CrossRef] [PubMed]

24. A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods **5**(8), 687–694 (2008). [CrossRef] [PubMed]

9. S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods **8**(4), 279–280 (2011). [CrossRef] [PubMed]

### 4.2 Identification of the molecule pairs with strong signal

^{2}. Processing images with higher molecule density is possible with SSM_BIC algorithm, although in this case it is necessary to increase signal intensity and make some modifications to the optical system. For example, for a molecule pair with 2000 detected photons from each molecule, the resolved distance reaches 125 nm for a pixel size of 50 nm (the blue lines in Fig. 5 ), which is much better than those with 500 detected photons and 100 nm pixel size (the black lines in Fig. 2b).

### 4.3 Identification of the molecule pair with non-equal emission

25. M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” Proc. Natl. Acad. Sci. U.S.A. **101**(17), 6462–6465 (2004). [CrossRef] [PubMed]

26. X. H. Qu, D. Wu, L. Mets, and N. F. Scherer, “Nanometer-localized multiple single-molecule fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. **101**(31), 11298–11303 (2004). [CrossRef] [PubMed]

27. S. H. DeCenzo, M. C. DeSantis, and Y. M. Wang, “Single-image separation measurements of two unresolved fluorophores,” Opt. Express **18**(16), 16628–16639 (2010). [CrossRef] [PubMed]

### 4.4 Computation load of SSM_BIC

**18**(11), 11867–11876 (2010). [CrossRef] [PubMed]

28. X. Q. Li, G. H. Shi, and Y. D. Zhang, “Time-domain interpolation on graphics processing unit,” J. Innovative Opt. Health Sci. **4**(1), 89–95 (2011). [CrossRef]

29. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods **7**(5), 373–375 (2010). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. W. Hell, “Far-field optical nanoscopy,” Science |

2. | B. Huang, H. Babcock, and X. Zhuang, “Breaking the diffraction barrier: super-resolution imaging of cells,” Cell |

3. | E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science |

4. | S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. |

5. | M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods |

6. | G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. |

7. | S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods |

8. | T. J. Gould, V. V. Verkhusha, and S. T. Hess, “Imaging biological structures with fluorescence photoactivation localization microscopy,” Nat. Protoc. |

9. | S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods |

10. | F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express |

11. | C. Hegde, M. F. Duarte, and V. Cevher, “Compressive sensing recovery of spike trains using a structured sparsity model,” in Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS), (Saint-Malo, France) (2009), pp. 13–16. |

12. | K. P. Burnham and D. R. Anderson, |

13. | M. Gu, |

14. | B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. |

15. | M. Elbaum and P. Diament, “SNR in photocounting images of rough objects in partially coherent light,” Appl. Opt. |

16. | Z. Gao, Y. Lai, and Z. Hu, “A generalized gradient projection method for optimization problems with equality and inequality constraints about arbitrary initial point,” Acta Appl. Math. Sin. |

17. | D. Mayne and E. Polak, “Feasible directions algorithms for optimization problems with equality and inequality constraints,” Math. Program. |

18. | R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. |

19. | T. W. Quan, P. C. Li, F. Long, S. Q. Zeng, Q. M. Luo, P. N. Hedde, G. U. Nienhaus, and Z. L. Huang, “Ultra-fast, high-precision image analysis for localization-based super resolution microscopy,” Opt. Express |

20. | P. N. Hedde, J. Fuchs, F. Oswald, J. Wiedenmann, and G. U. Nienhaus, “Online image analysis software for photoactivation localization microscopy,” Nat. Methods |

21. | R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. |

22. | Y. Cheng, “Mean shift, mode seeking, and clustering,” IEEE Trans. Pattern Anal. Mach. Intell. |

23. | S. Wolter, M. Schüttpelz, M. Tscherepanow, S. Van De Linde, M. Heilemann, and M. Sauer, “Real-time computation of subdiffraction-resolution fluorescence images,” J. Microsc. |

24. | A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods |

25. | M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” Proc. Natl. Acad. Sci. U.S.A. |

26. | X. H. Qu, D. Wu, L. Mets, and N. F. Scherer, “Nanometer-localized multiple single-molecule fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. |

27. | S. H. DeCenzo, M. C. DeSantis, and Y. M. Wang, “Single-image separation measurements of two unresolved fluorophores,” Opt. Express |

28. | X. Q. Li, G. H. Shi, and Y. D. Zhang, “Time-domain interpolation on graphics processing unit,” J. Innovative Opt. Health Sci. |

29. | C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.2960) Imaging systems : Image analysis

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: June 24, 2011

Revised Manuscript: August 4, 2011

Manuscript Accepted: August 11, 2011

Published: August 15, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Tingwei Quan, Hongyu Zhu, Xiaomao Liu, Yongfeng Liu, Jiuping Ding, Shaoqun Zeng, and Zhen-Li Huang, "High-density localization of active molecules using Structured Sparse Model and Bayesian Information Criterion," Opt. Express **19**, 16963-16974 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16963

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### References

- S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007). [CrossRef] [PubMed]
- B. Huang, H. Babcock, and X. Zhuang, “Breaking the diffraction barrier: super-resolution imaging of cells,” Cell 143(7), 1047–1058 (2010). [CrossRef] [PubMed]
- E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006). [CrossRef] [PubMed]
- S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006). [CrossRef] [PubMed]
- M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006). [CrossRef] [PubMed]
- G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010). [CrossRef] [PubMed]
- S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–505 (2011). [CrossRef] [PubMed]
- T. J. Gould, V. V. Verkhusha, and S. T. Hess, “Imaging biological structures with fluorescence photoactivation localization microscopy,” Nat. Protoc. 4(3), 291–308 (2009). [CrossRef] [PubMed]
- S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8(4), 279–280 (2011). [CrossRef] [PubMed]
- F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express 2(5), 1377–1393 (2011). [CrossRef] [PubMed]
- C. Hegde, M. F. Duarte, and V. Cevher, “Compressive sensing recovery of spike trains using a structured sparsity model,” in Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS), (Saint-Malo, France) (2009), pp. 13–16.
- K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. (Springer, 2002).
- M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
- B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46(10), 1819–1829 (2007). [CrossRef] [PubMed]
- M. Elbaum and P. Diament, “SNR in photocounting images of rough objects in partially coherent light,” Appl. Opt. 15(9), 2268–2275 (1976). [CrossRef] [PubMed]
- Z. Gao, Y. Lai, and Z. Hu, “A generalized gradient projection method for optimization problems with equality and inequality constraints about arbitrary initial point,” Acta Appl. Math. Sin. 12(1), 40–49 (1996). [CrossRef]
- D. Mayne and E. Polak, “Feasible directions algorithms for optimization problems with equality and inequality constraints,” Math. Program. 11(1), 67–80 (1976). [CrossRef]
- R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004). [CrossRef] [PubMed]
- T. W. Quan, P. C. Li, F. Long, S. Q. Zeng, Q. M. Luo, P. N. Hedde, G. U. Nienhaus, and Z. L. Huang, “Ultra-fast, high-precision image analysis for localization-based super resolution microscopy,” Opt. Express 18(11), 11867–11876 (2010). [CrossRef] [PubMed]
- P. N. Hedde, J. Fuchs, F. Oswald, J. Wiedenmann, and G. U. Nienhaus, “Online image analysis software for photoactivation localization microscopy,” Nat. Methods 6(10), 689–690 (2009). [CrossRef] [PubMed]
- R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002). [CrossRef] [PubMed]
- Y. Cheng, “Mean shift, mode seeking, and clustering,” IEEE Trans. Pattern Anal. Mach. Intell. 17(8), 790–799 (1995). [CrossRef]
- S. Wolter, M. Schüttpelz, M. Tscherepanow, S. Van De Linde, M. Heilemann, and M. Sauer, “Real-time computation of subdiffraction-resolution fluorescence images,” J. Microsc. 237(1), 12–22 (2010). [CrossRef] [PubMed]
- A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008). [CrossRef] [PubMed]
- M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” Proc. Natl. Acad. Sci. U.S.A. 101(17), 6462–6465 (2004). [CrossRef] [PubMed]
- X. H. Qu, D. Wu, L. Mets, and N. F. Scherer, “Nanometer-localized multiple single-molecule fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 101(31), 11298–11303 (2004). [CrossRef] [PubMed]
- S. H. DeCenzo, M. C. DeSantis, and Y. M. Wang, “Single-image separation measurements of two unresolved fluorophores,” Opt. Express 18(16), 16628–16639 (2010). [CrossRef] [PubMed]
- X. Q. Li, G. H. Shi, and Y. D. Zhang, “Time-domain interpolation on graphics processing unit,” J. Innovative Opt. Health Sci. 4(1), 89–95 (2011). [CrossRef]
- C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010). [CrossRef] [PubMed]

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