## Fast compressed sensing analysis for super-resolution imaging using L1-homotopy |

Optics Express, Vol. 21, Issue 23, pp. 28583-28596 (2013)

http://dx.doi.org/10.1364/OE.21.028583

Acrobat PDF (1746 KB)

### Abstract

In super-resolution imaging techniques based on single-molecule switching and localization, the time to acquire a super-resolution image is limited by the maximum density of fluorescent emitters that can be accurately localized per imaging frame. In order to increase the imaging rate, several methods have been recently developed to analyze images with higher emitter densities. One powerful approach uses methods based on compressed sensing to increase the analyzable emitter density per imaging frame by several-fold compared to other reported approaches. However, the computational cost of this approach, which uses interior point methods, is high, and analysis of a typical 40 µm x 40 µm field-of-view super-resolution movie requires thousands of hours on a high-end desktop personal computer. Here, we demonstrate an alternative compressed-sensing algorithm, L1-Homotopy (L1H), which can generate super-resolution image reconstructions that are essentially identical to those derived using interior point methods in one to two orders of magnitude less time depending on the emitter density. Moreover, for an experimental data set with varying emitter density, L1H analysis is ~300-fold faster than interior point methods. This drastic reduction in computational time should allow the compressed sensing approach to be routinely applied to super-resolution image analysis.

© 2013 Optical Society of America

## 1. Introduction

4. H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods **5**(5), 417–423 (2008). [CrossRef] [PubMed]

6. 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. S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A. **109**(35), 13978–13983 (2012). [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]

15. L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods **9**(7), 721–723 (2012). [CrossRef] [PubMed]

15. L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods **9**(7), 721–723 (2012). [CrossRef] [PubMed]

15. L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods **9**(7), 721–723 (2012). [CrossRef] [PubMed]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

16. M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. **20**(3), 389–403 (2000). [CrossRef]

18. D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory **54**(11), 4789–4812 (2008). [CrossRef]

## 2. Theory

*M*×

*N*, where

*M*and

*N*are the numbers of pixels in

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

*N*×

*N*matrix—an operation with complexity that scales as O(

*N*

^{3}). As

*N*is typically of order 10

^{3}– 10

^{4}, 10

^{9}– 10

^{12}floating point operations are needed for each iteration. In practice, reconstructing a ~40 µm × 40 µm STORM image with ~20 nm resolution using Eq. (1) and interior point methods takes about 30 minutes per camera frame on a high-end desktop PC. Since the typical movie used to construct a single STORM image contains thousands to tens of thousands of frames, it would take thousands of hours to reconstruct a typical STORM image. The long computational time has, thus, limited the use of compressed sensing in the reconstruction of STORM images to small fields of view, for example ~3 µm × 3 µm [15

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

19. A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010). [CrossRef]

*ε*there exists an equivalent value of

*λ*for which the solutions are identical. In practice, solving Eq. (2) is often computationally simpler than solving Eq. (1) which is the reason why many alternative algorithms have been reported to have faster computational speed than interior point methods [19

19. A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010). [CrossRef]

*ε*for the analysis of STORM data is simple. It can be done directly from the image data itself [15

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

*λ*for which Eq. (2) produces the equivalent solution to Eq. (1) is often not trivial since there is no known analytical connection between the noise parameter

*ε*and

*λ*for the general problem [21

21. E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. **31**(2), 890–912 (2009). [CrossRef]

*λ*before they can be applied to the analysis of STORM data.

*λ*may be particularly efficient for STORM imaging data: the L1-Homotopy (L1H) algorithm. This algorithm functions by exploiting a simple geometric property of the solution space of Eq. (2) to trace solutions of Eq. (2) from one value of

*λ*to another value. In particular, as the value of

*λ*is decreased, the balance shifts from favoring the sparsity of emitters to favoring the accuracy in the optimal solution, and this solution changes in a continuous, piece-wise-linear fashion as a function of

*λ*. Importantly, the slope of this solution changes only at well-defined ‘break-points’ where individual emitters are either added to, removed from, or moved to a different location within the solution, and the value of

*λ*at which each break-point occurs can be calculated from the solution at adjacent break-points. These properties are illustrated in Fig. 2 for a simple one-dimensional problem. The L1H algorithm begins at a large value of

*λ*where the solution is obvious,

*λ*from break-point to break-point until it reaches the desired value of

*λ*.

*K*—is small enough, we find that, in practice, the computational complexity of L1H is no longer dominated by the inversion of this matrix but rather by other matrix multiplications whose complexity scale roughly as O(

*N*×

*K*). By contrast, the matrix inversion typically dominates the cost of interior point methods, and its complexity scales as O(

*N*

^{3}). Thus, when the solution vector is sparse, as it is in STORM images,

*K*will be much less than

*N*and the computational cost of L1H should be dramatically lower than the cost of interior point methods.

*λ*values as part of its normal iteration, and L1H can be modified to identify the

*λ*that corresponds to the appropriate value of

*ε*. To accomplish this task, we introduced a simple stopping criterion into the L1H algorithm. After the identification of each break point, our algorithm computes the residual error between the reconstructed image at that break point and measured image, i.e. the L2 norm in Eq. (1). If that residual error is larger than the target error set by

*ε*, then the algorithm continues onto the next break point. If the residual error is less than

*ε*, then iteration stops, and the algorithm uses mid-point bisection to find the value of

*λ*and the intensity of the localized emitters for which the residual error is equal to

*ε*. This solution represents the final reconstructed image, and the final value of

*λ*corresponds to the value for which the solutions of Eqs. (1) and (2) are equal. Because calculation of the residual error involves

*K*×

*M*calculations, it does not significantly increase the computational cost of the L1H algorithm.

*λ*is a byproduct of its natural iteration. Thus, we reasoned that L1H should be the optimal approach for STORM image reconstruction. However, it is possible that when combined with an efficient search algorithm for

*λ*, other alternative compressed sensing algorithms might be comparable to L1H in terms of computational complexity. To test this possibility, we considered one of the more commonly used efficient compressed sensing algorithms, the Fast Iterative Shrinkage Thresholding Algorithm (FISTA) [22

22. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences **2**(1), 183–202 (2009). [CrossRef]

*N*×

*M*calculations per iteration, and, thus has a computational cost per iteration that is similar to L1H. However, each iteration in FISTA does not produce an exact solution to Eq. (2), as is the case for L1H, but rather serves only to improve the accuracy of an approximation to Eq. (2). Thus, in order for FISTA to be competitive with L1H, it must produce an approximation of the solution with sufficient accuracy in a relatively small number of iterations.

*λ*as identified by L1H, FISTA required 100 – 1000 times more iterations to converge to an approximate solution of a 7 pixel × 7 pixel data set than is required for L1H to solve the problem exactly. We expect similar limitations to arise for other iterative approaches such as gradient projection, proximal gradient, and augmented Lagrange multiplier methods [19

19. A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010). [CrossRef]

*λ*in addition to their natural iterative improvement of the solution to Eq. (2). Thus, L1H appears to be particularly well suited for the analysis of STORM data using compressed sensing.

## 3. Methods

*Algorithm and Image Analysis.*The L1H algorithm used here was written in C with a Python interface and is based on the approach described by Donoho and Tsiag [18

18. D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory **54**(11), 4789–4812 (2008). [CrossRef]

*λ*has been identified, as discussed above. The FISTA algorithm used here was also written in C with a Python interface following [22

22. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences **2**(1), 183–202 (2009). [CrossRef]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

23. M. C. Grant and S. P. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.0 beta,” http://cvxr.com/cvx (2013).

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

*Simulated Data.*For all simulated data, we randomly distributed emitters on an 8-fold up-sampled grid and generated the number of photons for each emitter from a lognormal distribution with a mean and standard deviation of 3000 and 1700 photons, respectively. A Gaussian PSF of width equal to one camera pixel was used to convolve the signal from individual emitters, and the photons from all up-sampled pixels within a low-resolution, camera pixel were summed. Finally, a uniform background of 70 photons was added to each camera pixel and the image was corrupted with Poisson distributed noise with a mean set by the number of photons in each pixel. In all simulations, the camera pixel is assumed to be 167 nm in size. These values were chosen to match our typical experimental data. Using this approach, 7 × 7 camera pixel images were generated, and these simulated images were analyzed by both the L1H and CVX methods to compare the accuracy of the reconstructions. We also generated 256 × 256 camera pixel images to compare the computation time of these two methods. Reported computation times are the average of three independent trials.

*Real Data.*To test the L1H algorithm on real data, we collected images of immunostained microtubules. BS-C-1 cells were fixed with 3% paraformaldehyde and 0.1% glutaraldehyde in phosphate buffered saline (PBS) for 10 minutes at room temperature and then were treated with 0.1% sodium borohydride for 7 minutes to reduce auto-fluorescence. After blocking with 3% w/v bovine serum albumin (BSA) and 0.1% v/v Triton X-100 in PBS, the microtubules were labeled with a mouse monoclonal anti-beta-tubulin primary antibody (Sigma, T5201) followed by a donkey anti-mouse secondary antibody (Jackson Labs, 715-005-150) conjugated with Alexa Fluor 405 and Alexa Fluor 647 [13

13. H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy **1**(1), 6–10 (2012). [CrossRef]

24. M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science **317**(5845), 1749–1753 (2007). [CrossRef] [PubMed]

25. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science **319**(5864), 810–813 (2008). [CrossRef] [PubMed]

## 4. Results

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

23. M. C. Grant and S. P. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.0 beta,” http://cvxr.com/cvx (2013).

^{−2}; however, the speed enhancement is substantially larger than that observed for the simulated data at a uniform density of 1 μm

^{−2}because of the large variation in the local emitter density observed in the real STORM data. Since the typical STORM image does not have a uniform emitter density, we anticipate this experimentally-derived speed enhancement to be representative of what will be achieved in the analysis of a wide range of real STORM data.

24. M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science **317**(5845), 1749–1753 (2007). [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]

13. H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy **1**(1), 6–10 (2012). [CrossRef]

^{−2}, the single-emitter localization algorithm begins to miss a substantial fraction of emitters, and above an intermediate density of ~3 emitters per μm

^{−2}the multi-emitter fitting algorithm also begins to miss a substantial number of emitters. However, L1H and CVX both reconstruct the vast majority of the emitters up to a density of 5-6 emitters per μm

^{−2}, as shown in Fig. 5(a). Thus, for the highest emitter densities, a compressed sensing approach is required to reconstruct most emitters.

27. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods **7**(5), 377–381 (2010). [CrossRef] [PubMed]

^{−2}. Beyond this density, a compressed sensing approach is superior. These results are consistent with those reported previously [15

**9**(7), 721–723 (2012). [CrossRef] [PubMed]

14. E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J. **102**(10), 2391–2400 (2012). [CrossRef] [PubMed]

**A**matrix to include PSFs of variable size and shape, which will in turn extend compressed sensing to the analysis of three dimensional data. The significant decrease in computational complexity afforded by L1H will partially offset the substantial increase in computational complexity of 3D implementations of compressed sensing. Finally, it may be possible to further increase computational speed by implementing L1H on GPUs or clusters of CPUs.

## 5. Conclusions

## Acknowledgments

## References and Links

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

2. | 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 |

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

4. | H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods |

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

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

7. | R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods |

8. | S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A. |

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. | S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods |

12. | T. Quan, H. Zhu, X. Liu, Y. Liu, J. Ding, S. Zeng, and Z.-L. Huang, “High-density localization of active molecules using Structured Sparse Model and Bayesian Information Criterion,” Opt. Express |

13. | H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy |

14. | E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J. |

15. | L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods |

16. | M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. |

17. | D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP |

18. | D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory |

19. | A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010). [CrossRef] |

20. | A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process. |

21. | E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. |

22. | A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences |

23. | M. C. Grant and S. P. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.0 beta,” http://cvxr.com/cvx (2013). |

24. | M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science |

25. | B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science |

26. | M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab Software for Disciplined Convex Programming, Version 1.0 beta 3,” Recent Advances in Learning and Control}, 95-110 (2006). |

27. | K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.2960) Imaging systems : Image analysis

(170.2520) Medical optics and biotechnology : Fluorescence microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 3, 2013

Revised Manuscript: November 4, 2013

Manuscript Accepted: November 6, 2013

Published: November 13, 2013

**Virtual Issues**

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

**Citation**

Hazen P. Babcock, Jeffrey R. Moffitt, Yunlong Cao, and Xiaowei Zhuang, "Fast compressed sensing analysis for super-resolution imaging using L1-homotopy," Opt. Express **21**, 28583-28596 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28583

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### References

- M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006). [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,” Science313(5793), 1642–1645 (2006). [CrossRef] [PubMed]
- S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J.91(11), 4258–4272 (2006). [CrossRef] [PubMed]
- H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008). [CrossRef] [PubMed]
- B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010). [CrossRef] [PubMed]
- S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011). [CrossRef] [PubMed]
- R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods10(6), 557–562 (2013). [CrossRef] [PubMed]
- S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012). [CrossRef] [PubMed]
- S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(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. Express2(5), 1377–1393 (2011). [CrossRef] [PubMed]
- S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9(2), 195–200 (2011). [CrossRef] [PubMed]
- T. Quan, H. Zhu, X. Liu, Y. Liu, J. Ding, S. Zeng, and Z.-L. Huang, “High-density localization of active molecules using Structured Sparse Model and Bayesian Information Criterion,” Opt. Express19(18), 16963–16974 (2011). [CrossRef] [PubMed]
- H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012). [CrossRef]
- E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012). [CrossRef] [PubMed]
- L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012). [CrossRef] [PubMed]
- M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000). [CrossRef]
- D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).
- D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory54(11), 4789–4812 (2008). [CrossRef]
- A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010). [CrossRef]
- A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).
- E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput.31(2), 890–912 (2009). [CrossRef]
- A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences2(1), 183–202 (2009). [CrossRef]
- M. C. Grant and S. P. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.0 beta,” http://cvxr.com/cvx (2013).
- M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007). [CrossRef] [PubMed]
- B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008). [CrossRef] [PubMed]
- M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab Software for Disciplined Convex Programming, Version 1.0 beta 3,” Recent Advances in Learning and Control}, 95-110 (2006).
- K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010). [CrossRef] [PubMed]

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