## Fast, bias-free algorithm for tracking single particles with variable size and shape

Optics Express, Vol. 16, Issue 18, pp. 14064-14075 (2008)

http://dx.doi.org/10.1364/OE.16.014064

Acrobat PDF (264 KB)

### Abstract

We introduce a fast and robust technique for single-particle tracking with nanometer accuracy. We extract the center-of-mass of the image of a single particle with a simple, iterative algorithm that efficiently suppresses background-induced bias in a simplistic centroid estimator. Unlike many commonly used algorithms, our position estimator requires no prior information about the shape or size of the tracked particle image and uses only simple arithmetic operations, making it appropriate for future hardware implementation and real-time feedback applications. We demonstrate it both numerically and experimentally, using an inexpensive CCD camera to localize 190 nm fluorescent microspheres to better than 5 nm.

© 2008 Optical Society of America

## 1. Introduction

1. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. **26**, 373–399 (1997). [CrossRef] [PubMed]

2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science **300**, 2061–2065 (2003). [CrossRef] [PubMed]

3. X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics,” Science **307**, 538–544 (2005). [CrossRef] [PubMed]

4. D. Weihs, T. G. Mason, and M. A. Teitell, “Bio-Microrheology: A Frontier in Microrheology,” Biophys. J. **91**, 4296–4305 (2006). [CrossRef] [PubMed]

5. P. Bahukudumbi and M. A. Bevan, “Imaging energy landscapes with concentrated diffusing colloidal probes,” J. Chem. Phys. **126**, 244702 (2007). [CrossRef] [PubMed]

6. H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, “Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes,” Langmuir **22**, 6826–6836 (2006). [CrossRef] [PubMed]

7. S. K. Sainis, V. Germain, and E. R. Dufresne, “Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces,” Phys. Rev. Lett. **99**, 018303 (2007). [CrossRef] [PubMed]

15. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. **88**, 2919–2928 (2005). [CrossRef] [PubMed]

16. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles,” Opt. Express **13**, 8069–8082 (2005). [CrossRef] [PubMed]

18. A. E. Cohen and W. E. Moerner, “Method for Trapping and Manipulating Nanoscale Objects in Solution,” Appl. Phys. Lett. **86**, 093109 (2005). [CrossRef]

19. H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. **88**, 223901 (2006). [CrossRef]

20. K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. **7**, 3535–3539 (2007). [CrossRef] [PubMed]

21. M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. **15**, 945–956 (2006). [CrossRef]

*no assumptions*about the object shape, making it robust and effective for localizing objects with varying sizes and complex shapes in the presence of uncharacterized background noise. Furthermore, the algorithm is computationally simple and can be executed with a few lines of code. Its performance is comparable to or exceeds full nonlinear least-squares minimization in many cases, while its execution time is orders of magnitude shorter. For all of these reasons, our algorithm is a promising candidate for implementation in signal-processing hardware for real-time applications.

## 2. Bias in the center-of-mass estimator

*S*with elements

*S*representing the total number of counts in pixel

_{jk}*P*with width Δ centered at location (

_{jk}*x*,

_{jk}*y*). [The pixel size Δ and coordinates (

_{jk}*x*,

*y*) are always considered in the object plane of the optical system, so that, for example, Δ is the actual CCD pixel size divided by the system magnification

*M*.]

*S*represents the experimental data from a single shot of the experiment,

*e.g.*a single CCD frame. In order to analyze, and later design, a position estimator algorithm, we must assume some underlying statistical model for such an image. For this purpose, we assume the signal from the particle is drawn from a Poisson distribution with spatially-dependent mean value

*N*(

_{S}*x*,

*y*)/Δ

^{2}, which represents the point-spread function of the optical system convolved with the shape of the fluorescent object. This mean value is explicitly normalized by the pixel area Δ

^{2}so that

*N*(

_{S}*x*,

*y*) represents a dimensionless number of counts. In order to calculate the mean number of counts in a particular pixel, averaged over an ensemble of individual images

*S*, we must integrate

*N*(

_{S}*x*,

*y*)/Δ

^{2}over the area of that pixel, as in Eq. (1a) below.

*∫∫P*denotes integration over the area of pixel

_{jk}*P*so that, for example, the mean count rate 〈

_{jk}*S*〉 is just the integral of the spatially-varying average count rate over the pixel area. Without the background contribution, Eqs. (1a)–(1b) represent familiar Poisson counting statistics, where the mean, 〈

_{jk}*S*〉, and variance, 〈

_{jk}*S*

^{2}

*〉-〈*

_{jk}*S*〉

_{jk}^{2}, of the counts in pixel

*P*are equal [22, 23]. Equations (1a)–(1b) give a prescription for calculating the statistics of the image

_{jk}*S*that includes the effects of pixelation, image truncation, background noise, and counting statistics.

*x*-position

*x*

_{0}:

*x*̂

*is nevertheless a nonlinear function of the image*

_{CM}*S*, making statistical calculations difficult. However, the statistics of a linearized approximation are accurate to order 𝓝

^{-3/2}, where 𝓝=

*∑*is the

_{jk}S_{jk}*total*number of counts in the image. The corresponding linearized approximation to Eq. (2) is

*S*satisfying (1a):

_{jk}*N*(

_{S}*x*,

*y*) superimposed on a spatially-varying, Poisson- or Gaussian-distributed background. The mean and mean-square errors (bias and variance) resulting from pixelation, truncation, shot noise, and background noise can each be derived from these expressions. For more complicated systems, such as electron-multiplying CCD cameras, the mathematical form of the noise term [Eq. (1b)] must be modified to accommodate signal-dependent noise introduced by on-chip gain. For the remainder of this paper, we will be concerned only with the mean value 〈

*x*̂

*〉, which determines the estimator bias; the noise (variance) terms are included as a reference.*

_{CM}*N*(

_{B}*x*,

*y*)=

*N*. We leave the

_{B}*noise*σ

^{2}

*(*

_{B}*x*,

*y*) on this background level unspecified, as it does not enter a calculation of the bias. Introducing the shorthand notation,

*B*, can be written as

_{CM}*N*(

_{B}*x*,

*y*)=

*N*, Eq. (4) and Eq. (6) are mathematically identical, and no approximations have yet been made. Let us now consider the different contributions to the bias in Eq. (6). The normalization factor ∑

_{B}

_{jk}N^{Δ}

*(*

_{S}*x*-

_{jk}*x*

_{0},

*y*-

_{jk}*y*

_{0}) represents the average number of detected counts from the particle itself, and is only a weak function of

*x*

_{0}and

*y*

_{0}as long as the particle image is not strongly truncated at the edges of the array. We may then denote this term by 〈𝓝

*〉 and neglect its functional dependence on (*

_{S}*x*

_{0},

*y*

_{0}). The first term in brackets on the right-hand side of Eq. (6) is a discretized analog of the continuous center-of-mass

*x*-coordinate of the center-of-mass of the image function

*N*(

_{S}*x*,

*y*), which we assume to be 0 [this condition can always be enforced through the definition of

*N*(

_{S}*x*,

*y*)]. We assume that the particle is near the center of the image and is not significantly truncated at the edges of the array. Violations of the approximate equality in Eq. (7) now correspond to estimator bias arising from pixelation and truncation of the underlying image. Assuming these are negligible for now, we are left with the following expression for the estimator bias

*〉 and 〈𝓝*

_{S}*〉 are the average number of signal and background counts in the entire image, respectively, and*

_{B}*x*̄ is the geometric center of the pixel array,

*i.e.*the unweighted average

*x*-coordinate. Equation (8) reveals that

*the estimator bias is proportional to the difference between the center of the pixel array*

*x*̄

*and the average estimate*〈

*x*̂

*〉. The constant of proportionality is the ratio of background to signal counts in the image. Recall that the estimator bias is unknown to the experimenter, since the underlying particle position*

_{CM}*x*

_{0}is unknown. However, the difference between the estimate

*x*̂

*and the center of the pixel array can be calculated in each shot of the experiment. In our algorithm described below, we exploit this fact in order to form an online estimate of the bias then correct it by truncating the pixel window to center the particle in the image array.*

_{CM}## 3. Virtual window center-of-mass algorithm

*x*̂

*can itself be estimated in real time through Eq. (8). The central concept of our Virtual Window Center-of-Mass (VWCM) estimator is to use this information to modify the image window in order to center it on the object and eliminate the estimator bias. The procedure is iterative, such that at the*

_{CM}*n*th iteration, we calculate the center-of-mass

*x*̂

^{(n)}

*then modify the image array by eliminating a portion of the image near one edge, effectively shifting the geometric center*

_{CM}*x*̄

^{(n)}of the window. The update rule that defines our iterative algorithm (explained below) is to truncate the window at the

*n*th iteration such that

*x*̄

_{(n+1)}=

*x*̂

_{(n)}

*.*

_{CM}*x*̄ any desired amount by discarding arbitrarily small portions of the image at one edge. Since we do not have infinitely fine resolution in practice, we may still approximate a sub-pixel truncation of the array by weighting the pixels and pixel coordinates along one edge of the image. For example, if we wish to translate the array center by a small amount

*δ*/2<Δ/2, we simply multiply the pixel intensities

*S*

*at the one edge by 1-*

_{jk}*δ*/Δ and redefine the coordinates along that edge by

*x*→

_{jk}*x*+δ. This procedure approximates the truncation of a region of width

_{jk}*δ*from the negative edge of the image, defining a virtual window shifted by

*δ*/2 as desired:

*x*̄→

*x*̄±

*δ*/2. To complete the algorithm, the user sets two termination conditions,

*ε*and

*n*, such that the algorithm terminates when |

_{max}*x*̄

^{(n+1)}-

*x*̄

^{(n)}|/Δ<

*ε*or the number of iterations exceeds

*n*. With these concepts, we can now precisely describe the VWCM algorithm, displayed graphically in Fig. 1:

_{max}*Virtual window center-of-mass (VWCM) algorithm*

*S*

^{(1)}=

*S*and coordinates (

*x*

^{(1)}

*,*

_{jk}*y*

^{(1)}

*)=(*

_{jk}*x*,

_{jk}*y*) correspond to the raw data.

_{jk}*n*=1, Calculate the center-of-mass

*x*̂

^{(n)}

*from the image*

_{CM}*S*

^{(n)}and coordinates

*x*

^{(n)}

*:*

_{jk}*y*̂

^{(n)}

*.*

_{CM}*S*

^{(n+1)}and new coordinate system (

*x*

^{(n+1)}

*,*

_{jk}*y*

^{(n+1)}

*) by truncating the previous image such that*

_{jk}*x*̄

^{(n+1)}=

*x*̂

^{(n)},

*y*̄

^{(n+1)}=

*y*̂

^{(n)}.

*x*̄

^{(n+1)}-

*x*̄

^{(n)}|<

*ε*or

*n*=

*n*.

_{max}*x*̄

^{(n+1)}=

*x*̂

^{(n)}

*corresponds to a shift of the image that centers the array on the current estimate of the particle position. Denoting the bias in the nth iteration by*

_{CM}*B*

^{(n)}

*=〈*

_{CM}*x*̂

^{(n)}

*〉-*

_{CM}*x*

_{0}, we find from Eq. (8) and the update rule

*x*̄

^{(n+1)}

*=*

_{CM}*x*̂

^{(n)}

_{CM}*〉 and 〈𝓝*

_{S}*〉 are the average number of counts arising from the signal and background respectively. Equation (9) shows that*

_{B}*the bias in the VWCM algorithm tends exponentially to zero with the number of iterations n*. In fact, the signal-to-background ratio 〈𝓝

*〉/〈𝓝*

_{S}*〉 changes at each iteration as the background and signal are truncated differently during the image-shifting procedure. However, the VWCM is designed to truncate more background than signal, so that 〈𝓝*

_{B}*〉/〈𝓝*

_{S}*〉*

_{B}*increases*at each iteration. Equation (9) is therefore a conservative estimate, providing a lower bound on the convergence rate. Regardless, the correction is small, and we have found Eq. (9) to give an accurate prediction of the bias-suppression rate for a wide range of parameters. The convergence rate depends on the signal-to-background ratio 〈𝓝

*〉/〈𝓝*

_{S}*〉, but the algorithm requires no knowledge of this quantity. In fact, the algorithm requires no input beyond the image*

_{B}*S*and pixel coordinates (

*x*,

_{jk}*y*), and termination conditions ε and

_{jk}*n*. Finally, note that the algorithm requires only simple arithmetic operations on the image and is therefore very fast, particularly when the signal-to-background ratio gives a satisfactory convergence rate.

_{max}## 4. Numerical simulations

10. R. Thompson, D. Larson, and W. Webb, “Precise Nanometer Localization Analysis for Individual Fluorescent Probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

*x*

_{0},

*y*

_{0},

*σ*,

_{x}*σ*,

_{y}*A*, and

*B*were the fit parameters. For both the Gaussian mask (see Ref. [10

10. R. Thompson, D. Larson, and W. Webb, “Precise Nanometer Localization Analysis for Individual Fluorescent Probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

*ε*=10

^{-3}and

*n*=200. Finally, we preprocessed every image array

_{max}*S*passed to the algorithms by subtracting the minimum value from the entire array. This simple procedure ensures that the counts in each pixel are nonnegative (a prerequisite for the VWCM algorithm) and removes spuriously large offsets, which severely degrade the CM algorithm, make the Gaussian fit more sensitive to its initial fit conditions, and require more iterations to reach convergence in the VWCM.

*S*.

## 5. Experimental results

*NA*1.2) and separated from the excitation by a dichroic filter. Images were obtained with a CCD camera operating at 30 frames per second. Using a three-axis piezoelectric stage, we displaced particles with nanometer precision, and for each image we located the object using the four algorithms discussed in section 4.

*µ*m at each step. The resulting images and position estimates are shown in Fig. 4. Despite the complicated, asymmetric particle shape, the VWCM algorithm tracks the particle motion with high fidelity. For data of this type, the initial parameters for the Gaussian fit and the mask size for the Gaussian mask algorithm would need to be tailored for each image in order to achieve satisfactory tracking. In contrast, the VWCM requires no adjustment for these (or any other) images.

## 6. Conclusions

## Acknowledgements

## References and links

1. | M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. |

2. | A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science |

3. | X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics,” Science |

4. | D. Weihs, T. G. Mason, and M. A. Teitell, “Bio-Microrheology: A Frontier in Microrheology,” Biophys. J. |

5. | P. Bahukudumbi and M. A. Bevan, “Imaging energy landscapes with concentrated diffusing colloidal probes,” J. Chem. Phys. |

6. | H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, “Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes,” Langmuir |

7. | S. K. Sainis, V. Germain, and E. R. Dufresne, “Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces,” Phys. Rev. Lett. |

8. | N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. |

9. | M. Cheezum, W. Walker, and W. Guilford, “Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles,” Biophys. J. |

10. | R. Thompson, D. Larson, and W. Webb, “Precise Nanometer Localization Analysis for Individual Fluorescent Probes,” Biophys. J. |

11. | J. Crocker and D. Grier, “Methods of Digital Video Microscopy for Colloidal Studies,” J. Colloid Interface Sci. |

12. | S. B. Andersson, “Position estimation of fluorescent probes in a confocal microscope,” in |

13. | T. Sun and S. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in |

14. | B. Carter, G. Shubeita, and S. Gross, “Tracking single particles: a user-friendly quantitative evaluation,” Phys. Biol. |

15. | V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. |

16. | A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles,” Opt. Express |

17. | M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” in |

18. | A. E. Cohen and W. E. Moerner, “Method for Trapping and Manipulating Nanoscale Objects in Solution,” Appl. Phys. Lett. |

19. | H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. |

20. | K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. |

21. | M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. |

22. | C. W. Gardiner, |

23. | N. G. van Kampen, |

24. | L. Novotny and B. Hecht, |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 19, 2008

Revised Manuscript: August 1, 2008

Manuscript Accepted: August 21, 2008

Published: August 26, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Andrew J. Berglund, Matthew D. McMahon, Jabez J. McClelland, and J. A. Liddle, "Fast, bias-free algorithm for tracking
single particles with variable size and
shape," Opt. Express **16**, 14064-14075 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-18-14064

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### References

- M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomol. Struct. 26, 373-399 (1997). [CrossRef] [PubMed]
- A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003). [CrossRef] [PubMed]
- X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005). [CrossRef] [PubMed]
- D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006). [CrossRef] [PubMed]
- P. Bahukudumbi and M. A. Bevan, "Imaging energy landscapes with concentrated diffusing colloidal probes," J. Chem. Phys. 126, 244702 (2007). [CrossRef] [PubMed]
- H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006). [CrossRef] [PubMed]
- S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007). [CrossRef] [PubMed]
- N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152 (1986). [CrossRef]
- M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001). [CrossRef] [PubMed]
- R. Thompson, D. Larson, and W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys. J. 82, 2775-2783 (2002). [CrossRef] [PubMed]
- J. Crocker and D. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Sci. 179, 298-310 (1996). [CrossRef]
- S. B. Andersson, "Position estimation of fluorescent probes in a confocal microscope," in Proceedings of IEEE Conference on Decision and Control (IEEE, 2007) pp. 2445-2450.
- T. Sun and S. Andersson, "Precise 3-D localization of fluorescent probes without numerical fitting," in Proceedings of IEEE Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181-4184.
- B. Carter, G. Shubeita, and S. Gross, "Tracking single particles: a user-friendly quantitative evaluation," Phys. Biol. 2, 60-72 (2005). [CrossRef] [PubMed]
- V. Levi, Q. Ruan, and E. Gratton, "3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells," Biophys. J. 88, 2919-2928 (2005). [CrossRef] [PubMed]
- A. J. Berglund and H. Mabuchi, "Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles," Opt. Express 13, 8069-8082 (2005). [CrossRef] [PubMed]
- M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control and micro-fluidics to steer individual particles," in Proceedings of IEEE Conference on Micro Electro Mechanical Systems (IEEE, 2005), pp. 855-858.
- A. E. Cohen and W. E. Moerner, "Method for Trapping and Manipulating Nanoscale Objects in Solution," Appl. Phys. Lett. 86, 093109 (2005). [CrossRef]
- H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006). [CrossRef]
- K. McHale, A. J. Berglund, and H. Mabuchi, "Quantum dot photon statistics measured by three-dimensional particle tracking," Nano Lett. 7, 3535-3539 (2007). [CrossRef] [PubMed]
- M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006). [CrossRef]
- C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).
- N. G. van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., 2001).
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

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