## Motion analysis of optically trapped particles and cells using 2D Fourier analysis |

Optics Express, Vol. 20, Issue 3, pp. 1953-1962 (2012)

http://dx.doi.org/10.1364/OE.20.001953

Acrobat PDF (1686 KB)

### Abstract

Motion analysis of optically trapped objects is demonstrated using a simple 2D Fourier transform technique. The displacements of trapped objects are determined directly from the phase shift between the Fourier transform of subsequent images. Using end- and side-view imaging, the stiffness of the trap is determined in three dimensions. The Fourier transform method is simple to implement and applicable in cases where the trapped object changes shape or where the lighting conditions change. This is illustrated by tracking a fluorescent particle and a myoblast cell, with subsequent determination of diffusion coefficients and the trapping forces.

© 2012 OSA

## 1. Introduction

1. J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics **5**(1), 7–8 (2011). [CrossRef]

2. D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef] [PubMed]

3. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**(13), 13563–13573 (2010). [CrossRef] [PubMed]

6. R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express **18**(11), 11785–11790 (2010). [CrossRef] [PubMed]

7. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

8. B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. **5**(8), 1266–1271 (1996). [CrossRef] [PubMed]

10. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. **33**(2), 156–158 (2008). [CrossRef] [PubMed]

12. D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. **51**(3), 249–265 (2010). [CrossRef]

14. P. J. Rodrigo, I. R. Perch-Nielsen, C. A. Alonzo, and J. Glückstad, “GPC-based optical micromanipulation in 3D real-time using a single spatial light modulator,” Opt. Express **14**(26), 13107–13112 (2006). [CrossRef] [PubMed]

15. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**(9), 2787–2809 (2004). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Fourier transform motion analysis

*F*

_{1}(

*k*) and

_{x},k_{y}*F*

_{2}(

*k*)

_{x},k_{y}*x*,Δ

*y*) of the particle from one frame to the next frame is calculated as the slope of the phase plane

## 3. Experimental results

### 3.1. Tracking of a 10 µm polystyrene bead

*-*view, showing the motion transversal to the trapping lasers (

*x*,

*y*), and in side

*-*view, showing motion along the

*z-*axis and

*x-*axis. The

*x-*axis is thus observed in both side

*-*and end

*-*view. The frame rate of the cameras was set to 400 Hz with a region of interest (ROI) of 576 × 576 pixels, and the two cameras are synchronized to within 17 nanoseconds. The method used to calibrate the imaging system, as well as more thorough description of the experimental setup, is described in previous work [13]. The motion of the trapped particle is followed for 600 seconds resulting in 240000 frames to be analyzed per camera. Using the FTMA method, we obtain the relative displacement of the particle between two successive frames.

*x-*component of the displacement vector (Δ

*x*,Δ

*y*) obtained from Eq. (3). The inset shows an example of the phase plane obtained from two consecutive images at 0.12 seconds. The phase plane has a falling slope pointing along the negative

*k*axis as indicated by the change in color from blue to yellow/orange in the inset. The measured displacements correspond to a fraction of a pixel, equivalent to ± 50 nm. In Fig. 1, we compared the FTMA method to a reference tracking algorithm (blue dots) where the particle position is obtained by convoluting each image with a filter function specific to the trapped particle [11]. Subsequently, the displacement is calculated by subtracting neighboring positions. Very good agreement between the two tracking methods is evident from Fig. 1. One can quantify the agreement by first assuming that the displacement obtained by the reference tracking algorithm is exact and then subtracting the displacement obtained from the two methods. This gives a Gaussian distribution centered on zero displacement with a standard deviation

_{x}-*σ*= 8 nm. The accuracy of the reference tracking method is described in more detail in [13] and is limited by the camera resolution, mechanical drift of the setup and the tracking algorithm. This amounts to an accuracy limited to a standard deviation

*σ*= 5 nm. In the present case, where a simple object is tracked under optimum lightning condition, the accuracy of the FTMA method is thus comparable to that of the reference method. In general, however, the accuracy of both methods is sensitively dependent on the experimental conditions. We have performed numerical tests of the FTMA method, investigating the standard error in determining the slope of the phase plane as function of the signal to noise ration of the image. The slope is directly proportional to the displacement, and we find that the standard error is inversely proportional to the signal to noise ratio of the image.

19. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**(3), 594–612 (2004). [CrossRef]

*x*- and

*y-*axis are

*κ*= 0.72(1) pN/µm and

_{x}*κ*= 0.73(1) pN/µm. The trap stiffness along the

_{y}*z-*axis is very low (

*κ*< 0.1 pN/µm) and will not be considered here. The diffusion coefficient was determined to be

_{z}*D*= 0.045(1) µm

^{2}s. When the particle is tracked using the FTMA method, we sample the displacement Δ =

*x*

_{i + 1}−

*x*

_{i}between two consecutive frames. The sampled displacement refers to the particle’s thermal (Brownian) motion inside the trapping potential defined by the two trapping laser beams. By solving the Langevin equation one can show that the Gaussian variance of the displacement is given bywhere

*D*is the diffusion coefficient given by

*D = k*

_{B}

*T/γ*,

*γ*is the friction coefficient,

*k*

_{B}is the Boltzmann constant, and

*T*is temperature.

*τ*is the sampling time given by

*τ*= 1/

*fps*, where

*fps*is the frame rate of the camera and

*f*is the corner frequency given by

_{c}*f*2π

_{c}= κ/*γ*, where

*κ*is the trap stiffness. The first term in Eq. (4) is equal to that of a free particle and the second, in the present case, negligible term is (through the corner frequency) dependent on the trapping potential. Consequently, the trap stiffness is poorly determined from the distribution of the particle displacements.

*x*−axis and Fig. 2(b) shows the corresponding power spectrum of the displacement, both obtained through the FTMA method.

*x*

^{2}> = 1.93 × 10

^{−4}µm

^{2}shown in Fig. 2(c), we obtain a diffusion coefficient of

*D*= 0.039 µm

^{2}s

^{−1}in fair agreement with the value

*D*= 0.045 µm

^{2}s

^{−1}obtained using the reference tracking method [11]. Similarly, we obtain a corner frequency of

*f*= 1.28 Hz, from the power spectrum of the displacement, corresponding to a trap stiffness of 0.75 pN/µm in excellent agreement with the value obtained using the reference tracking method. Using the FTMA method, we are thus able to retrieve the parameters describing the motion of the trapped bead, trap stiffness and diffusion coefficient, in full agreement with the values obtained using the reference tracking method.

_{c}### 3.2. Tracking of fluorescent beads

*x*,

*y*) at 100 fps and a light beam, travelling along the

*y-*axis from a supercontinuum light source (NKT Photonics, SuperK) was used to supply 1 mW of 580 nm light on the trapped bead at

*t*= 0.6 seconds. The displacements of the particle along the

*x-*axis and the

*y-*axis are shown in Fig. 3(b) as a function of time. The simultaneously measured fluorescence intensity is shown in Fig. 3(a). When the light source is turned on, the absorbed photons push the bead along the

*y-*axis, but motion along the

*x-*axis is also detected. Around

*t*= 0.6 seconds when the fluorescent laser is turned on, the approximation that the spectral shape of two consecutive images should remain constant is not valid. This could contribute to the transient motion observed along the

*x-*axis. However, the actual motion of the fluorescent sphere is a result of a non-trivial combination of the momentum transfer from the absorbed and emitted photons, bleaching of the dye, and the 3D characteristics of the optical trap. This will be described in more detail in a forthcoming paper. Tracking of the particle position with the reference method is not possible as a result of the abrupt changes in shape and illumination of the object.

### 3.3. Tracking of myoblast cells

*graft*-PEG (PLL-

*g*-PEG) solution was employed to coat the chamber’s surface since it spontaneously adsorbs onto negatively charged surfaces forming a stable polymeric monolayer [20

20. S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir **19**(22), 9216–9225 (2003). [CrossRef]

*k*= 0 µm

_{x}= k_{z}^{−1}in combination with a low pass filter that suppresses the high frequency noise in the image (

*k*> 0.5 µm

_{x}= k_{y}^{−1}). Subsequently, a threshold filter was used to suppress low amplitude frequencies, which are typically associated with noise. An example of the images is shown in Fig. 4 , where also the DC-, low pass-, and threshold filter are indicated. From the analysis we obtain the displacement of the cell along the

*x*- and

*y*-directions (end-view) and the displacement along the

*x*- and

*z*-directions (side-view). The

*x*-axis data from the side- and end-views show the expected correlation.

*x*-axis displacement (a) and the statistical analysis (b). The distribution of the displacements is Gaussian with a variance given in Eq. (4). From the analysis we obtain a diffusion coefficient of

*D*= 0.015 µm

^{2}s

^{−1}. This is less than half the value of the diffusion coefficient of the polystyrene sphere, in accordance with the fact that the myoblast cell is approximately twice the size of the polystyrene sphere. In the frequency analysis we determine a corner frequency along the

*x*-axis equal to

*f*= 0.62 Hz. From the corner frequency we obtain the trap stiffness for the

_{c}*x*-axis,

*κ*= 1.07 pN/µm.

_{x}*z*-axis, the optical trap is approximately a factor of 10 weaker than for the other two axes. However, due to the low frequency noise we are therefore unable to determine the corner frequency for the

*z*-axis. This is, however, a consequence of the properties of our trap rather than the FTMA method.

## 4. Discussion

8. B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. **5**(8), 1266–1271 (1996). [CrossRef] [PubMed]

*x*,Δ

*y*) as evident from Eq. (6). For multiple objects the function

*g*will contain several peaks, corresponding to the motion of the center of gravity of the particles and the motion of the individual particles relative to the center of gravity. Several factors, such as computational speed, noise, and resolution, influence the applicability of the FTMA method for multiple objects and these are presently being investigated. The FTMA method can also be used to obtain 3D motion. In the present case, where both end-view and side-view images are obtained, the method can be used to track the 3D displacement of the object. If a full 3D representation of the objects is available, the image can be transformed to spherical coordinates, and the FTMA method can be employed to detect angular displacement of the objects [8

8. B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. **5**(8), 1266–1271 (1996). [CrossRef] [PubMed]

21. G. M. Cortelazzo, L. Lucchese, and C. M. Monti, “Frequency domain analysis of general planar rigid motion with finite duration,” J. Opt. Soc. Am. A **16**(6), 1238–1253 (1999). [CrossRef]

## Acknowledgment

## References and links

1. | J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics |

2. | D. G. Grier, “A revolution in optical manipulation,” Nature |

3. | F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express |

4. | J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express |

5. | O. Otto, F. Czerwinski, J. L. Gornall, G. Stober, L. B. Oddershede, R. Seidel, and U. F. Keyser, “Real-time particle tracking at 10,000 fps using optical fiber illumination,” Opt. Express |

6. | R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express |

7. | J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. |

8. | B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. |

9. | J. P. Staforelli, E. Vera, J. M. Brito, P. Solano, S. Torres, and C. Saavedra, “Superresolution imaging in optical tweezers using high-speed cameras,” Opt. Express |

10. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. |

11. | C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” Proc. Int. Conf. Cybernetics Society |

12. | D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. |

13. | T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. |

14. | P. J. Rodrigo, I. R. Perch-Nielsen, C. A. Alonzo, and J. Glückstad, “GPC-based optical micromanipulation in 3D real-time using a single spatial light modulator,” Opt. Express |

15. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

16. | H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote |

17. | M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the Fourier domain,” IEEE Trans. Image Process. |

18. | T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

19. | K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. |

20. | S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir |

21. | G. M. Cortelazzo, L. Lucchese, and C. M. Monti, “Frequency domain analysis of general planar rigid motion with finite duration,” J. Opt. Soc. Am. A |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.0100) Image processing : Image processing

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: November 23, 2011

Revised Manuscript: January 2, 2012

Manuscript Accepted: January 2, 2012

Published: January 13, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Martin Verner Kristensen, Peter Ahrendt, Thue Bjerring Lindballe, Otto Højager Attermann Nielsen, Anton P. Kylling, Henrik Karstoft, Alberto Imparato, Leticia Hosta-Rigau, Brigitte Stadler, Henrik Stapelfeldt, and Søren Rud Keiding, "Motion analysis of optically trapped particles and cells using 2D Fourier analysis," Opt. Express **20**, 1953-1962 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-3-1953

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### References

- J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics 5(1), 7–8 (2011). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
- F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18(13), 13563–13573 (2010). [CrossRef] [PubMed]
- J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19(9), 8051–8065 (2011). [CrossRef] [PubMed]
- O. Otto, F. Czerwinski, J. L. Gornall, G. Stober, L. B. Oddershede, R. Seidel, and U. F. Keyser, “Real-time particle tracking at 10,000 fps using optical fiber illumination,” Opt. Express 18(22), 22722–22733 (2010). [CrossRef] [PubMed]
- R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express 18(11), 11785–11790 (2010). [CrossRef] [PubMed]
- J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001). [CrossRef] [PubMed]
- B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996). [CrossRef] [PubMed]
- J. P. Staforelli, E. Vera, J. M. Brito, P. Solano, S. Torres, and C. Saavedra, “Superresolution imaging in optical tweezers using high-speed cameras,” Opt. Express 18(4), 3322–3331 (2010). [CrossRef] [PubMed]
- M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33(2), 156–158 (2008). [CrossRef] [PubMed]
- C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” Proc. Int. Conf. Cybernetics Society 163–5 (1975).
- D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. 51(3), 249–265 (2010). [CrossRef]
- T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).
- P. J. Rodrigo, I. R. Perch-Nielsen, C. A. Alonzo, and J. Glückstad, “GPC-based optical micromanipulation in 3D real-time using a single spatial light modulator,” Opt. Express 14(26), 13107–13112 (2006). [CrossRef] [PubMed]
- K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef] [PubMed]
- H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001). [CrossRef]
- M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the Fourier domain,” IEEE Trans. Image Process. 15(7), 1965–1972 (2006). [CrossRef] [PubMed]
- T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000). [CrossRef] [PubMed]
- K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004). [CrossRef]
- S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003). [CrossRef]
- G. M. Cortelazzo, L. Lucchese, and C. M. Monti, “Frequency domain analysis of general planar rigid motion with finite duration,” J. Opt. Soc. Am. A 16(6), 1238–1253 (1999). [CrossRef]

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