## Fast feature identification for holographic tracking: the orientation alignment transform |

Optics Express, Vol. 22, Issue 11, pp. 12773-12778 (2014)

http://dx.doi.org/10.1364/OE.22.012773

Acrobat PDF (20413 KB)

### Abstract

The concentric fringe patterns created by features in holograms may be associated with a complex-valued orientational order field. Convolution with an orientational alignment operator then identifies centers of symmetry that correspond to the two-dimensional positions of the features. Feature identification through orientational alignment is reminiscent of voting algorithms such as Hough transforms, but may be implemented with fast convolution methods, and so can be orders of magnitude faster.

© 2014 Optical Society of America

1. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. **45**(16), 3893–3901 (2006). [CrossRef] [PubMed]

2. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505–1512 (2007). [CrossRef] [PubMed]

3. S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single
colloidal particles with video holographic
microscopy,” Opt. Express **15**, 18275–18282
(2007). [CrossRef]

4. F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. **92**, 95–99 (2009). [CrossRef]

5. F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta **48**, 109–115 (2009). [CrossRef]

6. G. Bolognesi, S. Bianchi, and R. Di Leonardo, “Digital holographic tracking of microprobes for multipoint viscosity measurements,” Opt. Express **19**, 19245–19254 (2011). [CrossRef] [PubMed]

7. F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter **7**, 6816–6819 (2011). [CrossRef]

8. H. Shpaisman, B. J. Krishnatreya, and D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. **101**, 091102 (2012). [CrossRef]

9. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

10. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express **19**, 4393–4398 (2011). [CrossRef] [PubMed]

9. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

11. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. **101**, 128301 (2008). [CrossRef]

13. J. Fung and V. N. Manoharan, “Holographic measurements of anisotropic three-dimensional diffusion of colloidal clusters,” Phys. Rev. E **88**, 020302 (2013). [CrossRef]

14. 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**, 8051–8065 (2011). [CrossRef] [PubMed]

9. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

1. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. **45**(16), 3893–3901 (2006). [CrossRef] [PubMed]

2. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505–1512 (2007). [CrossRef] [PubMed]

*λ*= 447 nm). Light scattered by the sphere interferes with the rest of the beam in the focal plane of a microscope objective (Nikon Plan Apo

*λ*, 100× oil immersion, numerical aperture 1.45). The objective, in combination with a tube lens, relays the interference pattern to a video camera (NEC TI-324A II) with an effective magnification of 135 nm/pixel. The intensity distribution recorded by the video camera is normalized by a background image [3

3. S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single
colloidal particles with video holographic
microscopy,” Opt. Express **15**, 18275–18282
(2007). [CrossRef]

**17**, 13071–13079 (2009). [CrossRef] [PubMed]

*b*(

**r**).

15. D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. **13**, 111–122 (1981). [CrossRef]

*𝒪*{

*N*

^{4}} in the number

*N*of pixels on the side of an

*N*×

*N*image [15

15. D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. **13**, 111–122 (1981). [CrossRef]

*𝒪*{

*N*

^{3}log

*N*} [16

16. C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. **24**, 683–694 (2013). [CrossRef]

*b*(

**r**)|, of the image’s gradient. Each pixel in the gradient image, ∇

*b*(

**r**), is associated with a direction, relative to the image’s

*x̂*axis. Figure 1(c) shows

*ϕ*(

**r**) for the image in Fig. 1(a). Each pixel therefore offers information that the center of a feature might lie somewhere along direction

*ϕ*(

**r**) relative to its position

**r**. Voting algorithms [9

**17**, 13071–13079 (2009). [CrossRef] [PubMed]

17. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

18. R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods **9**, 724–726 (2012). [CrossRef] [PubMed]

**17**, 13071–13079 (2009). [CrossRef] [PubMed]

18. R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods **9**, 724–726 (2012). [CrossRef] [PubMed]

*𝒪*{

*N*

^{3}}. Achieving this efficiency involves first identifying pixels with the strongest gradients, typically by imposing a threshold on |

*b*(

**r**)|.

*b*(

**r**) may be described with the two-fold orientational order parameter [19

19. B. I. Halperin and D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. **41**(2), 121–124 (1978). [CrossRef]

20. D. R. Nelson and B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B **19**(5), 2457–2484 (1979). [CrossRef]

*b*(

**r**)|

^{2}emphasizes contributions from regions with stronger gradients.

*ψ*(

**r**) with the two-fold symmetric transformation kernel, to obtain the orientation alignment transform The phase of

*K*(

**r**) complements the phase of

*ψ*(

**r**), as can be seen in the inset to Fig. 1(c). The integrand of Eq. (4) therefore is real-valued and non-negative along the line

**r′**−

**r**that is oriented along

*θ*=

*ϕ*(

**r′**), and is complex-valued along other directions. Real-valued contributions directed along gradients of

*b*(

**r**) accumulate at points

**r**in Ψ(

**r**) that are centers of symmetry of the gradient field, as illustrated schematically in the inset to Fig. 1(d). Complex-valued contributions, by contrast, tend to cancel out. Centers of symmetry in

*b*(

**r**) therefore are transformed into centers of brightness in

*B*(

**r**) = |Ψ(

**r**)|

^{2}, as can be seen in Fig. 1(d). The centroid of the peak then can be identified and located [17

17. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

*b*(

**r**) and thus would tend to have more influence over the position of centers of brightness in

*B*(

**r**). The factor of 1/

*r*in Eq. (3) ensures that all of the fringes in a sphere’s hologram contribute with equal weighting to the estimate for its centroid.

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

*b*(

**r**) are transformed into peaks in

*B*(

**r**) whose locations are identified by crosses superimposed on the original hologram.

*t*= 16.68 ms. The camera’s exposure time, 0.1 ms, is fast enough to avoid artifacts due to the particle’s motion [10

10. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express **19**, 4393–4398 (2011). [CrossRef] [PubMed]

25. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

26. T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

*r*(

_{j}*t*) is the sphere’s position along one of the Cartesian coordinates with

*r*

_{0}(

*t*) =

*x*(

*t*) and

*r*

_{1}(

*t*) =

*y*(

*t*), where

*D*is the diffusion coefficient along that direction, and where

_{j}*ε*is the error in the associated position measurement. Analyzing trajectories with Eq. (8) therefore provides a method to measure tracking errors [17

_{j}17. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

25. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

26. T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

*x̂*and

*ŷ*computed from the trajectories in Fig. 3(a) using Eq. (7). The error bars in Fig. 3(b) reflect statistical uncertainties. Although results along the two directions agree to within these uncertainties, least-squares fits to the Einstein-Smoluchowski prediction in Eq. (8) yield slightly different values for the particle’s diffusion coefficient:

*D*= 0.292 ± 0.002 μm

_{x}^{2}/s and

*D*= 0.281 ± 0.002 μm

_{y}^{2}/s. This discrepancy may be attributed to blurring along the

*ŷ*direction that arises when the even and odd scan lines are extracted from each interlaced video frame. The resulting loss of spatial resolution along

*ŷ*tends to suppress the apparent diffusivity along that direction [25

25. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

26. T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

*D*=

*k*/(6

_{B}T*πηa*) = 0.296 ± 0.002 μm

_{p}^{2}/s for a sphere of radius

*a*= 0.805 ± 0.001 μm [27

_{p}27. B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, and D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. **82**, 23–31 (2014). [CrossRef]

*η*= 0.912 ± 0.005 mPa s at absolute temperature

*T*= 297.1 ± 0.2 K.

*ε*= 8 nm and

_{x}*ε*= 9 nm, or roughly 0.06 pixel in each direction. This performance is comparable to the precision obtained with voting algorithms [9

_{y}**17**, 13071–13079 (2009). [CrossRef] [PubMed]

18. R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods **9**, 724–726 (2012). [CrossRef] [PubMed]

**17**, 13071–13079 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. |

2. | S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express |

3. | S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single
colloidal particles with video holographic
microscopy,” Opt. Express |

4. | F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. |

5. | F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta |

6. | G. Bolognesi, S. Bianchi, and R. Di Leonardo, “Digital holographic tracking of microprobes for multipoint viscosity measurements,” Opt. Express |

7. | F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter |

8. | H. Shpaisman, B. J. Krishnatreya, and D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. |

9. | F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express |

10. | L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express |

11. | Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. |

12. | K. Xiao and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. |

13. | J. Fung and V. N. Manoharan, “Holographic measurements of anisotropic three-dimensional diffusion of colloidal clusters,” Phys. Rev. E |

14. | 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 |

15. | D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. |

16. | C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. |

17. | J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. |

18. | R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods |

19. | B. I. Halperin and D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. |

20. | D. R. Nelson and B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B |

21. | J. Rubinstein, J. Segman, and Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. |

22. | T. J. Atherton and D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. |

23. | A. Savitzky and M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. |

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

25. | T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E |

26. | T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. |

27. | B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, and D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(350.4990) Other areas of optics : Particles

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 25, 2014

Revised Manuscript: May 5, 2014

Manuscript Accepted: May 12, 2014

Published: May 19, 2014

**Citation**

Bhaskar Jyoti Krishnatreya and David G. Grier, "Fast feature identification for holographic tracking: the orientation alignment transform," Opt. Express **22**, 12773-12778 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12773

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### References

- J. Sheng, E. Malkiel, J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45(16), 3893–3901 (2006). [CrossRef] [PubMed]
- S.-H. Lee, D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [CrossRef] [PubMed]
- S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007). [CrossRef]
- F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009). [CrossRef]
- F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009). [CrossRef]
- G. Bolognesi, S. Bianchi, R. Di Leonardo, “Digital holographic tracking of microprobes for multipoint viscosity measurements,” Opt. Express 19, 19245–19254 (2011). [CrossRef] [PubMed]
- F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011). [CrossRef]
- H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012). [CrossRef]
- F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009). [CrossRef] [PubMed]
- L. Dixon, F. C. Cheong, D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express 19, 4393–4398 (2011). [CrossRef] [PubMed]
- Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008). [CrossRef]
- K. Xiao, D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010). [CrossRef]
- J. Fung, V. N. Manoharan, “Holographic measurements of anisotropic three-dimensional diffusion of colloidal clusters,” Phys. Rev. E 88, 020302 (2013). [CrossRef]
- J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011). [CrossRef] [PubMed]
- D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. 13, 111–122 (1981). [CrossRef]
- C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. 24, 683–694 (2013). [CrossRef]
- J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
- R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods 9, 724–726 (2012). [CrossRef] [PubMed]
- B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. 41(2), 121–124 (1978). [CrossRef]
- D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19(5), 2457–2484 (1979). [CrossRef]
- J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991). [CrossRef]
- T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999). [CrossRef]
- A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).
- D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
- T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005). [CrossRef]
- T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005). [CrossRef]
- B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014). [CrossRef]

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