## High-speed autofocusing of a cell using diffraction patterns

Optics Express, Vol. 14, Issue 9, pp. 3952-3960 (2006)

http://dx.doi.org/10.1364/OE.14.003952

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### Abstract

This paper proposes a new autofocusing method for observing cells under a transmission illumination. The focusing method uses a quick and simple focus estimation technique termed “depth from diffraction,” which is based on a diffraction pattern in a defocused image of a biological specimen. Since this method can estimate the focal position of the specimen from only a single defocused image, it can easily realize high-speed auto-focusing. To demonstrate the method, it was applied to continuous focus tracking of a swimming paramecium, in combination with two-dimensional position tracking. Three-dimensional tracking of the paramecium for 70 s was successfully demonstrated.

© 2006 Optical Society of America

## 1. Introduction

1. J. H. Price and D. A. Gough, “Comparison of Phase-Contrast and Fluorescence Digital Autofocus for Scanning Microscopy,” Cytometry, **16**, 283–297 (1994). [CrossRef]

2. M. Subbarao and J.-K. Tyan, “Selecting the Optimal Focus Measure for Autofocusing and Depth-From-Focus,” IEEE Trans. Patternn Anal. Mach. Intell., **20**, 864–870 (1998). [CrossRef]

3. J.-M. Geusebroek, F. Cornelissen, A. W. M. Smeulders, and H. Greets, “ Robust Autofocusing in icroscopy,” Cytometry , **39**, 1–9 (2000). [CrossRef] [PubMed]

3. J.-M. Geusebroek, F. Cornelissen, A. W. M. Smeulders, and H. Greets, “ Robust Autofocusing in icroscopy,” Cytometry , **39**, 1–9 (2000). [CrossRef] [PubMed]

## 2. Depth-from-diffraction method

## 3. Experiments

### 3.1. Experimental set-up

*Paramecium caudatum*cells were adopted as specimens, which were cultured at 25 °C in a soy flour solution. Cells were collected together with the solution, filtered through a nylon mesh to remove debris, washed with mineral water without gas, and infused into a chamber. The chamber was a small tank with dimensions 15 mm × 15 mm × 150

*μ*m, made of a glass slide forming the base and pieces of cut cover glasses forming the side walls, which were bonded together using optical cement.

### 3.2. Extraction of diffraction pattern features

### 3.3. Profile measurement of the diffraction pattern features

*z*> 0 means the specimen is closer to the objective lens than the focal plane. The indexes were measured while the specimen was scanned from

*z*= - 50

*μ*m to

*z*= 50

*μ*m in 2 s. The frame rate was 1000 samples/s and a total of 2000 indexes were measured.

*z*> 0, the index of the outer fringe has a certain value, while that of the inner fringe is almost zero. In contrast, for

*z*< 0, the index of the inner fringe is larger than that of the outer fringe. In the region 0 ≤

*z*≤ 15

*μ*m, the index of the outer fringe is almost linearly dependent on

*z*, whereas in the region - 15 ≤

*z*≤ 0, the index of the inner fringe is almost linearly dependent on

*z*. Outside of these regions, there is no linear relationship. This is because, as the specimen moves away from the focal plane, the bright fringe becomes larger and exceeds the size of a ring-shaped mask used in the image processing.

*z*coordinate was estimated using the measured indexes of the bright fringes. Assuming that

*z*linearly depends on the index of the outer fringe when

*z*≥ 0 and the index of the inner fringe when

*z*< 0,

*z*could be estimated using

*z*̂ is the estimated

*z*,

*i*

_{outer}and

*i*

_{inner}are the indexes of the outer and inner fringes, and

*a*

_{1},

*a*

_{0},

*b*

_{1}, and

*b*

_{0}are coefficients calculated by fitting the measured profile of the indexes using a least squares method. The sets of coefficients (

*a*

_{1},

*a*

_{0}) and (

*b*

_{1},

*b*

_{0}) were calculated for 0 ≤

*z*8.0

*μ*m and -4.0

*μ*m <

*z*< 0, respectively.

*z*using Eq. (1), the sign of

*z*should be estimated first to determine which equation should be applied. According to the measured profile of the indexes, the index of the inner fringe was almost 0 when

*z*was positive. Thus, the sign of

*z*was estimated from the index of the inner fringe.

*z*< 15

*μ*m.

*μ*m, the sign of the specimen’s

*z*position can be estimated. When the specimen’s z position exceeds this range, its position cannot be estimated using the depth-from-diffraction method. In this case, other scanning-type focusing methods must be adopted.

### 3.4. Dynamic focusing applied to three-dimensional tracking of a swimming paramecium

*z*

_{d}, was determined to track the specimen on the focal plane using the specimen’s estimated z position. In this experiment,

*z*was given by

_{d}*z*is the current position on the Z-axis,

*i*

_{outer}and

*i*

_{inner}are, respectively, the indexes of the outer and inner fringes,

*k*(

*i*

_{outer}-

*i*

_{inner}) is the specimen’s estimated z position, and

*k*is a gain parameter used in the estimation, which is determined in ad-hoc way. Eq. (2) is based on the rough assumption that the specimen’s position is linearly dependent on the difference between the two indexes. This is an ad-hoc assumption which worked well for the purpose of focus tracking.

9. H. Oku, N. Ogawa, K. Hashimoto, and M. Ishikawa, “ Two-dimensional tracking of a motile microorganism allowing high-resolution observation with various imaging techniques,” Rev. Sci. Instrum., **76**, 034301 (2005). [CrossRef]

## 4. Discussion

*g*(

*ξ*,

*η*). The complex field across the

*z*= 0 plane placed just behind the object is represented by

*U*(

*x,y,z*). The object is observed using a microscope objective lens placed behind the object. Here, to simplify the problem, we assume that the magnification of the objective lens is 1, and the lens can perfectly project the inverted intensity distribution at the focal plane onto the image plane. That is, we neglected the aberrations and the effect of diffraction of the objective lens. Then, the intensity distribution

*I*(

*x,y*) of the image plane can be denoted as

*I*

_{0}is a constant intensity depending on the illumination light intensity. Note that this equation is valid also for negative z, that is, when the object is placed behind the focal plane. When the diffraction pattern

*I*(

*x,y*) depends uniquely on

*z*around

*z*= 0, the depth of the object can be estimated by extracting the features of the diffraction pattern, as described above.

*ξ*< 0, the phase and amplitude transmittance function

*g*(

*ξ*,

*η*) can be denoted as

*z*, we can estimate the absolute

*z*from the distance between intensity peaks.

*z*< 0 is the reversed profile of that for

*z*> 0. Thus, the sign of

*z*can be estimated when the object is a phase knife-edge. However, the diffraction patterns of the amplitude knife-edge for

*z*> 0 and

*z*< 0 are identical, which means that the sign of

*z*cannot be estimated in this case. These results show that the proposed method cannot be applied to all types of microscopic specimens. For example, typical micro electro mechanical systems (MEMS) on a common silicon substrate are opaque, and they cannot be focused using this method.

*z*can be estimated from the diffraction pattern. This indicates that we can estimate the positional relationship between the focal plane and the object from a single defocused image of the object when different patterns are formed depending on

*z*. This result suggests that the depth-from-diffraction method could be applied to not only spherical cells, as described in the first paragraph of this section, but also to flat transparent specimens, such as amoebae and squamous cells. However, the cell types that are suitable for use in this method remain the subject of a future study.

*z*, not √

*z*. The reason should be the coarse pixels of the imager. When

*z*is small, the distance between bright and dark fringe is always less than one pixel. Thus, the imager is less sensitive about the change of the distance between fringes. This should straighten the profile near

*z*= 0.

## 5. Conclusion

## References and links

1. | J. H. Price and D. A. Gough, “Comparison of Phase-Contrast and Fluorescence Digital Autofocus for Scanning Microscopy,” Cytometry, |

2. | M. Subbarao and J.-K. Tyan, “Selecting the Optimal Focus Measure for Autofocusing and Depth-From-Focus,” IEEE Trans. Patternn Anal. Mach. Intell., |

3. | J.-M. Geusebroek, F. Cornelissen, A. W. M. Smeulders, and H. Greets, “ Robust Autofocusing in icroscopy,” Cytometry , |

4. | M. Born and E. Wolf, |

5. | W. D. Nesse, |

6. | T. Tsuruta. |

7. | H. Toyoda, N. Mukohzaka, K. Nakamura, M. Takumi, S. Mizuno, and M. Ishikawa, “1ms column-parallel vision system coupled with an image intensifier; I-CPV,” in |

8. | Y. Nakabo, M. Ishikawa, H. Toyoda, and S. Mizuno, “1ms column parallel vision system and it’s application of high speed target tracking,” in |

9. | H. Oku, N. Ogawa, K. Hashimoto, and M. Ishikawa, “ Two-dimensional tracking of a motile microorganism allowing high-resolution observation with various imaging techniques,” Rev. Sci. Instrum., |

10. | J. W. Goodman, |

**OCIS Codes**

(100.5010) Image processing : Pattern recognition

(170.5810) Medical optics and biotechnology : Scanning microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: February 27, 2006

Revised Manuscript: April 12, 2006

Manuscript Accepted: April 13, 2006

Published: May 1, 2006

**Virtual Issues**

Vol. 1, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Hiromasa Oku, Masatoshi Ishikawa, Theodorus , and Koichi Hashimoto, "High-speed autofocusing of a cell using diffraction pattern," Opt. Express **14**, 3952-3960 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3952

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### References

- J. H. Price and D. A. Gough, "Comparison of Phase-Contrast and Fluorescence Digital Autofocus for Scanning Microscopy," Cytometry 16, 283-297 (1994). [CrossRef]
- M. Subbarao and J.-K. Tyan, "Selecting the Optimal Focus Measure for Autofocusing and Depth-From-Focus," IEEE Trans. Patternn Anal. Mach. Intell. 20, 864-870 (1998). [CrossRef]
- J.-M. Geusebroek, F. Cornelissen, A. W. M. Smeulders, and H. Greets, " Robust Autofocusing in Microscopy," Cytometry 39, 1-9 (2000). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 7th Edition (Cambridge University Press, Cambridge, 2002).
- W. D. Nesse, Introduction to Optical Mineralogy. (Oxford University Press, New York, 1991).
- T. Tsuruta. Oyo-kogaku (Applied Optics) I. (Baifukan, Tokyo, 1990). (in Japanese)
- H. Toyoda, N. Mukohzaka, K. Nakamura, M. Takumi, S. Mizuno, and M. Ishikawa, "1ms column-parallel vision system coupled with an image intensifier; I-CPV," in Proceedings of Symp. High Speed Photography and Photonics 2001, vol. 5-1, 2001, pp. 89-92 (in Japanese).
- Y. Nakabo, M. Ishikawa, H. Toyoda, and S. Mizuno, "1ms column parallel vision system and it’s application of high speed target tracking," in Proceedings of the IEEE International Conference on Robotics & Automation (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 650-655.
- H. Oku, N. Ogawa, K. Hashimoto, and M. Ishikawa, " Two-dimensional tracking of a motile microorganism allowing high-resolution observation with various imaging techniques," Rev. Sci. Instrum. 76, 034301 (2005). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics. (McGraw-Hill, Inc., Boston, Massachusetts, 1996).

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