## Multiple contrast metrics from the measurements of a digital confocal microscope |

Biomedical Optics Express, Vol. 4, Issue 7, pp. 1091-1103 (2013)

http://dx.doi.org/10.1364/BOE.4.001091

Acrobat PDF (4578 KB)

### Abstract

We describe various methods to process the data collected with a digital confocal microscope (DCM) in order to get more information than what we could get from a conventional confocal system. Different metrics can be extracted from the data collected with the DCM in order to produce images that reveal different features of the sample. The integrated phase of the scattered field allows for the three-dimensional reconstruction of the refractive index distribution. In a similar way, the integration of the field intensity yields the absorption coefficient distribution. The deflection of the digitally reconstructed focus reveals the sample-induced aberrations and the RMS width of the focus gives an indication on the local scattering coefficient. Finally, in addition to the conventional confocal metric, which consists in integrating the intensity within the pinhole, the DCM allows for the measurement of the phase within the pinhole. This metrics is close to the whole-field integrated phase and thus gives a qualitative image of the refractive index distribution.

© 2013 OSA

## 1. Introduction

2. C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta **24**(10), 1051–1073 (1977) [CrossRef] .

4. A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature **351**, 551–553 (1991) [CrossRef] .

5. G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. **24**, 811–813 (1999) [CrossRef] .

8. A. Goy and D. Psaltis, “Digital confocal microscope,” Opt. Express **20**, 22720–22727 (2012) [CrossRef] [PubMed] .

## 2. Digital data processing

9. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt. **39**, 4070–4075 (2000) [CrossRef] .

*z*coordinate) in the +

*z*and −

*z*direction using a standard Fourier beam propagation method [10

10. M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B **5**(3), 633–640 (1988) [CrossRef] .

*x*,

*y*and

*z*in the Fourier-transformed space depend on the focal length of the virtual tube lens of the digital microscope. Because this length is arbitrary, we simply consider that the imaging system has a magnification of one.

*z*

_{opt}of the effective focus along the

*z*axis, is where the 2D (

*x*−

*y*) correlation function between the ideal focus and the experimental beam reaches its maximum: where

*u*is the focus spot at the waist of the ideal probe beam,

_{i}*u*is the measured field and ★ denotes the correlation. In Fig. 2(d), we compare the

_{m}*c*(

*z*) function for an ideal beam and a scattered beam. The shift in

*z*, Δ

*z*, is defined by the distance between the maxima of these functions and is equal to

*z*

_{opt}by definition. The shift in

*x*and

*y*, Δ

*x*and Δ

*y*, are defined by the position of the center of mass of the experimental focus in the plane

*z*=

*z*

_{opt}: We define the width of the experimentally measured focus as: where the integral runs over the entire image plane. For an ideal Gaussian beam,

*w*is equal to the 1/

*e*-radius (in amplitude). The confocal signal through the dynamic pinhole is defined as: where

*p*(

*x*,

*y*) is the pinhole aperture function and

*a*is the pinhole diameter. Finally, the phase of the confocal signal (confocal phase) can also be measured. For this, we average the phase of the detected signal within the pinhole:

## 3. Results

*x*position of the focused spot with respect to the center of the nominal pinhole (i.e. the pinhole centered on an ideal beam not affected by scattering). Comparing Fig. 3(b) (the

*x*-image) with Fig. 3(a), we can clearly see that the basic shape of the cell is reproduced. Similarly, the

*y*-image obtained by plotting the

*y*position and shown in Fig. 3(c) forms a recognizable shape of the cell. In Fig. 3(d) we show the image corresponding to Δ

*z*. The image in Fig. 3(e) is the

*d*

_{⊥}-image obtained by plotting the Euclidian distance

*d*

_{3D}-image with

*d*

_{⊥}-image. At first glance they are contrast-inverted versions of each other. This can be explained as follows. In confocal microscopy, a region of the image is dark if the sample absorbs a lot, reflects or scatters light. In the cells we are imaging in this experiment, there is relatively little absorption or reflection. Therefore, dark areas are those that have some high spatial frequency content and scatter light. These same areas then are likely to have index gradients that steer the beam away from the detection pinhole and therefore a dark signal is plotted. On the

*d*

_{⊥}-image a large deflection from the pinhole translates into a bright pixel on the reconstructed image. On the other hand, a clear sample simply leaves the beam unaffected to pass through the digital pinhole. A strong (bright) signal is recorded on the confocal image and a dark signal in the

*d*

_{⊥}-image. This is generally true but other things may affect the system and, therefore, it is only approximately true that the

*d*

_{⊥}-image is the contrast-inverted image of the confocal image. The differences may contain interesting information. In order to explore this possibility, we plotted, in Fig. 4(a), the pixel values of the corresponding points in the

*d*

_{⊥}versus the confocal image. In such a plot if the two images were linear contrast inverted versions of each other then the relationship would be: where

*f̂*

_{DC}is the pixel value for the dynamic digital confocal normalized over its maximum and

*d̂*

_{⊥}the pixel value for the transverse distance normalized over the ideal beam waist

*w*

_{0}. A linear regression on the data of Fig. 4(a) is used to fit the model expressed by Eq. (7). We get

*α*= −2.13 and

*β*= −

*α*as the line is forced through the point (

*f̂*

_{DC}= 1,

*d̂*

_{⊥}= 0). The color-coded image shown in Fig. 4(d) shows the samples from the scatter diagram of Fig. 4(a) plotted back in the

*x*−

*y*coordinates of the image. The scatter diagram has been divided into three groups: The group above the regression line, in red, contains the pixels with a large beam deflection but only a small loss in the dynamic confocal signal. This suggest that, in these locations, the probe beam was only weakly scattered close to the focus but still deflected by larger scale structures. The group below the line, in blue, corresponds to the locations where stronger scattering occurred close to the focus, thus leading to a weaker dynamic confocal signal.

*f̂*

_{DC}> 0.93. In order to display the segregation of the data in a more continuous way, the data can be projected into a coordinate system (

*U*,

*V*) associated to the regression line. The variable

*U*is the distance along the line, from the point (

*f̂*

_{DC}= 1,

*d̂*

_{⊥}= 0) and the variable

*V*is the signed distance from the line (see Fig. 4(a)). Images in the

*U*and

*V*metrics are shown in Figs. 4(b) and 4(c) respectively. The

*U*-image represents the amount of scattering, i.e. the extent to which the beam has been altered through the sample. The

*V*-image gives information about the nature of the scattering, i.e. whether it consists in large scale deflection of the beam (e.g. close to the edge of the cell) or in smaller scale scattering close to the focus.

## 4. Tomographic interpretation of the data

*ϕ*(

*x*,

*y*), of the detected 2D field for each recorded hologram:

_{int}, exactly as we do for the digital confocal images we already discussed with the integrated phase being a different metric. The main difference with conventional optical phase tomography [11–16

16. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. **19**, 074009 (2008) [CrossRef] .

17. K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A **27**, 1347–1354 (2010) [CrossRef] .

18. K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt. **49**(13), 2529–2538 (2010) [CrossRef] .

*r*

^{2}where

*r*is the distance from the focus. Integration of the amplitude gives the absorption at the focus and the integration of the phase gives the refractive index at the focus.

*f*(

*x,y,z*), which corresponds to the absorption coefficient in optical absorption tomography and to the refractive index in the case of optical phase tomography. The X-ray transform is defined as the integral of

*f*(

*x,y,z*) along rays that all pass through point

**x**= (

*x,y,z*) where the probe beam is focused: where Θ ∈

*S*

^{2}is the direction vector of the ray. The signal we detect on the CCD camera in our experimental system is, in fact, equal to 𝒳(

*f*). The

*x*and

*y*components of the 3D vector Θ have on one-to-one correspondence to the two spatial coordinates on the CCD. In order to obtain an image of the sample from the measurements we need to invert Eq. (10). The X-ray transform can be inverted by following a two-step procedure. The first step consists in integrating 𝒳(

*f*)(

**x**, Θ) over all the directions Θ, on the half sphere

*S*

^{2}. From this integration, we get the function

*g*(

**x**): This can be rewritten from spherical coordinates (

*r,θ,ϕ*) as an integration in Cartesian coordinates

**x**= (

*x,y,z*), with

*r*= |

**x**|: The second step in the inversion of the X-ray transform is to invert the above equation in order to express

*f*(

*x,y,z*) as a function of

*g*(

*x,y,z*). It can be shown that this can be done by applying a ramp filter in the Fourier domain [20

20. N. S. Landkof, *Foundation of Modern Potential Theory* (Springer Verlag, 1972) [CrossRef] .

*κ*is the spatial frequency and ℱ denotes the Fourier transform. Because each direction Θ in the DCM corresponds to a pixel at some position (

*x,y*) in the detector plane, the integrated phase Φ

_{int}corresponds to function

*g*(

**x**) defined by Eq. (13). Using Eq. (14), the refractive index can simply be calculated from Φ

_{int}as:

*π*solid angle on the illumination side and 2

*π*on the detection side). The limited numerical aperture in our experimental system limits the resolution in a manner similar as what would happen for confocal imaging.

*π*. In this case, the detected two-dimensional phase function has to be unwrapped. Two-dimensional unwrapping is a difficult and time-consuming process that has to be carried out for each pixel, which is not practical in most experimental conditions. Instead, the phase integration metric can be approximated by the phase at the center the digital confocal pinhole. The principle of using the phase in confocal microscopy has been proposed previously in [21

21. S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy **107**, 196–201 (2007) [CrossRef] .

22. N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. **33**, 2074–2076 (2008) [CrossRef] [PubMed] .

*u*is given by the Fourier transform of the measured field

_{c}*u*= exp[

*jϕ*(

*x,y*)], the amplitude of which we consider as approximately constant. We have: where

*ϕ*is the phase of

*u*. The second equation holds because we consider the value of the field in the center of the pinhole, where

*k*=

_{x}*k*= 0. If the phase modulation is weak, i.e.

_{y}*ϕ*(

*x,y*) ≪ 1, we can write

*u*≈ 1 +

*jϕ*(

*x,y*) and we obtain: The phase in the center of the pinhole, i.e. the confocal phase, is thus approximately given by Φ

_{int}= ∬

*ϕ*(

*x,y*)

*dxdy*.

*z*-images (Fig. 7(b)) and the

*d*

_{⊥}-images (Fig. 7(c)). The confocal phase images were obtained by first calculating the complex optical field at the position of the virtual pinhole and then by measuring the phase of the pixels that falls within the pinhole, as defined by Eq. (6). In our experiment the pinhole was chosen to have the “optimum” diameter equal to one diffraction-limited spot.

23. R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988) [CrossRef] .

## 5. Conclusion

## References and links

1. | M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 (1961). |

2. | C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta |

3. | C. Sheppard and D. Shotton, |

4. | A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature |

5. | G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. |

6. | C. Yang and J. Mertz, “Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection,” Opt. Lett. |

7. | J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999). |

8. | A. Goy and D. Psaltis, “Digital confocal microscope,” Opt. Express |

9. | E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt. |

10. | M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B |

11. | G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis |

12. | F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. |

13. | F. Charrière, N. Pavillon, T. Colomb, and C. Depeursinge, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express |

14. | W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods |

15. | W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Extended depth of focus in tomographic phase microscopy using a propagation algorithm,” Opt. Lett. |

16. | M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. |

17. | K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A |

18. | K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt. |

19. | S. Helgason, |

20. | N. S. Landkof, |

21. | S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy |

22. | N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. |

23. | R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. |

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(180.1790) Microscopy : Confocal microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Microscopy

**History**

Original Manuscript: March 28, 2013

Revised Manuscript: May 30, 2013

Manuscript Accepted: June 3, 2013

Published: June 12, 2013

**Citation**

Alexandre S. Goy, Michaël Unser, and Demetri Psaltis, "Multiple contrast metrics from the measurements of a digital confocal microscope," Biomed. Opt. Express **4**, 1091-1103 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-7-1091

Sort: Year | Journal | Reset

### References

- M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 (1961).
- C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977). [CrossRef]
- C. Sheppard and D. Shotton, Confocal Laser Scanning Microscopy (BIOS Scientific Publishers, 1997).
- A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991). [CrossRef]
- G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett.24, 811–813 (1999). [CrossRef]
- C. Yang and J. Mertz, “Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection,” Opt. Lett.28, 224–226 (2003). [CrossRef] [PubMed]
- J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).
- A. Goy and D. Psaltis, “Digital confocal microscope,” Opt. Express20, 22720–22727 (2012). [CrossRef] [PubMed]
- E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt.39, 4070–4075 (2000). [CrossRef]
- M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B5(3), 633–640 (1988). [CrossRef]
- G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).
- F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett.31, 178–180 (2006). [CrossRef] [PubMed]
- F. Charrière, N. Pavillon, T. Colomb, and C. Depeursinge, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express14, 7005–7013 (2006). [CrossRef] [PubMed]
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4, 717–719 (2007). [CrossRef] [PubMed]
- W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Extended depth of focus in tomographic phase microscopy using a propagation algorithm,” Opt. Lett.33, 171–173 (2008). [CrossRef] [PubMed]
- M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008). [CrossRef]
- K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A27, 1347–1354 (2010). [CrossRef]
- K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt.49(13), 2529–2538 (2010). [CrossRef]
- S. Helgason, The Radon Transform, 2nd ed. (Birkhauser, 1999).
- N. S. Landkof, Foundation of Modern Potential Theory (Springer Verlag, 1972). [CrossRef]
- S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007). [CrossRef]
- N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett.33, 2074–2076 (2008). [CrossRef] [PubMed]
- R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.