## Resolution of oblique-plane images in sectioning microscopy |

Optics Express, Vol. 19, Issue 3, pp. 2662-2669 (2011)

http://dx.doi.org/10.1364/OE.19.002662

Acrobat PDF (1138 KB)

### Abstract

Live biological specimens exhibit time-varying behavior on the microscale in all three dimensions. Although scanning confocal and two-photon microscopes are able to record three-dimensional image stacks through these specimens, they do so at relatively low speeds which limits the time resolution of the biological processes that can be observed. One way to improve the data acquisition rate is to image only the regions of a specimen that are of interest and so researchers have recently begun to acquire two-dimensional images of inclined planes or surfaces extending significantly into the *z*-direction. As the resolution is not uniform in *x*, *y* and *z*, the images possess non-isotropic resolution. We explore this theoretically and show that images of an oblique plane may contain spectral content that could not have been generated by specimen features lying wholly within the plane but must instead arise from a spatial variation in another direction. In some cases we find that the image contains frequencies three times higher than the resolution limit for in-plane features. We confirm this finding through numerical simulations and experiments on a novel, oblique-plane imaging system and suggest that care be taken in the interpretation of such images.

© 2011 Optical Society of America

## 1. Introduction

*x-y*) plane of the microscope. A three-dimensional data set,

*I*(

*x*,

*y*,

*z*), may then be built by recording a series of through-focus images at different focal settings, or

*z*-positions, which can then be rendered in a number of ways to reveal details about the three-dimensional structure of the specimen. The confocal microscope is probably the best-known example of such an instrument where optical sectioning arises by introducing a pinhole aperture in front of the detector to obscure out-of-focus light and, as the focal spot is scanned across the plane, produce a clear image of the features lying in the focal plane [1]. Further techniques, using non-linear contrast mechanisms such as two-photon fluorescence [2

2. W. Denk, J. Strickler, and W. Webb “Two-photon fluorescence scanning microscopy,” Science **248**, 73–76 (1990). [CrossRef] [PubMed]

3. I. Freund and M. Deutsch “Second-harmonic microscopy of biological tissue,” Opt. Lett. **11**, 94–96 (1986). [CrossRef] [PubMed]

4. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. **70**, 922–924 (1997). [CrossRef]

*x-y*plane images at high speed. From the perspective of the specimen however, the

*x-y-z*axes of the system are usually arbitrary and depend on how it was mounted on the microscope. It is, therefore, common to find that features of interest in a specimen do not lie in the focal plane and that it would be preferable to acquire images from a plane inclined with respect to the focal plane.

5. W. Göbel and F. Helmchen “New angles on neuronal dendrites in vivo,” J. Neurophysiol. **98**, 3770–3779 (2007). [CrossRef] [PubMed]

7. P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. **103**, 257803 (2009). [CrossRef]

## 2. The theory of oblique-plane imaging

*θ*about the

*y*-axis. The two-dimensional intensity distribution recorded in this plane is given by where

*z*′

_{0}denotes the position of the oblique plane in the specimen and the co-ordinate system in Fourier space has been rotated so that

*m*′ =

*m*cos

*θ*+

*r*sin

*θ*,

*n*′ =

*n*and

*r*′ = −

*m*sin

*θ*+

*r*cos

*θ*.

*C*′ and

*T*′ therefore represent the OTF and object spatial frequency distributions in this rotated co-ordinate system. The spatial frequency content of the resulting oblique plane image can be found by taking the 2D Fourier transform of Eq. 3: with new spatial frequency variables

*n*″ and

*m*″. Integration in

*x*′ and

*y*′ then leads to a simplification: This is equivalent to a projection of the product of the object spatial frequency spectrum and the OTF for the system along the

*r*′ direction as defined in Fig. 1c, equal to −

*m*sin(

*θ*) +

*r*cos(

*θ*). With this knowledge we can now, for any oblique plane, carry out such a projection of the three-dimensional region of support to find the possible spatial frequency components that may appear in the final image. It is clear on geometric grounds, that the cutoff frequency along the rotational axis of the plane, here the

*y*-direction, is unaffected by the projection process. In contrast the cutoff,

*m*″

_{1}, along the

*x*′-direction, diminishes with higher angles. Simple geometry provides us with an expression for this cutoff in terms of

*θ*: This variation is plotted in Fig. 2 using solid lines for a 1.4 NA oil immersion lens and a 0.8 NA water dipping lens for

*θ*in the range (0, 90°).

**k**-vector lying wholly within the plane or whether it is a projection of it into the plane. Although the value of the out-of-plane component cannot be known exactly it is limited to a range of values constrained by the 3D OTF region of support. For instance, in Fig. 1c all

**k**-vectors shown in red have the same projection along the

*r*′-axis so will appear in the oblique plane image with the same spatial frequency. Any other

**k**-vector with the same projected value but lying outside the region of support will not appear in the final image.

**k**-vector lying wholly in the image plane. An example of this is shown in Fig. 1d. When imaging an oblique plane, spatial frequency

**k**

_{1}, lying in the

*x-y*plane, will appear in the final image with a projected magnitude, |

**k**

_{1}| cos(

*θ*), whereas spatial frequency

**k**

_{2}, that lies in the oblique plane and has the same length as the projection of

**k**

_{1}, would not appear since it falls outside the 3D OTF region of support (shaded) and so cannot be imaged by the microscope. This point is further emphasized in Fig. 2 where we show as dashed lines the limiting values of in-plane spatial frequencies which may appear in the image along the

*x*′-direction for planes oblique angles from 0° to 90°. Geometrically this cutoff,

*m*″

_{2}, is given by: As an example we note that for images taken on a 45° oblique plane with a 0.8 NA, water-immersion lens, spatial frequencies in the upper two thirds of the spectrum along the

*m*″-axis could not have arisen from specimen variation wholly in the inclined plane.

## 3. Numerical simulation of oblique-plane imaging

*m*″-direction to be particularly rich in Fourier content because of the large number of out-of-plane components which are projected onto the image plane. Fig. 3a shows a numerical simulation of the image that would be acquired from a 45° plane with a 1.15 NA, water-immersion objective with

*λ*= 850 nm. Alongside is shown the Fourier spectrum of this image with the projection of the 3D OTF region of support onto this plane. The spatial frequency content of the image matches the projected region of support with the limit in

*x*′ that of Eq. 6,

*m*″

_{1}. The frequency content outside arises as a numerical artifact from the discrete Fourier transform used in the simulation.

9. S. Vembu “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. **12**, 165–168 (1961). [CrossRef]

*m*″

_{1}angular variation as found with the noise simulation. However, the Fourier transform takes no account of the local distribution of spatial frequency content; the radial sinc function is built up from the Fourier integral over the entirety of the spherical shell and so oblique sections that do not contain the whole of the shell cannot contain the full spectrum. The

**k**-vectors observed in the oblique plane image must therefore arise from the plane itself. The in-plane cutoff,

*m*″

_{2}, is then expected to dominate leading to a greater truncation of the observed spectrum. This is borne out by Fig. 3b where the

*m*″

_{2}region of support determines the spatial frequency content in the image from a 45° plane through the simulated spherical shell. This is the effective support limit of the object-captured frequencies, and differs from that found in Fig.3a, where all regions of the OTF contribute to the final image.

## 4. Experimental verification of oblique-plane resolution phenomena

10. E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE **7569**, 756917 (2010). [CrossRef]

*z*-direction optically. The laser source used was a Ti:Sapphire laser (Tsunami, Spectra Physics), producing ultra-fast pulses with central wavelength 850 nm (Δ

*λ*∼ 50 nm, pulse length < 100 fs). Initially, this was expanded to form a plane wave with Gaussian width 5 mm and directed into the lateral scan unit (LSU), comprising two orthogonally mounted galvanometer mirrors (VM1000, Cambridge Technology, USA) that controlled the angular orientation of the wavefronts. From here the wavefronts were imaged into the pupil of L

_{1}in the ASU using a 4

*f*imaging system comprising two achromatic doublet lenses, with focal lengths 120 mm and 160 mm, to produce a magnification of 4/3. L

_{1}was an Olympus UApo/340, 40X, 0.9 NA dry objective, chosen for its favorable transmission characteristics at 850 nm. Light passing through L

_{1}was reflected off mirror M (PF03-03-P01, Thor labs) mounted on a flexible bridge assembly [10

10. E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE **7569**, 756917 (2010). [CrossRef]

_{2}using a further 4

*f*system of achromatic doublets, this time with focal lengths 150 mm and 200 mm, to produce a magnification of 4/3. It should be noted that the choice of this magnification is not trivial and for a full explanation of this parameter the reader is referred elsewhere [11

11. E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson “An optical technique for refocusing in microscopy,” Opt. Commun. **281**, 880–887 (2008). [CrossRef]

_{2}. A polarizing beam splitter (PBS) and quarter-wave plate (QWP) ensured all light entering the ASU was transmitted into the final stage of the system. A dichroic beamsplitter (DBS) and emission filter (EF) were used to separate fluorescence photons produced in the specimen in the range 400–720 nm for measurement on a digital PMT (P30PC-54, Sens-Tech). Photon counting and generation of galvanometer command voltage waveforms were performed through a reconfigurable I/O card with on-board FPGA (PCI-7830R, National instruments), operated from the Labview environment (National Instruments).

*r*′-direction. The experiment was then repeated with a commercially available 12

*μ*m spherical shell (F24634, Molecular probes, Invitrogen). For this case Fig. 5b shows in contrast the diminished spectrum arising from only the in-plane components, as observed in the numerical simulations.

*n*= 1.33). The step change in refractive index introduces increasing levels of spherical aberration when imaging deeper into the specimen, leading to the effect observed here. The spherical aberration can be incorporated into the simulation [12

12. O. Haeberlé “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. **216**, 55–63 (2003). [CrossRef]

13. D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. **34**2495–2497 (2009). [CrossRef] [PubMed]

## 5. Conclusion

14. G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. **298**, 1616–1625, (2010). [CrossRef]

## Acknowledgments

## References and links

1. | T. Wilson and C.J.R. Sheppard |

2. | W. Denk, J. Strickler, and W. Webb “Two-photon fluorescence scanning microscopy,” Science |

3. | I. Freund and M. Deutsch “Second-harmonic microscopy of biological tissue,” Opt. Lett. |

4. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. |

5. | W. Göbel and F. Helmchen “New angles on neuronal dendrites in vivo,” J. Neurophysiol. |

6. | N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyriéras, and E. Beaurepaire “Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy,” Science |

7. | P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. |

8. | M. Gu |

9. | S. Vembu “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. |

10. | E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE |

11. | E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson “An optical technique for refocusing in microscopy,” Opt. Commun. |

12. | O. Haeberlé “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. |

13. | D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. |

14. | G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. |

**OCIS Codes**

(180.1790) Microscopy : Confocal microscopy

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: September 8, 2010

Revised Manuscript: January 25, 2011

Manuscript Accepted: January 25, 2011

Published: January 27, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

C. W. Smith, E. J. Botcherby, and T. Wilson, "Resolution of oblique-plane images in sectioning microscopy," Opt. Express **19**, 2662-2669 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2662

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### References

- T. Wilson, and C. J. R. Sheppard, Theory and practice of Scanning Optical Microscopy (Academic Press, 1984).
- W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990). [CrossRef] [PubMed]
- I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986). [CrossRef] [PubMed]
- Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
- W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007). [CrossRef] [PubMed]
- N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010). [CrossRef] [PubMed]
- P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009). [CrossRef]
- M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996). [CrossRef]
- S. Vembu, “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. 12, 165–168 (1961). [CrossRef]
- E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010). [CrossRef]
- E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008). [CrossRef]
- O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216, 55–63 (2003). [CrossRef]
- D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009). [CrossRef] [PubMed]
- G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010). [CrossRef]

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