Resolution of Oblique-plane Images in Sectioning Microscopy References and Links

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. Nonlinear scanning laser microscopy by third harmonic generation, " Appl. Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy, " Science 329, 967–971 (2010). Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing, " Phys. Arbitrary-scan imaging for two-photon mi-croscopy, " Proc. An optical technique for refocusing in microscopy, " Opt. " Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Image-based adaptive optics for two-photon microscopy, " Opt. and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro, " Am.


Introduction
One of the most important advances in microscopy has been the introduction of optical sectioning which allows instruments to image a single slice of a volume specimen lying in the focal (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] and harmonic generation [3,4], have also been exploited to provide optical discrimination and hence produce optically sectioned images.
Using scanning galvanometer mirrors, sectioning microscopes are capable of capturing 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.
While it has long been possible to obtain such images by the post-processing of threedimensional image stacks, techniques have recently been developed so that oblique planes can be scanned directly and at high speed [5][6][7].This capability presents a significant advantage in the study of time-varying behavior of specimens where the dynamics are too fast to allow full three-dimensional stack acquisition.
It is timely therefore to consider the implication of oblique plane imaging on resolution: since the point spread function of a finite-aperture microscope is necessarily more elongated along the optical axis than in the focal plane it is to be expected that as the angle of inclination increases the image resolution becomes increasingly anisotropic.In this paper we explore, both theoretically and experimentally, two different mechanisms by which the resolution changes and draw attention to a counter-intuitive situation where the maximum spectral content of the image cannot possibly arise from spatial variations of the specimen which lie wholly in the oblique plane but rather from out-of-plane components that are projected into the plane.We later discuss the implications of these results for the field of oblique-plane microscopy.

The theory of oblique-plane imaging
We elect to base our discussion around imaging in a two-photon or ideal confocal, singlephoton microscope since, if we ignore the wavelength difference between the excitation light and fluorescence, the image formation properties of the two instruments may be described by the same mathematical expressions.More specifically they may both be described by an effective point spread function, |h(x, y, z)| 4 , where h(x, y, z) represents the amplitude point spread function of the imaging lens, which must be evaluated at the appropriate wavelength.To begin with, we consider the intensity measured when the point spread function is centered at some arbitrary (x, y, z) position in a specimen by way of the image formation process in Fourier space.Following Gu [8], we elect to describe the image formation in terms of the three-dimensional optical transfer function (3D OTF), C(m, n, r), and the spatial frequency content of the specimen, T (m, n, r), as I(x, y, z) = C(m, n, r) T (m, n, r) e i2π(mx+ny+rz) dm dn dr, (1) in which n, m and r represent spatial frequencies in the x, y, z directions respectively.The 3D OTF of a system with limited numerical aperture spans only a finite region of support in Fourier space (Fig. 1a), which limits the spatial frequency components of the specimen that can be imaged with the microscope.For a two-photon or confocal microscope the 3D OTF region of support [8] is given by where l = √ m 2 + n 2 and r are the normalized spatial frequency co-ordinates along the radial and axial directions respectively.For an illumination wavelength λ , an objective lens with semiacceptance angle α and an immersion medium with refractive index n, these normalized spatial frequency co-ordinates are related to the true spatial frequency co-ordinates by scaling factors [n sin α]/λ and [4n sin 2 (α/2)]/λ respectively.
To develop the theory for oblique-plane imaging we explore the situation shown in Fig. 1b, where the plane is rotated by angle θ about the y-axis.The two-dimensional intensity distribu-tion 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 • ).
It is important to remind ourselves that the presence of a particular spatial frequency in a two-dimensional image gives no indication as to whether this represents a k-vector lying wholly within the plane or whether it is a projection of it into the plane.Although the value of the outof-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.
An interesting consequence arises from this: in some cases a spatial frequency component can be present in an image that could not possibly have arisen from a 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, waterimmersion 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.

Numerical simulation of oblique-plane imaging
To illustrate these effects theoretically we begin by considering a specimen comprising fullspectrum, white noise with a high spatial frequency content in all directions.When imaging an oblique plane through this specimen we would expect the spatial frequency content along the m ′′ -direction to be particularly rich in Fourier content because of the large number of out-ofplane 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.
In a second example we simulate a different specimen, that of a thin, spherical shell whose three-dimensional Fourier transform is known to vary with the form of a radial sinc function [9].As the object spectrum extends broadly in all directions, this would initially suggest 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.

Experimental verification of oblique-plane resolution phenomena
To confirm these observations experimentally we built the two-photon imaging system shown in Fig. 4.An axial scan unit [10] (ASU) was incorporated to scan the focal spot in the z-direction optically.The laser source used was a Ti:Sapphire laser (Tsunami, Spectra Physics), producing ultra-fast pulses with central wavelength 850 nm (∆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] between two galvanometers (VM1000+, Cambridge Technology, USA) for linear motion.The reflected light passed back through the lens and the emerging wavefront was then re-imaged into the pupil plane of L 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].All experiments were performed using an Olympus UApo/340, 40X, 1.15 NA water objective for L 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).
As an example of a specimen containing many spatial frequency components in all directions, a fluorescent pollen grain was selected for oblique plane imaging.Fig. 5a shows an image taken along a 45 • plane together with the cutoff of Fig. 3a.The pollen grain contains, in this plane, frequency content in all directions leading to a fuller spectrum in the resulting image, with a region of support that is broadly matched in shape to the projection of the OTF along the r ′ -direction.The experiment was then repeated with a commercially available 12 µm 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.
In Fig. 5b it is observed that the deeper region of the imaged shell is more blurred than that nearer the surface.This is due to the microsphere being mounted in an optical cement with a refractive index of 1.52, in contrast to that of the immersion medium of the objective lens (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] or indeed corrected experimentally [13].

Conclusion
Increasingly researchers have found that, by tailoring scan geometries to a specific imaging problem, it is possible to acquire images from only the regions of interest in a specimen with much-improved temporal resolution compared with the acquisition and post-processing of a three-dimensional image stack.In this paper we have explored the ensuing limitation in resolution of such images by considering a plane inclined within a specimen volume.Our analysis, based on a consideration of the three-dimensional optical transfer function, has shown that the resolution becomes increasingly anisotropic with increasing angle of obliquity.We further showed that the inclined-plane images may contain spatial frequencies that cannot possibly have arisen wholly from in-plane spatial variations.For some configurations the maximum spatial frequency observed can be three times higher than the in-plane maximum.It is important therefore to exercise caution when analyzing images with the intent of extracting measurements of periodic variations in a particular plane [14].A corollary to this is that if the exact in-plane spatial variation is sought, post-processing with a digital filter derived from Eq. 7 would remove the misleading, higher-frequency modes.Likewise, we can take advantage of these supplementary modes when applying deconvolution techniques to the images.
While we have confined our analysis to inclined planes we note that, by taking a piecewise approach to the analysis, these conclusions are generalizable to all out-of-plane scan geometries, such as lines and curved surfaces.

Fig. 1 .
Fig. 1.(a) Region of support for the 3D OTF of the microscope.(b) Imaging oblique plane OP with angle θ .(c) When imaging an oblique plane, all k-vectors shown in red have the same projection along the r ′ -axis so will appear in the image with the same spatial frequency.(d) When imaging an oblique plane, spatial frequency, k 1 , will appear in the final image whereas a spatial frequency, k 2 , will not.

1 )Fig. 2 .
Fig. 2. Limiting values of spatial frequencies along the x ′ -direction for a 1.4 NA oil immersion objective and a 0.8 NA water immersion objective.Solid lines show the absolute spatial frequency cutoff which includes projections of out-of-plane spatial frequencies whereas dashed lines show the largest spatial frequencies that can be imaged of features lying wholly in the plane.a) b)

Fig. 3 .
Fig. 3. Numerical simulations of images from a 45 • oblique plane with a 1.15 NA water immersion objective and associated Fourier transforms.(a) White noise with a projection of the 3D OTF region of support overlain and (b) a spherical shell of diameter 12µm with the cut through the 3D OTF overlain.

Fig. 4 .
Fig. 4. Oblique plane scanning multi-photon microscope.The lateral scan unit (LSU) scans the focal spot laterally in the specimen and the axial scan unit (ASU) scans the spot axially.Oblique planes are scanned in three dimensions by using a combination of the LSU and ASU.

Fig. 5 .
Fig. 5. Experimental images from a 45 • oblique plane with a 1.15 NA water immersion objective and their associated Fourier transforms.(a) A fluorescent pollen grain with a projection of the OTF overlain and (b) a spherical shell with diameter 12µm with the cut through the OTF overlain.