2.1.

Theory

For clarity, we adopt here a similar notation to that used by Gustafsson et al.⁷ In fluorescence microscopy, the observed raw data $D(\mathbf{r})$ are the convolution of the sample-emitted fluorescence $E(\mathbf{r})$ and microscope’s detection point spread function (PSF) $H(\mathbf{r})$

The emitted fluorescence is the product of the fluorescently labeled sample structure $S(\mathbf{r})$ and the excitation PSF $I(\mathbf{r})$

This can also be written in frequency space as $\tilde{E}(\mathbf{k})=(\tilde{S}\otimes \tilde{I})(\mathbf{k})$, where $\tilde{S}(\mathbf{k})$ and $\tilde{I}(\mathbf{k})$ are the Fourier transforms of $S(\mathbf{r})$ and $I(\mathbf{r})$.

The latticeSIM excitation pattern $I(\mathbf{r})$ is formed by the interference of multiple beams at the back pupil of the excitation objective. In the case of the hexagonal lattice pattern, the six beams at the back pupil [Figs. 1(a) and 1(b)] will interfere and form an excitation pattern at the sample plane that is the squared magnitude of the Fourier transform of the pupil function [Fig. 1(c)]. Similarly, the excitation OTF $\tilde{I}(\mathbf{k})$ is the Fourier transform of $I(\mathbf{r})$ or the autocorrelation function of the pupil function [Fig. 1(d)]. As is the case for conventional 3D SIM, the excitation pattern can be expressed as a finite sum of “$m$” components, each of which can be separable into functions that depend solely on the lateral or axial coordinates in real space or frequency space, respectively

Fig. 1

Overview of latticeSIM imaging. (a) Image of the latticeSIM setup. Six beams are emitted by the excitation objective (left) and form a hexagonal interference pattern. The detection objective (right) collects the emitted fluorescence. The red and orange boxes highlight the positions of the pupil and sample planes, respectively. (b) The excitation pattern at the back pupil plane of the excitation objective. (c) The excitation PSF of the hexagonal pattern that illuminates the sample. It is the squared amplitude of the Fourier transform of the pupil function. The inset shows an $x$-averaged intensity profile along the $z$ axis. (d) The excitation OTF of the hexagonal pattern. It is a Fourier transform of the excitation PSF and an autocorrelation of the pupil function. Different lateral orders ($m$) are coded by color. Different axial orders ($n$) are listed as well. (e) Detection PSF $H(\mathit{r})$. (f) Widefield detection OTF $\tilde{O}(\mathit{k})$. It is a Fourier transform of the detection PSF. (g)–(i) SIM imaging and reconstruction process. In the SIM images $\tilde{D}(\mathit{k})$ (g), information components for different lateral orders are shifted to be centered at $k_x=0$. In SIM reconstruction, these components are separated (h) and then shifted back to their correct position in frequency space (i). (j) Schematic of the overlap regions between different orders in ${\tilde{D}}_m(\mathit{k})$, left: $m=0$ and $m=1$; middle: $m=0$ and $m=2$; right: $m=1$ and $m=2$.

For hexagonal lattice illumination, the excitation OTF has a signal at five lateral frequencies and $m$ ranges from $-2$ to $+2$, as shown in Fig. 1(d).

Rewriting Eq. (1) with Eq. (2) and the separated components in the excitation PSF (3), it becomes

In the equation above with a spatially invariant PSF, ${\mathbf{r}}^{\prime }$ stands for the reference frame of the specimen, and $\mathbf{r}\text{-}{\mathbf{r}}^{\prime }$ stands for the reference frame of the objective lenses. When operating the microscope, the excitation pattern is kept fixed relative to the detection objective focal plane, while the sample is translated along the “$z$” direction to acquire a 3D volume. Therefore, the axial components of the excitation pattern $I_m$ follow the same reference frame as the objectives (same as $H(\mathbf{r}-{\mathbf{r}}^{\prime })$) rather than the sample, and $I_m(z^{\prime })$ can be replaced by $I_m(z-z^{\prime })$. We refer the reader to a full discussion of Gustafsson et al.⁷ for more information about the coordinate reference frames. Equation (4) can then be written as

And its Fourier transform $\tilde{\mathrm{D}}(\mathbf{k})$ is

Therefore, Eq. (6) becomes

Equations (6) and (7) indicate that the resolution is improved by two methods. First, resolution is increased axially through ${\tilde{O}}_m(\mathbf{k})=\tilde{O}(\mathbf{k})\otimes {\tilde{I}}_m(\mathbf{k})$. The overall OTF becomes a series of transfer functions that are the convolution between the widefield detection OTF and the axial part of each excitation OTF order $m$. And second, sample information components $\tilde{S}(\mathbf{k})$ have been shifted laterally into the detection envelope by the vector $\mathit{m}\mathbf{p}$ [Fig. 1(g)]. Note that because the high-resolution axial components are already in their correct locations in frequency space (because the sample is scanned axially as described above), the SIM reconstruction process will need to separate the different “$m$” lateral components of ${\tilde{D}}_m(\mathbf{k})$ [Fig. 1(h)] and then shift the high-frequency components laterally back to their original positions in frequency space [Fig. 1(i)], thus yielding improved resolution (see Gustafsson et al.⁷ for details).

In conventional SIM imaging, the fundamental frequency $\mathbf{p}$ and the lateral starting phase $\phi$ may be different from when the transfer functions, ${\tilde{O}}_m(\mathbf{k})$, are measured and when the sample is imaged. Thus, these parameters are typically fit after data acquisition and during the reconstruction process. This is done by examining the overlap regions among different $m$ components of ${\tilde{D}}_m(\mathbf{k})$ in frequency space.

More specifically, once the separated information components of the data have been isolated and shifted to their correct locations in frequency space, the frequency information of the sample $\tilde{S}(\mathbf{k})$ should be identical in the overlapping regions. However, as noted in Eq. (7), the sample information $\tilde{S}(\mathbf{k})$ in the observed data terms ${\tilde{D}}_m(\mathbf{k})$ has also been scaled and phase shifted by a corresponding order-specific transfer function ${\tilde{O}}_m(\mathbf{k})$ as a result of the physical observation process. Thus, the shifted information components ${\tilde{D}}_m(\mathbf{k}+m\mathbf{p})$ are not directly comparable to each other. To compensate for this, we multiply the observed data components ${\tilde{D}}_m(\mathbf{k}+m\mathbf{p})$ by the corresponding transfer function of the other information component in the overlap region. For example, in the overlap regions between $m=0$ and $m=1$, we compare

If we first assume that ${{\tilde{O}}^{\prime }}_m(\mathbf{k})={\tilde{O}}_m(\mathbf{k})$, which means that the transfer functions when taking the sample image were identical to those obtained when taking the calibration images, then at the correct shift vector $\mathbf{p}$, Eqs. (8) and (9) above will be identical in theory except for a constant phase offset $\phi$. In practice and in the presence of noise, $\mathbf{p}$ is determined as the value that maximizes the cross-correlation within the overlapping region. At the correct value for $\mathbf{p}$, the starting phase $\phi$ associated with ${\tilde{D}}_m(\mathbf{k})$ can be determined by examining the ratio between Eqs. (8) and (9)

Note that this ratio is independent of sample information and is constant throughout the entire overlap region. Therefore, $\phi$ can be extracted using complex linear regression upon the two terms, and the slope carries the information of the starting phase $\phi$.

In contrast to conventional single-objective SIM, in latticeSIM the axial components in ${\tilde{I}}_m(\mathbf{k})$ may also acquire a phase term that is dependent on $\delta z$—the physical offset between the detection and excitation reference planes. In this case, assuming that the illumination pattern along z also follows a harmonic distribution, the axial components of the excitation pattern can be defined as:

Here, $\mathbf{q}$ is the fundamental frequency along the axial direction, and $n$ is a function of $m$. Specifically, as shown in Fig. 1(d)

Fig. 2

Excitation and SIM OTFs with different axial offsets. (a) Excitation PSF $I(\mathbf{r})$ in real space. The number at the left indicates the offset from the simulated detection objective focal plane. (b) Excitation OTF $\tilde{I}(\mathbf{k})$ in frequency space. The first row shows the magnitude. The rows underneath show the phase under different axial offsets. (c) Separated SIM transfer functions ${\tilde{O}}_m(\mathbf{k})$ [same as Fig. 1(h)] showing the magnitude and phase under different excitation pattern axial offsets.

However, the transfer functions ${\tilde{O}}_m(\mathbf{k})$ are not typically accessible in raw images, and as with the lateral starting phase above, the axial offset may vary from when the transfer functions are calibrated at the beginning of an experiment and when a given image is acquired. For example, as shown in Eq. (7), the phase of the raw images ${\tilde{D}}_m(\mathbf{k})$ composed of eight beads [rightmost column in Fig. 3(a)] is a mixture of illumination and sample information [Figs. 3(a) and 3(b)]. Therefore, to retrieve the axial misalignment, we need to first cancel out the sample information $\tilde{S}(\mathbf{k})$ from the observed image $\tilde{D}(\mathbf{k})$. This can be achieved by the same approach described in Eq. (10). If we include an axial offset which leads to phase ramp in ${\tilde{I}}_m(k_z)$ as described in Eq. (11), and assume that there may be a different axial offset $\delta z$ and $\delta z^{\prime }$ when imaging the sample or calibration beads respectively, then Eq. (10) then becomes

Fig. 3

Residual phase in OTF overlaps reflects the axial offset. (a) Columns 1 to 3: Magnitude of the zeroth information component (${\tilde{D}}_0$), the first SIM transfer function (${{\tilde{O}}^{\prime }}_1$) after being shifted to the correct position in frequency space, and the magnitude of the product of the two. Here, the SIM image is composed of eight beads. Columns 4 to 6: Magnitude of the first information component in the SIM image after being shifted to the correct position in frequency space (${\tilde{D}}_1$), the zeroth SIM transfer function (${{\tilde{O}}^{\prime }}_0$), and the magnitude of the product of the two. (b) The phase of the corresponding columns in panel (a). In this case, both the SIM image ${\tilde{D}}_m(\mathbf{k})$ and transfer functions (${{\tilde{O}}^{\prime }}_m(\mathbf{k})$) have been simulated with a $0.4\pi$ axial offset between the excitation and detections. The rightmost column shows the residual phase in the overlapped region as measured via the ratio in Eq. (12). Panel (c) is the same as panel (b) but in this case, the transfer function ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$, does not have the corresponding $0.4\pi$ axial offset as ${\tilde{D}}_m(\mathbf{k})$. (d) Residual phase as computed via Eq. (12) for different axial offsets between the excitation and detection planes in simulated SIM images (different columns) and simulated transfer functions (different rows).

Note that here, we have expanded ${\tilde{O}}_m(\mathbf{k})=\tilde{O}(\mathbf{k})\otimes {\tilde{I}}_m(k_z)$. We also use ${\tilde{I}}_m(k_z)$ and ${{\tilde{I}}^{\prime }}_m(k_z)$ to denote the axial component of illumination when imaging the sample and the calibration beads, respectively. Here, we assume there is no phase contributed from the widefield detection OTF, as it can be corrected by proper alignment or adaptive optics. Note that ${\mathrm{R}}_{m_1,m_2(m_1\ne m_2)}$ is valid only in regions where ${\tilde{D}}_{m_1}(\mathbf{k}+m_1\mathbf{p}){{\tilde{O}}^{\prime }}_{m_2}(\mathbf{k}+m_2\mathbf{p})$ and ${\tilde{D}}_{m_2}(\mathbf{k}+m_2\mathbf{p}){{\tilde{O}}^{\prime }}_{m_1}(\mathbf{k}+m_1\mathbf{p})$ are non-zero, which are indicated by the magnitude of the products shown in Fig. 3(a) ($m_1=0$ and $m_2=1$). For a specific combination of $(m_1,n_1)$ and $(m_2,n_2)$, the residual phase, which is the phase of $R_{m_1,m_2(m_1\ne m_2)}$, can be expressed as

Equation (13) is the sum of two terms. The first term is determined by the axial offset between the excitation and detection foci, will be zero if $\delta z=\delta z^{\prime }$ [Figs. 3(b) and 3(c)], and is a function of $k_z$ (because it contains the $k_z$-oriented vector). The second term is due to the lateral phase $\phi$ in the illumination pattern and is a constant across the overlap region [the same as Eq. (10)]. This has two important implications for the reconstruction process. The first is that it creates a need to explicitly account for the axial phase offset during reconstruction. For light-sheet illumination, the delta functions of the hexagonal lattice in the pupil are extended into lines or Gaussian profiles along the $k_z$ direction, corresponding to the axial confinement of the light sheet at the sample. This complicates the simplified version of the residual phase shown in Eq. (13) and makes it challenging to derive an analytical solution. In practice, to extract the axial offset $\delta z$ from the sample images, we can collect a gallery of transfer functions with different known axial offsets $\delta z^{\prime }$ and then identify for which measured offset ${\mathrm{\Phi }}_{m_1,m_2}$ reaches a minimum value at $\delta z^{\prime }=\delta z$ [Fig. 3(d)]. The corresponding transfer functions can then be used to reconstruct the SIM images and will yield minimized artifacts due to axial misalignment between the excitation and detection focus. The second implication of Eq. (13) is that it shows that the conventional method for estimating the lateral phase offset $\phi$ as described in Eq. (10) via complex linear regression will fail as the residual phase ${\mathrm{\Phi }}_{m_1,m_2}$ is no longer constant in the overlap region as assumed by conventional SIM reconstruction. As illustrated in Fig. 4, the residual phase $\mathrm{\Phi }$ is constant only when the first term in $\mathrm{\Phi }$ is zero ($\delta z=\delta z^{\prime }$, [Fig. 4(a)]. Whenever $\delta z\ne \delta z^{\prime }$, $\mathrm{\Phi }$ will depend on $\mathbf{q}(n_1-n_2)$. This means that the ratio $R_{m_1,m_2}$ is no longer a constant and ${\tilde{D}}_{m_1}(\mathbf{k}+m_1\mathbf{p}){{\tilde{O}}^{\prime }}_{m_2}(\mathbf{k}+m_2\mathbf{p})$ is no longer linearly related to ${\tilde{D}}_{m_2}(\mathbf{k}+m_2\mathbf{p}){{\tilde{O}}^{\prime }}_{m_1}(\mathbf{k}+m_1\mathbf{p})$. Therefore, complex linear regression applied in conventional SIM reconstruction will fail. In Secs. 3.1–3.4, we address both of these aspects, by generating a two-dimensional (2D) map of the residual phase ${\mathrm{\Phi }}_{m_1,m_2}$ that is composed of $\delta z$ and $\phi$ and use the minimum to simultaneously extract the axial and the lateral offset needed for proper reconstruction [Fig. 5(a)]. This method can also be extended to another illumination lattice (e.g., square lattice) used in lattice SIM (Fig. S1 in the Supplementary Material).

Fig. 4

Mixing between axial and lateral phase terms in the overlap regions. (a) Residual phase computed via Eq. (12) for a single bead image when both the SIM image ${\tilde{D}}_m(\mathbf{k})$ and simulated transfer functions ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$, have a $0.4\pi$ axial offset between the excitation and detection objectives. Different columns show the effects of different lateral offsets in the excitation pattern $\tilde{I}(\mathbf{k})$ used when simulating the data ${\tilde{D}}_m(\mathbf{k})$ and transfer functions ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$. Panel (b) is the same as panel (a) but when utilizing a simulated transfer function with no axial offset.

Fig. 5

Retrieving the axial offset between excitation and detection planes in simulated data. (a) 2D maps of the summed residual phase in the overlap regions with different simulated axial (vertical axis) and lateral (horizontal axis) offsets in the transfer functions ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ used for reconstruction. The simulated applied axial offset in the SIM image ${\tilde{D}}_m(\mathbf{k})$ is shown on the top of each image. (b) Line profiles in panel (a) under different SNRs assuming that the lateral offset between ${\tilde{D}}_m(\mathbf{k})$ and ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ is zero. Different colors indicate different applied axial offsets in the SIM image as indicated by the dashed lines in panel (a). Blue: no axial offset, red: $0.4\pi$ axial offset, orange: $0.8\pi$ axial offset. (c) Line profiles in panel (a) under different SNR when the lateral offset between ${\tilde{D}}_m(\mathbf{k})$ and ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ is either zero (blue) or $-\pi$ (green). This illustrates the residual phase differences between the optimal pattern and the local minimum that is one-half period off both axially and laterally. (d) Reconstructed SIM image of four beads that were simulated with a $-0.4\pi$ axial offset between the excitation illumination and the detection focal plane. The different columns show the effects of reconstructing with transfer functions that assume no axial offset (left), the correctly identified axial offset (middle), and an axial offset that is one-half period off laterally and axially (right). The red curves show the intensity line profile along the red dashed lines.

2.2.

Simulation and Image Analysis

2.3.

Experiments

3.1.

Minimizing Residual Phase Yields the Transfer Functions with the Correct Axial Offset in Simulation

To test whether our methodology can reliably extract the axial misalignment of the excitation pattern from simulated images, we first simulated a transfer function library ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ and bead images ($\tilde{D}(\mathbf{k})$) with different excitation pattern axial offsets. We quantify the sum of the residual phase in the overlap region between different orders in ${\mathrm{\Phi }}_{01}$ and ${\mathrm{\Phi }}_{12}$ as the metric to minimize. As shown in Eq. (12), the phase of ${\mathrm{\Phi }}_{01}$ and ${\mathrm{\Phi }}_{12}$ will be a function of axial offset $\delta z^{\prime }$ and the lateral offset $\phi$. To determine both the axial offset and lateral phase simultaneously, we search for the minimum in ${\mathrm{\Phi }}_{01}$ and ${\mathrm{\Phi }}_{12}$ in a 2D residual phase map corresponding to each pairing of axial and lateral offset [Fig. 5(a)]. As illustrated in Fig. 5(b), the axial offset corresponding to the minimum value in the 2D residual phase map matches the misalignment that we introduced in the simulation. Moreover, we noticed that as the SNR decreases, the modulation depth of the residual phase array also decreases, making it harder to distinguish the true light-sheet axial focus from the position that is one full period off [Fig. 5(b)].

Another caveat of this methodology is that the aligned [blue line in Fig. 5(c)] and pi offset along axial and lateral direction [green line in Fig. 5(c)] are both local minima in the 2D phase map. The difference between the two minima also decreases as the SNR decreases [Fig. 5(c), blue and green curves]. This means that with increasing noise, our method will have trouble distinguishing between transfer functions that are laterally and axially offset by half of a period rather than honing in on the correct axial offset. We next tested how this will affect the performance of SIM reconstruction. We simulated an image of eight beads whose excitation axial focal plane is $0.4\pi$ below the detection focal plane, and we reconstructed this simulated image with the transfer functions that generated the minimum variance in residual phase ($-0.4\pi$ offset) or the transfer functions that are a half a period off axially ($0.6\pi$ offset). As shown in Fig. 5(d), both reconstructed images show minimum artifacts compared with the image reconstructed with calibration transfer functions that did not account for axial misalignment. For axially periodic illumination patterns, there will be no difference between the two minima as they represent different nodes in the lattice. However, the Gaussian envelope associated with the light-sheet confinement differentiates these two local minima and may cause noticeable artifacts for tightly confined beams. Therefore, in practice to minimize this effect in the experimental data, we limited the lateral offset search range to a minimum between $-\pi /2$ and $\pi /2$.

3.2.

Validation with Fluorescent Beads

After testing the approach via simulations, we next experimentally validated it with fluorescent beads. We deliberately introduced a known axial offset between the lattice light sheet and the detection objective focal plane and tested whether the minimum position in the 2D phase map could recover the applied offset. As illustrated in Fig. 6(a), the minimum position in the 2D phase map within the $-\pi /2$ to $\pi /2$ lateral phase offset range [indicated by the red asterisks in Fig. 6(a)] matches the introduced axial offset, thus demonstrating that our simulated results also transfer over to experimental datasets. In latticeSIM, the shape of the illuminated lattice will vary as the beam diverges along the propagation direction of the beam. Therefore, after validating that the method works when the sample is centered along the propagation direction of the excitation pattern, we next tested the performance when the sample is displaced along this axis [Figs. 6(b)–6(h)]. We applied a known axial offset upon the illumination light sheet (the value of the $x$-axis in Figs. 6(c), 6(e) and 6(g). We then extracted the axial position of the minimum in the 2D phase map and compared the predicted axial offsets (blue circles in Figs. 6(c), 6(e), and 6(g)] to the ideal fit of the experimental offsets (red dashed lines). We found that at the propagation focal position [Fig. 6(c), orange line in Fig. 5(b)] or $5\mu \mathrm{m}$ off from the propagation focal position [Figs. 6(e), green line in Fig. 5(b)], we can successfully retrieve the axial offset within the range of $1.2\mu \mathrm{m}$ ($1.1\pi$) above or below the detection focal plane (one full lattice period along the axial direction is $2.13\mu \mathrm{m}$), as the blue circles matched well with the red dashed lines. We investigated the performance of SIM reconstruction using the predicted transfer function from our methods. As shown in Figs. 6(d) and 6(f), the raw lattice light-sheet images with a 600 nm ($0.56\pi$) axial offset between the focal plane of the excitation and the detection objectives have clear ghost copies [YZ max intensity projection (MIP) in the first column of Figs. 6(d) and 6(f)], and similar effects were observed in the SIM images that were reconstructed with “zero-offset” transfer functions [the second column of Figs. 6(d) and 6(f)]. The reconstructed images using transfer functions that were generated with the computationally predicted excitation offset ($y$ values of blue circles in Figs. 6(c) and 6(e), reconstructed images shown in the third column of Figs. 6(d) and 6(f)] yield a less aberrated image. Moreover, images reconstructed with transfer functions that are half a period off axially and laterally from the predicted positions [green crosses in Figs. 6(a) and 6(c) inset] yield comparable quality [Figs. 6(d) and 6(f), fourth column].

Fig. 6

Retrieving the axial offset between excitation and detection planes in experimental data with point-like objects. (a) 2D maps of the summed residual phase in the overlap regions with different experimentally obtained axial (vertical axis) and lateral (horizontal axis) offsets in the transfer functions ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ used for reconstruction. The applied axial offset in the experimentally obtained SIM data is shown at the top of each image. The red (*) shows the minimum residual phase measured between $-0.5\pi$ and $0.5\pi$ lateral offsets. The green (x) shows the local minima that are located one-half period off axially and laterally. (b) Different offsets in the excitation pattern along the beam propagation direction. Orange: no offset, green: $5\mu \mathrm{m}$ offset, blue: $10\mu \mathrm{m}$ offset. (c) Comparison between the applied (horizontal axis) and predicted (vertical axis) axial offset between the excitation and detection focal planes in SIM image taken with the sample located at the center of the beam [orange line in panel (b)]. The inset shows the illumination pattern $I(\mathbf{r})$. Red (*) and green (x) correspond to the two minima in panel (a). $\text{Scale bar}=5\mu \mathrm{m}$. (d) Raw and reconstructed SIM images of fluorescent beads taken under the same settings as in panel (c) and with a $-600\mathrm{nm}$ axial offset between the excitation and detection focal planes. The first column shows the raw image, the second column shows the image reconstructed with transfer functions using the correctly predicted axial offset ($-600\mathrm{nm}$), the third column shows the image reconstructed with transfer functions whose excitation and detection focal planes are aligned, the fourth column shows the image reconstructed with transfer functions that utilize an axial offset that is one-half a period off from the predicted value (i.e., 500 versus $-600\mathrm{nm}$). $\text{Scale bar}=2\mu \mathrm{m}$. Panels (e) and (f) are the same as panels (c) and (d) but images are taken with $5\mu \mathrm{m}$ offset from the beam center along the propagation direction. Panels (g) and (h) are the same as panels (c) and (d), but images are taken with $10\mu \mathrm{m}$ offset from the beam center along the propagation direction. The red curves show the intensity line profile along the red dashed lines.

However, we also noticed that when imaging with an excitation pattern that is $10\mu \mathrm{m}$ away from the focus along the beam propagation direction [Figs. 6(g) and 6(h), the blue line in Fig. 6(b)], the predicted axial offset locked in on one that is a half period off laterally and axially [Fig. 6(g)]. We suspect that this is because the aberrations in the excitation pattern for this condition actually represent a combination of a linear phase ramp due to the axial beam offset and a defocus phase term due to the propagation offset. A failure to account for the defocus phase term likely reduced the ability to distinguish between the two local minima that are one-half period away from each other. Furthermore, as in Fig. 6(h), images reconstructed with transfer functions having the correct axial offset [third column in Fig. 6(h)] or half a period off [fourth column in Fig. 6(h)] both yield symmetric side lobes in the YZ MIP. However, we also noticed that, even at the corrected axial focus, these reconstructed images were more aberrated compared with the ones in Figs. 6(d) and 6(f) where the sample was at the propagation focus of the light sheet. We believe that this is likely because the structured illumination pattern is more distorted compared with the ones at the focal position or $5\mu \mathrm{m}$ off from the focal position. Therefore the weighting among different frequency components of ${\tilde{O}}_m(\mathbf{k})$ is different from ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$, which will yield artifacts during the Wiener deconvolution step in image reconstruction. However, the fact that the side lobes are symmetric in Fig. 6(h) still indicated that our method can still successfully predict the amount of axial offset given the raw image.

3.3.

Validation with Adherent Cells

To benchmark the performance of our method on biological samples, we imaged two subcellular structures in RPE cells: mitochondria and nuclei. As illustrated in Figs. 7(a) and 7(b), our method can successfully retrieve the introduced axial excitation offset in these two structures over the range of $-1.2\mu \mathrm{m}$ to $+1.2\mu \mathrm{m}$ ($-1.1\pi$ to $+1.1\pi$). Similar to our benchmark with beads, raw lattice light-sheet images taken with a 600 nm offset between the excitation focal plane and the detection focal plane show ghost copies in the YZ MIP, and the same ghost copies appeared in the reconstructed images that were processed with “zero-offset” transfer functions [Figs 7(c) and 7(d), first and second columns]. Moreover, using transfer functions with the correctly predicted axial offset again yielded an image with minimal aberration [Figs. 7(c) and 7(d), third column]. The same trends can be observed in linecuts of the image power spectrums where the SIM image reconstructed with the corresponding transfer functions [with or without axial offset between the excitation and detection focal planes, red and blue curves in Fig. 7(e)] have better high-frequency support compared with SIM images that were reconstructed with incorrect transfer functions. Overall, these results highlight that our method can successfully retrieve the axial offset from the raw latticeSIM images and yield reconstructions with minimal aberrations even when imaging complex 3D structures in cells.

Fig. 7

Demonstration of the method with adherent cells. (a), (b) Comparisons between the experimentally applied (horizontal axis) and computationally predicted (vertical axis) axial offset between the excitation and detection focal plane when imaging cellular mitochondria (a) and the cell nucleus (b). (c) Raw and reconstructed SIM images of mitochondria taken with the cell aligned at beam propagation focus [orange line in Fig. 6(b)] and with a $-600\mathrm{nm}$ axial offset between the excitation and detection focal planes. The first column shows the effective dithered lattice image by summing the images from each of the five lateral phase steps, the second column shows the image reconstructed with transfer functions whose excitation and detection focal planes are aligned, and the third column shows the image reconstructed with transfer functions that used a computationally identified axial offset of $-600\mathrm{nm}$. $\text{Scale bar}=5\mu \mathrm{m}$. Insets show a zoomed-in view of the red box, inset $\text{scale bar}=2\mu \mathrm{m}$. Panel (d) is the same as panel (c) but with images of the cell nucleus. (e) Linecuts of the power spectrums for the five-phase summed raw images (black), and images that were acquired and reconstructed under the following settings: ${\tilde{D}}_m(\mathbf{k})=0\mathrm{nm}$ and ${{\tilde{O}}^{\prime }}_m(\mathbf{k})=0\mathrm{nm}$ (red); ${\tilde{D}}_m(\mathbf{k})=-600\mathrm{nm}$ and ${{\tilde{O}}^{\prime }}_m(\mathbf{k})=0\mathrm{nm}$ (green); ${\tilde{D}}_m(\mathbf{k})=-600\mathrm{nm}$ and ${{\tilde{O}}^{\prime }}_m(\mathbf{k})=-600\mathrm{nm}$ (blue). (f) Comparisons between the applied (circles) and predicted (crosses) axial offsets between the excitation and detection focal plane in images of mitochondria taken with different laser power at the back pupil of the excitation objective. The test was run at three different experimentally applied offsets: $-600\mathrm{nm}$ (orange), 0 nm (cyan), and $+600\mathrm{nm}$ (purple).

Another factor that will affect the performance of our method is SNR. Because SNR may be low when imaging biological samples where phototoxicity and photobleaching need to be minimized, we tested our methods with cellular mitochondria using different laser powers. As shown in Fig. 7(f), the accuracy of our approach did not degrade substantially as the applied laser power decreased. This indicated that our method is robust across SNR, at least over the range measured between 4 and 25.

3.4.

Validation with Fluorescent Beads in 3D Hydrogel

Thus far, we have validated that our method can posteriorly extract the axial offset of the excitation pattern from raw SIM images of beads and adherent cells. However, these structures are still relatively thin. To address how our algorithm would perform in thicker samples, we next tested its performance using fluorescent beads embedded within thick 3D hydrogels. Because beads in the imaged hydrogels span multiple locations along the light-sheet propagation direction and to exclude possible aberrations from this aspect (covered in Fig. 6), we only include beads illuminated by $\pm 5\mu \mathrm{m}$ within the focus along the light-sheet propagation direction when computing the axial offset of the beam. To quantitatively validate whether the prediction from our methodology is correct and assess additional artifacts from imaging through the hydrogel, we also independently measured the offset between the excitation focus (determined by the axial profile of the lattice light sheet) and the detection focus (determined by the axial profile of the bead illuminated by widefield illumination). As shown in Fig. 8(a), our method can correctly extract the sample-induced excitation axial offset within $5\mu \mathrm{m}$ below the hydrogel surface. However, our method tends to overestimate this offset for beads more than $10\mu \mathrm{m}$ below the hydrogel surface. This mismatch is likely attributed to the distortion of the illumination pattern (and possibly the detection PSF) caused by the hydrogel refractive index mismatch with the media [Fig. 8(b)]. The excitation aberration becomes visually apparent when the depth is larger than $10\mu \mathrm{m}$. Therefore, the assumption of our method that, other than axial or lateral translations, the same excitation pattern and detection PSFs are applied to take both the transfer function library ${{\tilde{O}}^{\prime }}_m(\mathbf{k})$ and the sample images $\tilde{D}(\mathbf{k})$ is no longer valid.

Fig. 8

Demonstration of method in 3D hydrogels. (a) Comparison between the experimentally measured (red dashed lines) and computationally predicted (blue circles) axial offset between the excitation and detection focal plane in images of fluorescent beads taken at different depths in a 3D polyacrylamide hydrogel. In this case, no offset was experimentally applied, but the hydrogel introduced a shift in the excitation pattern that varies as a function of depth. (b) Experimentally measured excitation pattern ($I(\mathbf{r})$) taken at different depths in the hydrogel. $\text{Scale bar}=5\mu \mathrm{m}$.