Structured illumination imaging with quasi periodic patterns

Abstract

Structured illumination microscopy (SIM) is a well established method for optical sectioning and superresolution. The core of structured illumination is using a periodic pattern to excite image signals. This work reports a method for estimating minor pattern distortions from the raw image data and correcting these distortions during SIM image processing. The method was tested with both simulated and experimental image data from two-photon Bessel light sheet SIM. The results proves the method is effective in challenging situations, where strong scattering background exists, SNR is low and the sample structure is sparse. Experimental results demonstrate restoring synaptic structures in deep brain tissue, despite the presence of strong light scattering and tissue-induced SIM pattern distortion.

Graphical Abstract

Two-photon Bessel SIM light sheet image of neurons in mouse brain. A new SIM processing method removes background and fully corrects artifacts due to distorted SIM pattern in deep tissue. x

1 |. INTRODUCTION

Structured illumination microscopy (SIM) is a well established method for optical sectioning^[1] and superresolution^[2]. The core of structured illumination is using a periodic pattern to excite image signal. Harmonic spatial frequencies of the periodic pattern play key roles in SIM. In superresolution SIM, they demodulate high spatial frequency image signals to frequencies lower than the diffraction limit and make these signals detectable^[3–5]. In optical section SIM, they provide means to remove out-of-focus light^[1] and clear blurry scattering background in deep tissue light sheet images^[6].

The original SIM theory assumes the pattern has perfect periodicity^[2,3]. Irregularity in the illumination pattern will cause artifacts in image results. Experimentally, the pattern is alway subjected to aberration, either from the optical system^[7] or caused by refraction at index mismatched interfaces^[8], all of which can affect the truthfulness of the result. To solve this problem, several groups proposed iterative numerical methods for estimating pattern distortions^[9,10]. New paradigms of SIM with random speckle patterns were also attempted^[11]. Adaptive optics guided by a two-photon-excitation guide star was previously used to correct pattern distortions in SIM^[8,12]. All these methods require either more computation power, more exposures or more complex hardware.

This work reports a method for estimating SIM pattern distortions from the raw image data and correcting distortions during image processing. Our method works within the framework of the SIM theory and reconstructs the correct image non-iteratively with a modified SIM image processing algorithm. It does not require a guide star or extra exposures, nor does it significantly increase the computation time.

Our method applies to incoherent SIM experiments that have quasi periodic patterns, which are the case for most SIM instruments. Incoherent SIM, which captures intensity-only images from phase-less objects, such as fluorescent samples, consists of the most of SIM applications. The method does not apply to phase sensitive SIM^[13]. Most SIM instruments operate with quasi periodic patterns that are neither fully random nor perfectly periodic. Our method provides a solution for removing artifacts due to minor pattern distortions in these instruments.

In theory, the method can be applied to both optical section and super-resolution SIM. However, because it requires a nominal pattern period wider than the diffraction limit, the method is not an ideal solution for super-resolution SIM, due to the fact that widening the pattern period causes a decrease in the resolution. In optical section SIM, such as light sheet SIM imaging, pattern periods are typical much wider than the diffraction limit, and the requirement is naturally met. Therefore, the result part of this paper will focus on simulation and experimental results of light sheet SIM imaging.

2 |. THEORY

2.1. SIM with quasi periodic patterns

The original SIM theory starts by considering an object $O(\overrightarrow{x})$ being illuminated by a periodic pattern^[2,3]. Images are captured by a microscope with a point spread function of $PSF(\overrightarrow{x})$

$$I(\overrightarrow{x})=[O(\overrightarrow{x})\times P(\overrightarrow{x})]\otimes PSF(\overrightarrow{x})$$ (1)

where

$$P(\overrightarrow{x})=\sum _n{a}_n\text{exp}(in{\overrightarrow{k}}_0\cdot \overrightarrow{x}+in\varphi )$$ (2)

represents the sample’s linear (first order, $n=1$) and nonlinear (higher orders, $n>1$) responses to the periodic illumination pattern, whose base spatial frequency is ${\overrightarrow{k}}_0$ and phase offset is $\varphi$.

Experimentally, the pattern can be generated in two ways: in wide-field illuminated SIM, the pattern is typically generated by diffracting the collimated beam with a grating^[3,14–16]; in laser scanning light sheet SIM, the pattern is generated by modulating the scanning speed or the intensity of a scanning laser^[6,17]. In all cases, the phase offset $\varphi$ of a pattern is tightly controlled before aberration is introduced, and thus, unaffected by aberration. For example, in wide-field superresolution SIM, the sinusoidal pattern is generated by interfering $\pm 1$ orders of the same illumination beam diffracted from a grating. The phase offset $\varphi$ of the pattern shifts from 0 to $2\pi$ as the grating moves laterally by a grating period. These $\pm 1$ diffraction beams may pick up different waveform distortions before reaching the focusing plane, which means Eq. 2 needs to be modified to

$$P(\overrightarrow{x})=\sum _n{a}_n\text{exp}\left[in{\overrightarrow{k}}_0\cdot \overrightarrow{x}+in\psi (\overrightarrow{x})+in\varphi \right]$$ (3)

where $\psi (\overrightarrow{x})$ is the wavefront mismatch between $\pm 1$ diffraction beams due to aberration. The wavefront mismatch causes the pattern deviating from its perfect periodicity. However, the control over the pattern’s overall phase offset $\varphi$ is unaffected.

Fourier transform of Eq.1 yields

$$\tilde{I}(\overrightarrow{k},\varphi )=\sum _n{a}_n\text{exp}(in\varphi ){\tilde{O}}_n(\overrightarrow{k}+n{\overrightarrow{k}}_0)OTF(k)$$ (4)

where $OTF(\overrightarrow{k})$ is the optical transfer function, and

$$\begin{gathered} {\tilde{O}}_n(\overrightarrow{k})=ℱ\{O(\overrightarrow{x})\text{exp}[in\psi (\overrightarrow{x})]\} \\ =ℱ\{O(\overrightarrow{x})\}\otimes ℱ\{\text{exp}[in\psi (\overrightarrow{x})]\} \end{gathered}$$ (5)

Here, $ℱ$ refers to Fourier transform.

The SIM theory requires $2N+1$ exposures being taken while shifting the pattern phase offset $\varphi$ from 0 to $2\pi$ in $2N+1$ equal steps. N equals to 1 in linear SIM and higher than 1 in nonlinear SIM. Fourier transforming $2N+1$ raw images yields a system of linear equations given by Eq. 4, with the value of $\varphi$ stepping through a full $2\pi$ cycle in ${\varphi }_m=\frac{2\pi m}{2N+1}$ intervals. The coefficient matrix of the system of linear equations is therefore ${\mathbf{M}}_{mn}=\text{exp}(imn\frac{2\pi }{2N+1})$. Solving the linear equation system yields $2N+1$ pieces of frequency domain images of the same object:

$${\tilde{R}}_n(\overrightarrow{k})={\tilde{O}}_n(\overrightarrow{k}+n{\overrightarrow{k}}_0)OTF(\overrightarrow{k}),n=-N,N$$ (6)

${\tilde{R}}_n(\overrightarrow{k})$ can be numerically shifted back in the spatial frequency domain by $-n{\overrightarrow{k}}_0$, which yields

$${\tilde{R}}_n^{\prime }(\overrightarrow{k})={\tilde{O}}_n(\overrightarrow{k})OTF(\overrightarrow{k}-n{\overrightarrow{k}}_0),n=-N,N$$ (7)

Because $OTF(\overrightarrow{k})$ has its band limit set by the diffraction, ${\tilde{R}}_n^{\prime }(\overrightarrow{k})$ is limited within a circular region in the frequency domain. The region is off-centered by $n{\overrightarrow{k}}_0$ for $n\ne 0$. When $\psi (\overrightarrow{x})$ is a constant zero, i.e. the pattern is ideal, there is ${\tilde{O}}_n(\overrightarrow{k})=ℱ[O(\overrightarrow{k})]$. Thus each order of ${\tilde{R}}_n^{\prime }(\overrightarrow{k})$ reveals a portion of the frequency domain image. Merging multiple orders of ${\tilde{R}}_n^{\prime }(\overrightarrow{k})$ expands the frequency space of the image and yields a super-resolution image of the object. When $\psi (\overrightarrow{x})$ is spatially varying, ${\tilde{R}}_n^{\prime }(\overrightarrow{k})$ is compromised by effects of a distorted pattern. Without correction, the resulting image will contain artifacts.

2.2 |. Measuring pattern distortion from raw image data

Eq. 5 and Eq. 7 indicate that the raw image data does contain information about $\psi (\overrightarrow{x})$. To extract the information from the raw image data, several conditions have to be met: first, SIM is applied to intensity-only imaging application, i.e. not used for phase imaging, and both the object $O(\overrightarrow{x})$ and the $PSF(\overrightarrow{x})$ are real valued; second, we assume ${\overrightarrow{k}}_0$, the nominal frequency of the pattern is less than the spatial cutoff frequency K_cutoff of the imaging system OTF, so that the bandwidth of $OTF(\overrightarrow{k}\pm {\overrightarrow{k}}_0)$ encircles a low frequency area around the zero frequency (Figure 1a); third, the pattern distortion is minor and slow varying, i.e. the pattern remains quasi periodic, and the spatial frequency of exp $[i\psi (\overrightarrow{x})]$ does not exceed $k_{cutoff}-k_0$.

FIGURE 1.

Measuring and correcting pattern distortions in SIM. (a) By selecting a relaxed pattern period, the nominal pattern frequency $k_0$ moves into the bandwidth of OTF, leaving enough bandwidth $\left(k_{cutoff}-k_0\right)$ for measuring the pattern distortion. (b) SIM acquires raw images with patterns going through phase shifting steps ${\varphi }_m=\frac{2m\pi }{2N+1}+{\overrightarrow{k}}_0\cdot \overrightarrow{x}$ (green scattered points). When the pattern is distorted, phase sampling points are shifted by $\psi (\overrightarrow{x})$ (red sampling points). The amount of shifting is spatially varying.

When these conditions are met, a circular 2D low pass filter $L(\overrightarrow{k})$, whose cutoff frequency $k_l\le k_{cutoff}-k_0$, can be applied to ${\tilde{R}}_{\pm 1}^{\prime }(\overrightarrow{k})$. Inverse Fourier transform can then be applied to $L(\overrightarrow{k})R_{\pm 1}^{\prime }(\overrightarrow{k})$ to get

$$\begin{gathered} O_{\pm 1}^{\prime }(\overrightarrow{x})={ℱ}^{-1}\left[L(\overrightarrow{k})R_{\pm 1}^{\prime }(\overrightarrow{k})\right] \\ =ℒ\{O(\overrightarrow{x})\text{exp}[\pm i\psi (\overrightarrow{x})]\otimes PSF(\overrightarrow{x})\} \end{gathered}$$ (8)

where $ℒ$ represents the low pass filtering. In the case of a quasi periodic pattern, because $\psi (\overrightarrow{x})$ is slow varying, there is

$$O_{\pm 1}^{\prime }(\overrightarrow{x})\simeq \text{exp}[\pm i\psi (\overrightarrow{x})]ℒ[O(\overrightarrow{x})\otimes PSF(\overrightarrow{x})]$$ (9)

Since both the object $O(\overrightarrow{x})$ and the $PSF(\overrightarrow{x})$ are real, the phase distortion map can then be calculated as

$$\psi (\overrightarrow{x})=\pm angle[O_{\pm 1}^{\prime }(\overrightarrow{x})]$$ (10)

Equation 10 calculates the pattern distortion phase map from the raw image data and, as detailed in the next section, provides the base for correcting pattern distortions during SIM image processing.

Equation 10 is valid only when all frequency components of the pattern phase distortion map exp $[i\psi (\overrightarrow{x})]$ fall within the bandwidth of the low pass filter $ℒ$, whose cutoff frequency K_l cannot exceed $k_{cutoff}-k_0$. The condition is usually met in optical section SIM, where the pattern frequency is typically much smaller than the diffraction limited spatial frequency, leaving an enough frequency bandwidth for measuring the phase distortion.

2.3 |. Correcting pattern distortion during SIM processing

Once the shape of the distorted pattern is known, adaptive optics could be used to correct distortions. However, an easier way is to correct pattern distortions during image processing.

The SIM experiment requires a pattern being phase shifted in $2N+1$ steps, and after each shift an exposure being taken. When the pattern is ideal, for an image pixel at $\overrightarrow{x}$, the pixel intensity $I(\overrightarrow{x},{\varphi }_m)$ is measured at $2N+1$ pattern phase sampling points ${\varphi }_m=2m\pi /(2N+1)+{\overrightarrow{k}}_0\cdot \overrightarrow{x}$. When the pattern is distorted, the pixel intensity $I(\overrightarrow{x},{\varphi }_m^{\prime })$ is measured at ${\varphi }_m^{\prime }=2m\pi /(2N+1)+\psi (\overrightarrow{x})+{\overrightarrow{k}}_0\cdot \overrightarrow{x}$, which is shifted by an offset of $\psi (\overrightarrow{x})$ from the ideal situation (Figure 1b). The function relationship between pixel intensities vs. pattern phases is still sufficiently and evenly samples, but the start point of sampling varies irregularly across the field of view and has to be taken in account during image processing (Figure 1b).

Rewriting Eq. 4 in the spatial domain yields

$$\begin{gathered} I(\overrightarrow{x},{\varphi }_m)=\sum _n{a}_n\text{exp}(in{\varphi }_m)\times \\ \left\{O(\overrightarrow{x})\text{exp}(in\overrightarrow{x}\cdot {\overrightarrow{k}}_0)\text{exp}[in\psi (\overrightarrow{x})]\right\}\otimes PSF(\overrightarrow{x}) \end{gathered}$$ (11)

Because $\psi (\overrightarrow{x})$ is slow varying compared to the size of PSF, there is

$$\begin{gathered} I(\overrightarrow{x},{\varphi }_m)\approx \sum _n{a}_n\text{exp}(in{\varphi }_m)\times \\ \left[O(\overrightarrow{x})\text{exp}(in\overrightarrow{x}\cdot {\overrightarrow{k}}_0)\otimes PSF(\overrightarrow{x})\right]\times \text{exp}[in\psi (\overrightarrow{x})] \end{gathered}$$ (12)

Eq. 12 is still a system of linear equations, except that the coefficient matrix in Eq. 4 is spatially invariant, whereas the coefficient matrix in Eq. 12 is spatially varying as $\mathbf{M}{(\overrightarrow{x})}_{nm}=\text{exp}\left(inm\frac{2\pi }{2N+1}\right)\text{exp}[in\psi (\overrightarrow{x})]$. Solving Eq. 12 yields

$$R_n^{Cor}(\overrightarrow{x})=\left[O(\overrightarrow{x})\text{exp}(in{\overrightarrow{k}}_0)\right]\otimes PSF(\overrightarrow{x})$$ (13)

whose Fourier transform are

$${\tilde{R}}_n^{Cor}(\overrightarrow{k})=\tilde{O}(\overrightarrow{k}+n{\overrightarrow{k}}_0)OTF(\overrightarrow{k})$$ (14)

$R_n^{Cor}(\overrightarrow{x})$ are free of artifacts from the distorted pattern and can be merged in the frequency domain to form a truthful image.

The above derivation shows that once the pattern distortion is known, artifacts can be avoided by making the processing coefficient matrix M spatially varying in phase by $\text{exp}[in\psi (\overrightarrow{x})]$, and processing raw SIM images in the spatial domain instead of the frequency domain.

3 |. RESULTS

The theoretical derivation shows measuring pattern distortions and correcting their artifacts in incoherent SIM can be carried out in a straight forward process, as long as two conditions are met: 1) the distortion is relatively minor, thus the phase distortion map $\psi (\overrightarrow{x})$ is relatively slow varying; 2) the pattern period is relaxed, leaving a sufficient spatial bandwidth within the OTF to accommodate the frequency band of $\text{exp}[i\psi (\overrightarrow{x})]$. Both conditions are typically met in optical section SIM.

3.1 |. Simulation

The optical section SIM simulation first used an ideal multi-spoke target (Figure 2a), took an idea periodic strip-shape pattern and added a phase distortion map, which contains a mixture of spherical, astigmatism and coma aberrations (Figure 2b). The phase distortion was amplified from a maximal value of half period to two periods in multiple rounds of simulation, which allowed us to test the method with increasingly distorted patterns (Figure 2c). Simulation conditions were set to match typical experimental conditions as: an NA0.8 objective lens for collecting emission images, a $0.25\mathrm{\mu }\text{m}$ pixel size, a $2\mathrm{\mu }\text{m}$ FWHM wide two-photon beam and a $6\mathrm{\mu }\text{m}$ nominal pattern period. The low pass filter used in simulation has the function shape of an NA0.75 OTF, which has a cutoff frequency at $\frac{1}{k_{cutoff}}-\frac{1}{6\mu m}$. The same filter was later used to process experimental data.

FIGURE 2.

Light sheet SIM images simulated with distorted patterns. (a) The spoke-shape simulation target. The target size is $86\mathrm{\mu }m$. (b) The phase distortion pattern used for simulation. (c) Distorted patterns. Results from three cases, with the maximal pattern distortion increasing from a half period to two periods respectively, are presented. (d) Phase maps recovered from simulated raw images. The index bar is in the unit of period. (e) Images processed by the standard SIM method. (f) Corrected images processed by the new method. (g) A more realistic simulation target, with fibers and point structures sparsely distributed in the view. (h) To further test the method, the simulation added out-of-focus background and noise in raw images. Intensity scale represents photon counts in simulated raw images. (i) Recovered phase maps from noisy and sparse raw images. The index bar is in the unit of period. (j) Images processed by the standard SIM method. (k) Corrected SIM images processed by the new method.

Raw images were generated by simulating a $N=2$ nonlinear SIM process, which consists of applying distorted illumination patterns to the spoke pattern object and shifting the illumination in 5 steps. The resulting SIM image, processed with the standard SIM processing method, shows signal losses aggravating as pattern distortions increase (Figure 2e). For the extreme case of two-period distortion, i.e. strips in the pattern were shifted by two periods in some part of the view, the target shape is unrecognizable in the standard SIM result.

Using the phase measurement method proposed in Section 2.2, phase distortion maps were successfully recovered in areas covered by the object(Figure 2c). Following the phase-corrected SIM process method described in Section 2.3, corrected SIM images were obtained. The spoke-shape target was fully recovered in all cases (Figure 2f). Some low-frequency image artifacts can be seen in the two-period distortion case, although the phase map measured in this case is still correct. These results indicate that for the extreme case of two-period distortions, the bandwidth $k_{cutoff}-k_0$ available is sufficient for measuring the phase map; however, the slow-varying $\psi (\overrightarrow{x})$ approximation we adapted during image processing (Equation 12) is no longer fully valid. Despite that, the corrected image is still significantly better than its counterpart processed with the standard SIM method.

To test the method in a realistic setting, the simulation was repeated with a target containing randomly generated fibers distributed sparsely in the field of view (Figure 2g). Out-of-focus background and Poisson noise were added into simulated raw images (Figure 2h). We first processed raw images with the standard SIM algorithm and added additional filtering steps to remove residue out-of-focus background in order to avoid strip-shape artifacts in the processed image^[6]. Resulting SIM images exhibit artifacts due to pattern distortions (Figure 2j). Structures appear dimmer and broken in some parts of the field of view. These artifacts become more severe as the pattern distortion increases. With the new method, phase distortion maps were recovered in areas with structures despite the present of out-of-focus background and noise (Figure 2i). In areas containing no structures, phase values are random. However, these values did not affect the correction of SIM images (Figure 2k), because the new method corrects the phase distortion in the spatial domain. The correction consistently improved the image quality in all cases, although some signal loss can still be seen in the case of two-period distortions.

It worths noting that extreme artifacts seen in the two-period distortion case were never observed by us experimentally. In our experience with deep tissue two-photon light sheet SIM, images processed by the standard SIM method never lose the overall structure, artifacts are mostly localized and only noticeable on fine structures. Our estimations are, in our deep tissue light sheet SIM imaging experiments, the amount pattern distortions is most likely to be less that a full period, and the method presented here is sufficient for correcting these distortions.

3.2 |. Experiment: Two-photon light-sheet SIM imaging of brain tissue

We applied the method to experimental SIM image data of Thy1:YFP mouse brain slice taken by a two-photon light sheet imaging system^[6]. The system (Figure 3) uses a scanning Bessel beam, which is provided by a Ti:Sapphire laser and shaped into a Bessel beam by a spatial light modulator, to illuminate the sample. The Bessel beam, after passing through the water immersion excitation objective lens, has a $2-\mathrm{\mu }m$ wide two-photon excitation width and propagates along the x-axis to illuminate an over $200\mathrm{\mu }m$ wide field of view. A y-axis scanner moves the beam to form a x-y plane illumination. The z-axis scanner shifts the plane in depth. The emission imaging path uses a NA0.8 objective lens to collect fluorescence emission. An electric tunable lens provides adjustable focusing in the emission imaging path.

FIGURE 3.

Schematic of the two-photon Bessel light sheet system. SLM: Spacial light modulator. ETL: Electric tunable lens.

The sample tissue was scanned twice, first with a uniform plane illumination generated by sweeping a 170mW Bessel beam across the field of view every 100ms, second with the SIM illumination, which was generated by hopping the same beam in steps. A total of 5 exposures were needed to reconstruct a single SIM image. A $200\times 150\times 86\mathrm{\mu }{m}^3$ volume of brain tissue was scanned at $1-\mathrm{\mu }m$ depth steps. The raw SIM image set was processed slice-by-slice twice, first by the standard SIM method, and second by the new method, which calculated a pattern phase distortion map for each individual slice and correct effects of distortions during slice image processing. Within each slice, fluorescence signals were extremely sparse and mostly from scattered point-like structures (Supplementary Movie 1, a fly-through movie of the 3D volumetric image taken with a uniform plane illumination).

Plane illumination images also shows the sample produced low signal to noise ratio (SNR) and contained strong scattering, which adds a blurry background to the image (Figure 4 a–b, volume projection images, and Supplementary movie 1, slice fly-through). The image taken with the SIM illumination and processed by the standard SIM algorithm with background filtering (Figure 4d–e and Supplementary movie 2) is free of scattering. But its over all image signal is less than the image processed with the new method (Figure 4g–h and Supplementary movie 3). The zoomed-in 3D image (Figure 4f) of a dendrite (white dashed box in Figure 4d and e) shows the signal loss in the standard SIM image is caused by artifacts. The dendrite structure in Figure 4f has significant signal loss along the center. The fiber appears broken and split. These artifacts are likely cause by SIM patterns being distorted after traveling a long distance in heterogeneous tissue. The same SIM raw image set, after being processed by the new method, generated a continuous dendrite structure and revealed individual spines (Figure 4i), which are not visible in the plane illumination image (Figure 4c).

FIGURE 4.

Experimental images of Thy1:YFP mouse brain slice, taken with two-photon light sheet. All volumetric images were taken with a 170 mW Bessel beam and 100ms exposure time. The image plane was shifted in $1\mathrm{\mu }m$ depth step during 3D scan. A total of 86 layers were acquired. (a-b) X-Z and X-Y projections of a set of 3D image of Thy1:YFP mouse brain tissue, taken with uniform plane illumination. The dashed box highlights a dendrite section at approximately $60\mathrm{\mu }m$ deep. (c) Zoomed-in 3D rendering of the dendrite section, taken from the plane illumination volumetric image set. (d-e) X-Z and X-Y projections of 3D images of the same tissue, taken with SIM illumination and processed with standard SIM algorithm. (f) Zoomed-in 3D rendering of the dendrite section, taken from the standard SIM volumetric image set. (g-h) X-Z and X-Y projections of 3D images from the same SIM raw data, processed with the new method. (i) Zoomed-in 3D rendering of the dendrite section, taken from the corrected SIM volumetric image set processed by the new method.

4 |. DISCUSSION

The original SIM method was built on the assumption that a perfectly periodic illumination pattern can be engineered. In reality, such assumption often does not hold. Light sheet SIM uses a focused beam to illuminate across the tissue. Aberration from optics components and index-mismatched interfaces within the sample can alter the illumination beam’s propagating mode. The situation could change as the beam scans to different portions of the tissue sample, as biological samples have irregular shapes and heterogeneous structures. As the result, the standard SIM processing method often produces artifacts in light sheet SIM.

In this paper, we reported a method for dealing with quasi-periodic SIM patterns containing minor distortions, and removing artifacts with during SIM image processing. The method was tested with both simulation and experiments. The results prove our method is effective in challenging situations such as deep tissue imaging, where strong scattering background exists, SNR is low and the sample structure is sparse. The method truthfully recovered fine neuron structures in mouse brain tissue, where the traditional SIM process failed. The method presented here in theory could be applied to both optical section SIM and superresolution SIM. However, because it requires the illumination pattern period to be wider the diffraction limit, our method is best suitable for optical section SIM, where the illumination pattern is typically wide and SIM is used to for the purpose of removing out-of-focus or scattering background, not for the purpose of reaching a resolution beyond the diffraction limit.

A such requirement is needed for allocating a spatial bandwidth to measure irregularities in SIM patterns. An image system has a limited spatial bandwidth, which is set by the NA. Measuring the SIM pattern and the sample structure in the same time with the same imaging system means two tasks have to share the total bandwidth. The optical section SIM naturally does not make the full use its bandwidth, because superresolution is not needed. The “unused” bandwidth in optical section SIM carries information about the SIM pattern, which can be extracted with a simple processing algorithm presented here.

In superresolution SIM experiments, however, the full bandwidth ideally should be dedicated to imaging the sample. Measuring the SIM pattern through the same imaging system will require sacrificing the resolution. It is possible to measure the SIM illumination distortion with an auxiliary imaging system, such as a two-photon guide star. Adaptive optics was previously used to correct the distortion. The pattern correcting method presented in this work could be used to achieve the same effect at a much lower cost.