# Efficient Multifocal Structured Illumination Microscopy Utilizing a Spatial Light Modulator
## 2 Methods
### Method for the Reconstruction of MSIM Images  
In conventional linear SIM, the equation governing an SIM image formation for the detected fluorescence intensity can be approximately described as  
\[D(r)=\left\{I_{s}(r)\times S(r)\right\}\otimes h(r), \tag{3}\]  
where \(I_{s}(r)\) is the illumination intensity, \(S(r)\) is the sample fluorescence distribution and \(\otimes\) represents the convolution operation, and \(h(r)\) is the detection point spread function. Normally, SIM utilizes sinusoidal striped illumination patterns to elicit the fluorescence. In the frequency space, Equation (1) becomes  
\[D(k)=I_{0}\times\left\{S(k)+\frac{m}{2}S(k+k_{0})e^{-i\varphi}+\frac{m}{2}S(k- k_{0})e^{i\varphi}\right\}\times h(k), \tag{4}\]  
where \(I_{0}\), \(m\), \(k_{0}\) and \(\varphi\) are the peak illumination intensity, modulation depth, spatial frequency and phase of the illumination patterns, respectively. The first term (\(S(k)\times h(k)\)) in Equation (4) denotes all the spatial frequencies observed by the conventional microscope while the two remaining terms contribute to the spatial resolution enhancement. To disentangle these three overlapping frequency components, one can capture three raw images using illumination patterns with three different phases and then solve a linear function. By recombining these shifted frequency components to their proper locations in the Fourier domain and carrying out inverse Fourier transform, SIM image reconstruction is accomplished. As for MSIM, the illumination patterns mainly contain the frequency in the X and Y directions. Therefore, the detected images in the Fourier domain can be expressed as  
\[D(k)=I_{0}\times\left\{ \begin{array}{c}S(k)+\frac{m}{2}S(k+k_{x})e^{-i\varphi _{x}}+\frac{m}{2}S(k-k_{x})e^{i\varphi_{x}}\\ +\frac{m}{2}S\Big{(}k+k_{y}\Big{)}e^{-i\varphi_{y}}+\frac{m}{2}S\Big{(}k-k_{y} \Big{)}e^{i\varphi_{y}}\end{array} \right\}\times h(k), \tag{5}\]  
where \(\varphi_{x,\sigma\;y}\) and \(k_{x,\sigma\;y}\) are the spatial frequency and the phase of the illumination patterns along the X or Y direction, respectively. To reconstruct the super-resolved images, we perform the reconstruction column by column and row by row separately based on the standard SIM reconstruction procedure. As is shown in Figure 3a, the final super-resolved image is obtained by superimposing the images reconstructed in the three columns and three rows. Figure 3b shows the schematic of the reconstruction procedure for the MSIM data. To extract the illumination parameters, we place a reflective mirror at the focal plane and nine raw multifocal images, D\({}_{1}\)-D\({}_{9}\), are subsequently acquired. Illumination parameters along the Y direction can be extracted out by fitting the data profiles of the same positions (e.g., three red dashed lines in Figure 3a) of D\({}_{1}\)-D\({}_{3}\), while illumination parameters along the X direction can be extracted out by fitting the data profiles of the same positions (e.g., three black dashed lines in Figure 3a) of D\({}_{1}\), D\({}_{4}\) and D\({}_{7}\), respectively. \(I_{0}\), \(m\) and \(k_{0}\) are the average of the corresponding fitting results while the three different phases (\(\varphi_{j,\;X\;\sigma\;Y}\), where \(j=1,\,2,\,3\)) are retained. During reconstruction in a certain column or row, our method is essentially based on the reconstruction procedure of the standard SIM utilizing the illumination parameters along the Y or X direction. More details of the reconstruction procedure are previously illustrated in our previous work [30]. According to Equation (2) and combined with the extracted illumination parameters, a system of three independent equations can be constructed and solved to separate the three overlapping frequency components. A super-resolved image can be obtained by an inverse Fourier transform of the recombination of these frequency components shifted to their origin locations in the Fourier domain.