Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ

Super-resolved structured illumination microscopy (SR-SIM) is an important tool for fluorescence microscopy. SR-SIM microscopes perform multiple image acquisitions with varying illumination patterns, and reconstruct them to a super-resolved image. In its most frequent, linear implementation, SR-SIM doubles the spatial resolution. The reconstruction is performed numerically on the acquired wide-field image data, and thus relies on a software implementation of specific SR-SIM image reconstruction algorithms. We present fairSIM, an easy-to-use plugin that provides SR-SIM reconstructions for a wide range of SR-SIM platforms directly within ImageJ. For research groups developing their own implementations of super-resolution structured illumination microscopy, fairSIM takes away the hurdle of generating yet another implementation of the reconstruction algorithm. For users of commercial microscopes, it offers an additional, in-depth analysis option for their data independent of specific operating systems. As a modular, open-source solution, fairSIM can easily be adapted, automated and extended as the field of SR-SIM progresses.


SR-SIM of U2OS osteosarcoma cells stained for tubulin
shifted band ̃2 (blue) and the common region (magenta), defined here by ℎ( ⃗ ), ℎ( ⃗ + ) > 0.05, i.e. both bands' OTFs are above a certain threshold. Right: Composite spatial representation of only frequencies from the common region, with 0 (green) and 2 (blue), base dataset U2OS actin. See Supplementary Note 1 for how the cross-correlation of this band overlap is used in the SR-SIM parameter estimation process.

Supplementary Note 1: Parameter estimation and reconstruction
In super-resolved structured illumination microscopy a sample is illuminated with a series of harmonic illumination intensity patterns. The image sequence ( ) acquired by the microscope is given by 1 Here ( ) denotes the fluorophores' response to light, i.e. the property to be measured. M denotes the number of harmonics, so = 1 for two-beam, and = 2 for three-beam illumination. is the modulation light wave vector, i.e. the pattern orientation and spacing.
denotes the phase, the modulation depth of the pattern. h( ) is the point-spread function, which causes the resolution limit. Switching to Fourier space, each harmonic illumination pattern transforms to delta-peaks at (± ) and, using their folding and translation properties all bands ̃(⃗ ) in ( ⃗ ) can be extracted. This step is referred to as band separation.
Afterwards, the bands ̃(⃗ ) are moved to their correct position ± in Fourier space, added up and transformed back to a high resolution image in real space. The image assembly in Fourier space is usually performed through a Wiener filter: The filtering compensates for the frequency dampening introduced by ℎ( ⃗ ), the parameter dampens the degree of compensation especially in regions where ℎ is low. It should thus be set in accordance to the SNR of the input data. ( ⃗ ) is the apodization, compensating for ringing artifacts.
A SIM reconstruction thus amounts to Fourier-transforming the input, carrying out bandseparation, shifting the bands to , summing them up through (e.g.) a Wiener filter, and transforming the result back to a high resolution image. The resolution gain is given by the length | |, which is approximately 3 limited to the same cut-off as ℎ( ⃗ ) for linear SIM 4 . Thus, SIM typically doubles the resolution in comparison to a wide-field measurement.

Parameter estimation:
The SIM reconstruction introduced so far needs correct parameters (pattern spacing and orientation , phases ) to be carried out, which are typically 5 extracted from the input data. A reliable algorithm to obtain this estimation is often much more involved than the reconstruction itself. The method employed be fairSIM follows the method by Gustaffson et. al. 4 . Because the original publication provides little detail on the parameter estimation, we briefly lay out the mathematical background here. In general, the use of cross-and autocorrelation of frequency components for SIM parameter estimation is documented in a number of recent publications 4, 5, 6 .
Assuming equi-distant phases ′ , differing from correct phases = ′ + Δ only by a global offset Δ , the band-separation step is carried out.
For linear SIM 6 , the separated and OTF-corrected bands ̃ will have common, overlapping regions (see Supplementary Fig. 8 for an illustration).With ⃗ ′ denoting only components within these regions, ̃(⃗ ′ + ) and ̃0 ( ⃗ ′ ) should only differ by a constant, complex factor ′ . As only holds for the correct shift vector , the cross-correlation 3 For an exact number, wavelength change (Stokes-shift) has to be taken into account, and TIRF illumination will further shift the limit somewhat. 4 Non-linear SIM completely circumvents the limit by using effects such as photo-switching, depletion, or twophoton excitation. 5 Stable systems, especially SLM-based, should yield rather constant reconstruction parameters over time. However, retrieving a parameter estimate from data should often be much easier than characterizing them from the experimental properties. It might then be used to run multiple reconstructions. 6 For non-linear SIM, at least neighboring bands −1 , will overlap (otherwise, frequencies are missing from the reconstruction), thus the process can be carried out iteratively. ) limited to a region common to both band, will reach a maximum at the correct . This can be used in an iterative search to find , and implemented effectively, to sub-pixel precision, via the Fourier shift theorem. FairSIM provides visual feedback of this fit process, so the user can check it for plausibility. Now, at the correct , the equivalence given above holds, so ′ is found as and yields both, the global phase offset Δ = arg( ′ ) and -in principle -the pattern modulation depth = | ′ |.