High-speed blind structured illumination microscopy via unsupervised algorithm unrolling

Domain: physics
Published: 2026-01-23
Editorial outcome: minor_revision

Abstract
Blind structured illumination microscopy (blind-SIM) is a valuable tool for achieving super-resolution without the need for known illumination patterns. However, in its current formulation the algorithm requires many iterations to converge, leading to long inference times and limited use for real-time or video-rate imaging. We present unrolled blind-SIM (UBSIM), an algorithm which integrates a learnable neural network inside the unrolled iterations of the blind-SIM algorithm. UBSIM delivers a reconstruction speed two to three orders of magnitude faster than that of current iterative blind-SIM methods, while achieving similar resolution and image quality. Furthermore, we demonstrate that UBSIM can be trained in an unsupervised manner that reduces hallucinations and produces superior generalization capability when compared to benchmark super-resolution networks. We test UBSIM experimentally on live cells and present video-rate super-resolution imaging up to 50 Hz. Using our method, we observe dynamic remodeling of the endoplasmic reticulum with high spatiotemporal resolution.

Introduction
Structured illumination microscopy (SIM) is a widely used super-resolution (SR) technique that illuminates a sample with spatially non-uniform light to recover high-resolution information1,2. While SIM has a moderate two-fold resolution improvement when compared to other SR techniques such as STORM or STED, it requires only a handful of raw images to generate an SR frame, and thus provides high temporal resolution from both the data acquisition perspective and the SR reconstruction perspective. Over the past decade, SIM has been demonstrated as an attractive SR method for cases where video-rate imaging is needed or when exposure time is limited due to photobleaching3–5.

Various methods of SIM have been demonstrated using either periodic or random illumination patterns6,7. Standard periodic illumination SIM requires precise knowledge of the illumination patterns and their relative locations to correctly retrieve the frequency domain information of the object; thus, it is highly sensitive to these parameters. Any mismatch between the assumed and actual illumination patterns leads to artifacts and degraded performance8. Therefore, standard SIM systems need to be carefully calibrated and maintained for various imaging conditions9, which limits their practical applications in biological research.

To overcome this limitation, the blind-SIM algorithm was first developed by Mudry and colleagues to utilize randomized illumination patterns for SR image reconstruction10. As an iterative, model-based reconstruction method, blind-SIM solves jointly for the object and illumination patterns based on constraints of illumination statistics. Several other iterative blind-SIM algorithms have since been developed based on various constraints11–14. A thorough comparison of our work with the existing blind-SIM literature is provided in the supplementary material (Supp. Table 1). Blind-SIM has proven to be a powerful technique that has allowed the design of SIM systems which rely on random illumination patterns that do not require calibration and are easier and cheaper to build and maintain.

While blind-SIM has greatly reduced the hardware constraints of SIM, it comes with the tradeoff of a slow, iterative reconstruction process. Even with GPU-based hardware acceleration, the time for a single reconstructed frame can range from tens of seconds to minutes, depending on the image size and number of input frames. Thus, reconstruction of videos becomes prohibitively long, and real-time reconstruction is challenging. Ideally, users of an SR system should be able to rapidly reconstruct a high-resolution image on the fly in order to evaluate a sample immediately. Widespread adoption and use of SR technologies by target groups, such as the biology community, relies on the ease of use of such systems.

Recently, data-driven learning-based techniques such as deep learning have become widely used for SR image reconstruction. These methods often have much faster inference times than traditional iterative reconstruction algorithms and can produce higher quality results in challenging conditions, such as shorter exposure times with lower signal-to-noise ratios15–18. However, there are several critical limitations to deep neural networks (DNNs) that prevent their widespread use for SR microscopy.

Firstly, most methods using DNNs for SR rely on supervised training strategies where datasets of matched low and high-resolution images are needed. Generating a training set of experimental, paired images is not typically feasible; therefore, simulated data is generated using various types of objects combined with the physical model of the imaging process. Yet, it is often unclear how well these models will perform when applied to an experimental system where the types of objects may differ from the objects the model is trained on. It is well documented that DNNs can overfit to training data and hallucinate when applied to new, unseen data19–21. This lack of generalizability is a major concern with the use of DNNs in biological imaging.

Secondly, neural networks are so-called black box systems and are not readily interpretable. Unlike iterative algorithms, when trained in an end-to-end manner, DNNs typically do not explicitly incorporate any knowledge of the physical imaging model22. Thus, it is hard to determine whether a DNN is truly learning the more general inverse process of the forward model or simply relying on image statistics from the training set to predict the most likely object features.

There has been growing interest in combining physics with DNNs for image reconstruction23–26. One promising technique, known as algorithm unrolling (or unfolding), uses trainable parameters within the framework of iterative algorithms27–30. In algorithm unrolling, each iteration of an iterative algorithm becomes a layer in a DNN, and multiple layers are connected end-to-end, forming the network. In this way, the number of layers in the network is equivalent to running an iterative algorithm for the same number of iterations. Within each layer, learnable parameters can be included, and the entire network can be trained via backpropagation to update these parameters. Unrolled algorithms have been shown to be interpretable, generalizable, and require fewer parameters and training data than traditional DNNs.

In this paper, we propose a new unrolled algorithm we name unrolled blind-SIM (UBSIM). UBSIM produces a two-fold improvement in resolution with a reconstruction speed that is two to three orders of magnitude faster than iterative blind-SIM in both simulations and experiments. Furthermore, we demonstrate that UBSIM generalizes to unseen data far better than reference SR DNNs. Lastly, to illustrate its applicability to biological research, we perform live-cell imaging of endoplasmic reticulum dynamics with a high-speed SIM acquisition setup utilizing a synchronized DMD illumination system. Using UBSIM, we capture the real-time movement and collapse of endoplasmic reticulum network structures.

Results
Derivation of unrolled algorithm
To begin, we describe the unrolled blind-SIM algorithm following the derivation of the iterative algorithm developed by Mudry and colleagues in their paper10. We adopted the same notation for consistency. For structured illumination microscopy, the forward model for each sub-frame M is:

M = (Iρ) * h (1)

where I is the illumination pattern, ρ is the object and h is the point spread function of the detection objective. The aim of SR imaging is to recover ρ given the set of diffraction-limited images M. In traditional SIM the illumination patterns I are known and only ρ is solved for. However, in blind-SIM, we must jointly solve for ρ and I as the illumination patterns are unknown. This is achieved by making an additional assumption that all the illumination patterns on average sum to a uniform field:

Σ L l=1 I_l ≈ L I_0 (2)

Where L is the number of sub-frames and I_0 is a constant equal to the average intensity received by the sample. The error in the uniform approximation drops at a rate of 1/sqrt(L). See Supplementary Section S3 for derivation.

To ensure positivity, I_l and ρ are rewritten as the square of auxiliary variables such that:

I_l = i_l^2
ρ = ξ^2 (3)

The cost function to evaluate potential solutions for the object and illumination patterns can be written as:

F(ξ, i_1, ..., i_L) = Σ L l=1 ||M_l - (ξ^2 i_l^2) * h||^2 + ||M_L - ξ^2 I_L * h||^2 (4)

with:

I_L = L I_0 - Σ L l=1 i_l^2 (5)

To solve for the object and illumination patterns, the gradients of the cost function with respect to the object and illumination patterns are calculated, and the solutions are updated by a weighted step:

ξ^n = ξ^{n-1} + α^n g^{n,ξ}
i_{l,n} = i_{l,n-1} + α^n g^{l,n,i} (6)

where α is the step size, g is the gradient, and n is the iteration number. This update constitutes one iteration of standard gradient descent and is repeated for either a fixed number of iterations or until the reduction in the cost function plateaus. The Polak-Ribière conjugate gradient method31, used by Mudry and colleagues, as well as Nesterov acceleration32, can also be used to significantly accelerate optimization. Details of the implementation of these methods are included in Supplementary Section S4.

Inspired by recent work in algorithm unrolling, we propose using a trainable neural network which operates on the gradient as follows:

ξ^n = ξ^{n-1} + α^n CNN(g^{n,ξ})
i_{l,n} = i_{l,n-1} + α^n CNN(g^{n,i,l}) (7)

where CNN is a convolutional neural network for which the weights are learned to minimize the cost function across a set of training data. For our algorithm, only one set of weights was learned for one CNN, which was applied recursively at each iteration.

Figure 1 shows a block diagram for UBSIM. The network consists of a fixed number of iteration blocks, which serve to update the object and illumination patterns in a similar manner to a traditional iterative algorithm. In each iteration block, the residual from the cost function is used to calculate the gradients for the object and each illumination pattern. The gradients (stacked as channels in a tensor object) are then used as inputs to a CNN. The CNN's scaled outputs are added to the original inputs to obtain the updated object and illumination patterns. The CNN operates directly on the gradients similar to how a preconditioner33,34 or update direction calculation31,32,35 is used to accelerate convergence for iterative gradient descent algorithms. In this way, the use of the network can be thought of as a type of learned preconditioner36,37 or learned gradient descent38–41 for which there are examples from literature.

We employed an unsupervised training process to optimize the weights of the CNN in the unrolled neural network. We used the cost function loss of equation 4 to update the CNN weights during the training step rather than a loss function directly comparing the output object and ground truth object. This has the added benefit of removing the need for ground truth training data and helps with generalization.

For the unrolled network, we selected a modified U-Net architecture with 2 rescaling layers and skip connections42 (Supplementary Section S5). We chose this architecture because it is widely used for many image processing tasks and has been shown to perform well in denoising. Furthermore, with less than 250,000 parameters, this model is still relatively small compared to many state-of-the-art super-resolution models. More details on the training process can be found in the methods section and supplementary material.

Evaluation on simulated data
To evaluate the performance of UBSIM, we first tested the algorithm on a set of simulated fluorescence images that were generated from experimental microscopy data43 (see supplementary material, S7). Simulated data provided a ground truth to evaluate the model's performance and accuracy.

In Fig. 2a we quantitatively compared the image resolution improvement using several standard image quality metrics on a set of test data which were not included in the training process. The mean squared error (MSE), structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR) all showed significant improvement for the UBSIM results compared to the diffraction-limited widefield images, indicating an improvement in image fidelity. To estimate the improvement in spatial resolution, we employed the widely used image decorrelation analysis method44. The UBSIM results on average showed a resolution improvement of approximately 2-fold, which matches the expectation, as the simulated illumination speckles were also diffraction-limited with the same numerical aperture (NA) as the detection objective.

Comparisons of reconstruction results on four types of cellular structures are shown in Fig. 2b. The cell structures used were microtubules (MT), clathrin-coated pits (CCP), endoplasmic reticulum (ER), and F-actin filaments. The results from UBSIM revealed structural features that were not visible from the widefield images and matched well with the ground truth images. For reference, the results from iterative blind-SIM are also displayed. While iterative blind-SIM provided a similar level of resolution improvement, the main advantage of UBSIM is the reduced inference time and greater resistance to noise overfitting (see Supplementary Section S8). Specific examples of structural feature recovery are presented in Fig. 2c,d. Two regions from the microtubule and CCP examples in Fig. 2b are displayed in Fig. 2d. UBSIM was able to reveal closely spaced MT filaments and the ring-like structures in CCPs. Line profiles taken from both regions in Fig. 2c show close alignment with the ground truth and demonstrate the ability of UBSIM to resolve sub-diffraction features.

Comparison of inference time
In Fig. 3 we quantitatively compared the speed for unrolled and iterative blind-SIM. For iterative blind-SIM, we ran evaluations using regular gradient descent as well as two accelerated methods: nonlinear conjugate gradient descent31 and gradient descent with Nesterov momentum32. Additional information on the implementation of these methods is included in Supplementary Section S4. There are two ways we assessed the speed: (1) the number of iterations to reach a target loss and (2) the time taken for the computation. For each image in the test dataset, we first calculated the final cost function value for the result from UBSIM. For this paper we chose a UBSIM model with six iteration blocks (see Supplementary Section S10). Next, we ran the iterative blind-SIM methods until their cost functions reached the same level as the UBSIM result and recorded the time and number of iterations. For a fair comparison, all algorithms were written in PyTorch and run on the same GPU (Nvidia RTX 1080 Ti).

We found that UBSIM was on average 3 orders of magnitude faster than the gradient descent implementation and 2 orders of magnitude faster than the conjugate gradient descent and Nesterov acceleration methods in terms of computation time. UBSIM required 2 orders of magnitude fewer iterations than gradient descent and 1 order of magnitude fewer than the accelerated methods. A single 256 × 256 pixel image took approximately 10-20 seconds to reconstruct for gradient descent, 1-3 seconds for the accelerated methods, and 10 milliseconds for UBSIM. The UBSIM computation time improvement was relatively faster than the iteration improvement due to differences in how PyTorch implements the two functions. UBSIM has a fixed number of iterations, and thus PyTorch uses a static graph, whereas iterative blind-SIM has a variable iteration number and requires a dynamic graph. While absolute computation time will vary with hardware, the GPU used here is a consumer-grade card four generations older than current, and thus is of negligible cost for many labs. With UBSIM, real-time image reconstruction for video-rate blind-SIM is possible. In this work, we used a raw image frame rate of 100 fps for our high-speed imaging demonstration. The highest speed experimental data in this paper produced SR images at 50 frames per second, which required a 20 ms capture time. Therefore, even with modest computation power, real-time SR video reconstruction is possible. The improvement in speed greatly increases the practical usability of blind-SIM for potential users such as biologists, as they can now quickly view SR imaging results rather than waiting hours for reconstruction of a single video.

Evaluation of generalization performance
We tested the generalization capability of UBSIM and three common SR networks by training them on three separated datasets and cross-comparing their performance on each. The SR networks we chose to compare with are the U-Net42, Enhanced Deep Residual Network (EDSR)45, and Residual Channel Attention Network (RCAN)46. All of these networks have been widely used for SR image reconstruction. We generated three simulated datasets consisting of ER, CCPs, and digits from the MNIST47 dataset. Each of these objects has distinct structures which are different from each other. The U-Net, EDSR, and RCAN networks were all trained in a supervised manner where the MSE loss between the predicted and ground truth data was used to update the network weights. UBSIM was trained in the same unsupervised method as detailed earlier. The networks were trained on the ER, CCP, and MNIST datasets separately, and then each trained model was applied to test sets from each object type.

A qualitative comparison of the networks’ generalization ability is shown in Fig. 4a. In this figure, we compared the output of each trained model when applied to a single example from the ER test dataset. The top row shows results for each network type trained on ER data. As expected, when the training and test data were drawn from the same distribution, all networks produced images that show increased resolution and matched well with the ground truth image. However, when the models were tested on out-of-distribution data, a clear difference emerged between the unrolled network and the standard models. The middle and bottom rows display the model outputs on the ER test data when trained on CCP and MNIST data, respectively. UBSIM produced results which were visually similar to the results produced from the ER-trained model and again showed high fidelity to the ground truth image. Conversely, the results from U-Net, EDSR, and RCAN all showed signs of hallucination-based artifacts. For example, when trained on CCP data, the standard models produced results that have dot-like characteristics that are similar to those seen in the CCP data. When trained on the MNIST-based dataset, the models again showed artifacts that are characteristic of the dataset, appearing overly saturated with uniform object intensity. Cropped regions from these comparisons are displayed in Fig. 4b.

We quantitatively compared UBSIM to standard networks using the Learned Perceptual Image Patch Similarity (LPIPS)48 metric for each network and combination of training and test datasets. The LPIPS metric is a perceptual image metric that compares the differences in the extracted features of images from a pretrained convolutional neural network (VGG49 in our case) and has been shown to match human perception well. We chose a perceptual metric as it is highly sensitive to changes in object structure, indicating if artifacts are occurring.

Figure 4c contains a matrix for each network architecture with every combination of training and test data. Diagonal elements of the matrix represent cases where the training and test data types are the same, whereas all the other elements represent cases where they differ. Each element in the matrix contains the average LPIPS-based similarity score for 100 images in a test dataset. Here, we define the similarity score as 1 - LPIPS, so that the value of 1 indicates perfect similarity.

Generalization performance can be determined by assessing the change in similarity score across each column of the matrix. The figure shows that UBSIM maintains relatively consistent similarity scores across the different test object data types, while the other SR networks exhibited larger changes when applied to new objects.

Live-cell imaging of actin
Next, we demonstrated the ability of UBSIM to produce SR images on experimental live-cell data. We employed an automated system to control the projection of random, diffraction-limited speckle illumination patterns onto our samples using a digital micromirror device (DMD). The DMD was synchronized with the camera to ensure that the patterns did not move and were not averaged out during each image exposure period. More details of the experimental setup can be found in Supplementary Section S11.

We first imaged live COS-7 cells, fibroblast-like cells derived from monkey kidney tissue, expressing mNeonGreen-tagged Lifeact, a short peptide which binds F-actin50. The results are shown in Fig. 5. The UBSIM model was trained with the same simulated dataset as Fig. 2 and contained an even mix of the 4 cell structure types.

UBSIM resulted in a 1.94x resolution improvement over the widefield images, as confirmed by the image decorrelation metric. The estimated resolution dropped from approximately 300 nm to 150 nm after the unrolled network. Furthermore, resolution improvement was indicated by increased high-spatial-frequency content in the normalized logarithmic Fourier magnitude spectra of the widefield and UBSIM images. We were able to resolve closely spaced actin filaments that could not be seen in the widefield images (Fig. 5b,c). Additionally, the results from UBSIM matched well with the results obtained from iterative blind-SIM, indicating that hallucinations were not occurring due to the mismatch between the simulated data used to train the model and the experimental data on which it was tested. As iterative blind-SIM has no learned components, it can serve as a good unbiased benchmark to test against when there is sufficient signal-to-noise ratio.

Video imaging of ER dynamics
To demonstrate video-rate imaging using UBSIM, we imaged the spatiotemporal dynamics of ER remodeling in real-time. The ER regulates protein synthesis, folding, and modifications51 and is a highly dynamic organelle which forms a dense network with multiple structural elements, including cisternae (sheet-like structures), and tubules52,53. The dense network and rapid movements of the ER make it a particularly challenging organelle to image54.

To demonstrate the ability of UBSIM to capture the dynamic movements of the ER in live cells, we first used COS-7 cells expressing ER-stagRFP55, a red fluorescent protein targeted to the cytoplasmic side of the ER via a transmembrane signal peptide56. Figure 6a–e presents the UBSIM imaging results. We modulated the illumination patterns at 20 fps and used 20 frames for reconstruction. To visualize dynamics, we used a rolling-frame reconstruction method with a 2-frame step to achieve an effective reconstructed video speed of 10 fps. Because this method includes a large overlap in frames between reconstructions, it introduces temporal smoothing, but can be appropriate for ER where the dynamics of interest (compartment remodeling) occur on timescales of seconds and are generally slower than the total SR frame integration time57,58. Figure 6e, shows the dynamics of a tubule region in the ER which could not be seen from the widefield images (Fig. 6d). ER tubules are generally peripherally located and exhibit membrane curvature as opposed to the sheet-like cisternae structures52. Over the course of approximately two seconds, the tubule collapsed as the ER network began to remodel and form a new tubule structure. Additional videos of the changing ER networks are included as supplementary material (Supplementary Movies 1–5).

Figure 6f–j presents another set of the UBSIM imaging results for ER, this time expressing ER-mNeonGreen (a green fluorescent protein). The raw images were captured at 100 fps. Using the same rolling window and number of input frames the reconstructed SR video is at 50 fps. Example video reconstructions are included in the supplementary material. We observed the rapid collapse of a tubule structure in just under one second as shown in Fig. 6i,j. ER remodeling, like what we observed here in our test cases, can function to maintain homeostasis and is dynamically regulated in response to changing cellular conditions or stress59. Our approach can thus capture the complex and rapid remodeling of the ER and can be utilized in the future to characterize ER dynamics under various stressed and diseased conditions.

Discussion
We present the unrolled blind-SIM algorithm (UBSIM), which can reconstruct SR images from unknown illumination patterns with much faster speeds than current methods. By incorporating learnable parameters with the unrolled, physics-based iterations of the blind-SIM algorithm, we can achieve high-resolution images with many fewer iterations, thus reducing the inference time by orders of magnitude. UBSIM combines the benefits of neural networks (fast inference time) and iterative algorithms (no data bias, physics-based interpretability).

Using simulated data, we verified the fidelity of UBSIM results against ground truth images and quantified the speed improvement. Additionally, we studied the effect of training data composition on image reconstruction and found that UBSIM generalizes better to unseen data than benchmark SR networks. We further demonstrated live-cell imaging with UBSIM and video-rate imaging with up to a 50Hz reconstructed frame rate.

Our current demonstration results in a 2× improvement in resolution, as we used diffraction-limited speckle patterns for illumination. However, this improvement is fundamentally limited by the resolution of the illumination patterns, not by the UBSIM algorithm. In future work, we aim to combine high-index substrates with UBSIM to further improve the resolution below 100 nm.

While UBSIM improves on current blind-SIM algorithms, several limitations must be considered by potential users. Firstly, UBSIM requires training before it can be used for inference, which creates an upfront time cost. For one-off reconstructions, iterative methods, despite their longer reconstruction times, may be preferred. However, for cases that include repeated image reconstruction, such as video imaging or repeated experiments, UBSIM can provide a significant time advantage. Secondly, UBSIM, like all blind-SIM methods, requires multiple sub-frames for each reconstructed SR frame, which can affect image acquisition speeds. In our work we utilize a sliding-frame reconstruction approach in addition to our high-speed illumination system to boost acquisition speed. However, this approach can cause temporal smoothing and is not suitable for cases where the sample movement is generally faster than the effective integration time for a single SR frame. Further improvements in blind-SIM acquisition speed require either a reduction in sub-frames per reconstructed image or increased speckle modulation and camera capture rates.

We envision UBSIM as a useful tool for those in the SR community interested in low-cost, flexible high-speed imaging. UBSIM can enable real-time, video-rate super-resolution with modest hardware requirements. Additionally, as UBSIM does not require knowledge of illuminations patterns, users can produce SR video without the need for calibrated patterns. Lastly, UBSIM's superior generalization performance allows users to have confidence in the fidelity of reconstructed images without having to carefully consider whether their training dataset matches the test data. Going forward, UBSIM can be harnessed to study the spatiotemporal dynamics of subcellular organelles, like the ER, and uncover the complex biology of the cell.

Methods
Algorithm code and training
The UBSIM network and iterative blind-SIM codes were written in PyTorch 2.1.2 and Python 3.10.14. A simulated dataset was generated using images from the BioSR dataset43 and the physical forward model for speckle-based incoherent structured illumination. The simulated data was generated using code written in MATLAB release R2021b. Code and the datasets are provided for open-source use on GitHub and Zenodo, respectively (UBSIM code is available at https://github.com/Zach-T-Burns/Unrolled-blind-SIM, datasets and simulation code are available at https://zenodo.org/records/17852915). Models were trained on an Nvidia A6000 Ada GPU, and inference testing was performed on an Nvidia RTX 1080 Ti GPU. For a typical training run, models used an initial learning rate of 1e-4 with a reduction upon plateauing and were trained for 200 epochs, taking about 2 h. Specific details on datasets, physical model parameters, and training details for each trained model instance can be found in the supplementary material (Section S6).

Optical setup
Live-cell imaging was conducted on a customized Olympus IX83 inverted microscope. A 488nm CW laser (Coherent Genesis MX488−1000 STM) and a 532nm CW laser (Laser Quantum Ventus 532) were combined and coupled into a multimode fiber (Thorlabs, core diameter: 50 μm, NA 0.22). The fiber-coupled laser was then collimated and projected to a DMD (Texas Instruments DLP4710LC), which generates random illumination patterns. The random patterns were then projected onto the sample plane using a tube lens and an objective lens (Olympus 60X, 0.8 NA) to produce diffraction-limited speckle illumination. The fluorescence signal was detected with a water immersion objective (Olympus 60X, 1.2 NA, effective 1.0 NA in thick sample) and bandpass filters (Chroma ET520/40m for mNeonGreen, Semrock FF01-588/21-25 for stag-RFP). The detected images were captured with a sCMOS camera (Hamamatsu ORCA Flash 4.0 V3), which was synchronized with the DMD so that each frame is captured with a stabilized speckle illumination pattern. The software HCImage version 4.4.5 was used for image collection.

Cell growth and preparation
COS-7 cells used in this study were acquired from ATCC (CRL−1651). Cells were cultured in Dulbecco’s modified Eagle medium (DMEM; Gibco 11995-065) including 4.5 g/L glucose, 4 mM L-glutamine, 1.5 g/L sodium bicarbonate, and 1 mM sodium pyruvate, which was supplemented with 10% fetal bovine serum (FBS, Gibco 26140-079), and 1% (v/v) penicillin/streptomycin (Pen/Strep, Gibco 15140−122). Mycoplasma contamination was tested routinely in cells and found to be negative. For experiments, cells were plated onto glass slides two days prior to imaging. 16-24 hours after initial seeding at ~70% confluency, cells were transfected with ER-mNeonGreen, ER-stagRFP, or Lifeact-mNeonGreen using Polyjet In Vitro DNA Transfection Reagent (SL100688, SignaGen Laboratories). Cells were incubated overnight, then the transfection medium was replaced with modified Hank’s balanced salt solution, including 1x HBSS (diluted from 10X HBSS; Gibco, 14065) with 2 g/L glucose at pH 7.4 for imaging.

Constructs
LifeAct-mNeonGreen was a gift from Dorus Gadella60 (Addgene plasmid #98877; http://n2t.net/addgene:98877; RRID:Addgene_98877). ER-stagRFP was generated as previously reported55. ER-mNeonGreen was generated by restriction enzyme cloning. mNeonGreen was sub-cloned into the ER-stagRFP construct, replacing stagRFP, using the BamHI and XbaI restriction sites. All constructs were verified by forward and reverse sequencing.

Imaging conditions
For both ER-stagRFP and ER-mNeonGreen imaging, the image plane laser intensity is maintained at 200 W/cm2. The DMD and sCMOS camera are synchronized such that a new speckle illumination is projected only after the current frame is readout on the sCMOS to prevent distortion in images. For a 512 × 512 raw video at 100 fps, the readout time per frame is 5 ms, and the actual exposure time for fluorescence detection is 5 ms.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript presents an unrolled version of the Blind-SIM (UBSIM) algorithm aimed at reducing inference time. The\nresults are clearly described, and the authors have commendably shared the code and the training dataset, which enhances\nreproducibility and transparency.\nHowever, I have several concerns and suggestions for improvement:\n1. Novelty Claim: The authors assert that their approach is the first to achieve video-rate blind-SIM imaging at 50 Hz. While\nthis is an impressive claim, it may be overstated given that other algorithms have been developed to address the same\nproblem (e.g., ref [1] below). A more thorough discussion of related work is needed to contextualize this contribution.\n2. Dataset Size and Reconstruction Validity: The use of only 20 images for reconstruction seems inadequate for ensuring\nthe validity of equation (2). While this number might suffice for sparse samples, it is likely insufficient for denser samples,\nwhere the signal complexity increases. A discussion on the limitations of this approach for varying sample densities would\nstrengthen the manuscript.\n3. Temporal Smoothing: The authors employ a shift of only two images between consecutive reconstructions, resulting in 18\nimages being common across adjacent reconstructions. This overlap introduces a temporal smoothing effect, which may\nartificially enhance the perception of temporal resolution. The implications of this overlap should be explicitly discussed, as it\ncould undermine the claim of achieving true video-rate performance.\n4. Training Requirements: While unrolled algorithms are known to generalize better and require less training data, the need\nfor training is still a limitation. This aspect is understated in the manuscript and should be more thoroughly addressed to\nprovide a balanced perspective on the advantages and limitations of the proposed approach.\nIn summary, while the proposed UBSIM algorithm represents a relevant and promising contribution, several claims and\nmethodological choices require further justification or discussion. Addressing these concerns would enhance the\nmanuscript's rigor and provide a clearer understanding of the algorithm's true capabilities and limitations.\n[1] Giroussens, Guillaume, et al. \"Fast super-resolved reconstructions in fluorescence random illumination microscopy\n(RIM).\" IEEE Transactions on Computational Imaging (2024).", "reviewer_2": "In this work, the authors propose to unroll the Blind SIM algorithm of Mudry et al. Numer-\nical experiments show that the proposed unrolled version achieves the same quality in a much\nlower computational time.\nI have several concerns regarding this work (detailed below). First, it appears that the\nimplementationoftheoriginalmethodbyMudryetal. isnotaccuratelyreproduced. Second,the\nonly benefit of the proposed unrolled method is the improved speed. However, before turning to\nunrolled networks, there exist numerous standard optimization algorithms that are significantly\nfasterthanthesimplegradientdescentbaselineusedhere(i.e.,aproperlyimplementedconjugate\ngradient method, an (accelerated) gradient descent with a sound line search, or even second-\norder methods). The true interest of the proposed unrolling approach would be justified only\nif it demonstrated improvements over these stronger baselines. Lastly, since the publication of\nMudry et al., non-iterative methods for speckle blind SIM have been proposed and should also\nbe considered as baselines for a fair comparison.\nFor all these reasons, I believe that this manuscript is not suitable for publication in Nature\nCommunications.\nComments\n• The constraint (2) is used to obtain a closed system with L unknowns and L equations.\nHowever, how is I determined? The value assigned to it will influence the reconstructed\n0\nimage.\n• In (4), the authors present a gradient descent algorithm to minimize the data fidelity\nterm (3) (without any regularization). Then, in (5), they propose an unrolled version by\ninserting a CNN around the gradient step. The authors interpret this CNN as a proximal\noperator. However, this interpretation is unusual and not clearly justified. Typically,\nin algorithm unrolling where a network is used in place of a prox, one starts from a\nwell-established proximal algorithm and replaces each proximal step with a learned CNN.\nIn (5), it is unclear which underlying algorithm is being unrolled. If the CNN is meant to\nrepresentaproximaloperator, theresultingstructuredoesnotcorrespondtoanystandard\nalgorithm I am aware of. A common and well-understood strategy is to unroll a proximal\ngradient algorithm, where the CNN is applied around the entire gradient descent step,\nthus acting as a learned prior or regularization term. In the current setup, given how the\nCNNispositioned, itwouldbemorenaturaltointerpretitasapreconditionerratherthan\na proximal operator. This distinction needs to be clarified by the authors.\n• The iterative algorithm considered is a simple gradient descent, which is well known to\nconvergeslowly, witharateofO(1/k). However, itwouldbestraightforwardtoimplement\nan accelerated version using Nesterov’s method, requiring only minor modifications to the\ncode and achieving a faster convergence rate of O(1/k2). Based on my experience with\nimaging inverse problems, such an acceleration significantly improves convergence speed.\nacceleration, in order to better highlight the relevance and advantages of the proposed\nunrolled method.\n• Inthesupplementarysection5,theauthorsshowthatconjugategradientisnotfasterthan\na simple gradient descent, which is very surprising to me. How is the step size α chosen?\nFrom the provided code I can see that the value of α is fixed by hand and reduced along\nwith iteration in a very heuristic manner... In contrast, in Mudry et al. (and in general in\nconjugate gradient), it is optimized at each iteration. Also what about the non-negativity\nconstraint imposed (through squaring the relevant variables) in Mudry et al.?\n• The blind SIM iterative algorithm proposed by Mudry et al. is now more than 10 years\nold. Since then, faster non-iterative alternatives have been proposed such as\nGiroussens et al (2024). Fast super-resolved reconstructions in fluorescence random illu-\nmination microscopy (RIM). IEEE Transactions on Computational Imaging.\nTypos\n• ”Widespread adaption and use of SR technologies by target groups” → adoption ?\n• InSuppfile", "reviewer_6": "WeexaminedthereconstructionresultsofUBSIManditerative\nblind-SIM when there is a low signal-to-noise ratio. As can be seen in Supplementary\nFigure 5” → It is Supplementary Figure 6.\n• Page 14 line 301: ”On axis elements of the matrix” → On diagonal elements ?\n• Page 17, line 350: ”... make it a particularly challenging organelle to image” → the word\norganelle seems misplaced"}, "author_response": "Reviewer #1\n1. Comment: “Novelty Claim: The authors assert that their approach is the first to achieve\nvideo-rate blind-SIM imaging at 50 Hz. While this is an impressive claim, it may be\noverstated given that other algorithms have been developed to address the same problem\n(e.g., ref [1] below). A more thorough discussion of related work is needed to contextualize\nthis contribution.”\nResponse:\nWe thank the reviewer for their comment and recognize the importance of making accurate\nand nuanced novelty claims for this work. We would like to first note that the specific claim\nof “video-rate blind-SIM imaging at 50 Hz” depends on the experimental configurations for\nhigh-speed data acquisition, while the algorithm provides real time super-resolution display.\nThere are two notions of speed in regard to super-resolution imaging. One is the acquisition\nspeed which describes how long it takes to collect the frames for the imaging system and is\nlimited by hardware. The other is reconstruction speed which refers to how fast a super-\nresolved image can be reconstructed from raw data via an algorithm.\nOur work has novelty in regard to both notions of speed. The unrolled algorithm method\nenables exceptional reconstruction speed that is orders of magnitude faster than other\nalgorithms for random speckle imaging. Additionally, we were able to achieve high image\nacquisition speeds using a DMD to create speckle patterns that are synchronized with a\nsCMOS camera. This type of reconstruction speed, combined with experimental live-cell\nvideo data, has not been demonstrated for blind-SIM imaging.\nHowever, as we recognize there have been other traditional SIM imaging implementations\nwith high capture speed [Ref. 1], we removed the wording “for the first time” from the\nabstract with regard to capture speed to avoid confusion.\nWe have modified the main text to clarify the paper’s contributions in terms of both\nacquisition and reconstruction speed.\n[Revision in the main text]\nWe present unrolled blind-SIM (UBSIM), an algorithm which integrates a learnable neural\nnetwork inside the unrolled iterations of the blind-SIM algorithm. UBSIM delivers a\nreconstruction speed two to three orders of magnitude faster than that of current iterative\nblind-SIM methods, while achieving similar resolution and image quality. Furthermore, we\ndemonstrate that UBSIM can be trained in an unsupervised manner that reduces\nhallucinations and produces superior generalization capability when compared to benchmark\nsuper-resolution networks. We test UBSIM experimentally on live cells and present video-\nrate super-resolution imaging up to 50 Hz. Using our method, we observe dynamic\nremodeling of the endoplasmic reticulum with high spatiotemporal resolution.\n[Revision in the main text]\nStructured illumination microscopy (SIM) is a widely used super-resolution (SR) technique\nthat illuminates a sample with spatially non-uniform light to recover high-resolution\ninformation. While SIM has a moderate two-fold resolution improvement when compared to\nother SR techniques such as STORM or STED, it requires only a handful of raw images to\ngenerate a SR frame, and thus provides high temporal resolution from both the data\nacquisition perspective and also SR reconstruction perspective.\n[Revision in the main text]\nSeveral other iterative blind-SIM algorithms have since been developed based on other\nvarious constraints. A thorough comparison of our work with the existing blind-SIM\nliterature is provided in the supplementary material (Supp. Table 1).\n[Revision in the main text]\nIn this paper, we propose a new unrolled algorithm we name Unrolled Blind-SIM (UBSIM).\nUBSIM produces a two-fold improvement in resolution with a reconstruction speed that is\ntwo to three orders of magnitude faster than iterative blind-SIM in both simulations and\nexperiments. We demonstrate that UBSIM generalizes to unseen data far better than\nreference SR DNNs. Lastly, to demonstrate its applicability in biological research, we\nperform live cell imaging of endoplasmic reticulum dynamics in live cells with a high-speed\nSIM acquisition setup utilizing a synchronized DMD illumination system. Using UBSIM,\nwe\nobserve the movement and collapse of endoplasmic reticulum network structures in real time.\nThe supplementary material is also modified to include a table comparing our paper to other\nblind-SIM papers in the literature:\n[Revision in the supplementary material]\nS1: Comparison with current blind-SIM literature\nTitle Method Live Cell Video Limitations\ndata? data?\nStructured illumination Iterative No No • Slow reconstruction even\nmicroscopy using physics- (fixed with conjugate gradient\nunknown speckle based model tissue only) acceleration\npatterns1 • No dynamic, or live cell\n(2012) imaging demonstrated\nStructured illumination Iterative No No • Slow iterative\nfluorescence physics- (fixed cell reconstruction method\nmicroscopy with based model only) • No live cell or video data\ndistorted excitations\nusing a filtered blind-\nSIM algorithm2\n(2013)\nFluorescent microscopy Iterative No No • Slow iterative\nbeyond diffraction limits physics- (fixed cell reconstruction method\nusing speckle based mode only) • No live cell or video data\nillumination and joint\nsupport recovery3\n(2013)\nStructured illumination Iterative No No • Slow iterative\nmicroscopy with physics- reconstruction method\nunknown patterns and a based model • No video or live cell\nstatistical prior4 demonstrations\n(2017)\nJoint Reconstruction Iterative No No • Requires regularization\nStrategy for Structured physics- parameter tuning\nIllumination based model • Accelerated iterative\nMicroscopy With method still requires more\nUnknown Illuminations5 iterations than UBSIM\n(2017) • No video or live cell\ndemonstrations\nOn the Superresolution Iterative No No • Purely theoretical paper\nCapacity of Imagers physics- • Slow iterative\nUsing based model reconstruction method\nUnknown Speckle • No live cell or video data\nIlluminations6\ndemonstrations\n(2018)\nSuper-resolved live-cell Iterative Yes Yes • Slow iterative\nimaging using random physics- reconstruction\nillumination based model\nmicroscopy7\n(2021)\nDeep learning for blind Deep Yes No • Limited experimental\nstructured illumination learning evaluation\nmicroscopy8 • No rigorous exploration\n(2022) of generalizability\n• Requires supervised\ntraining scheme\nFast super-resolved Iterative No No • Accelerated iterative\nreconstructions in physics- method\nfluorescence based model • Resolution improvement\nrandom illumination cannot reach 2x in non-\nmicroscopy (RIM)9 noiseless demonstrations\n(2024) • Limited quantitative\nresolution evaluation\n• No experimental live cell\ndata\nSupplementary Table 1: Summary of current blind-SIM papers and their limitations compared\nto this work.\nAdditionally, we thoroughly read the specific reference provided by the reviewer\n(Giroussens, Guillaume, et al. \"Fast super-resolved reconstructions in fluorescence random\nillumination microscopy (RIM).\" IEEE Transactions on Computational Imaging (2024).)\n[Ref. 2] to assess whether this work effects any of our paper’s claims of novelty. While the\nwork does introduce an accelerated algorithm for the related method of variance-based\nRandom Illumination Microscopy, we find that the paper does not affect the novelty of\nour paper or improve upon our UBSIM algorithm for the following reasons:\n[1] In terms of algorithmic reconstruction speed, the paper introduces a non-iterative\nreconstruction method termed RIM-CF. However, the authors explain that this\nmethod, while better than Wiener deconvolution in some cases, cannot reach the full\n2x resolution improvement and that iterative methods are needed for full resolution\ndoubling. See comments from the text such as “Obviously, even in a favorable\nscenario, RIM-CF remains based on the approximation (12) and modeling errors\ndeteriorate the resolution gain, e.g., RIM-CF cannot reach the theoretical resolution\nlimit one can expect from RIM in a noise-free setting. An iterative solver, based on a\nmore accurate model is required to reach this limit.”, “RIM-CF is basically a non-\niterative regularized deconvolution” (section III. A non-iterative estimator, RIM-CF),\nand “RIM-CF is a fast, FFT-based inversion of the standard deviation that provides a\nsuper-resolved estimate of the sample. However, because it is based on an\napproximation, this estimator cannot achieve the maximal theoretical resolution gain\nfor RIM.” (Conclusion) Furthermore, this lack of resolution gain from the non-\niterative method can be clearly seen in Figure 2. Here, even in a noiseless\ndemonstration case, RIM-CF is unable to reach full resolution improvement\ncompared to RIM-STD/RIM-VAR.\n[2] The paper also introduces an iterative algorithm RIM-STD which is an accelerated\nversion of the earlier developed RIM-VAR algorithm. However, there are several\naspects in which this algorithm does not match our paper’s performance. Firstly, the\nnumber of iterations needed for convergence in the noiseless 2D imaging case (Figure\n2) is 100 iterations which is still an order of magnitude more than our UBSIM\nalgorithm. Secondly, it appears that RIM-STD only can reach a full 2x resolution\nimprovement in highly idealized, simulated scenarios. In Figure 3, RIM-STD\nachieves the full resolution improvement under the “asymptotic variance” case where\nessentially the variance in perfectly known. However, when the empirical variance\nmust be calculated, which is what would happen in real-world imaging, the\nperformance of RIM-STD is much worse. Even with 1,000 sub-images and a very\nhigh SNR (30 dB) RIM-STD is unable to achieve the full resolution improvement\nprovided by the speckle and in fact it is stated that “RIM-CF and RIM-STD now reach\ncomparable results in terms of quality”. That means in the experimental case, RIM-\nSTD does not perform any better than a form of regularized deconvolution and cannot\nreach full resolution improvement.\n[3] We find the paper has several issues in how resolution is determined and presented.\nThe only figures for which quantitative resolution improvement or intensity cross\nsections are provided for are Figures 1 and 2 which are both simulated target objects\nwith noiseless data collection. However, this is problematic for two reasons: (1) in the\ncase of zero noise (infinite SNR) super-resolution is somewhat trivial. Using the\nknown PSF, the super-resolution object can be perfectly recovered via bandwidth\nextrapolation for many synthetic objects [see Refs. 3,4,5], and (2) for Figure 1 a\nsingle point source is used as the resolution measurement object. This again is a\ntrivial problem as there exist many algorithms that can arbitrarily sharpen a single\npoint source and thus does not provide an accurate measure of resolution. In\nmicroscopy, resolution is defined as the distance at which two closely spaced point\nsources can be distinguished (Abbe, Rayleigh) and not the FWHM of a single point\nsource. For example, simple Lucy-Richardson deconvolution can arbitrarily sharpen a\nsingle point’s width in a similar manner with few iterations, however it is not a true\nsuper-resolution method as it will fail to resolve closely spaced point sources in an\nenvironment with realistic noise levels.\n[4] Among the provided figures, only figure 4 contains reconstruction on experimental\ndata. The data is of sparse fixed fluorescent beads and there is no reconstruction on\nfixed or live cell data that could provide dynamic behavior.\n[5] We find that the paper does not contain any video reconstructions, only single frame\nreconstructions (see figures 1- 4, no supplementary available).\nTo conclude, the paper “Fast super-resolved reconstructions in fluorescence random illumination\nmicroscopy (RIM)” is not comparable to our paper as it presents (1) a non-iterative algorithm\nthat is only slightly better than deconvolution and cannot reach full resolution improvement even\nin noiseless conditions, and (2) an accelerated iterative algorithm that improves upon RIM-VAR,\nbut cannot reach full resolution improvement in any realistic scenario with noise and is still 1\norder of magnitude slower than our algorithm. The paper mainly focuses on highly idealized\nsimulations and does not provide live-cell or video imaging.\n2. Comment: “Dataset Size and Reconstruction Validity: The use of only 20 images for\nreconstruction seems inadequate for ensuring the validity of equation (2). While this number\nmight suffice for sparse samples, it is likely insufficient for denser samples, where the signal\ncomplexity increases. A discussion on the limitations of this approach for varying sample\ndensities would strengthen the manuscript.”\nResponse:\nWe thank the reviewer for their comment and recognize that importance of clarifying for the\nreader the relationship between number of sub-frames and image reconstruction performance\nin blind-SIM imaging.\n!\nFor fully developed speckle the contrast of L summed frames will be reduced at a rate of\n√#\n(see J. W. Goodman, “Some fundamental properties of speckle,” Journal of the Optical\nSociety of America (1976). [Ref. 6]). We can derive an expression for the RMS error by\ndoing a bias and variance analysis of the uniform intensity estimator. Starting from:\n#\n!𝐼 ≈ 𝐿𝐼\n$ %\n$&!\nWe define the fractional error as:\n∑# 𝐼 −𝐿𝐼\n$&! $ %\n𝐸 =\n$ 𝐿𝐼\n%\nFor fully developed speckle, the intensity follows an independent and identically distributed\nexponential random variable with mean 𝐼 and variance 𝐼'.\n% %\nThe expected value of the error (bias) is then:\n𝔼(∑# 𝐼 )−𝐿𝐼 ∑# 𝔼(𝐼 )−𝐿𝐼 𝐿𝐼 −𝐿𝐼\n$&! $ % $&! $ % % %\n𝔼(𝐸 ) = = = = 0\n$ 𝐿𝐼 𝐿𝐼 𝐿𝐼\n% % %\nAnd the variance of the error is:\n∑# 𝐼 −𝐿𝐼 ∑# 𝐼 𝑉𝑎𝑟(∑# 𝐼 ) 𝐿𝐼' 1\n𝑉𝑎𝑟(𝐸 ) = 𝑉𝑎𝑟0 $&! $ % 1 = 𝑉𝑎𝑟0 $&! $ −11 = $&! $ = % =\n$ 𝐿𝐼 𝐿𝐼 (𝐿𝐼 )' 𝐿'𝐼' 𝐿\n% % % %\nMean squared error can then be calculated as the decomposition:\n1\n𝑀𝑆𝐸(𝐸 ) = 𝐵𝑖𝑎𝑠(𝐸 )' +𝑉𝑎𝑟(𝐸 ) =\n$ $ $ 𝐿\nGiving a RMSE of:\n1\n𝑅𝑀𝑆𝐸 =\n√𝐿\nThus, the residual error in the uniformity constraint will decrease with an inverse square\nrelationship with the number of added frames.\nPractically, the result of this error will lead to (1) a multiplicative shading of the resulting\nimage and (2) higher noise which will result in artifacts [Refs. 7,8]. The resolution, however,\nis mainly determined by the OTF and the spatial frequencies of the illumination patterns.\nThe results of smaller frame number can be seen in Mudry et al.’s original paper [Ref. 7]\n(Supplementary Figure 4) where a smaller number of frames are used for reconstruction of\nthe Siemen star object than the main text (note that the Siemen star is a relatively dense\nobject). The resulting reconstruction still has the same resolution as periodic SIM with 9\nknown frames. However, there is additional granularity introduced by multiplicative shading\nand noise.\nUltimately, the number of frames is a parameter that must be determined by the user with\ndiminishing returns as the number of frames increases. There are other blind-SIM papers\nwhich have demonstrated reconstruction in the range of 20-40 frames [Refs. 8,9].\nWe clarified the relationship between number of frames and uniformity error to make it\nclearer to the reader. Additionally, we added Supplementary Section S3 which includes the\nerror derivation shown here along with a discussion on the effects on reconstruction results.\n[Revision in the supplementary material]\nIn this section, an error analysis for the uniformity assumption of the blind-SIM algorithm is\npresented. The bias, variance, and root mean square error of the estimator is derived as a\nfunction of the number of sub-frames used. We also discuss how error in the uniformity\nassumption affects the reconstructed image.\nThe uniformity assumption is defined as:\n#\n!𝐼 ≈ 𝐿𝐼\n$ %\n$&!\nThe assumption is that sum of all illumination patterns is the same as the average intensity\nmultiplied by the number of frames.\nThe fractional error is then defined as:\n∑# 𝐼 −𝐿𝐼\n$&! $ %\n𝐸 =\n$ 𝐿𝐼\n%\nFor fully developed speckle10, the intensity follows an independent and identically distributed\n(i.i.d) exponential random variable with mean 𝐼 and variance 𝐼'.\n% %\nThe expected value of the error (bias of the estimator) is then:\n𝔼(∑# 𝐼 )−𝐿𝐼 ∑# 𝔼(𝐼 )−𝐿𝐼 𝐿𝐼 −𝐿𝐼\n$&! $ % $&! $ % % %\n𝔼(𝐸 ) = = = = 0\n$ 𝐿𝐼 𝐿𝐼 𝐿𝐼\n% % %\nThe variance of the estimator is:\n∑# 𝐼 −𝐿𝐼 ∑# 𝐼 𝑉𝑎𝑟(∑# 𝐼 ) 𝐿𝐼' 1\n𝑉𝑎𝑟(𝐸 ) = 𝑉𝑎𝑟0 $&! $ % 1 = 𝑉𝑎𝑟0 $&! $ −11 = $&! $ = % =\n$ 𝐿𝐼 𝐿𝐼 (𝐿𝐼 )' 𝐿'𝐼' 𝐿\n% % % %\nMean squared error can then be calculated from the decomposition:\n1\n𝑀𝑆𝐸(𝐸 ) = 𝐵𝑖𝑎𝑠(𝐸 )' +𝑉𝑎𝑟(𝐸 ) =\n$ $ $ 𝐿\nGiving a root mean squared error of:\n1\n𝑅𝑀𝑆𝐸(𝐸 ) =\n$\n√𝐿\nTherefore, the residual error in the uniformity constraint will decrease with an inverse square\nrelationship with the number of sub-frames used for reconstruction. Adding more frames will\nalways result in reduced error, however at a diminishing rate of return.\nThe presence of error is the uniformity constraint will lead to (1) a multiplicative shading of\nthe resulting image and (2) higher noise which will result in granular artifacts1,4. The\nresolution of reconstructed image is ultimately determined by the spatial frequency support\nof the optical transfer function (OTF) and the illumination speckle. However, artifacts from\nshading and lowered signal to noise ratio from fewer sub-frames may reduce apparent\nresolution. Ultimately, the number of sub-frames used is a parameter to be chosen by the user\nthat will cause a tradeoff between shading accuracy and temporal resolution.\n[Revision in the main text]\nThis is achieved by making an additional assumption that all the illumination patterns on\naverage sum to a uniform field:\n#\n!𝐼 ≈ 𝐿𝐼\n$ %\n$&!\nWhere 𝐿 is the number of sub-frames and 𝐼 is a constant equal to the average intensity\n%\n!\nreceived by the sample. The error in the uniform approximation drops at a rate of . See\n√#\nSupplementary Section S3 for derivation.\n3. Comment: “Temporal Smoothing: The authors employ a shift of only two images between\nconsecutive reconstructions, resulting in 18 images being common across adjacent\nreconstructions. This overlap introduces a temporal smoothing effect, which may\nartificially enhance the perception of temporal resolution. The implications of this overlap should be\nexplicitly discussed, as it could undermine the claim of achieving true video-rate\nperformance.”\nResponse:\nWe thank the reviewer for their comment and agree that the sliding-frame window can\nintroduce temporal smoothing and that it is important to make this point clear to the reader.\nOur approach can introduce a smoothing effect but can be appropriate when the sample\nmovement is generally not faster than the SR frame integration time [Refs. 10,11]. We\nincluded a paragraph in discussion section that mentions the limitations of such an approach.\n[Revision in the main text]\nWe modulated the illumination patterns at 20 fps and used 20 frames for reconstruction. To\nvisualize dynamics, we used a rolling-frame reconstruction method with a 2-frame step to\nachieve an effective reconstructed video speed of 10 fps. Because this method includes a\nlarge overlap in frames between reconstructions, it introduces temporal smoothing, but can\nbe appropriate for ER where the dynamics of interest (compartment remodeling) occur on\ntimescales of seconds and are generally slower than the total SR frame integration time57, 58.\n[Revision in the main text]\nSecondly, UBSIM, like all blind-SIM methods, requires multiple sub-frames for each\nreconstructed SR frame which can affect image acquisition speeds. In our work we utilize a\nsliding-frame reconstruction approach in addition to our high-speed illumination system to\nboost acquisition speed. However, this approach can cause temporal smoothing and is not\nsuitable for cases where the sample movement is generally faster than the effective\nintegration time for a single SR frame. Further improvements in blind-SIM acquisition speed\nrequire either a reduction in sub-frames per reconstructed image or increased speckle\nmodulation and camera capture rates.\n4. Comment: “Training Requirements: While unrolled algorithms are known to generalize\nbetter and require less training data, the need for training is still a limitation. This aspect is\nunderstated in the manuscript and should be more thoroughly addressed to provide a\nbalanced perspective on the advantages and limitations of the proposed approach.”\nResponse:\nWe thank the reviewer for their comment and agree that the need for training is a limitation\nof the method. There is a tradeoff between the upfront time needed to train the model versus\nthe time savings provided by UBSIM that should be considered by a potential user. For one-\ntime reconstructions of a single image, it would make sense to use an iterative reconstruction\nmethod as the longer inference time is still much shorter than training the model. However,\nfor applications where a user will be doing many repeated image reconstructions, such as\ntaking videos, or fixed setups where repeated experiments are being run, UBSIM can provide\na significant advantage. Additionally, compared to other learning-based reconstruction\nmethods, UBSIM does not need to be retrained depending on the object type due to its strong\ngeneralization capability.\nA likely scenario for deciding between reconstruction methods would be using iterative\nmethods for initial testing of a setup and parameters and switching to UBSIM for\nrepeated/video imaging afterwards.\n[Revision in the main text]\nWhile UBSIM provides improvements over current blind-SIM algorithms, there are several\nlimitations that must be considered by a potential user. Firstly, UBSIM requires training\nbefore inference use, which creates an upfront time cost. For one-off reconstructions,\niterative methods, even with their longer reconstruction times, may be preferred. However,\nfor cases that include repeated image reconstruction, such as video imaging or repeated\nexperiments, UBSIM can provide a significant time advantage.\nReviewer #2\n1. Comment: “The constraint (2) is used to obtain a closed system with L unknowns and L\nequations. However, how is I0 determined? The value assigned to it will influence the\nreconstructed image.”\nResponse:\nWe thank the author for the comment and realize this can be a source of confusion. We set I\n0\nthe same way as described in the original paper from Mudry et. al. [Ref. 7] “I is a constant\n0\nequal to the average intensity received by the sample” (see supplementary material for\n“Structured Illumination microscopy using unknown speckle patterns”, section 1. Blind-SIM\nand deconvolution algorithms). In order to make this clearer for the reader, we added\nadditional sentence to the manuscript in the section “Derivation of unrolled algorithm.”\n[Revision in the main text]\nThis is achieved by making an additional assumption that all the illumination patterns on\naverage sum to a uniform field:\n#\n!𝐼 ≈ 𝐿𝐼 (2)\n$ %\n$&!\nWhere 𝐿 is the number of sub-frames and 𝐼 is a constant equal to the average intensity\n%\nreceived by the sample\nIn our code, 𝐼 is estimated by the following procedure. At each pixel, the average recorded\n%\nsignal over L frames is the product of the average illumination intensity and fluorophore\ndensity: 𝑆(𝑥) = 𝐼 𝜌(𝑥). Thus, the effective 𝐼 , after normalization, can be found by\n% %\naveraging the input frames tensor along the frame dimension and taking the maximum of the\nper-pixel average intensity. This can be seen in the provided code in the initialization of\nvariable I0 in Iterative_Blind_SIM() and Unrolled_Blind_SIM().\n2. Comment: “In (4), the authors present a gradient descent algorithm to minimize the data\nfidelity term (3) (without any regularization). Then, in (5), they propose an unrolled version\nby inserting a CNN around the gradient step. The authors interpret this CNN as a proximal\noperator. However, this interpretation is unusual and not clearly justified. Typically, in\nalgorithm unrolling where a network is used in place of a prox, one starts from a well-\nestablished proximal algorithm and replaces each proximal step with a learned CNN. In (5),\nit is unclear which underlying algorithm is being unrolled. If the CNN is meant to represent a\nproximal operator, the resulting structure does not correspond to any standard algorithm I am\naware of. A common and well-understood strategy is to unroll a proximal gradient algorithm,\nwhere the CNN is applied around the entire gradient descent step, thus acting as a learned\nprior or regularization term. In the current setup, given how the CNN is positioned, it would\nbe more natural to interpret it as a preconditioner rather than a proximal operator. This\ndistinction needs to be clarified by the authors.”\nResponse:\nWe thank the reviewer for their comment and agree with their interpretation of our method. It\nis true that our approach is different from many unrolled algorithm implementations as in our\nmethod the CNN operates directly on the gradient instead of operating on the object after a\ngradient step has already been applied as a proximal step does. This indeed is more similar to\ngradient preconditioning. This approach could also be interpreted as a form of learned\ngradient descent where the update direction function is learned. The CNN operates on the\ngradient to calculate an update direction similar to how the conjugate gradient method\ncalculates an update direction using the gradient. We agree with the author that our use of\nproximal operator was incorrect and that we should clarify as well as provide relevant\nreferences for these interpretations.\n[Revision in the main text]\nThe gradients (stacked as channels in a tensor object) are then used as inputs to a CNN. The\nscaled outputs of the CNN are added to the original inputs to get the updated object and\nillumination patterns. The CNN operates directly on the gradients similar to how a\npreconditioner32, 33 or update direction calculation31,34,35 is used to accelerate convergence for\niterative gradient descent algorithms. In this way, the use of the network can be thought of as\na type of learned preconditioner36, 37 or learned gradient descent38,39,40.41 for which there are\nexamples from literature.\n3. Comment: “The iterative algorithm considered is a simple gradient descent, which is well\nknown to converge slowly, with a rate of O(1/k). However, it would be straightforward to\nimplement an accelerated version using Nesterov's method, requiring only minor\nmodifications to the code and achieving a faster convergence rate of O(1/k2). Based on my\nexperience with imaging inverse problems, such an acceleration significantly improves\nconvergence speed. I believe that Fig. 3 should be complemented with a comparison\nincluding Nesterov acceleration, in order to better highlight the relevance and advantages of\nthe proposed unrolled method.”\nResponse:\nWe thank the reviewer for their comment and agree that including a comparison to an\naccelerated method such as gradient descent with Nesterov momentum strengthens the paper\nand provides a stronger baseline to compare our unrolled method against. We have revised\nFigure 3 of our manuscript to include results using gradient descent with Nesterov\nmomentum. Implementation of the method can be viewed in our attached code as well.\nOur results show that the Nesterov method reduces both the average iterations and\ncomputation time by approximately one order of magnitude over plain gradient descent.\nHowever, UBSIM still requires one order of magnitude less iterations and is two orders of\nmagnitude faster in computation time compared to the Nesterov method. Thus, our unrolled\nmethod still displays a strong speed improvement over accelerated iterative methods.\n[Revision in the main text]\nThe Polak-Ribière conjugate gradient method31, used by Mudry and colleagues, as well as\nNesterov acceleration32, can also be used to significantly accelerate optimization. Details of\nthe implementation of these methods are included in Supplementary Section S4.\n[Revision in the main text]\nFig 3: Inference time comparison of iterative methods versus unrolled blind-SIM. a, Comparison of\ncomputation time for image reconstruction. b, Comparison of normalized loss versus iteration number. Lines are the\naverage values for 100 test set images. Dashed black line marks the average normalized loss for the unrolled\nnetwork after 6 iterations.\n[Revision in the main text]\nIn Fig. 3 we quantitatively compared the speed for unrolled and iterative blind-SIM. For\niterative blind-SIM, we ran evaluations using regular gradient descent as well as two\naccelerated methods: nonlinear conjugate gradient descent31 and gradient descent with\nNesterov momentum32. Additional information on the implementation of these methods is\nincluded in Supplementary Section S4.\n[Revision in the main text]\nWe found that UBSIM was on average 3 orders of magnitude faster than the gradient descent\nimplementation and 2 orders of magnitude faster than the conjugate gradient descent and\nNesterov acceleration methods in terms of computation time. UBSIM required 2 orders of\nmagnitude fewer iterations than gradient descent and 1 order fewer iterations than the\naccelerated methods. A single 256 x 256 pixel image took approximately 10-20 seconds to\nreconstruct for gradient descent, 1-3 seconds for the accelerated methods, and 10\nmilliseconds for UBSIM.\n4. Comment: “In the supplementary section 5, the authors show that conjugate gradient is not\nfaster than a simple gradient descent, which is very surprising to me. How is the step size Œ±\nchosen? From the provided code I can see that the value of Œ± is fixed by hand and reduced\nalong with iteration in a very heuristic manner... In contrast, in Mudry et al. (and in general in\nconjugate gradient), it is optimized at each iteration. Also what about the non-negativity\nconstraint imposed (through squaring the relevant variables) in Mudry et al.?”\nResponse:\nWe thank the reviewer for their comment and agree that for a fair comparison with the\nnonlinear conjugate gradient method a more rigorous line search method is needed. In our\noriginal implementation we used a simple backtracking line search that dropped step size by\n½ when the cost function did not decrease. This did not yield a significant difference from\ngradient descent. In order to achieve the true acceleration of the conjugate gradient method\nwe implemented an inexact line search method that optimizes at each step for strong Wolfe\nconditions. The strong Wolfe conditions ensure a (1) sufficient decrease in the cost function\nand (2) strong curvature of the descent direction. [Ref. 12]. Details on the new\nimplementation of the conjugate gradient method and line search are provided in\nSupplementary section S4. Additionally, the provided code has been updated to include the\nnew line search method.\nThe results for the new line search method are shown in the updated Figure 3 along with\ngradient descent, Nesterov acceleration, and UBSIM. The conjugate gradient method\nprovides an approximately 1 order reduction in number of iterations and a 1 order\nimprovement in compute speed. The conjugate gradient method provides a similar speed\nimprovement as the Nesterov accelerated implementation. The conjugate gradient method\nrequires less iterations, although is slightly longer in compute time on average due to the\nmore complex optimization at each step.\nThe results show that UBSIM is still 1 order faster than the conjugate gradient method in\nterms of iterations and 2 orders faster in terms of compute time.\n[Revision in the main text]\nThe Polak-Ribière conjugate gradient method31, used by Mudry and colleagues, as well as\nNesterov acceleration32, can also be used to significantly accelerate optimization. Details of\nthe implementation of these methods are included in Supplementary Section S4.\n[Revision in the main text]\nFig 3: Inference time comparison of iterative methods versus unrolled blind-SIM. a, Comparison of\ncomputation time for image reconstruction. b, Comparison of normalized loss versus iteration number. Lines are the\naverage values for 100 test set images. Dashed black line marks the average normalized loss for the unrolled\nnetwork after 6 iterations.\n[Revision in the main text]\nIn Fig. 3 we quantitatively compared the speed for unrolled and iterative blind-SIM. For\niterative blind-SIM, we ran evaluations using regular gradient descent as well as two\naccelerated methods: nonlinear conjugate gradient descent31 and gradient descent with\nNesterov momentum34. Additional information on the implementation of these methods is\nincluded in Supplementary Section S4.\n[Revision in the main text]\nWe found that UBSIM was on average 3 orders of magnitude faster than the gradient descent\nimplementation and 2 orders of magnitude faster than the conjugate gradient descent and\nNesterov acceleration methods in terms of computation time. UBSIM required 2 orders of\nmagnitude fewer iterations than gradient descent and 1 order fewer iterations than the\naccelerated methods. A single 256 x 256 pixel image took approximately 10-20 seconds to\nreconstruct for gradient descent, 1-3 seconds for the accelerated methods, and 10\nmilliseconds for UBSIM.\nIn regard to the reviewer’s concern over the non-negativity constraint (Also what about the\nnon-negativity constraint imposed (through squaring the relevant variables) in Mudry et\nal.?”) we would like to assure them that in our original provided code we do implement the\nnon-negativity constraint through variable squaring. We realize there may have been\nconfusion based on our description in the section “Derivation of unrolled algorithm” where\nwe do not mention this step. To make this clearer, we have added text to this section\nexplaining how variable squaring is used to ensure non-negativity.\n[Revision in the main text]\nTo ensure positivity, 𝐼 and 𝜌 are rewritten as the square of auxiliary variables such that:\n$\n𝐼 = 𝑖'\n$ $\n𝜌 = 𝜉' (3)\n5. Comment: “The blind SIM iterative algorithm proposed by Mudry et al. is now more than\n10 years old. Since then, faster non-iterative alternatives have been proposed such as [1]\nGiroussens et al (2024). Fast super-resolved reconstructions in fluorescence random illumination\nmicroscopy (RIM). IEEE Transactions on Computational Imaging.”\nResponse:\nWe thank the reviewer for their comment and agree that it is important to compare our work\nwith other blind-SIM papers that have been published since Mudry et al. We have added\nSupplementary Section S1 (which is referenced in the introduction section) to compare our\nwork to other blind-SIM papers that have been published in terms of speed, method, and\ntypes of experimental data used. From our findings UBSIM is state of the art in terms of\nreconstruction speed. In addition, our work is the only paper to demonstrate high-speed\nreconstruction along with video live cell imaging while most other blind-SIM papers have\nonly shown single-frame bead or fixed cell reconstructions.\n[Revision in the main text]\nSeveral other iterative blind-SIM algorithms have since been developed based on other\nvarious constraints. A thorough comparison of our work with the existing blind-SIM\nliterature is provided in the supplementary material (Supp. Table 1).\n[Revision in the supplementary material]\nS1: Comparison with current blind-SIM literature\nTitle Method Live Cell Video Limitations\ndata? data?\nStructured illumination Iterative No No • Slow reconstruction even\nmicroscopy using physics- (fixed with conjugate gradient\nunknown speckle based model tissue only) acceleration\npatterns1 • No dynamic, or live cell\n(2012) imaging demonstrated\nStructured illumination Iterative No No • Slow iterative\nfluorescence physics- (fixed cell reconstruction method\nmicroscopy with based model only) • No live cell or video data\ndistorted excitations\nusing a filtered blind-\nSIM algorithm2\n(2013)\nFluorescent microscopy Iterative No No • Slow iterative\nbeyond diffraction limits physics- (fixed cell reconstruction method\nusing speckle based mode only) • No live cell or video data\nillumination and joint\nsupport recovery3\n(2013)\nStructured illumination Iterative No No • Slow iterative\nmicroscopy with physics- reconstruction method\nunknown patterns and a based model • No video or live cell\nstatistical prior4 demonstrations\n(2017)\nJoint Reconstruction Iterative No No • Requires regularization\nStrategy for Structured physics- parameter tuning\nIllumination based model • Accelerated iterative\nMicroscopy With method still requires more\nUnknown Illuminations5 iterations than UBSIM\n(2017) • No video or live cell\ndemonstrations\nOn the Superresolution Iterative No No • Purely theoretical paper\nCapacity of Imagers physics- • Slow iterative\nUsing based model reconstruction method\nUnknown Speckle • No live cell or video data\nIlluminations6\ndemonstrations\n(2018)\nSuper-resolved live-cell Iterative Yes Yes • Slow iterative\nimaging using random physics- reconstruction\nillumination based model\nmicroscopy7\n(2021)\nDeep learning for blind Deep Yes No • Limited experimental\nstructured illumination learning evaluation\nmicroscopy8 • No rigorous exploration\n(2022) of generalizability\n• Requires supervised\ntraining scheme\nFast super-resolved Iterative No No • Accelerated iterative\nreconstructions in physics- method\nfluorescence based model • Resolution improvement\nrandom illumination cannot reach 2x in non-\nmicroscopy (RIM)9 noiseless demonstrations\n(2024) • Limited quantitative\nresolution evaluation\n• No experimental live cell\ndata\nSupplementary Table 1: Summary of current blind-SIM papers and their limitations compared\nto this work.\nWe also would like to specifically address the paper mentioned by the reviewer (Giroussens\net al (2024). Fast super-resolved reconstructions in fluorescence random illumination\nmicroscopy (RIM). IEEE Transactions on Computational Imaging) [Ref. 2]. We have\nthoroughly read the paper to assess whether this work effects any of our paper’s claims of\nnovelty. While the work does introduce an accelerated algorithm for the related method of\nvariance-based Random Illumination Microscopy, we find that the paper does not affect\nthe novelty of our paper or improve upon our UBSIM algorithm for the following\nreasons:\n[1] In terms of algorithmic reconstruction speed, the paper introduces a non-iterative\nreconstruction method termed RIM-CF. However, the authors explain that this\nmethod, while better than Wiener deconvolution in some cases, cannot reach the full\n2x resolution improvement and that iterative methods are needed for full resolution\ndoubling. See comments from the text such as “Obviously, even in a favorable\nscenario, RIM-CF remains based on the approximation (12) and modeling errors\ndeteriorate the resolution gain, e.g., RIM-CF cannot reach the theoretical resolution\nlimit one can expect from RIM in a noise-free setting. An iterative solver, based on a\nmore accurate model is required to reach this limit.”, “RIM-CF is basically a non-\niterative regularized deconvolution” (section III. A non-iterative estimator, RIM-CF),\nand “RIM-CF is a fast, FFT-based inversion of the standard deviation that provides a\nsuper-resolved estimate of the sample. However, because it is based on an\napproximation, this estimator cannot achieve the maximal theoretical resolution gain\nfor RIM.” (Conclusion) Furthermore, this lack of resolution gain from the non-\niterative method can be clearly seen in Figure 2. Here, even in a noiseless\ndemonstration case, RIM-CF is unable to reach full resolution improvement\ncompared to RIM-STD/RIM-VAR.\n[2] The paper also introduces an iterative algorithm RIM-STD which is an accelerated\nversion of the earlier developed RIM-VAR algorithm. However, there are several\naspects in which this algorithm does not match our paper’s performance. Firstly, the\nnumber of iterations needed for convergence in the noiseless 2D imaging case (Figure\n2) is 100 iterations which is still an order of magnitude more than our UBSIM\nalgorithm. Secondly, it appears that RIM-STD only can reach a full 2x resolution\nimprovement in highly idealized, simulated scenarios. In Figure 3, RIM-STD\nachieves the full resolution improvement under the “asymptotic variance” case where\nessentially the variance in perfectly known. However, when the empirical variance\nmust be calculated, which is what would happen in real-world imaging, the\nperformance of RIM-STD is much worse. Even with 1,000 sub-images and a very\nhigh SNR (30 dB) RIM-STD is unable to achieve the full resolution improvement\nprovided by the speckle and in fact it is stated that “RIM-CF and RIM-STD now reach\ncomparable results in terms of quality”. That means in the experimental case, RIM-\nSTD does not perform any better than a form of regularized deconvolution and cannot\nreach full resolution improvement.\n[3] We find the paper has several issues in how resolution is determined and presented.\nThe only figures for which quantitative resolution improvement or intensity cross\nsections are provided for are Figures 1 and 2 which are both simulated target objects\nwith noiseless data collection. However, this is problematic for two reasons: (1) in the\ncase of zero noise (infinite SNR) super-resolution is somewhat trivial. Using the\nknown PSF, the super-resolution object can be perfectly recovered via bandwidth\nextrapolation for many synthetic objects [see Refs. 3,4,5], and (2) for Figure 1 a\nsingle point source is used as the resolution measurement object. This again is a\ntrivial problem as there exist many algorithms that can arbitrarily sharpen a single\npoint source and thus does not provide an accurate measure of resolution. In\nmicroscopy, resolution is defined as the distance at which two closely spaced point\nsources can be distinguished (Abbe, Rayleigh) and not the FWHM of a single point\nsource. For example, simple Lucy-Richardson deconvolution can arbitrarily sharpen a\nsingle point’s width in a similar manner with few iterations, however it is not a true\nsuper-resolution method as it will fail to resolve closely spaced point sources in an\nenvironment with realistic noise levels.\n[4] Among the provided figures, only figure 4 contains reconstruction on experimental\ndata. The data is of sparse fixed fluorescent beads and there is no reconstruction on\nfixed or live cell data that could provide dynamic behavior.\n[5] We find that the paper does not contain any video reconstructions, only single frame\nreconstructions (see figures 1- 4, no supplementary available).\nTo conclude, the paper “Fast super-resolved reconstructions in fluorescence random illumination\nmicroscopy (RIM)” is not comparable to our paper as it presents (1) a non-iterative algorithm\nthat is only slightly better than deconvolution and cannot reach full resolution improvement even\nin noiseless conditions, and (2) an accelerated iterative algorithm that improves upon RIM-VAR,\nbut cannot reach full resolution improvement in any realistic scenario with noise and is still 1\norder of magnitude slower than our algorithm. The paper mainly focuses on highly idealized\nsimulations and does not provide live-cell or video imaging.\n6. Comment: “Widespread adaption and use of SR technologies by target groups\" ‚ à\nadoption ?”\nResponse:\nWe agree and have changed the typo.\n7. Comment: “In Supp file, after Fig S6: \"We examined the reconstruction results of UBSIM\nand iterative blind-SIM when there is a low signal-to-noise ratio. As can be seen in\nSupplementary Figure 5\" à It is Supplementary Figure 6.”\nResponse:\nWe have modified the text to ensure that it corresponds to the correct supplementary figure\nand to follow the correct numbering convention for figures in the supplementary material.\nThere has been a change to the order since revision as well.\n8. Comment: “Page 14 line 301: \"On axis elements of the matrix\" à On diagonal elements ?”\nResponse:\nWe agree and have changed the text.\n9. Comment: “Page 17, line 350: \"... make it a particularly challenging organelle to image\" à\nthe word organelle seems misplaced”\nResponse:\nEndoplasmic reticulum is considered an organelle. It is referenced as such in biology\ntextbooks (see: The Cell: A Molecular Approach [Ref. 13]) as well as the references provided\nin the manuscript [Refs. 14-17]). An organelle is “A small structure in a cell that is\nsurrounded by a membrane and has a specific function”\n(https://www.cancer.gov/publications/dictionaries/cancer-terms/def/organelle)\nReferences:\n1. Markwirth, A., et al. Video-rate multi-color structured illumination microscopy with\nsimultaneous real-time reconstruction. Nat. Commun. 10, 4315 (2019).\n2. Giroussens, G., et al. Fast super-resolved reconstructions in fluorescence random\nillumination microscopy (RIM). IEEE Trans. Comput. Imaging, (2024).\n3. Goodman, J. W. Introduction to Fourier Optics (Roberts and Company Publishers, 2005).\n4. Lindberg, J. Mathematical concepts of optical superresolution. J. Opt. 14, 083001 (2012).\n5. Bertero, M. & De Mol, C. Super-Resolution by Data Inversion, Vol. 36 (Elsevier, 1996).\n6. Goodman, J. W. Some fundamental properties of speckle. JOSA 66, 1145-1150 (1976).\n7. Mudry, E. et al. Structured illumination microscopy using unknown speckle patterns. Nat.\nPhotonics 6, 312–315 (2012)\n8. Yeh, L.-H., Tian, L. & Waller, L. Structured illumination microscopy with unknown\npatterns and a statistical prior. Biomed. Opt. Express 8, 695–711 (2017).\n9. Mangeat, T., et al. Super-resolved live-cell imaging using random illumination\nmicroscopy. Cell Rep. Methods 1, (2021).\n10. Huang, X., et al. Fast, long-term, super-resolution imaging with Hessian structured\nillumination microscopy. Nat. Biotechnol. 36, 451-459 (2018).\n11. Zhao, W. et al. Sparse deconvolution improves the resolution of live-cell super-resolution\nfluorescence microscopy. Nat. Biotechnol.40, 606-617 (2022).\n12. Nocedal, J., & Wright, S. Numerical Optimization. (Springer, New York, 2006).\n13. Cooper, G.M., & Adams, K. The cell: a molecular approach. (Oxford Univ. Press,\nNew\nYork, 2022).\n14. Schwarz, D. S. & Blower, M. D. The endoplasmic reticulum: structure, function and\nresponse to cellular signaling. Cell. Mol. Life Sci. 73, 79–94 (2016)\n15. Pain, C. & Kriechbaumer, V. Defining the dance: quantification and classification of\nendoplasmic reticulum dynamics. J. Exp. Bot. 71, 1757–1762 (2020)\n16. Shibata, Y., Voeltz, G. K. & Rapoport, T. A. Rough sheets and smooth tubules. Cell 126,\n435–439 (2006)\n17. Costantini, L. & Snapp, E. Probing endoplasmic reticulum dynamics using fluorescence\nimaging and photobleaching techniques. Curr. Protoc. Cell Biol. 60, 21–7 (2013)"}, {"round": 2, "reviewer_comments": {"reviewer_2": "Despite my initially somewhat negative opinion of this work during the first round of review, I must admit that the substantial\nrevisions made by the authors, along with their detailed responses, have led me to change my mind. I would like to\ncongratulate the authors on this very thorough revision, in which they have effectively addressed all of my concerns. For\nthese reasons, I recommend that the paper be accepted for publication."}, "author_response": ""}]

Decision letter