Non‐Invasive Super‐Resolution Imaging Through Scattering Media Using Object Fluctuation

Introducing super‐resolution techniques to imaging through scattering media potentially revolutionizes the technical analysis for many exotic applications, such as cell structures behind biological tissues. The main challenge is scattering media's inhomogeneous structures, which scramble the light path and create noise‐like speckle patterns, hindering object's visualization even at a low‐resolution level. Here, a computational method is proposed relying on the object's spatial and temporal fluctuation to visualize nanoscale objects through scattering media non‐invasively. Taking advantage of the optical memory effect and multiple frames, the point spreading function (PSF) of scattering media is estimated. Multiple images of fluctuating objects are obtained by deconvolution; then, the super‐resolution image is achieved by computing the higher‐order cumulants. Non‐linearity of high order cumulant significantly suppresses artifacts in the resulting images and enhances resolution by a factor of N$\sqrt N $ , where N is the cumulant order. The proof‐of‐concept demonstrates a resolution of 266 nm at the 6th‐order cumulant with numerical aperture (NA) of 0.42, breaking the diffraction limit by a factor of 2.45. An adaptive approach is also demonstrated for imaging through dynamic scattering media. The non‐invasive super‐resolution speckle fluctuation imaging (NISFFI) presents a nanoscopy technique with straightforward imaging hardware configuration to visualize samples behind scattering media.


Introduction
Most biological tissues are strongly scattering media, which scramble light propagation paths, posing a considerable challenge for imaging (Figure 1a).A camera typically captures just speckle images, which are seemingly random noises.Various techniques based on the optical memory effect [1,2] in thin scattering media have recently been introduced to overcome this challenge.[5] The object can also be more deterministically computed using a deconvolution algorithm [6][7][8] if the media's point spreading function (PSF) is known typically by an invasive measurement.By multiplexing multiple uncorrelated PSFs in a single speckle image, it is possible to achieve single-shot multispectral [9] or multi-view imaging [10] with the help of scattering media.These snapshot approaches have successfully recovered the object, but the best achievable resolution is the diffraction limit (Figure 1b) quantified as either 0.61 NA −1 (Rayleigh criterion) or 0.51 NA −1 (full-width at halfmaximum, FWHM), where  is wavelength and NA is the numerical aperture of imaging optics. [11]maging beyond the optical diffraction limit for transparent samples is mature.It has revolutionized technical analysis in various fields, from biological to physical sciences.Several examples are either using single fluorescence emitters like stochastic optical reconstruction microscopy, STORM, [12,13] and photo-activated localization microscopy, PALM, [14] or sharpening a known PSF such as stimulated emission depletion, STED, [15] nonlinear structure illumination microscopy, SSIM, [16] and super-resolution optical fluctuation imaging, SOFI. [17]The significance of these optical nanoscopy techniques is to provide detailed information at electron microscopes' resolution in alive samples such as cells or transparent biological tissues. [18,19]The transparency requirement for optical imaging often needs to engage advanced techniques to engineer biological tissues, such as the primate-optimized uniform clearing method (PuClear) for only in vitro samples, unfortunately. [19]Expanding the capability   [11] that utilizes speckle correlation to image non-invasively through strongly scattering media at 100-nm resolution.Although the SOSLI's resolution limit is merely governed by the signal-to-noise ratio (SNR), the requirement of the sparse individual blinking emitters-only one emitter per diffraction-limited area turns on at a time -poses a considerable challenge for current single quantum or molecular emitters to achieve a practical SNR.
In this work, we present a non-invasive super-resolution speckle fluctuation imaging (NISSFI) technique, which allows non-invasive visualization of objects behind scattering media at super-resolution.The NISSFI relies on two things: the independent intensity fluctuation from an object and the speckle frames' correlation.The former presents a similar object prepared for the traditional SOFI [17] technique, but the scattering media pose substantial challenges compared to transparent samples in the conventional method.The latter, thanks to the optical memory effect of scattering media, has been utilized in SOSLI [11] ; however, the hidden object can have denser emitters per diffraction-limited area to boost the SNR.The object's intensity fluctuates spatially and temporally (Figure 1c), that is, uncorrelated in time and space.A camera captures multiple fluctuating speckle frames, allowing us to recover various images of the uncorrelated fluctuating object.A super-resolution image is achieved by calculating the higher-order cumulants (Figure 1d).The simulation and experimental results show that NISSFI can break the diffraction limit and achieve resolution improvement by a factor of √ N, where N is the cumulant order.We experimentally demonstrate our NISSFI's resolution at 266 nm FWHM for a cumulant order of 6 with NA = 0.42, while the diffraction limit has FWHM of 646 nm.NISSFI can also be used in an adaptive mode to image through dynamic scattering media (chicken eggshell membrane), decorrelating from 1.0 to 0.2 during the image acquisition.

Principle and Simulation of NISSFI
Our principle and simulation are shown in Figure 2. A sample consists of several intensity-fluctuating dots with different interdistances (300, 250, and 200 nm).The size of each dot is one pixel of 25 nm.When passing through scattering media, light from the sample forms speckle frames I n that are the convolution of the samples O n with a PSF as shown in Equation ( 1), thanks to the incoherent nature of light and the optical memory effect of scattering media. [3]A sequence of fluctuating speckle frames I n is captured as the dots fluctuate (Figure 2a).
When the number of frames is large enough, the averaged frame Ī is considered as the speckle frame from the fully bright object.A phase-retrieval (PR) algorithm, which is a mature technique in previous studies, [3][4][5]11] is used to recover the fully bright object (Figure 2b). Certinly, the reconstruction Õ has artifacts and cannot break the Rayleigh diffraction limit of 342 nm (286 nm FWHM), which is simulated for wavelength  = 532 nm and NA = 0.95, defined by the aperture on scattering media and distance from the object to scattering media.We then can estimate the PSF from the deconvolution of the averaged speckle frame Ī with the recovered object Õ: In the simulation with an SNR of 10 dB and 16 bits quantization, our estimated PSF (Figure 2b), though with a lot of noise, can have a very high correlation (exceeding 0.90) with the actual PSF (Figure 2a).The correlation between actual PSF and estimated PSF at different SNR scenarios are mainly affected by PR and deconvolution parameters.With the estimated PSF, a series of fluctuating object images Õn (Figure 2b) are derived by deconvolution from corresponding fluctuating speckle frames I n : The estimated PSF and deconvolution keep all the fluctuating object images Õn aligned to each other, while the PR result is blind to absolute position and central flip.While the absolute position is not essential, as it only affects the final image's position, the orientation information is crucial in estimating the PSF.Only one central flip orientation allows the sub-sequence deconvolution to successfully derive a series of fluctuating object frames (Figure 2b).We have to try two possibilities and choose the correct one.
The series of fluctuating object images Õn recovered above are of low resolution (diffraction-limited) and exhibit some artifacts.However, the higher-order cumulants of these images can break the diffraction limit and remove artifacts substantially, as shown in Figure 2c.The first-order cumulant image (i.e., the mean image) is blurry.We can barely resolve 300 nm gap between the two dots.However, after subtracting the mean, we calculate higher-order cumulants and reveal the object much clearer.The second-order cumulant (C 2 , also understood as variance) and the 4th order cumulant (C 4 ) break the diffraction limit, resolving the 250 nm and 200 nm gap between dots very well.More interestingly, a lot of artifacts in deconvolution images and also in the PR image are significantly suppressed as the result of the nonlinear effect in higher-order cumulants.We also simulate the conventional SOFI, where the fluctuating object is imaged directly by an objective lens of equivalent NA (0.95) in a transparent environment (i.e., no scattering media).The lower row in Figure 2c shows a comparable resolution of conventional SOFI with our NISFFI.The intensity along the horizontal line indicated by dashed arrows in Figure 2b,c shows the significant resolution improvement for both techniques as expected (Figure 2d). Figure 2e presents the full width at half maximum (FWHM) of the point as a function of the cumulant order.Both techniques show the FWHM reduction by a factor of √ N, where N is cumulant order.It is no supprise that NISSFI and SOFI show very similar results; and both first cummulant order images (mean images) present a typical diffraction limit resolution (FWHM of 286 nm) due to the linear signal processing technique.For higher-order cumulants, C 2 and C 4 images, the FWHM of NISSFI can reach 205 and 147 nm, respectively.With the simulation model, a resolution of 100 nm or even smaller can be achieved with cumulant orders >8.By estimating the PSF non-invasively, the scattering media become a scattering lens, allowing us to utilize conventional super-resolution techniques and break the diffraction limit in imaging through scattering media.

NISSFI Demonstration with Fluctuating Emitters
To demonstrate our NISSFI technique, we fabricate a 3-hole sample by a focus ion beam technique. [20]Three holes of 100 nm are milled through a thin gold film on a glass substrate (Figure 3a).We illuminate the sample from the back with a green laser beam passing through a rotating diffuser (a pseudo-thermal source) to create three bright nanometer spots.Here, the use of a pseudo incoherent light source, as opposed to a coherent laser source, is crucial for achieving intensity superposition on the camera, which is the principle underlying Equation (1).Narrow spectral bandwidth of the light source is also useful to enhance the speckle contrast.The sample is mounted on a linear piezo stage to move 60 steps with a step size of 100 nm to cover 6 μm space.At each position, we illuminate the sample with 50 random laser intensity levels, and the camera captures 50 corresponding speckle frames.We randomly pick 1 out of 50 images in each position, then superpose these 60 images (corresponding to 60 sample positions) to form a single speckle image.The resulting image (Figure 3b) represents a speckle image captured from a 3-line object, which comprises fluctuating emitters, similar to typical samples for the conventional SOFI technique.We repeat the random picking and superposing process to create 300 speckle images for NISSFI demonstration.
From the set of 300 speckle images, we follow our signal processing pipeline as mentioned in principle (Figure 2).These speckle images have very similar patterns but spatially different intensities, as Figure 3c shows.With the highest SNR, the averaged speckle pattern is used to reconstruct the diffraction-limit object (Figure 3f) by phase-retrieval.Unlike SOSLI with samples of sparse emitters, i.e., one emitter per diffraction limit area, [11] NISSFI can utilize denser emitters to gain more SNR in actual applications but cannot localize the emitters to the pixel level.Nevertheless, the PSF (Figure 3d,e) can be estimated and used in the subsequent deconvolution to reveal the temporally and spatially fluctuating object.Figure 3g shows a series of deconvolution images, presenting the diffraction limit resolution of the fluctuating object.We calculate higher-order cumulants of these intensity-fluctuating images to display in Figure 3h.The mean image (the first-order cumulant) shows the diffraction limit of the fully bright object (no fluctuation) with the line's FWHM of 625 nm (NA = 0.42, similar to our SOSLI demonstration in reference 11).The two lines with 500 nm distance cannot be resolved in PR and mean images, but appear resolvable in the C 2 image, even better in C 4 image, and higher-order cumulants such as C 12 images.The FWHM of the line reduces from 625 nm to 441 nm and 188 nm for the 1st (also known as mean), 2nd, and 12th order cumulant, respectively, following the reduction factor of √ N as shown in Figure 3j.While the image quality is very good at C 2 and C 4 , it is noticeable that higher-order cumulant images become more uneven intensity, which is well understood in SOFI technique. [17]A lot more number of raw images and high SNR are needed to achieve uniform high-order-cumulant images.And similar to the conventional SOFI technique, C 4 and C 6 are reasonable for NISSFI.

NISSFI for Non-Labeling Samples
Unlike samples decorated with fluorescent materials, the nonlabeling samples do not emit light, challenging many imaging techniques.Here, we demonstrate NISSFI technique to enhance the imaging resolution for non-labeling samples embedded in scattering media.In many cases, objects can be imaged in transmission mode where the imager and illuminator are at opposite sides, one side is thin scattering media, and the other is thick scattering media, such as objects under the skin.Certainly, we will do imaging through the thin side to utilize the memory effect, therefore, illuminating from the thick side.A pseudo-incoherent light source made by focusing a laser beam on a rotating ground glass diffuser is used for illumination.Light passing through the thick scattering media creates speckle patterns on the sample, which is a metal film with a transmission pattern made by a focus ion beam (FIB) technique. [20]The sample is a stair-like object, as shown in the SEM image of Figure 4a, where the line width is 300 nm, and the gaps between lines are 300, 400, and 600 nm.The focused laser point is moved on the rotating ground glass diffuser, parallel to the thick scattering media, to create randomly changing speckle illumination on the sample.This makes a temporally and spatially fluctuating object in the transmission mode.The speckle size of the illumination pattern needs to be as small as possible to maintain the independent fluctuation of the points on the object.We focus a laser beam tightly into the surface of a rotating diffuser to create a tiny pseudo-thermal source and utilize a thick scattering media (5 mm thick polyurethane foam) to achieve our speckle size of ≈700 nm.Thicker scattering media allow larger scattering angles, i.e., higher spatial frequency (smaller size) for illumination speckle patterns.However, thicker media reduce the fluctuation effect due to the incoherent nature of the light source.Note that fluctuating illumination speckle patterns can be also achieved by utilizing dynamic scattering media instead of moving the tiny light source.26] Light from the fluctuating object passes through the lens or scattering media.The camera captures 200 fluctuating object frames (Figure 4b) for lens or scattering media, respectively.The results of NISSFI and the conventional SOFI with these two sets of images are presented in Figure 4d,e, respectively.Both results show similar performance with breaking the diffraction limit at higher cumulant.Similar to PR reconstruction image, the mean image cannot resolve the closest two lines (300 nm gap), but it is a lot smoother and fewer artifacts because of the deconvolution process and averaging over a large number of images.The NISSFI results in Figure 4d show super-resolution at higher-order cumulants.The closest two lines are resolved gradually in higher-order cumulants.These results are the same as traditional SOFI results (imaging the sample with lens, no scattering media) in Figure 4e.The uneven intensity distribution is attributed to imperfect sample fabrication and especially to the distortion of higher-order cumulants. [17]We plot the normalized intensity across the line indicated Figure 4f,g by dashed arrows in Figure 4d,e, respectively.Obviously, the intensity profiles (line 1) of mean results (red) do not really show the two closest lines.However, two peaks gradually become distinguishable at higherorder cumulants.The FWHM of both experiments is shown in Figure 4h, which are consistent with the √ N resolution improvement as presented in the simulation above.The estimated NA of the current imaging through a thin scattering medium is 0.37 (i.e., 733 nm FWHM).We also present the NISSFI results for other samples in Figures S1 and S2 (Supporting Information).Again, imaging results at higher-order cumulants are sharper and break the diffraction limit of imaging optics.

Adaptive NISSFI Through Dynamic Scattering Media
A fluctuation object can also be achieved by a static light source illuminating through dynamic scattering media, which naturally happens in most practical applications.However, dynamic scattering media imply a dynamic PSF as well, [11] and our static PSF approach in Figure 2 will not work.We propose here an adaptive approach utilizing the robustness of the deconvolution method to do NISSFI through dynamic scattering media.We utilize the data in the previous SOSLI study, [11] in which an object comprised of multiple blinking dots is imaged through dynamic scattering media (chicken eggshell membrane).We captured 309 speckle frames, during which chicken eggshell membrane decorrelates, [11] that is, its PSF's correlation reduces from 1.0 to 0.2.In order to make a fluctuating object for NISSFI rather than a blinking object in SOSLI, we superpose together ten continuous speckle frames with random weights to generate a single speckle frame.Within ten successive frames, the membrane decorrelation is insignificant, and the new superposed frame is virtually equivalent to the speckle frame captured from a fluctuating object in a single shot.With the rolling window of ten frames, we generate 300 new speckle frames for NISSFI from the 309 original SOSLI speckle frames, where the membrane's PSF correlation reduces gradually from 1.0 to 0.2.
With the new 300 fluctuating speckle frames, we calculate their autocorrelation then averaging these 300 autocorrelation images.This averaged autocorrelation represents the autocorrelation of the fully illuminated object.Then the PR image can be achieved as Figure 5a.We take the average of the first 20 frames to represent the speckle image of the fully illuminated object while the decorrelation is not significant.Then the estimated PSF can be obtained, allowing to do deconvolution for the fluctuating object from fluctuating speckle frames.The deconvolution image will bring more and more artifacts for later fluctuating frames (Figure 5b) due to chicken eggshell membrane decorrelation.For example, for the 50th/150th/250th fluctuating frame, the reconstructed images have more noise because the PSF correlation decrease to 0.64/0.43/0.26,respectively.Such artifacts do not allow to reconstruct the correct object with higher-order cumulant (Figure 5c).Obviously, the NISSFI with a static PSF strategy does not work for dynamic scattering media.
However, the PSF can be re-estimated as needed by deconvolution between PR reconstruction (Figure 5a) and the averaged frame of any 20 successive speckle images, during which the decorrelation is not significant.Here, we re-estimate PSF after every 50 frames so that the membrane correlation only decreases from 1.0 to 0.64, and the deconvolution images maintain high quality, as shown in Figure 5d.By doing this re-estimate PSF, our adaptive NISSFI always operates in the high correlation time windows of dynamic scattering media.The final higher-order cumulant images are shown in Figure 5e, which is significantly better than the static NISSFI result (Figure 5c).Higher cumulant orders show higher resolution as expected.
The key point for adaptive NISSFI is the PSF re-estimation when the medium is not decorrelated too much.In very fast decorrelation media, we might not be able to get the averaged frame of multiple speckle frames with high-correlation PSF for this estimation.Instead, we can fully illuminate the sample with a large light source for PR and PSF estimation in a single shot.Then immediately after full illumination, a point-like light source is used to get fluctuating illumination for deconvolution.The process repeats continuously with a pair of full and fluctuating illumination.The PR is used for the first time only, while PSF estimation is done for every full illumination frame using the first PR image.By using this extreme adaptive NISSFI approach, we can recover images at high resolution with only the requirement of some scattering media correlation within the short time of taking the frame pair.

Discussion
NISSFI consists of multiple steps, the most critical of which is the reconstruction of low-resolution image of a hidden object, especially a complex object, by PR (Section S3, Supporting Information).To enhance SNR for the input of PR algorithm, we always utilize averaging strategy, which is the averaging speckle frame for static scattering media and the averaging autocorrelation for dynamic scattering media.In addition, the total number of speckles should be large enough to guarantee the autocorrelation of a finite speckle image is approximately equivalent to the object's autocorrelation.Alternatively, bi-spectrum is also a reconstruction method tolerated with low SNR. [27]Previous studies show that ≈1 × 10 6 speckle grains will have a satisfactory result. [3,28]However, in order to break the diffraction limit by high order cumulants, we will need to significantly oversample the speckle size, leading to a reduced number of speckles on a finite-size imaging sensor.Therefore, there is a trade-off between approximate autocorrelation (for better phase retrieval) and resolution improvement.The sparsity of the sample also relaxes the requirement of a large speckle number.In the present study, the number of speckles (sensor pixel number /speckle size) is ≈2.2 × 10 4 and 8.5 × 10 4 in simulation and experiment, respectively.A camera with more pixels and a smaller pixel size will be useful in practical experiments.
For the unlabeled object, its highest resolvable frequency is decided by the illumination NA, which is limited by the scattering angle of the front scattering media.Labeling objects with fluorescent molecules or quantum dots will be immune from illumination NA [21,23] and also guarantee the uncorrelated intensity fluctuation like conventional SOFI. [17]But a lower SNR for such sample in imaging through scattering media could be a challenge for any computational approach.Even the typical resolution gain for conventional SOFI (without scattering media) is limited to analysis of the low cumulant order (4th to 6th) only due to SNR. [24][25][26] Another limitation of our current approach is the field of view (FOV).Capturing desired speckles only from the object well within the memory effect is not practical in many applications.Recently studies showed combining ultrasound probe, [29] connecting PSF [30] and cross-correlation [31] are feasibility approach to achieve a large FOV.
The NA of non-invasive imaging through scattering media is not straightforward to determine.Using a tiny aperture to limit the NA as small as 0.05, we can calculate this through the angle from center of the object plane to the edge of the aperture, similar to conventional lenses. [11]But when the aperture is larger, many other factors will also determine NA, such as the scattering power and position of the media, the configuration of optics to capture speckle patterns to the camera.Unlike imaging with lens, we do not have a pre-determined object plane with a fixed NA value; in fact, we can image an object in any planes with different values of NA.Therefore, in our work, we have to estimate the NA of the whole imaging system for the object plane used in each experiment.The phase-retrieval approach is currently the best technique for doing non-invasive imaging through scattering media with a single-shot measurement.Certainly, optical diffraction is the physical factor limiting the resolution.One can estimate the NA from the phase-retrieval result, but the artifacts presented in this image affect the precision.Here, with the advantage of multishot imaging, the mean image (or the first-order cumulant image) cleans the artifact in the phase-retrieval result and allows us to estimate NA from the smallest feature more accurately.We would like to note that no matter how we estimate NA non-invasively, the enhancement of higher-order cumulant images compared to lower order ones is valid, similar to SOFI's principle. [17]ur NISSFI inherits the behaviors of the conventional SOFI technique.The higher-order cumulants are known for emphasizing the bright pixels while suppressing lower intensity pixels due to the nonlinear nature of the technique.On one hand, it helps us to suppress the artifacts, which is obviously lower intensity, but on the other hand, it can distort the object if there is any intensity difference.Our simulation and experimentally prepared samples with equal intensity (i.e., a binary sample) make the technique easier to succeed.In addition, we use auto cumulants with zero time lag ( = 0) to simplify calculation in our current simulation and experiment; other higher-order cumulants algorithms like cross-cumulants, [32] bSOFI, [33] fSOFI [34] could also be suitable for imaging through scattering media.
In summary, we present the non-invasive super-resolution imaging through scattering media based on speckle fluctuation.As long as the intensity of a nanoscale object fluctuates in a random fashion spatially and temporally, we capture multiple fluctuating speckle frames.Our post-processing algorithm reconstructs the diffraction-limited images of the fluctuating object, and a super-resolution image is achieved by calculating the higher-order cumulants.The resolution can be enhanced by a factor of √ N, that can easily break the diffraction limit by simply calculating higher-order cumulant (N).With NA = 0.42 (Rayleigh diffraction limit of 773 nm or FWHM of 646 nm), we can resolve a line width of 266 nm with the 6th cumulant order.Our adaptive NISSFI approach allows super-resolution imaging through dynamic scattering media.

Experimental Section
Experiment Setup: Different illumination approaches aimed to achieve object fluctuation.A pseudo-thermal source was used, which was created by focusing 532 nm laser beam on a rotating diffuser (Thorlabs, DG10-1500).The light source was directly illuminated by the object (for fluctuating emitters, Figure 3) or highly scattered by thick scattering media (5 mm thickness polyurethane foam, Figure 4).The thick scattering media was almost touching the object to ensure as small illumination speckle grains as possible.
For the imaging part, the scattering media was ground glass (Thorlabs DG10-120), translucent tape (Scotch Magic tape), or chicken eggshell membrane placed ≈0.2 to 1 mm away from the object.A microscope objective was used to collect more light into a CCD (Andor iKon-L, 2048 × 2048 pixels).Adjusted the distances among objects, scattering media, microscope objective, and CCD camera to get desired speckle sizes and magnification values.Here, the CCD was placed at ≈30 cm from the microscope objective and the objective was 1-2 mm away from the scattering media.Though the magnification in the setup corresponded to the sampling of 100 or 125 nm/pixel on the object plane, the nature of non-invasive imaging did not provide the magnification because of the unknown distance from the scattering media to the sample.In fact, the object can always be resolved by viewing angles from scattering media's view.
The transmission nanoscale objects were fabricated by depositing a 250-nm-thick gold film on ITO glass, and then various shapes were milled using the focus ion beam (FIB) technique.
Simulation: The simulation frames (Figure 2) were generated with 25 nm/pixel (sampling frequency 1/25 nm −1 ).The NA of the imaging system was set as 0.95, which determined the coherent transfer function (CTF, presented as a low pass filter with a cut-off frequency of NA/0.61).Then, a random phase pattern was added to the CTF and converted it into PSF (speckle pattern) by a Fourier transform.The PSF's minimum speckle size (diffraction-limit) is 342 nm (13.7 pixels).The fluctuating object sequence was a binary object illuminated by other speckle frames (i.e., speckle illumination).These illumination speckle frames were generated similarly to PSF but with different random phase pattern each time.The final 1000 speckle frames (2048 × 2048 pixels) on the camera were convolution between PSF and fluctuating object sequence.
Data Processing: The center 2048 × 2048 or 1536 × 1536 pixels (depends on practical speckle image) from the averaged frame were used for PR algorithm.For simulation, the whole 2048 × 2048 pixels were used.The PR reconstruction image was centered and rotated to the correct direction tentatively.Zeros were padded to the reconstructed image to the corresponding speckle frames size before estimating PSF.
PR algorithm follows the previous studies [3][4][5]11] to constrain the reconstruction image by using "Hybrid Input-Output (HIO)" and "Error reduction" algorithms. For IO algorithm, the feedback parameter for controlling convergence was set from 2 to 0 in steps of 0.02.For each feedback parameter value, there were 40 iterations.For "Error reduction" algorithms, there were also 40 iterations to reduce residual noise.
Deconvolution method is the standard Wiener-deconvolution as presented in Equation ( 4), and the speckle frames utilized are raw data without processing.
where I is the speckle frames,  and  −1 is the Fourier transform and its inversion, respectively, (.) c is the complex conjugate, and  is noise to signal ratio.To estimate PSF, A and B are estimated PSF and PR reconstructed image; to calculate the fluctuating object, A and B are fluctuating object and the estimated PSF. Certainly, Wiener deconvolution approach implies that the object has non-negligible frequency content in all spatial frequencies of interest.Such condition is typically hold for samples with blinking or fluctuating point sources.The higher-order cumulant calculation is auto-cumulants as Equation (5) shows, and time lag  is set to zero for simplifying computation.
where C n and G n are nth-order cumulants and nth-order correlations respectively.It is suggested to refer to previous studies [17,35] for more details.
The FWHM was calculated from Gaussian fitting curves.

Figure 1 .
Figure 1.Schematic of NISSFI.a) The scattering media scramble the light path, causing object information loss in comparison to transparent media.b) The recovered image with diffraction-limit.c) The object intensity fluctuates temporally.d) The recovered super-resolution image by NISSFI.e) The light from fluctuating object (O 1 , O 2 , …, O n ) is scattered by the scattering media, collected by objective lens, and resulting in a sequence of speckle fluctuation frames (I 1 , I 2 , …, I n ) in the CCD.
of super-resolution imaging to translucent samples or scattering media will open many opportunities for not only biological applications but also various science and technology applications.Recently, Wang et al. introduced a stochastic optical scattering localization imaging (SOSLI) technique

Figure 2 .
Figure 2. Principle and simulation results.a) A temporal fluctuating object is convolved with a PSF, resulting in a sequence of fluctuating speckle frames.Their averaged frame can be utilized to reconstruct the object by Phase-Retrieval algorithm (PR).b) The estimated PSF by deconvolution of the averaged frame with the reconstructed image.A sequence of fluctuating frames can be derived from the estimated PSF and fluctuating speckle frames by deconvolution.c) The first row is NISSFI results (Mean image, C 2 : 2nd order cumulants, C 4 : 4th order cumulants) from fluctuating frames in (b); The second row is conventional SOFI technique where the object is imaged by lens without scattering media.d) The intensity profile along the line as indicated by the dashed arrows in (b) and (c).Lines indicate NISSFI.Squares indicate SOFI.e) The full-width at half-maximum (FWHM) at different cumulant orders of NISSFI and SOFI, respectively.Scale bar: 500 nm.

Figure 3 .
Figure 3. NISSFI with fluctuating emitters.a) The 100 nm three-bright-point sample moves along one direction.b) A typical speckle frame.c) Zoom in the same position of different fluctuating speckle frames, similar structure but different details.d,e) The estimated PSF. f) PR reconstruction.g) A sequence of deconvolution frames.h) Mean, 2nd, 4th and 12th order cumulants images.i) The intensity profile along the line as indicated by the dashed line in (h): red-Mean; green-C 2 ; blue-C 4 .j) The FWHM of different cumulant orders.Scale bar: 20 μm in (b) and (d), 2.0 μm in others.

Figure 4 .
Figure 4. Experiment resolution improvement.a) Schematic of the experiment setup.The laser is focused on thin and rotating scattering media.A nanoscale sample (SEM image) is speckle illuminated, then imaged by lens or scattering media.b) The fluctuating frames imaged by lens for SOFI technique.c) The fluctuating speckle frames imaged by scattering media for NISSFI technique.d,e) Mean, 2nd and 8th order cumulants image from NISSFI and SOFI, respectively.f) The intensity profile along the line indicated by dashed arrows in (d).g) The intensity profile along the line indicated by dashed arrows in (e).red-Mean; green-C 2 ; blue-C 8 .h) The FWHM of different cumulant orders.Scale bar: 25 μm in (c) and 1.0 μm in others.

Figure 5 .
Figure 5. Adaptive NISSFI through dynamic scattering media.a) PR reconstruction from averaged frame of first 50 speckle frames.b,d) The 50th, 150th, and 250th deconvolution frames from origin estimated PSF and re-estimated PSF respectively.The chicken eggshell membrane is decorrelated, at the 50th, 150th, and 250th speckle frame, its PSF correlation with the initial PSF decreases to 0.64, 0.43, and 0.26, respectively.(b) has more noise and artifacts than (d).c) The higher-order cumulants image of NISSFI.e) The higher-order cumulants image of adaptive NISSFI.Scale bar: 65 μm.