Large field-of-view non-invasive imaging through scattering layers using fluctuating random illumination

Non-invasive optical imaging techniques are essential diagnostic tools in many fields. Although various recent methods have been proposed to utilize and control light in multiple scattering media, non-invasive optical imaging through and inside scattering layers across a large field of view remains elusive due to the physical limits set by the optical memory effect, especially without wavefront shaping techniques. Here, we demonstrate an approach that enables non-invasive fluorescence imaging behind scattering layers with field-of-views extending well beyond the optical memory effect. The method consists in demixing the speckle patterns emitted by a fluorescent object under variable unknown random illumination, using matrix factorization and a novel fingerprint-based reconstruction. Experimental validation shows the efficiency and robustness of the method with various fluorescent samples, covering a field of view up to three times the optical memory effect range. Our non-invasive imaging technique is simple, neither requires a spatial light modulator nor a guide star, and can be generalized to a wide range of incoherent contrast mechanisms and illumination schemes.

With the purpose of proving that our technique can be applied in imaging beyond the ME, different diffusers with different scattering properties are tested. We estimate the ME range of each diffuser by exploring the correlation between the speckle patterns generated while laterally moving an emitter (a small fluorescent bead). As the source moves, the correlation between speckle patterns gradually decreases, reaching a minimum correlation when the distance is longer than the ME range. Measuring this distance allows estimating the ME range for each diffuser. As reported in Fig.3, the ME range is approximately 50 µm and 20 µm corresponding to diffuser #1 and diffuser #2, respectively. Our non-invasive technique not only can retrieve objects within ME range as shown in Fig.2a of the manuscript, which is possible to be recovered via autocorrelation approaches, but also can reconstruct an extended object, which could not be retrieved with autocorrelation approaches. As shown in Fig.2b of the manuscript, the size of the reconstructed extended object is around 58 µm × 22 µm with the diffuser #2.

Supplementary III. Fingerprint-based reconstruction
As described in the main text, it is possible to reconstruct the whole object as long as the ME patches have some overlaps. To present the details of FBR, we experimentally choose a fluorescent object which contains 3 fluorescent beads and display  Figure 1. NMF rank estimation procedure. (a) Simulation results with P fluorescent sources. The triangle symbol marks the minimum mean value of the RMSR, the solid line stands for the mean value of the RMSR over 12 times with the random initialization value of NMF, and the asterisk is the error bar that indicates its standard deviation in both positive and negative direction. The minimum mean RMSR value is achieved when the estimated rank ρ equals the true rank P of the system. (b) Experimental results with a fluorescent object which contains P = 11 beads. (c) Estimating the rank of a fluorescent object shown in (b), the solid line stands for the mean value of the RMSR over 12 repetitions, the triangle symbol marks the minimum mean value of the RMSR, and the error bars indicate the standard deviation of the RMSR. the various results from the pairwise deconvolution, o i,k , and the different partial images, O k . As shown in Fig.4, the O k of the emitter k can be recovered by choosing w k as the PSF. By looking at the maximum value of those o i,k , the shift r i,k between fingerprints w i and w k can be retrieved, as shown in Fig.1c of the manuscript. In practice, the fingerprints coming from two emitters which are beyond the ME range will not provide the relative position information of their emitters (as they will be totally uncorrelated). Thus, it is necessary to infer whether the fingerprints w i and w k are within or beyond the ME range. In our method, we study α = max{o i,k } max{o k,k } as a function of relative distance, where max{o i,k } stands for maximum value of o i,k . A given threshold, α tres of α is introduced to evaluate it. For example, if α is greater than α tres , w i and w k belong to the same ME range. Otherwise, they belong to different ME ranges, and their relative position cannot be retrieved. As shown in Fig.5, we experimentally investigate the maximum value of various results of pairwise deconvolutions as a function of distance between emitters. We set α tres to 0.01. The full workflow of technique is depicted in Alg.1.

Supplementary IV. Producing random illumination using a SLM
Our technique retains the potential of employing a SLM for producing random illumination. The SLM can produce a large number of independent speckle illuminations. Experimentally, we replace the rotating diffuser with the SLM and perform experiments on fluorescent point-like objects and continuous volumetric objects by generating random illuminations with the SLM. The reconstruction is presented in Fig.6. To prove that it is possible to recover a more reliable image with more patterns, we show the reconstruction with the different number of patterns in Fig.7. Note that the results as shown in Fig.6 and Fig.7 are just tests to recover with the illumination patterns produced by SLM that allows generating a higher diversity of patterns than our current rotating diffuser, but it is not a limitation of our technique, as we could use different rotating diffusers to get more independent illumination patterns.

Supplementary V. Deconvolution method and comparison with cross-correlation approach
Once the fingerprints have been retrieved by using the NMF algorithm, the relative position between emitters can be obtained by looking at the correlation between their fingerprints. For emitters within the ME range, the fingerprints will be highlycorrelated, laterally shifted speckle patterns, while faraway emitters will present uncorrelated fingerprints. Conventionally, the lateral shift is measured by doing a cross-correlation between fingerprints. When both are correlated, a peak appears at the cross-correlation, where its distance from the center provides the shift between them. However, this method presents some drawbacks. The main issue comes from the common background envelope on the fingerprints, which even when filtered, can raise to a background in the cross-correlation, partially masking the peaks (see Fig.8.d). Moreover, when the two fingerprints are not 100% correlated, the peak tends to broaden, hindering localization accuracy. Even though it is possible to post-process this result in order to "clean" the cross-correlation (see Fig.8.e), automating the task for different scattering media, ME ranges, and signal-to-noise ratios is not a trivial process.
The rationale behind using the deconvolution approach is that a lateral shift can be expressed as the convolution of the image with a delta positioned at a distance from the center of the image equal to the shift. Then, if an image S, is a shifted version of another image, I, shifted by an amount (x 0 , y 0 ), we have S = I δ(x − x 0 , y − y 0 ). In that case, the shift can be retrieved by deconvolving S and I. While there are many deconvolution algorithms available, we opted for a minimization procedure adding a prior on the expected image characteristics. We chose a Total Variation (TV) minimization approach, as it promotes either delta-like or flat solutions (x) with constant background, as we would expect from neighbouring or distant emitters: arg min This approach evades the cross-correlation background problem (see Fig. 8.f), and allows to retrieve the final image of the object just by adding the results from all the fingerprint deconvolutions, with no additional post-processing (Fig.8.i). The minimization problem is solved by using the augmented Lagrangian method [5]. The implementation we used can be found at [6].      Figure 9. Experimental setup. The expanded 532 nm laser beam illuminates the rotating diffuser and the modulated light is imaged on the back focal plane of objective 1 (OB1). The object is placed in the focal plane of OB1. Camera1 is located in the imaging plane of the fluorescent imaging system that is employed to capture fluorescent speckle. The passive controlling part is made of objective 2 (OB2), tube lens 2 (TL2), and camera 2. DM: dichroic mirror, SF: spectral filter, Scat.: scattering medium.