Abstract
In biological microscopy, light scattering represents the main limitation to image at depth. Recently, a set of wavefront shaping techniques has been developed in order to manipulate coherent light in strongly disordered materials. The Transmission Matrix approach has shown its capability to inverse the effect of scattering and efficiently focus light. In practice, the matrix is usually measured using an invasive detector or lowresolution acoustic guide stars. Here, we introduce a noninvasive and alloptical strategy based on linear fluorescence to reconstruct the transmission matrices, to and from a fluorescent object placed inside a scattering medium. It consists in demixing the incoherent patterns emitted by the object using lowrank factorizations and phase retrieval algorithms. We experimentally demonstrate the efficiency of this method through robust and selective focusing. Additionally, from the same measurements, it is possible to exploit memory effect correlations to image and reconstruct extended objects. This approach opens up a new route towards imaging in scattering media with linear or nonlinear contrast mechanisms.
Introduction
Propagation of light in materials with refractive index inhomogeneities, such as biological tissues, results in scattering. In such disordered media, ballistic light exponentially decreases with penetration depth, which limits the scope of conventional optical microscopy. At depth, coherent light is affected by multiple scattering and produces very complicated interference patterns, known as speckle^{1}. Recent years have witnessed many advances in the ability to coherently manipulate this figure of interferences owing to the availability of spatial light modulators (SLMs)^{2,3}. In particular, they enable refocusing light to a diffractionlimited spot^{4}. These techniques often rely on optimizing the incident wavefront such that it maximizes a feedback signal emitted from the target focus point. A key constraint for biological imaging at depth, is that the measurement of the feedback has to be noninvasive. Several strategies using acoustics or nonlinear fluorescent guidestars have been proposed for this purpose^{5}. However, most of them are only able to form a single focus at a given output position^{6,7,8,9}, which limits the acquisition speed or the fieldofview.
Deterministic focusing of light on multiple targets is optimally achieved with a transmission matrix (TM) that linearly relates the input field to the output field^{10}. However, its noninvasive measurement remains very challenging. Indeed, even if each target has its own optical response, what is measured in epidetection is the backscattered emission, thus spatially and temporally mixed. To overcome this limitation, optics has been combined with acoustics to coarsely locate each target^{11,12}, which enables the reconstruction of a TM but requires complicated acoustooptical setups. Another powerful approach relies on the measurement of a timegated matrix in reflection^{13,14}, but is based on retroreflected ballistic photons, hence limited in depth.
Linear fluorescence remains an essential technique in microscopy because systems are fairly inexpensive and easy to handle. As such, it remains a staple tool in biology and biomedical sciences. It has enabled imaging of cells and submicroscopic cellular components with high spatial resolution, specificity, contrast and speed. Combined with lightsheet or structured illumination microscopy, linear fluorescence allows sectioning and imaging at moderate depth^{15,16}. Although imaging fluorescent objects through thin scattering media can be done thanks to the memory effect^{17,18}, a general method to focus on fluorescent objects at depth, and image them if they extend beyond the memory effect is still missing.
Here, we report on a robust TM approach for fluorescence imaging through a relatively strong scattering medium, in a noninvasive way. The technique relies on shining a sequence of known wavefronts on a fluorescent object hidden behind a scattering medium, and collect in reflection the corresponding lowcoherence fluorescent speckles backscattered by the medium. From this set of inputoutput information, we are able to computationally retrieve both (i) the ingoing fieldTM for the excitation light, and (ii) the outgoing intensityTM for the fluorescence light. We demonstrate robust and selective focusing across all the object, both on beads and on more complex fluorescent objects. Finally, if the medium exhibits limited memory effect, we show that an image of the object can be retrieved, even when its size exceeds the memory effect range.
Results
The experimental apparatus is depicted in Fig. 1a. A coherent beam of light is first modulated in phase by an SLM and directed through a scattering medium onto a fluorescent object made of several emitters. To describe both the ingoing and outgoing light propagation, we use a transmission matrix formalism. A fieldTM, denoted T, connects the input field E_{in} (specifically the phase pattern displayed onto the SLM) to the field at the position of the N targets. Thus, the speckle intensity in the plane of the fluorescent object reads \({\left{E}_{{\mathrm{exc}}}\right}^{2}={\leftT{E}_{{\mathrm{in}}}\right}^{2}\). Once excited, each target fluoresces proportionally to its illumination. This lowcoherence signal is backscattered by the medium and can be noninvasively measured with a camera placed in reflection. It can be written as \({I}_{{\mathrm{out}}}=W{\left{E}_{{\mathrm{exc}}}\right}^{2}\), where W is an intensityTM, linking the N targets to the D pixels of the camera via their respective fluorescent eigenpatterns. We define here eigenpatterns, as all the independent speckles, each single target generates on the camera. It is worth stressing that the measurement is made in reflection only, thus entirely noninvasive. A control camera placed on the far side of the sample allows to monitor the excitation patterns \({\left{E}_{{\mathrm{exc}}}\right}^{2}\) at the object plane.
Our technique relies on exciting the sample consisting of the scattering medium and the fluorescent object with a variety of p = 1, …, P random input phase patterns E_{in}(p) and collecting the fluorescence responses I_{out}(p) reflected on the same side. For all p = 1, …, P, I_{out}(p) can be written as:
I_{out}(p) corresponds to a low contrast speckle because, first, the fluorescence emission is broadband and not polarized, second, the N beads generate N different speckles that partially average out^{1}. Nevertheless, the decrease in contrast due to the number of beads is relatively slow and scales as \(\sqrt{2/N}\) in the case of linear fluorescence (see Supplementary Note 8). The overall output \({I}_{{\mathrm{out}}}\in {{\mathbb{R}}}_{+}^{D\times P}\) can be written as a rankN product of two positive matrices I_{out} = WH (with N ≪ D, P), where both \(H={\leftT{E}_{{\mathrm{in}}}\right}^{2}\) and W are real positive matrices. This corresponds exactly to the framework of Nonnegative Matrix Factorization (NMF) that we, therefore, use to estimate W and H from I_{out}, see Fig. 1b. Thanks to its robustness and interpretability, this framework has already been applied in many settings^{19}, including the fluorescent readout of neuronal activity^{20,21}, but has never been associated with wavefront shaping yet. It is interesting to note that similar approaches based on leading eigenvector decompositions (without the nonnegativity assumption) have already been used in optics^{22} and in combination with acoustics^{23} for imaging at depth. In a second step, a phase retrieval (PR) algorithm allows retrieving the ingoing fieldTM, T^{24,25}, from H. This computational problem has been studied extensively for the case of random matrices and is in theory solvable when the number of measurements (here, P) is a few times larger than the dimension of the unknown (here, N_{SLM})^{26,27,28}.
Experimentally, we first performed the measurement with fluorescent objects made of 1μm beads, placed behind holographic diffusers. Once a series of input patterns have been displayed and the corresponding fluorescence images recorded, the matrix I_{out} is factorized into two lowrank matrices thanks to NMF. The rank r of the factor matrices is the main input parameter required to run the algorithm. In principle, r should correspond to the number of independent sources N in the system, which in this specific case corresponds to the number of resolvable targets N since the speckle grain matches the size of the bead. Note that N is unknown in such reflection configuration. Nevertheless as discussed in Methods, an upper bound for r can be easily estimated from I_{out}, and is sufficient to identify all the N targets. An additional step of phase retrieval estimates the fieldTM T that links the SLM pattern to the plane of the beads.
Focusing light using phase conjugation provides a method to check the quality of the NMF + PR pipeline. On Fig. 2, this focusing capability is shown with images monitored on the control camera, but also noninvasively from the observation of backscattered fluorescence. When light is successfully focused on a target, the spatial variance of the fluorescence speckle increases^{7}. Since we typically overestimate the rank r > N, the NMF may generate spurious eigenpatterns (which do not focus the illumination) and duplicates. Looking at the spatial variance and the correlation between fluorescent patterns allows to identify both (see Supplementary Note 4). We thus validate the ability to accurately reconstruct the ingoing and outgoing transmission matrices and deterministically focus on every beads.
The doubleTM reconstruction can be applied in principle whatever the depth and scattering properties of the medium, as long as it provides a measurable fluorescent speckle. In the following, we show that, if there is some memory effect (ME), the technique allows not only focusing but also fluorescence imaging at depth by looking at the correlations between fluorescent eigenpatterns.
In essence, two beads within the ME should exhibit translated fluorescent patterns with a shift equal to their relative distance. By crosscorrelating the fluorescence patterns, which are recorded in epi while displaying the r focusing patterns onto the SLM, it is possible to retrieve a distance map between all the beads. In Fig. 3, we show an example of such reconstruction with an object much larger than the ME range (see Supplementary Note 5). Interestingly, computing successively these pairwise correlations between close targets allows retrieving the full object, well beyond a single ME patch. All the beads are thus reconstructed as long as their ME patches have some overlap. In Fig. 3 we show reconstruction of an object extending approximately three times the ME range. Note that using directly the W patterns from the NMF works but did not provide here as good results (see Supplementary Note 6).
Finally, to demonstrate that our technique can also be used with continuous volumetric objects, we tested it on biological objects, here fluorescencestained pollen grains. The whole process, from the acquisition to the reconstruction, is similar to what is presented in Fig. 3. Figure 4a–d shows fluorescent images of three different pollen grains taken without the diffuser. The blue lines are contours of the reconstructed image at 10% of the maximum intensity, showing that the highintensity features of the object are faithfully retrieved. The full reconstructed images are presented in Fig. 4e–h. While the reconstruction appears grainy, we do retrieve the main and brightest features of the pollen seeds. With N_{SLM} = 256 pixels (Fig. 4e–g), the SNR of the focus is not very high, around 10–20, but sufficient to connect the epidetected fluorescent speckle to the eigenpattern of the focused emitter and reconstruct the shape of the pollen seeds. Increasing the number of SLM pixels to N_{SLM} = 1024 as in Fig. 4h did not improve significantly the results.
Discussion
Several conditions should be met in order to successfully operate our noninvasive technique. A first point is the required number of patterns to accurately retrieve the doubleTM. As detailed in Supplementary Note 3, reconstructing W can be done even with a low number of patterns related to the complexity of the object (number of separate emitters). On the other hand, T is recovered through an additional step of phase retrieval which requires a larger number of patterns, related to the number of SLM pixels. For example, experiments of Figs. 2 and 3 require P ≃ 15,000 patterns, which at 50 Hz (limited by the exposure time of the camera) corresponds to ~5 min. Diminishing the number of SLM pixels as in Fig. 4 where N_{SLM} = 256 reduces the acquisition time to few tens of seconds, which should be compatible with stability time of exvivo biological tissues^{29}.
Another important aspect is the complexity of the object that can be reconstructed. Here we demonstrate focusing and imaging on multiple beads, but also on continuous and even volumetric objects. Reconstructions up to around 50 focus positions have been performed in this work. One limitation is the contrast of the measured speckle, that decreases with the complexity (number of separate emitters) of the object, but only with a mild squareroot dependency. For further insight, we may refer to the Supplementary Information of^{7} where a study about the contrast measurement in presence of noise was carried out and^{19} where the effect of noise on the NMF is investigated. In tissues, a general problem is the background fluorescence, that could be tackled via appropriate sparse staining or acoustic tagging of a small region^{30,31}.
Regarding imaging, our technique is limited by the spatial sparsity of the object rather than its size that can be much larger than the memory effect range. We propose in the Supplementary Note 7, an alternative algorithm based on MultiDimensional Scaling (MDS) that offers some advantage in terms of noise robustness. But still, the major limitation is that isolated targets (without correlations with others) cannot be correctly located.
It is important to note we only reconstruct a 2D projection of an object, even though the object might be volumetric. Strategies for 3D reconstruction after propagation in scattering media have been proposed^{32,33}, but require specific experimental configurations. Additionally, techniques to reduce background fluorescence^{30,31} may be useful to reduce the necessity of a full 3D reconstruction.
We focused here on linear fluorescence contrast, but the technique should readily generalize to any incoherent linear mechanisms, such as spontaneous Raman. Nonlinear incoherent processes should also be possible (as shown in Supplementary Note 8 for 2photon fluorescence), which should benefit from a higher contrast and lower background, at the cost of a lower overall signal.
In conclusion, we have presented a completely noninvasive computational strategy to characterize light propagation in and out of a scattering medium based on linear fluorescence feedback only. It allows both focusing at depth and, providing some memory effect is present, imaging of an extended object. The method is very simple, robust, and provides a promising route towards deep fluorescence imaging beyond the ballistic regime. It should be applicable to a large variety of contrast mechanisms.
Methods
Experimental setup
A continuouswave laser (λ = 532 nm, Coherent Sapphire) is expanded on a phaseonly MEMS SLM (KiloDM segmented, Boston Micromachines), such that all the N_{SLM} = 1024 segments can be used. Once modulated, the beam is directed through the illumination objective (Zeiss W “PlanApochromat” 20× , NA 1.0) to excite the fluorescent object made of orange beads (540/560 nm, Invitrogen FluoSpheres, size 1.0 μm) or pollen seeds (Carolina, Mixed Pollen Grains Slide, w.m.) placed on top of the scattering medium. The excitation beam (diameter < 6 mm) underfills the objective back aperture (diameter 20 mm) which reduces the actual illumination NA. It results that the speckle grain size at the fluorescent object plane is around 1 μm. The SLM is imaged to the back focal plane of the microscope objective. The scattering medium is not the same in all the experiments in order to control the memory effect. In the experiment presented in Fig. 2 we use a ground glass (Thorlabs, DG10), in Fig. 3 we use two holographic diffusers (Newport 1^{∘} + Newport 10^{∘}) and in Fig. 4 only one holographic diffuser (Newport 1^{∘}).
Part of the 1photon fluorescence emission is backscattered by the medium and epidetected on a first camera: CAM1 (sCMOS, Hammamatsu ORCA Flash). Recording of the P fluorescence images is the slowest step throughout the acquisition process; it is between 20 and 50 Hz depending on the scattering medium and the fluorescent sample. Once acquired, raw images are cropped (such that one image contains roughly few tens of speckle grains). Then a high pass Gaussian filter removes the background which significantly improves the contrast. The corresponding data form a matrix I_{out} which is later processed with the algorithm to reconstruct the two TMs. We use a dichroic mirror shortpass 550 nm (Thorlabs) and two other filters (F): a 532 nm longpass (Semrock) and a 533nm notch (Thorlabs). Additional spectral filters can be used before light reaches CAM1 in order to narrow the spectral width of the detected fluorescence and increase the contrast of the fluorescent speckle. To the same end, a polarizer can be inserted, enhancing the contrast by a factor \(\sqrt{2}\). A second microscope objective (Olympus “MPlan N” 20×, NA 0.4), placed in transmission, provides an image of the plane of the beads, onto a CCD camera CAM2 (Allied Vision, Manta). This part of the setup is for passive control only. It allows us to correctly position the beads using a white light source (Moritex, MHAB 150 W), but also to monitor illumination speckles \({\left{E}_{{\mathrm{exc}}}\right}^{2}\).
In the first experiment presented in Fig. 2, P = 15,360 different random inputs are generated and corresponding fluorescence images of size D = 50 × 52 = 2600 pixels were recorded. In the second experiment presented in Fig. 3, P = 14336 different random inputs were generated and corresponding fluorescence images of size D = 70 × 64 = 4480 pixels were recorded on the camera in epi. In the third experiment presented in Fig. 4, P = 5120 different random inputs were generated and corresponding fluorescence images were recorded on the camera in epi.
The experimental setup is shown in Supplementary Note 1.
NMF + PR algorithm
Before factorizing I_{out} with the NMF algorithm, the rank r of the lowrank factor matrices needs to be determined. The latter is related to the number of fluorescent beads in the sample and is not known in our reflection configuration. We estimate it by looking at the residual error ∣∣I^{fluo} − WH∣∣_{F} as a function of the rank r. Its plot should have a typical change of slope, as described in ref. ^{34}. It provides a good estimate for the rank of I_{out}. However, when the number of targets N ⪆ 10 we experimentally observe that the change of slope cannot be determined with good accuracy. As detailed in Supplementary Note 2, we decided to take the upper bound and remove spurious values afterwards.
For the NMF, we use the nnmf Matlab function with default parameters. In particular, matrices for the initialization are random.
For the PR we use an algorithm very similar to ref. ^{26}, involving a spectral method to obtain a good initial estimate for the subsequent gradient descent iterations, except that we use a refined spectral initialization to speed up the convergence^{27,28}.
Simulation codes are available at: https://github.com/laboGigan/NMF_PR
Data availability
All relevant data are available from the authors upon request.
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Acknowledgements
We thank Claudio Moretti for fruitful discussions and constructive comments. This research has been funded by the European Research Council ERC Consolidator Grant (Grant SMARTIES  724473). S.G. is a member of the Institut Universitaire de France.
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A.B., J.D. and S.G. conceived the idea. A.B. and J.D. wrote the Matlab code to collect and process experimental data. A.B. performed the experiments and analysed the experimental data. All authors discussed the results and commented on the paper.
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Boniface, A., Dong, J. & Gigan, S. Noninvasive focusing and imaging in scattering media with a fluorescencebased transmission matrix. Nat Commun 11, 6154 (2020). https://doi.org/10.1038/s41467020196968
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DOI: https://doi.org/10.1038/s41467020196968
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