In this manuscript, we introduce a new on-chip fluorescent imaging modality that utilizes a compressive sampling based algorithm to achieve ~10µm spatial resolution for imaging of sparse objects over an ultra-large field of view of >8 cm² without the use of any lenses or mechanical scanning. In this approach (see Fig. 1 ), fluorescent samples, positioned e.g., within a micro-channel, are pumped through a prism interface, where the excitation light (in the form of propagating waves) faces total internal reflection after exciting the entire sample volume. The fluorescent emission from the excited micro-objects is then collected using a dense fiber-optic faceplate and is delivered to a large format opto-electronic sensor array without the use of any lenses. This detected lensfree fluorescent image does not have any physical magnification and therefore has a field-of-view (FOV) that is equivalent to the detector active area, covering >8 cm² (see Fig. 1). On the other hand, because of its lensless operation, the detected fluorescent spots are significantly enlarged and distorted due to free space propagation, large pixel size, as well the as the faceplate operation. We demonstrate that these distortions can be significantly eliminated by using a digital reconstruction/decoding algorithm that is based on compressive sampling theory to achieve ~10µm spatial resolution across the entire detector active area. When compared to our earlier lensfree fluorescent imaging report [19], our new results present an improvement of ~5 fold in spatial resolution without a trade-off in FOV, which we attribute to the use of the fiber-optic faceplate and the compressive sampling based numerical processing. Furthermore, with this new on-chip platform, we also demonstrate lensfree fluorescent imaging of vertically stacked micro-channels, all in parallel, further increasing the throughput of fluorescent on-chip imaging.

This on-chip fluorescent imaging platform, being lensfree and compact, together with a powerful compressive sampling algorithm behind it, would especially be valuable for high-throughput imaging of cells within micro-fluidic chips. Specifically, it would be quite important for cytometry applications involving e.g., rare-cell analysis in large area micro-fluidic devices. In such rare-cell analysis applications [20–23], the target cells of interest (e.g., circulating tumor cells in whole blood samples) occur at extremely low concentrations which make their corresponding images highly sparse, providing an important connection to compressive sampling. Therefore, the presented fluorescent on-chip imaging technique, having an ultra-wide FOV of >8 cm² within a lensfree compact platform, would be quite valuable to rapidly screen large area micro-fluidic devices to quantify the capture of specific cells at extreme rare statistics, providing a useful tool for high-throughput cytometry. Furthermore, as we demonstrate in this work, our platform also permits simultaneous imaging of vertically stacked micro-channels on the same chip, which is another step forward towards increasing the fluorescent imaging throughput. Our approach could also potentially impact micro-array imaging technologies by providing lensless quantification of hundreds of thousands of micro-spots, all in parallel, within a compact chip.

For the applications that are of interest to this work, such as wide-field fluorescent cytometry, rare cell analysis and high-throughput micro-array imaging, one can in general assume that is sparse to start with, such that only S coefficients of are non-zero, where S<<N. This assumption is further justified with our unit magnification lensless geometry since most cells of interest would not be over-sampled due to limited spatial resolution, restricting the value of S for a practical . Therefore, the sparsity of is the first connection to compressive sampling, as it is an important requirement of its underlying theory [1–3].

In a lensfree fluorescent imaging platform as shown in Fig. 1(a), uniquely determines the intensity distribution that is impinging on the detector-array. For each non-zero element of , a wave is transmitted, and after passing through different layers on the chip it incoherently adds up with the waves created by the other fluorescent points within the sample volume. Therefore, one can write the intensity distribution right above the detector plane (before being measured/sampled) as: where represents the 2D wave intensity right before the detector plane that originated from the physical location of . The analytical form of can be derived for any particular lensfree geometry such as the one presented in Fig. 1(a). However, from a practical point of view, it can easily be measured for each object plane by using e.g., small fluorescent particles, which is the approach taken in this work.

Without the use of a faceplate in Fig. 1(a), it is straightforward to see that the functional form of for a given object plane is space invariant. This is equivalent to say that , where is the incoherent point-spread function (psf) of the system for a given object layer, and denotes the physical location of . Note that in this definition, has no relationship to the pixel size at the detector since Eq. (1) describes the intensity right before the sampling plane. The same space invariance property also holds with a dense fiber-optic faceplate as shown in Fig. 1(a) since there is a significant gap between the sample and faceplate planes, and a similar gap between the bottom surface of the faceplate and the detector plane. Therefore for our lensfree fluorescent imaging geometry of Fig. 1(a), with or without the faceplate operation, one can in general write:

For multiple layers of fluorescent objects, a similar equation could also be written where the incoherent point-spread function of different layers are also included in the summation.

Equation (2) relates the “already” sparse fluorescent object distribution ( ) to an optical intensity distribution that is yet to be sampled by the detector array. The representation basis provided by is surely not an orthogonal one since it is based on lensfree diffraction. This is not limiting the applicability of compressive decoding to our work since is assumed to be already sparse, independent of the representation basis. On the other hand, the fact that does not form an orthogonal basis limits the spatial resolution that can be compressively decoded, since for closely spaced values, the corresponding would be quite similar to each other for a given detection signal to noise ratio (SNR). This is related to the restricted isometry property [1,2] of our system as will be discussed later on; however its physical implication is nothing new since it is already known that we trade off spatial resolution to achieve wide-field lensfree fluorescent imaging with unit magnification.

Next, sampling of at the detector-array can be formulated as: where represents the sampling/measurement basis; m=1:M denotes the m^th pixel of the detector-array with center coordinates of ( , ); and represents the pixel function, which can be approximated to be a detection constant, K, for |x|,|y|≤W/2 (assuming a square pixel size of W) and 0 elsewhere, |x|,|y|>W/2. In this notation, the fill-factor of the detector-array together with the quantum efficiency etc are all lumped into K. Note that in this work, we have used W=9 µm and W=18 µm (through pixel binning).

With these definitions, the lensfree fluorescent imaging problem of this manuscript can be summarized as such: based on M independent measurements of , we would like to estimate the sparse fluorescent source distribution, , at the sample.

To give more insight, Eq. (3) models a hypothetical near-field sampling experiment, where each pixel of the CCD measures part of . For an arbitrary intensity distribution impinging on the detector array, a few pixel values ( ) can surely not represent the entire function. However, if the sampled intensity profile at the detector plane is created by a sparse distribution of incoherent point sources located in the far-field, then much fewer pixels can potentially be used to recover the source distribution based on compressive decoding. For this decoding to work efficiently, each pixel should ideally detect “some” contribution from all the values, which implies the need for a relatively wide point spread function. However since spreading of the fluorescence also decreases the signal strength at the detector plane, the optimum extent of the point spread function is practically determined by the detection SNR. On one extreme, if the same sparse source distribution ( ) was hypothetically placed in direct contact with the CCD pixels, this would not permit any compressive decoding since each incoherent point source can now only contribute to a single pixel value. For instance two sub-pixel point sources that are located on the same pixel would only contribute to that particular pixel, which would make their separation physically impossible regardless of the measurement SNR. However, the same two sub-pixel point sources could be separated from each other through compressive decoding if they were placed some distance above the detector plane, such that more pixels could detect weighted contributions of their emission.

Since we are considering non-adaptive imaging here (i.e., no a priori information about the possible x-y locations of the fluorescent particles/cells), we have not used a sub-set of the pixel values (I_m) to reconstruct . Therefore, for a single layer of object, using a unit magnification as in Fig. 1, we have N x d ² = M x W². In this work, to claim a spatial resolution of ~10µm at the object plane, we used d = 2-3 µm, which implies N ≥ 9M for W=9 µm. For some experiments, we have also used a pixel size of W=18µm with d=2µm, implying N=81M. Furthermore, for multi-layer experiments (to be reported in the next section) where 3 different fluorescent channels were vertically stacked and simultaneously imaged in a single snap-shot, we had N=27M, which all indicate compressive imaging since the number of measurements (M) are significantly smaller than the number of reconstructed points (N).

As already known in compressive sampling literature, the effectiveness of the decoding process to estimate in our technique should also depend on the maximum spatial correlation between and for all possible m=1:M and i=1:N pairs. Accordingly, this maximum spatial correlation coefficient defines the measure of incoherence between sampling and representation bases, which can then be related to the probability of accurately reconstructing from M measurements [1–3]. For a given object plane, because of the shift invariant nature of both and , this coherence calculation is equivalent to calculation of the correlation between the pixel function and the incoherent point-spread function p(x,y). The smaller the correlation between these two spatial functions is, the more accurate and efficient the compressive decoding process gets. Based on this, a smaller pixel size would further help in our lensfree on-chip scheme by reducing this maximum correlation coefficient, i.e., increasing incoherence between and .

Thus, we can conclude that the primary function of compressive sampling in this work is to digitally undo the effect of diffraction induced spreading formulated in Eqs. (1)–(2) through decoding of lensfree image pixels indicated in Eq. (3). Such a decoding process, however, can also be done physically rather than digitally, through the use of a lens (as in conventional fluorescent microscopy at the cost of reduced FOV) or through the use of a faceplate as we demonstrate in this work. The use of the faceplate in Fig. 1 partially decodes the diffraction induced spreading, which also relatively increases the correlation between and p(x,y), since p(x,y) gets narrower and stronger with the faceplate. Despite this relatively increased coherence between the sampling and representation bases, the improvement in the detection SNR with the faceplate enables better measurement of p(x,y) as well as values, which then improves the accuracy of the compressive decoding process in terms of achievable spatial resolution. This will be further quantified in experimental results presented in the next section.

Finally, we would like mention that the above analysis could be also done using a different set of measurement and representation bases without changing the end conclusions. In the above analysis, we did not include the diffraction process as part of the measurement, and therefore the measurement basis only involved the pixel sampling at the detector-array. As an alternative notation, we could have also used for the representation basis, which implies that Ψ is an identity matrix. This is not a surprising choice since the object, is already sparse and therefore the sparsifying matrix can be seen as an identity matrix. Based on this definition of the representation basis, the measurement basis will now need to include both the diffraction and the pixel sampling processes. Following a similar derivation as in Eq. (3), the measurement basis now becomes: . As expected, the correlation behavior between and for all possible m and i pairs remains the same as before, yielding the same set of conclusions that we arrived using the previously discussed choice of bases.

While it is just a matter of notation, with this new pair of bases, it is also easier to qualitatively relate the spatial resolution to restricted isometry property (RIP) of the system. RIP is a measure of the robustness of sparse signal reconstruction for N>M and S<<N [1–3]. For this new choice of bases, RIP holds if all the possible subsets of S columns taken from are nearly orthogonal to each other. Assuming that the pixel size is much narrower than the incoherent psf of the object layer of interest, we can then approximate:

First, the comparison provided in Figs. 2(a) and 2(b) indicates that the fiber-optic faceplate in our imaging geometry (Fig. 1) significantly reduces the diffraction induced spreading of the fluorescent signatures of the objects. Specifically, based on Fig. 2 we conclude that the FWHM of the fluorescent signatures at the detector plane is now reduced by ~5 fold, from ~180 µm down to ~36 µm using the faceplate (note that except the faceplate thickness, all the other vertical distances are kept the same in both configurations - with and without the faceplate - to provide a fair comparison). This improvement is quite important as it permits a better detection SNR and a higher spatial resolution to be achieved, which will be further quantified in the next figures.

The physical function of the faceplate used in our experiments is to collect the fluorescent emission from the specimen with an effective numerical aperture of ~0.3 and to guide it to the detector array. However, since the fluorescent emission from the objects spreads with an effective numerical aperture of 1 over the air gap above the faceplate, several oblique fluorescent rays (corresponding to higher angles than the acceptance NA of each fiber) remain unguided. These unguided rays (which undergo various partial reflections over multiple fiber cross-sections) are also detected at the sensor plane and are incoherently superimposed onto the fluorescent signal that is guided through the core of each fiber. However, since the thickness of the faceplate is relatively large (~1 cm), the contribution of these unguided fluorescent waves is weaker than the guided fluorescent signal.

Therefore, the faceplate in our on-chip imaging geometry, even though significantly reduces the signal spreading at the detector plane as shown in Fig. 2, also brings its own distortion to the recorded images by creating a unique incoherent point-spread function (psf) at the detector plane. The exact spatial form of this 2D incoherent point-spread function is determined by the faceplate periodicity and lattice, numerical aperture of the individual fibers, the distance between the sample plane and the upper surface of the faceplate, as well as the distance between the exit plane of the faceplate and the detector array. Once all these parameters are fixed in our imaging geometry (see Fig. 1 for details), the resulting psf for a given object plane is easily measured using e.g., small diameter fluorescent particles that are imaged at a low concentration. Quite importantly, the physical gap (~50-100 µm) between the sample and the faceplate planes, together with the gap between the faceplate and the detector planes (~75-100 µm) ensure that this incoherent point spread function is space invariant all across our imaging FOV, which enables the use of a single point spread function for decoding of each object plane as already highlighted in the previous section.

After this brief discussion on the use of the faceplate, let us now expand on the compressive decoding results of our imaging technique. As discussed in the previous section, the main function of the compressive sampling theory in our work is to recover the distribution of the fluorescent points that created the 2D lensless image sampled at the detector array. Knowing the incoherent psf of our system for each object layer, for an arbitrary distribution of fluorescent sources (within e.g., a single micro-channel or a stack of vertical channels), one can easily calculate the expected lensfree image at the detector-array. Using this fact, through a compressive sampling algorithm [26] we optimize the distribution of the fluorescent sources at the object volume based on a given 2D lensless fluorescent measurement. The choice of this compressive decoder [26] is highly suitable for the presented wide FOV fluorescent imaging platform since it is especially designed for sparse signal recovery from large data sets.

To be more specific, our reconstruction/decoding process can be formulized as an l₁-regularized least squares problem (LSP) [26], such that: where β > 0 is a regularization parameter (typically between 1 and 10); I_det is the detected raw fluorescent image at the sensor-array (in a vector form); represents the 2D convolution matrix based on the incoherent point spread function of the system; is the fluorescent source distribution that creates the lensfree image at the detector plane; and represents the l_p norm of vector . For multiple micro-channels that are vertically stacked (see e.g., Figs. 7 , ​,8),8), there is a separate for each source layer. The compressive decoder used in this work [26] is based on truncated Newton interior-point method and rapidly provides a sparse solution ( ) for Eq. (4) especially for large-scale data sets using a non-negativity constraint, which is surely satisfied for fluorescent imaging in general.

The experimental results that illustrate and quantify the performance of this compressive decoding and reconstruction process are summarized in Figs. 2–8. Specifically, Figs. 2(c3)–2(f3) demonstrate the compressive decoding results of the raw lensfree images acquired with a faceplate. Note that the same compressive sampling algorithm also performs very well even without the use of a faceplate to resolve rather closely spaced fluorescent particles from each other as highlighted in Fig. 3 . However, Fig. 3 also compares the performance of the faceplate for decoding of the same field-of-view, which clearly illustrates the superior performance of faceplate based decoded images in terms of resolving closely spaced fluorescent particles without any reconstruction artifacts or ambiguities (e.g., refer to the arrows in Fig. 3).

We further quantified the spatial resolution of this compressive decoding process by resolving several closely packed fluorescent micro-particles from each other. Figures 4 , ​,55 and ​and66 summarize our experimental results which clearly demonstrate resolving two fluorescent particles that are separated by g~10µm (center to center). Figure 4 and Fig. 5 are obtained using 10µm and 2µm diameter fluorescent particles, respectively; and the pixel size in the decoded images is 3µm and 2µm, which implies N = 9M and N~20M for Fig. 4 and Fig. 5, respectively. In Fig. 6, we further reduced M by pixel binning at the detector-array such that the pixel size became W = 18µm. This implies N = 81M, which still permitted reconstruction of 2µm particles that were separated by g~12µm (center to center) as shown in Fig. 6. Note that even if the detector array was to directly sample these fluorescent objects in their near-field (hypothetically), the spatial resolution would not be equivalent to ~10µm due to the large pixel size (W = 9µm or 18µm). The computation times of these decoded images are fairly small ranging between 0.1min to 0.5min using an Intel Centrino Duo Core, 1GHz PC.

Another important dimension of the presented lensfree approach is its ability to reconstruct the distribution of fluorescent micro-objects located at multiple micro-channels that are stacked vertically. This enables to decode the 2D lensfree fluorescent image at the detector-array into a 3D distribution through compressive sampling, which is especially important to further increase the throughput of fluorescent on-chip imaging. A proof of concept demonstration of this capability is presented in Figs. 7 and ​and88 where the fluorescent micro-particles (10µm diameter) located at two and three different micro-channels, respectively, were imaged and decoded all in parallel. In these two experiments the fluorescent channels were vertically separated by 50 and 100 µm, respectively. In Fig. 7 the comparison of the decoding results with and without the faceplate is also provided for the same sample volume, which illustrates the superior performance of the faceplate case for resolving overlapping fluorescent signatures from each other.

In general, lensfree on-chip imaging (both transmission [14,15,27,28] and fluorescent [19]) can be thought as an effective compressive sampling encoder. Recent results on “compressive holography” provide an important proof of this for lensfree holographic imaging where a 2D array of pixels at the detector-array was used to reconstruct a 3D volume of voxels without ambiguities [14,15]. Therefore one of the main contributions of these recent results was to use compressive sampling theory to handle the ill-posed problem of 3D image reconstruction from lensfree in-line holograms. This important contribution remains true regardless of the physical magnification embedded in the lensless imaging system. However, the effectiveness of compressive sampling and the encoding process at the detector array even further increases if there is no physical magnification in the lensless system, such that the detection process now becomes either under- or critically-sampled (as in lensfree holography with unit fringe magnification [27]) or highly ambiguous due to significant spatial overlap and distortion at the detector plane (as in lensfree fluorescent imaging of this work). Therefore, as further discussed in the previous section, the primary function of the compressive decoding process in our work is to remove this spatial ambiguity and distortion due to lensfree and magnification-free operation of our wide-field fluorescent imaging platform.

Finally, we would like to also provide a quantitative comparison of the presented compressive sampling based reconstruction approach over some of the existing deconvolution methods that could potentially be used for the same purpose. One such numerical recipe is the Lucy-Richardson algorithm [29–31] which relies on the knowledge of the incoherent point spread function to iteratively converge to the maximum-likelihood estimation of the fluorescence source distribution based on the detected image. This algorithm is not restricted to sparse objects only and has been shown to be quite effective converging within typically a few hundred iterations to the source distribution [19]. A comparison of the performance of this algorithm against compressive sampling based reconstruction is demonstrated in Fig. 4 (bottom row), which clearly indicates the advantages of the compressive decoding approach especially in terms of spatial resolution. Figure 4 demonstrates that, for the same set of lensless fluorescent measurements, the compressive decoder achieves ~10µm resolution, while the Lucy-Richardson deconvolution algorithm achieves ~20µm. This behavior is intuitively expected since the Lucy-Richardson algorithm does not use the sparsity of the object as an optimization constraint. Besides resolution, another important difference between the two approaches is that unlike compressive sampling which can easily perform multi-layer reconstruction for sparse objects as illustrated in Figs. 7–8, the Lucy Richardson algorithm would need to be significantly modified to handle 3D reconstruction. On the other hand, conventional deconvolution algorithms that do not rely on a sparsity constraint (such as the Lucy Richardson algorithm) could potentially better handle highly dense fluorescent samples, where the sparsity condition of the object is no longer satisfied.