1. Introduction

The ability to record wide field-of-view (FOV) images without compromising the spatial and temporal resolution is of crucial importance for imaging science. For example, high-throughput microscopy allows to high-content quantitative analysis of multiple events in a large population of cells for many applications, such as drug discovery personalized genomics, cancer diagnostics, and drug development [1, 2]. However, biological study processes are in a large majority based on fluorescence imaging, which requires fluorescent dyes and fluorescent proteins as bio-markers. This induces photo-toxic effects and perturbs the native cellular behavior. In addition, the photobleaching of fluorophores prevent long term time-lapse imaging (over hours) of live cells. To proceed unbiased long-term high-content studies, label-free techniques, such as quantitative phase imaging (QPI) approaches are preferred due to their noninvasive and nontoxic properties [3,4].

During the last decades, various QPI techniques have been developed based on different imaging principles, such as digital holography (DH) [5–7], lensless microscopic imaging [8–10], transport-of-intensity equation (TIE) [11,12], differential phase contrast (DPC) [13, 14], and Fourier ptychographic microscopy (FPM) [15,16]. All these approaches have been widely used to recover quantitative phase information of live cells and tissues with different imaging performance in terms of lateral resolution, FOV, and imaging speed [12,16–19]. Among others, DPC provides promising non-interferometric QPI capability due to its advantages of higher imaging efficiency (only 4 images required) and less sensitivity to low frequency noise (higher response at low frequencies) [20–25].

By utilizing asymmetric illumination patterns, the DPC method creates a complex optical transfer function (OTF) to convert the invisible phase information into measurable intensity signal, which reflects the quantitative phase information of the specimen [17,18,20–22]. Furthermore, recent advances in LED lighting and digital display technology provide new opportunities for active digital illumination control. By integrating a programmable LED array or liquid crystal display (LCD) panel light into the illumination/detection path of the microscope, one gains the flexibilities to produce sophisticated illumination patterns or dynamically switchable aperture functions with no moving parts [12,24,26,27], which makes DPC imaging more convenient to implement. In a LED-array-based DPC system, 4 half-circle illumination patterns are created and corresponding intensity images are recorded. Then, under weak object assumption and through image formation of partially coherent imaging, DPC weak OTF is derived based on the illumination pattern and pupil function. Finally, by utilizing weak OTF, phase can be separated from intensity, and quantitative phase measurement can be achieved by deconvolution based on the weak phase transfer function (PTF) [24,28–30]. If the illumination numerical aperture (NA) is equal to or greater than the objective NA, the lateral resolution of DPC imaging can theoretically reach 2× of the coherent diffraction limit.

In a typical DPC microscopic system, in order to correctly implement the DPC deconvolution and recover a phase image with expected resolution, the pixel size of the imaging sensor should be small enough to prevent spatial aliasing/undersampling [32]. However, microscope cameras are in general designed to have a large pixel size to improve the photo-sensitivity and maintain a large FOV so that the intensity information transmitted by the optical system cannot be adequately sampled or digitized. On the other hand, using a image sensor with smaller pixel size or adding a magnification camera adapter to the camera can resolve the undersampling at the expense of a reduced FOV. For example, the use of a 10× 0.4 NA_obj objective lens with a commonly used 5.5 megapixel SCMOS camera (PCO.edge 5.5, pixel size 6.5 μm) would necessitate at least a 2× camera adapter to reduce the pixel size by half to avoid pixel aliasing. This would unfortunately sacrifice three quarters of imaging FOV and, therefore, result in sub-optimal use of the space-bandwidth product (SBP) of the microscopic imaging system.

As discussed above, the tradeoff between resolution and imaging FOV poses a major obstacle for DPC to achieve high-throughput QPI. To solve this tradeoff, we introduce a new variation of quantitative DPC microscopic approach, termed anti-aliased DPC (AADPC), that uses several aliased intensity images under asymmetric illuminations to recover wide-field aliasing-free phase images. AADPC firstly recovers an initial phase estimate by DPC-like deconvolution in the Fourier domain based on the system’s weak PTF. Then the obtained initial phase map is further refined by alternative projection algorithm between space and frequency domain to overcome pixel-aliasing and improve the imaging resolution. In order to get better imaging performance, we analyze the data redundancy requirements of AADPC. Based on our analysis results, an optimal illumination scheme is proposed to obtain a sub-pixel resolution with using only 4 images (same as conventional DPC). We experimentally verify that AADPC can achieve a half-pitch imaging resolution of 345 nm, corresponding to 1.88× of the theoretical Nyquist-Shannon sampling resolution limit imposed by the sensor pixel size. The high-speed, high-throughput quantitative phase imaging capabilities of AADPC are also demonstrated by imaging HeLa cells mitosis in vitro, achieving the theoretical resolution corresponding to 2NA_obj across a wide FOV of 1.77 mm² at 25 fps.

It should be noted that some pixel super-resolution methods based on angle diversity have been proposed to solve the problem of pixel aliasing in different microscopy methods. For example, the angle diversity and aperture synthesis was introduced to lensless imaging methods to further improves the imaging resolution [8,10]. But due to the large pixel size of the camera, lensless imaging still suffers from the effect of pixel aliasing and cannot achieve the theoretical imaging resolution ($\frac{\lambda }{2}$). Moreover, the requirement for high degree of coherent light source makes some algorithms that can reduce the number of captured images, such as the multiplexing strategy, unable to adapt to lensless imaging [33]. However, in this work, we introduce a multiplexing iterative method for phase recovery in AADPC, which leads to a wide-field, anti-aliased and high robustness method without increasing the number of captured images in traditional DPC. Our method is proved to be able to obtain the resolution of incoherent theoretical limit ($\frac{\lambda }{2{\mathit{NA}}_{\mathit{obj}}}$) in camera field size. The remainder of this paper is organized as follows: In Section 2, we briefly introduce the principle of DPC and explain the causes and effects of pixel aliasing problems. By analyzing the characteristics of the PTF of DPC, our method is introduced in Section 3 to solve the problem of pixel aliasing and obtain a wide-field and anti-aliased quantitative phase result. Meanwhile, we analyze the data redundancy requirements of AADPC method and give an optimal illumination solution. Experimental results are presented in Section 4, and this paper ends with discussions and conclusions in Section 5.

2. Principle and problem

2.1. DPC and PTF for phase retrieval problem

Consider a thin pure phase sample with complex transmission function t(r) = e^iϕ(r) [34], where r = (x, y) denotes the coordinates of the sample and ϕ(r) characterizes the phase of the sample. If we adopt the weak phase approximation, the complex transmission function of the sample can be approximated as t(r) ≈ 1 + iϕ(r) [29,35]. When the sample is illuminated with spatially partially coherent illumination, the intensity spectrum under complex illumination pattern can be interpreted as an incoherent superposition of all intensity spectra arising from all light source points (spatially coherent illuminations with different illumination angles). So we firstly analyze the sample intensity spectrum distribution for a bright-field image under single angle oblique illumination. The complex field at the sample plane from a single angle illumination is the product of the illumination function L(r) and sample’s transmission function t(r). Thus, the Fourier spectrum of the transmitted complex wave-front in camera plane is [36]

Next, we take the Fourier transform on both sides of Eq. (6), and the resultant Fourier spectrum of the phase contrast image is

2.2. Effect of pixel aliasing on DPC reconstruction

For a microscope imaging system, the achievable imaging resolution is generally determined by several system parameters, including the illumination wavelength λ and the numerical apertures of the illumination NA_ill and objective NA_obj. For a partially coherent imaging system like DPC, the maximum spatial cutoff frequency of the system corresponds to twice the numerical apertures of the objective $\frac{2*{\mathit{NA}}_{\mathit{obj}}}{\lambda }$, which determines the maximum imaging resolution of the system (so-called diffraction-limited resolution). However, the maximum imaging resolution of a microscope can only be achieved if the pixel size of the imaging sensor is smaller than that required to adequately sample the raw intensity image. According to Nyquist–Shannon sampling theorem, the imaging pixel size should at least be half the size of the smallest feature one wish to record. This imposes another resolution limit (so-called Nyquist limit) representing the maximum sampling frequency provided by the camera $f_{\mathit{cam}}=\frac{\mathit{mag}}{2*d_{\mathit{cam}}}$, where mag denotes the magnification of the microscopy system and d_cam denotes the pixel size of the camera. In order to reproduce the diffraction-limited resolution of the imaging system, the pixel size of the camera should satisfy the Nyquist−Shannon sampling criterion, i.e., the condition f_obj < f_cam has to be satisfied. Otherwise, the imaging resolution of the system will be limited by the camera rather than the imaging optics, which means that there exists pixel aliasing in captured images. Pixel aliasing exists in a wide variety of microscopic imaging methods, and the effect of pixel aliasing on resolution is especially severe in lensless imaging [9,10]. But unlike lensless imaging, the degree of aliasing in DPC imaging is generally up to 2 times (2×2 binning). This is because most microscopic systems are usually able to meet the sampling frequency of coherent resolution limit (NA_obj), but less than the incoherent resolution limit (NA_obj). So we only discuss the case where the pixel aliasing is no more than 2 times in this paper.

Figure 1 shows the effect of pixel aliasing on DPC reconstruction using simulations. The simulation parameters were chosen to realistically model a DPC platform, with an incident illumination wavelength of 525 nm, an image sensor with pixel size of 6.5 μm, and an 10× objective lens with NA_obj of 0.4. We simulated the use of the central 21 × 21 LEDs in the array placed 80.5 mm beneath the sample, and the distance between adjacent LEDs is 4 mm. Thus, those spatial cutoff frequencies can be numerically determined (f_obj = 1.5238μm⁻¹ > f_cam = 0.7692μm⁻¹). All of the following simulations are performed using these parameters. Here, we consider the most general two-axis illumination of the traditional DPC. The LED illumination patterns are shown in the leftmost column of Fig. 1. When pixel aliasing exists, the frequency components in the shaded part of Figs. 1.(a1) and (a2) are aliased into the central low frequency part, so that the camera’s spatial cutoff frequency is below the incoherent diffraction limit of the objective, making it difficult to implement the DPC de-convolution based on the full-resolution PTF [Figs. 1(b1) and 1(b2)]. Under such conditions, an up-sampled version of DPC provides a simple solution based on the assumption that the raw images are acquired by a camera with a pixel size that is $\frac{1}{n}$ times the size of the real camera pixel (interpolate each raw image before DPC deconvolution). Thus, the cutoff frequency of the camera will be magnified n times, which allows the system to satisfy the Nyquist–Shannon sampling criterion. Under this assumption, the captured images should be interpolated to enlarge the size of the image by n times, and the PTF is calculated by setting camera pixel size to $\frac{d_{\mathit{cam}}}{n}$. By using the up-sampled DPC, the deconvolution can be correctly implemented based on the full-resolution PTF. However, significant loss in imaging resolution and quality can still be clearly observed when compared the retrieved phase [Fig. 1(d)] with the ground truth [Fig. 1(c)]. The simulation result suggests that pixel aliasing does deteriorate the phase reconstruction quality of DPC, which cannot be simply addressed by up-sampling or interpolation.

 

Fig. 1 Effect of pixel aliasing on DPC reconstruction. (a1), (a2), (b1), (b2) are PTFs of DPC in the cases of pixel aliasing and without pixel aliasing; (c) Ideal image with 2NA_obj resolution; (d) phase reconstruction result with pixel aliasing.

Download Full Size | PDF

3. Anti-aliased quantitative DPC

3.1. Analysis of different PTFs for robust DPC reconstruction

Before introducing the algorithm of AADPC, the phase transfer characteristics of DPC are firstly analyzed. In Fig. 2, we illustrate the illumination diagrams of single LED at different angles and their corresponding PTFs. As shown in Fig. 2(a1), when on-axis illustration is used, the frequency shift vector is zero, and the corresponding two pupil apertures completely cancel each other out. In this case, values of the PTF are all zero, which means that no phase information can be transferred into intensity and recorded by camera. This is the reason why pure-phase object can hardly be observed under normal bright field illumination, such as unstained live cells. With illumination angle increases, the frequency shift vector gradually increases, so the high frequency phase information can be recovered but the paraxial region still cannot. When the NA_ill increase to NA_obj, the two shifted apertures are just tangent, and there is no coincide region. In this case, the two apertures reach the highest frequency and both the high frequency and low frequency phase components can be recovered. As a result, the illumination angle when NA_ill equals NA_obj in the LED array has the best phase transfer characteristics.

 

Fig. 2 PTF with half-circle illumination, continuous half-annular illumination, and discrete half-annular illumination. (a1)–(a3) are PTFs under single LED at different angles; (b1)–(b3) PTFs under half-circle illumination, continuous half-annular illumination and discrete half-annular illumination; (c1)–(c3) Quantitative curves of PTFs corresponding to each illumination pattern.

Download Full Size | PDF

Based on this theory, we analyze the phase transfer characteristics for three complex illumination patterns of DPC imaging. These three complex illumination patterns are shown by black wire frames in Fig. 2, namely half-circle illumination, continuous half-annular illumination, and discrete half-annular illumination. Among these illumination patterns, continuous half-annular illumination and discrete half-annular illumination adopt the LEDs under the angles that NA_ill = NA_obj. According to the solution method described in Section 2.1, we can get the PTFs under these three illumination patterns, as shown in Figs. 2(b1)–2(b3). To quantitatively analyze these three PTFs, the blue and red lines are used to characterize the responses of the PTFs. As shown in Figs. 2(c1)–2(c3), the peak value of PTF under half-circle illumination is about 0.8, while it can reach 1 under two half-annular illumination patterns. A larger value of the PTF means that the corresponding frequency components can be better transferred and converted into phase information. Besides, compared with the PTF of half-circle illumination, the low-frequency responses of the PTFs under two half-annular illuminations become steeper, which give better phase contrast for low-frequency components. Moreover, the high frequency phase contrast is slightly better than the half-circle case. Thus, the half-annular DPC measurement provides a better PTF for measuring and recovering high quality phase imaging results using DPC based method with higher accuracy and robustness. Furthermore, compared with the continuous half-annular illumination, the corresponding high-frequency and low-frequency responses of PTF under the discrete half-annular illumination are further enhanced (though the PTF becomes step-shaped), suggesting a higher phase reconstruction quality. Besides, the discrete half-annular illumination is more convenient to perform illumination mode de-multiplexing and associated anti-aliasing than the continuous half-annular illumination, as will be discussed in details in Subsection 3.2.

The response of PTF determines the imaging performance of DPC, but the phase reconstruction result is also affected by imaging noise. In order to further compare the three illumination patterns under a more realistic condition, we simulated four captured images with same level Gaussian noise. The simulation parameters were chosen to realistically model an DPC platform, with an incident illumination wavelength of 525 nm, an image sensor with pixel size of 3.25 μm, and an 10× objective with NA_obj of 0.4. In general, the low frequency components near the origin and the high frequency components approaching 2NA_obj both have the PTF values which extremely close to zero, thus the noise of the frequency components corresponding to these regions may be excessively amplified during deconvolution, leading to phase reconstruction errors. So one will very likely to get phase reconstruction results as shown in Figs. 3(a1)–3(a3). Obviously, half-annular illumination is not sensitive to noise compare with half-circle illumination. At the same noise level, half-annular illumination can achieve better phase reconstruction results. By introducing a regularization parameter α can effectively stabilize the deconvolution. It sets the low-value region in the denominator of Eq. (10) to a larger value to avoid noise amplification. In order to study the effect of the regularization parameters α on the final phase reconstruction results, we performed DPC with a regularization parameter under different illumination patterns. As shown in Fig. 3(b1), although the noises in phase reconstruction result obtained under half-circular illumination are suppressed, the low frequency phase component cannot be recovered accurately. Finding a more strictly proper regularization parameter α is very important for half-circle DPC to achieve reliable phase reconstructions under different noise conditions. The phase reconstruction results of two half-annular are shown in Figs. 3(b2) and 3(b3). As can be seen, the regularization parameter is helpful to suppress the artifacts, and the phase information can be more reliably recovered. Compared to half-circular illumination, half-annular illumination shows better tolerance to the choice of regularization parameter. Besides, the absolute values of PTF under discrete half-annular illumination remain larger than 0.05 over all the frequency support region, there is almost no need for regularization. These simulation results suggest that the discrete half-annular illumination pattern provides the highest frequency response of PTF as well as the best robustness to noise among the three.

 

Fig. 3 Phase reconstruction results with noise under half-circle illumination, continuous half-annular illumination, and discrete half-annular illumination. (a1)–(a3) Phase reconstruction results with noise; (b1)–(b3) Phase reconstruction results with regularization parameters α = 0.05.

Download Full Size | PDF

3.2. Basic principle of AADPC

In order to improve the anti-aliasing ability of DPC and obtain a high-throughput phase result, we propose an AADPC method based on half-annulus discrete illumination and de-multiplexing iteration. Compared to traditional half-circle DPC, AADPC uses half-annulus discrete illumination patterns for image acquisition, resulting in better imaging performance in terms of resolution, robustness, and contrast. On the other hand, compared to continuous half-annulus illumination, it greatly reduces the number of LEDs, making different angles information easier to be de-multiplexed. Therefore, a high-resolution and wide-FOV phase reconstruction result can be obtained.

Figure 4 shows the image acquisition process of the AADPC method. Assuming that N LEDs are used for illumination (in Fig. 4, N = 6 is taken as an example), these LEDs are evenly distributed on the annulus corresponding to the NA_obj, as shown in Fig. 4(a). If we set the location of LED which is marked number 1 in Fig. 4 as the starting position, and then mark all remaining LEDs sequentially in a counter-clockwise direction. All LEDs can be marked as [1, 2, 3,· · · , N]. In the beginning, all LEDs are lit at the same time to collected an image I_B as a uniform intensity image [Fig. 4(b)]. This image only needs to be collected once for subsequent calculations. When collecting each image, $\frac{N}{2}$ LED are lit to illuminate the sample, which means that each image contains information at $\frac{N}{2}$ illumination angles. Each illumination pattern is postponed one LED position backwards so that there are a total of N illumination multiplexed patterns. Noted that these N multiplexed illumination patterns are symmetrical with $\frac{N}{2}$ different axes, we only need to collect one image in each axis direction, and another image could be get according to the uniform intensity image I_B. Therefore, only $\frac{N}{2}$ illumination patterns are needed in AADPC method. As shown in Figs. 4(c1)–(c3), when m-th image is acquired, the illumination LEDs with the serial number m(m = 1, 2, · · · , $\frac{N}{2}$) is taken as the starting point and the serial number [m, m + 1, m + 2, · · · , $m+\frac{N}{2}$] are simultaneously lit, then the captured image is recorded as I_m. Finally, a total of $\frac{N}{2}$ multiplexed images are collected, which are shown in Figs. 4(d1)–(d3).

 

Fig. 4 Image acquisition process of the AADPC method. (a) and (b) are uniform intensity illumination scheme and uniform intensity image I_B; (c1)–(c3) LED illumination patterns of AADPC method; (d1)–(d3) Captured images.

Download Full Size | PDF

In Fig. 5, we show the flowchart of the complete AADPC algorithm. The reconstruction process of AADPC algorithm is divided into two stages. In the first stage, an initial phase is obtained by the upsampled DPC based on the system’s PTF under discrete half-annular illumination. The red dotted box shows the phase initialization stage, which recovers the quantitative phase value ϕ₀ of the sample based on the deconvolution described in Section 2.1. Then ϕ₀ is served as the initialization phase and the uniform intensity image I_B to get the initialization HR spectrum O(u, v). The second stage is the de-multiplexing iteration, which is inspired by the multiplexing and coherent-state decomposition schemes in FPM [16,31,32]. It uses the alternate projection algorithm between spatial domain and frequency domain so that the different frequency components corresponding to each image can be stitched together and create a HR spectrum. The de-multiplexing iteration process is displayed in the green dotted box. When iterating the j-th image, the first step is to extract sub-apertures spectrum o_j(u, v), o_j+1(u, v), · · · , o_j+k(u, v) (where $k=\frac{N}{2}$) of the j-th image $I_j^m$ corresponding to multiple angle illuminations from the HR spectrum, and then perform inverse Fourier transform for these sub-apertures to obtain multiple target images $\sqrt{I_{j,1}^t}{e}^{i{\varphi }_{j,1}},\sqrt{I_{j,2}^t}{e}^{i{\varphi }_{j,2}},\cdots ,\sqrt{I_{j,k}^t}{e}^{i{\varphi }_{j,k}}$. Second, add the intensity of these target images to obtain the sum matrix $I_j^t$. Third, calculate coefficient matrix C_j based on formula $C_j=\frac{I_j^m}{I_j^t}$, and get updated images $\sqrt{C_j{I}_{j,1}^t}{e}^{i{\varphi }_{j,1}},\sqrt{C_j{I}_{j,2}^t}{e}^{i{\varphi }_{j,2}},\cdots ,\sqrt{C_j{I}_{j,k}^t}{e}^{i{\varphi }_{j,k}}$. Fourth, the updated image of each sub-aperture is sequentially used to modify the corresponding spectral region in the HR spectrum. Since we assume a pure phase object, an extra uniform intensity constraint for the full spectrum at the end of each iteration could be introduced, and after that the frequency content of rest opposite aperture regions can be updated automatically. Therefore, intensity constraint is imposed on the full spectrum as the following fifth step so that the other half of the sub-aperture spectrum is filled. It should be noted that the uniform intensity image used to constrain can be obtained from the average intensity of the I_B. Lastly, the entire process is repeated for all intensity measurements, and iterated for several times until the solution converges. In the end, we will get an extended HR spectrum. By inverse transforming the HR spectrum, a HR phase reconstruction result with resolution corresponding to 2NA_obj can be obtained.

 

Fig. 5 Algorithm flowchart of AADPC.

Download Full Size | PDF

3.3. Data redundancy analysis and optimal illumination strategy

In AADPC, there must be sufficient data redundancy to guarantee accurate spectrum recovery in de-multiplexing process. The reconstruction quality can be strongly influenced by the spectrum overlapping percentage R_overlap of the adjacent angled illuminations [31,32]. This is because a certain amount of R_overlap is required to ensure the iterative reconstruction algorithm converge to the correct solution. As reported in [32], when there is no pixel aliasing in iteration process, only R_overlap > 31.8% is enough to guarantee a reasonable and convergent result. When the system exists pixel aliasing, a larger R_overlap is needed to obtain the desired image quality. Similar to these cases under the scenario of FPM, the data redundancy of AADPC should be explored.

In order to analyze the requirement of R_overlap in AADPC, we simulated and compared the phase imaging performance under different R_overlap. In AADPC, R_overlap is directly determined by the number of LED N, so the phase reconstruction results under different N values are analyzed to explore the requirement of R_overlap. Here, N was chosen to be an even number between 4 and 12 (4, 6, 8, 10, 12), within which the number of captured images will not be excessively increased. In Figs. 6(a1)–(a5), we show the LED illumination patterns with different values of N, where the position of the white LED is labeled as number 1. The AADPC phase reconstruction results with different values of N and the corresponding illumination patterns are shown in Figs. 6(b1)–(b5). It can be seen that when N equals to 4, severe distortions can be obversed in the reconstruction result, and the high frequency features cannot be recovered. This is because R_overlap is only 18%, which is insufficient for de-multiplexing and de-aliasing. Using a larger N can increase R_overlap proportionally, thus more features in the resolution target become recognizable. In order to explore the appropriate range of N, the root-mean-square error (RMSE) values, which are calculated by using Fig. 1(c) as a reference, are shown in Fig. 6. It can be found that RMSE values tend to a stable value as N increases. Moreover, when N is greater than 8, the decrease of RMSE is insignificant and the RMSE values all stay below 0.02, which is considered to be independent of the algorithm error and is only the error generated by the calculation process. Therefore, only if N is no less than 8, a reasonable phase reconstruction result can be obtained. We can conclude that when pixel aliasing exists in DPC, N needs to be selected as no less than 8 in order to guarantee R_overlap ≥ 52%.

 

Fig. 6 Phase reconstruction results under different number of LEDs (N). (a1)–(a5) The LED illumination patterns under different N; (b1)–(b5) Phase reconstruction results with different N; (c1)–(c5) Partial enlarged images of (b1)–(b5); (d1)–(d5) HR spectrum under different N.

Download Full Size | PDF

Besides the R_overlap discussed above, the reconstruction quality of AADPC can also be influenced by the number of multiplexed illumination patterns. In the iterative reconstruction process of AADPC, the number of multiplexed illumination patterns determines the degree of de-multiplexing, which also affects the imaging resolution. In order to get an appropriate number of multiplexed illumination patterns, we compared the phase reconstructions under different number of captured images. In Fig. 7, we show the phase reconstruction results obtained by gradually reducing the number of captured images under different N values. It can be found that with the number of captured image decreases, the resolution and the quality of the phase reconstruction degrades accordingly. We use the same color to outline the phase reconstruction results obtained with the same number of captured images. It is also shown that the error of phase reconstruction gradually increases with an increasing in N. In order to quantitatively assess the phase reconstruction quality, the RMSE curves with different number of captured images are plot in Figs. 7(f). Obviously, reducing of the number of captured images results in increase of the RMSE curves, indicating a deterioration of the phase reconstruction result. This result is expectable because the reduction in data redundancy increases the difficulty of illumination de-multiplexing, leading to larger phase reconstruction errors. In Fig. 7(g), we further plot RMSE curves for the phase reconstruction results of same number of captured images versus the values of N. As can be seen, with the increase of N, the phase RMSEs obtained with same number of images slightly reduce. In order to better understand this phenomenon, we simulated the multiplexing method of 4 images with different values of N, as shown in Figs. 7(h1)–(h3). It is noticeable that, as the value of N increases, the quality of the reconstruction result gradually decreases. So we can get the conclusion that the AADPC method can obtain the optimal results by capturing $\frac{N}{2}$ images for total N LEDs.

 

Fig. 7 Phase reconstruction results under different number of captured images. (a)–(e) Phase reconstruction results obtained by reducing different number of captured images under different N values; (f) and (g) are RMSE curves; (h1)–(h3) are phase reconstruction results with 4 images under different N.

Download Full Size | PDF

Based on the above analysis, we know that AADPC requires the R_overlap ≥ 52% to guarantee sufficient data redundancy. In this case, the number of illumination LEDs N should be no less than 8. Meanwhile, in order to achieve illumation de-multiplexing and associated anti-aliasing, at least 4 images need to be acquired for phase recovery. Based on the above considerations, the optimal illumination strategy for AADPC is proposed: we use 8 LEDs to create 4 half-annulus discrete illumination patterns, and capture 4 corresponding images for phase recovery. This optimal strategy not only overcomes pixel-aliasing and improves the phase resolution, but also has the least number of captured images which does not increase the number of captured images as in traditional DPC. It should be noted that if a severer pixel aliasing effect occurs, a higher spectrum overlapping percentage and more captured images should be required for super-resolution reconstruction.

4. Experimental results

4.1. Phase resolution target

To quantitatively evaluate the resolution enhancement of AADPC, we firstly measured a real pure phase resolution target [Quantitative Phase Microscopy Target (QPT^TM), Benchmark Technologies Corporation, USA]. Our setup was built based on a commercial inverted microscope (IX73, Olympus), in which the original built-in illumination source is replaced by a programmable LED array (1.67 mm) and a condenser (a cemented doublet with the focal length of 50 mm). The LED array was placed at the front focal plane of this condenser. In our experiments, only the green LED elements (center wavelength of 525 nm) are used to illuminate the sample in asymmetrical manners. The corresponding images are captured by an objective lens with a magnification of 10× and a NA of 0.4 (Olympus, UPLANSAPO 10×, 0.4NA) and finally digitalized by a SCMOS camera (PCO.edge 5.5, pixel size of 6.5 μm, 100 fps). Under such system parameters, the spatial cutoff frequency f_obj of the objective lens is 1.5238 μm⁻¹ and the spatial cutoff frequency f_cam of the camera is 0.7692 μm⁻¹. It can be found that f_cam is less than f_obj, which does not satisfy the Nyquist–Shannon sampling criterion, i.e., pixel aliasing exists in captured images. In experiments of this paper, we use the optimal scheme proposed in section 3.3 for phase reconstruction, and all images are iterated 5 times.

The quantitative phase images obtained by different methods are shown in Fig. 8. Figure 8(a) shows the observed image under the incoherent bright field illumination. Because the the resolution target is a pure phase object, the image only shows very little contrast. We chose the region in Fig. 8(b) as the region of interest to solve the phase using up-sampled half-circular DPC and the proposed AADPC. Figure 8(c) shows the quantitative phase image obtained by the up-sampled half-circular DPC. The smallest resolvable unit is Group 9, Element 5, which corresponds to a half-pitch resolution about 615 nm. This verified that for the up-sampled DPC, the imaging resolution is fundamentally limited by the imaging pixel size (650 nm), instead of the diffraction limit of the imaging optics. However, when the proposed AADPC was applied, the resolution of the phase reconstruction is obviously improved, and Group 10, Element 4 can be clearly resolved, as shown in Fig. 8(e). To further distinguish the highest achievable resolution, line profiles of resolution target Group 10, Element 1–4 along x-axes and y-axes (corresponding to half-pitch resolution of 488 nm, 435 nm, 388 nm, 345 nm respectively) are extracted and shown in Figs. 8(d) and 8(f). It can be seen that AADPC can achieve a half-pitch resolution of 345 nm, which corresponds to 1.88× of the theoretical Nyquist−Shannon sampling resolution limit imposed by the sensor pixel size (650 nm).

 

Fig. 8 Phase reconstruction results on Quantitative Phase Microscopy Target (QPT^TM). (a) and (b) are observed results under the bright field; (c),(d) Phase reconstruction results with half-circular DPC; (e),(f) Phase reconstruction results with AADPC.

Download Full Size | PDF

4.2. Unstained HeLa cells

At last, we implemented AADPC to image unstained HeLa cells, which can be regarded as pure phase objects approximately. HeLa cells were seeded in a 35 mm glass-bottom Petri dish and placed in the 37°C incubator of the microscope with 5% CO₂ for long-term QPI. In order to observe the quantitative phase changes of cell division process, we captured images within two hours, and used these images for AADPC phase recovery. The experimental setup generally followed the parameters of the previous experiment, except that a larger spacing LED array with a pitch of 2 mm is used. The phase reconstruction result of one frame in the resulting large-SBP phase reconstruction video is shown in the Fig. 9(a) (the corresponding video are presented in Visualization 1). Two selected zoom-ins regions are shown in Figs. 9(b) and 9(c), respectively. In the two zoom-ins of the phase images, subcellular features, such as cytoplasmic vesicles and pseudopodium, are clearly observed. The line profile on one cell indicated by the yellow arrow in Fig. 9(b) demonstrates a valley between two closely spaced features with center-to-center distance of 665 nm, indicating the phase imaging resolution is better than this value. By using only 4 half-annular illuminations per reconstruction, AADPC achieved a full-pitch resolution of 665 nm over a wide FOV of 1.77 mm², with a imaging speed of 25 fps, corresponding to a high-throughput with SBP of 399 megapixels per second. In addition, we show the phase reconstruction results on different time scales in Figs. 9(d1)–9(d5), which show different stages of cell division. During the cell division process, the cocytes divide into two nucleus, and the nucleolus and nuclear membrane gradually appear to form two daughter cells. These results demonstrate that AADPC successfully relaxes the pixel resolution requirement in DPC, allowing for wide-field aliasing-free dynamic QPI over an extended period of time.

 

Fig. 9 Phase reconstruction results on HeLa cells (Visualization 1). (a) Large-SBP Phase reconstruction result; (b),(c) Phase of two selected zoom-ins regions; (d1)–(d5) Phase reconstruction results on different time scales.

Download Full Size | PDF

5. Conclusions and discussions

In this paper, we have proposed a new variation of quantitative DPC approach, termed AADPC to solve the tradeoff between imaging resolution and FOV in conventional DPC, to achieve high-throughput QPI. Different from traditional DPC method that uses half-circle illumination, AADPC uses discrete annular illumination patterns for image acquisition, which not only provides a theoretical basis for the coherent de-multiplexing iteration, but also strongly boosts the response of the PTF for both low and high spatial frequencies. After recovering an initial phase estimate based on deconvolution, alternating projection algorithm is followed to overcome pixel-aliasing and improve the imaging resolution. In addition, we have analyzed the data redundancy of AADPC method, which depends on the number of illumination LEDs and the number of multiplexed illumination patterns. It has been found that in AADPC, R_overlap ≥ 52% is required to satisfy the requirement of spectrum overlap percentage. Meanwhile, for total N LEDs, $\frac{N}{2}$ images must be acquired to achieve a reasonable reconstruction quality. Based on these analytical results, we presented an optimal illumination pattern and de-multiplexing iteration strategies, which uses 8 LEDs for illumination and acquire 4 images to get desired phase reconstruction quality. The capability of resolution enhancement of AADPC has been quantitatively verified by measuring a phase resolution target, verifying that AADPC can achieve a half-pitch imaging resolution of 345 nm, corresponding to 1.88× of the theoretical Nyquist–Shannon sampling resolution limit imposed by the sensor pixel size. AADPC has also been applied to high-speed, high-throughput QPI of live HeLa cells mitosis in vitro, achieving a full-pitch lateral resolution of 665 nm across a wide FOV of 1.77mm² at 25 fps.

There are several aspects that need to be further improved in the proposed method, which we will leave for future consideration. First, in this work, we only recover the phase distribution for pure phase objects. However, it should be noted that the de-multiplexing iterative process of AADPC also can be used to retrieve complex amplitude distributions of objects to obtain high-resolution intensity and phase simultaneously. But for such cases, the results about the data redundancy analysis in our paper are no longer applicable. This is because when the intensity and phase of the sample are simultaneously recovered, the amount of data required will be almost doubled. Second, in Section 3.1, we quantitatively assess the performance of different illumination patterns based on the responses of their PTFs. The PTFs are calculated by Eq. (9), which is normalized by the total illumination intensity (intensity of the source integrated over the pupil), thus demonstrating the ”contrast” of the image (relative strengths of information-bearing portion of the image and the ever-present background). But in some cases (e.g., the sample is poorly illuminated by a low brightness LED array), it is better to consider the unnormalized PTF [39], which can give the absolute value of the image signal and determine its strength relative to the noise level (the discrete annular illumination is obviously no longer the optimal choice). How to handle these cases is an another interesting direction for further investigation.