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Abstract
Quantitative phase imaging (QPI), such as digital holography, is considered a promising tool in the field of life science due to its noninvasive and quantitative visualization capabilities without the need for fluorescence labeling. However, the popularity of QPI systems is limited due to the cost and complexity of their hardware. In contrast, Zernike phase-contrast microscopy (ZPM) has been widely used in practical scenarios but has not been categorized as QPI, owing to halo and shade-off artifacts and the weak phase condition. Here, we present a single-image phase retrieval method for ZPM that addresses these issues without requiring hardware modifications. By employing a rigorous physical model of ZPM and a gradient descent algorithm for its inversion, we achieve single-shot QPI with an off-the-shelf ZPM system. Our approach is validated in simulations and experiments, demonstrating QPI of a polymer microbead and biological cells. The quantitative nature of our method for single-cell imaging is confirmed through comparisons with observations from an established QPI technique conducted through digital holography. This study paves the way for transforming non-QPI ZPM systems into QPI systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phase-contrast microscopy, invented by Frits Zernike, has been widely used in biomedical fields for decades due to its simple implementation, which consists of a standard bright-field microscope with a condenser annulus and a phase ring attached [1]. Although it provides high-contrast images of unstained transparent specimens, such as cells and tissues, Zernike phase-contrast microscopy (ZPM) inevitably suffers from the halo and shade-off artifacts caused by the finite radial width of the phase ring, leading to its intrinsically non-quantitative nature [2]. Furthermore, although it is recognized that ZPM is nearly quantitative in the weak phase range for a condenser annulus and a phase ring with infinitely narrow radial widths [3], most biological specimens fall outside of this range because a single cell already induces a phase delay of a few rad, as shown in Fig. 1.
Fig. 1. An unambiguous phase range of $\pi$ and a weak phase range around 0 rad in ZPM. The solid sinusoidal curve represents a contrast map for negative phase-contrast. The blue bins show a histogram of a typical single-cell phase image.
On the other hand, quantitative phase imaging (QPI) is anticipated to play a crucial role in quantitative life science, such as applications for cellular dry mass or growth rate analyses [4,5]. In general, however, the optical hardware in QPI systems tends to be more complex compared with ZPM, resulting in less popularity in the fields of biomedical and life sciences to date. For example, digital holography (DH) is recognized as the gold standard technique for QPI, reconstructing an object-induced phase map from an interference pattern between the object and reference light [6,7]. However, its interferometric optical setup with a highly coherent light source poses challenges for practical applications.
To exploit applicability of ZPM for quantitative biomedical analysis by addressing its aforementioned issues, a numerical approach for suppressing the artifacts of ZPM has been demonstrated [8]. However, this method only considers the weak phase range, and the capability of quantitative phase retrieval for larger phases remains unexplored. Alternative approaches have involved hardware modifications to the ZPM setup, enabling quantitative phase retrieval beyond the weak phase range [9,10]. For instance, spatial light interference microscopy (SLIM) is one of the representative techniques that realizes QPI [11–13]. However, this technique requires an additional optical unit containing a spatial light modulator and relies on multiple phase-shifted measurements for the purpose of phase retrieval, which compromises the simplicity and robustness of ZPM.
In this study, we present a numerical method for quantitative phase retrieval from a single ZPM image without requiring hardware modification. We revisit a physical model of ZPM and develop an efficient phase retrieval algorithm that robustly works for single-adherent-cell images. As our technique does not necessitate any hardware modifications, it can be applied to any commercially available ZPM systems, thereby enabling effortless QPI. This method may also be useful for other phase-contrast imaging modalities, such as X-ray or electron beam imaging [14,15].
2. Method
In order to determine the available range for phase retrieval without ambiguity, it is essential to reconsider the physical model of ZPM. The intensity $i^{\pm }$ of images acquired with positive and negative-contrast ZPMs, comprising a condenser annulus and a phase ring with infinitely narrow radial widths, can be written as a function of the object-induced phase delay $\theta$, given by
Although we showed the potential for overcoming the weak phase range in ZPM without hardware modification, the issue of the halo and shade-off artifacts still remains because the condenser annulus and the phase ring are not infinitely narrow in practice. To solve this issue, a numerical propagation model that rigorously describes the spatially partially-coherent light from the condenser annulus is needed, which would impose a tremendous computational cost. We address this problem using a stochastic gradient descent approach called compressive propagation (CP) proposed in our recent work [16]. In CP, spatially incoherent light is described as a set of random wavefronts. The computational cost of incoherent light propagation in inverse problems can be reduced through the use of the CP framework. This framework incorporates the concept of stochastic gradient descent, which has been widely utilized in deep learning for effectively reducing computational costs [17].
The forward model of ZPM based on CP is illustrated in the upper part of Fig. 2. In this model, a number of two-dimensional random wavefronts $w_m$ passing through the condenser annulus $c$ is inverse Fourier transformed by the condenser lens. Here $m\in \{1,2,\ldots,M\}$ is the index of the random wavefronts. Then, the illumination field $l_m$ just before the object is written as
where $\mathcal {F}^{-1}[\bullet ]$ denotes the inverse Fourier transform. We assume a two-dimensional phase object $f$ as follows: where $\theta$ is the two dimensional phase delay induced by the object. The illumination field $l_m$ is modulated by the object field $f$. Then, the resultant field is filtered by the phase ring $p$ on the pupil plane. This process is written as where $g$ is the complex amplitude field on the sensor plane and $\mathcal {F}[\bullet ]$ denotes the Fourier transform. The captured intensity image of ZPM is described asTo perform gradient descent for phase retrieval in ZPM as shown in Fig. 2, we define the error function $e$ as
where the accent mark of $\widehat {\bullet }$ denotes estimated variables during the gradient descent process and $\|\bullet \|_2$ is the $\ell _2$ norm, respectively. The partial derivative of the error $e$ in Eq. (7) with respect to the estimated phase vector $\widehat {\theta }$ based on the chain rule is written asAdditionally, we introduced a regularization process based on the alternating direction method of multipliers (ADMM), as shown in Fig. 2 [19]. In this case, the updating step in Eq. (11) is rewritten with auxiliary two dimensional phase distributions $v^{(k)}, u^{(k)}$ at the $k$-th iteration as
3. Numerical demonstration
We numerically demonstrated the proposed method as shown in Fig. 3. In this simulation, the pixel count of the phase object $\theta$, the condenser annulus $c$, and the phase ring $p$ was set to $128\times 128$. The inner and outer radii of both the condenser annulus and the phase ring were 15 and 20, respectively. The negative (bright) phase-contrast was supposed and the transmittance of the phase ring $\alpha$ was set to 0.5. In this case, the unambiguous phase range for the phase retrieval shown in Fig. 1 was from −0.46 rad to 2.68 rad. The Shepp–Logan phantoms, having phase ranges of 1.5 rad, 2.5 rad, and 3.5 rad, were used as the phase objects. The ZPM images $i$ in the simulations were calculated using the forward model described in Eqs. (3)–(6), where the number of random wavefronts $M$ was set to 1000. The learning rate of the Adam optimizer was set to 0.008, and the tuning parameter $\rho$ in the ADMM was set to 20.
Fig. 3. Simulations at different noise levels. (a) Reconstruction RMSEs at different measurement SNRs for phase objects with three phase ranges. (b) Visual representations of phase retrievals at measurement SNRs of 10 dB and 30 dB for the phase object with a phase range of 2.5 rad. The maximum value on the colorbar was 2.5 rad for both the phase image and the PR-ZPM images.
The plot in Fig. 3(a) illustrates the relationship between the measurement signal-to-noise ratio (SNR) and the reconstruction error. We apply additive white Gaussian noise at different SNRs to the ZPM images. The phase retrieval results from these noise-applied ZPM images were evaluated using the root-mean-square error (RMSE) between the original phase objects and the phase retrieved ZPM (PR-ZPM) images. In the plot, the centers and heights of the error bars represent the averages and standard deviations of the RMSEs from ten trials, each with a different random seed.
The plot shows that the acceptable noise level was about 30 dB when the phase range of the object was smaller than 2.68 rad, which is the upper bound of the unambiguous phase range. If the phase range was larger than 2.68 rad, the phase retrieval algorithm failed even at high measurement SNRs, due to the ambiguity in phase retrieval as shown in Fig. 1. The phase object with a phase range of 2.5 rad, the noise-applied ZPM images with SNRs of 10 dB and 30 dB, and their PR-ZPM images are shown in Fig. 3(b). These results show the visual impacts of measurement noise on the reconstructions.
4. Experimental demonstration
We experimentally demonstrated our phase retrieval method with an off-the-shelf ZPM system. This system consists of a commercial microscope (Olympus IX73) equipped with a 525-nm LED (Thorlabs SOLIS-525C), a condenser annulus (Ph2), a negative (bright) phase-contrast objective (UPlanFLN 40x/0.75NHPh2) and a CMOS image sensor (Basler acA2440-75um). Our choice of negative (bright) phase-contrast guarantees full coverage of the range of interest (0–a few rad) with the unambiguous phase range, as shown in Fig. 1. In this experimental ZPM, the half-pitch spatial resolution is 260 nm, and its field of view is 176 µm $\times$ 211 µm. These imaging parameters, in principle, remain unchanged after the phase retrieval process.
To build a numerical physical model that accurately describes our implemented ZPM system, it is essential to carefully set the model parameters, such as the radii and the radial widths of the condenser annulus and phase ring, as well a the transmittance of the latter. In our case, we experimentally measured these parameters using the following procedure (see [11] for more details). We imaged the back aperture of the objective onto the camera plane both with and without condenser annulus. In the former case, the size of the condenser annulus was determined, while in the latter case, the radius and transmittance of the phase ring $\alpha = 0.26$ could be characterized from the attenuated area of the phase ring. The inner and outer radii of the condenser annulus and the phase ring were calculated as ratios to the numerical aperture of the objective lens, specifically as 0.31 and 0.40, and 0.33 and 0.38, respectively. In this experimental condition, the unambiguous phase range for the phase retrieval depicted in Fig. 1 was calculated to be from −0.25 rad to 2.89 rad. Objects’ silhouettes were segmented from ZPM images by detecting and dilating edges [22] and were utilized as the initial guesses for the phase retrieval. In the Adam optimizer, we set the learning rate to 0.016.
4.1 Imaging of a microbead
First, we validated the quantitative performance of our method using a microbead made of poly methyl methacrylate (PMMA) with an average diameter of 6.33 $\mu$m (MMA5000, Phosphorex). The bead was immersed in refractive-index matching oil (refractive index: 1.48, SHIMAZU). An image of a bead captured by the ZPM, along with a PR-ZPM image, are displayed in the top left and middle of Fig. 4, respectively. In this experiment, the tuning parameter $\rho$ in the ADMM was set to 50. The cross-section profiles of the bead in both the ZPM and PR-ZPM images are shown at the bottom of Fig. 4. These results demonstrate that the halo and shade-off artifacts on and around the bead were successfully eliminated by the phase retrieval algorithm.
Fig. 4. Phase retrieval of a bead image captured with ZPM. The top panels display images of ZPM, PR-ZPM, and DH from left to right. The bottom panels present the cross-section profiles of the bead images at the dotted lines.
To compare the PR-ZPM image with a reference quantitative phase image, we measured the bead with a homemade DH system based on the common-path broadband diffraction phase microscopy technique [23], equipped with a 532-nm laser for illumination, an objective (LUCPLFLN40X, 40x/0.6), and the same image sensor as employed in the ZPM system. The captured image and the cross-section profile are displayed in the right of Fig. 4. The half-pitch spatial resolution of DH images is 440 nm. Details on our DH system are described in our recent work [24]. The phase delay of the bead measured by DH is within the unambiguous phase range of 0–$\pi$ rad, ensuring single-shot phase retrieval without ambiguity. The RMSE between the phase delays on the bead of the PR-ZPM and DH images is 0.02 rad.
4.2 Imaging of biological cells
Next, we conducted imaging of single cells using our quantitative ZPM method. The first and second left columns in Fig. 5 represent ZPM images of fixed COS-7 cells and their PR-ZPM images, where the former are captured with our ZPM system. As depicted in the figure, the phase retrieval results exhibit minimal residual halo and shade-off artifacts. For this experiment, the tuning parameter $\rho$ in the ADMM was set to 20. The model parameters remained consistent with those in the experiment measuring a microbead. Note that model and algorithmic parameters were fixed for the phase retrieval of all the cell images presented in Fig. 5.
Fig. 5. Phase retrieval of COS-7 cell images captured with ZPM. The first and second left columns show ZPM and their phase-retrieved (PR-ZPM) images, respectively. The third and fourth columns show reference quantitative phase images independently captured with DH and the absolute difference between PR-ZPM and DH images, respectively. Phase values of DH images are determined relative to their background images.
We compared the PR-ZPM images with quantitative phase images independently captured with the homemade DH system, which was also used for the microbead measurement [24]. As depicted in Fig. 5, the phase values of all DH images lie between 0 and approximately 2 rad, which largely surpasses the weak phase range but falls within the unambiguous phase range of $\pi$ for single-shot phase retrieval. The rightmost column in Fig. 5 shows the absolute difference between PR-ZPM and DH images. The RMSEs between the PR-ZPM and DH images within the intracellular regions, which are determined by the cells’ silhouettes used for the initial guesses in the phase retrieval, are 0.18 rad, 0.25 rad, 0.25 rad, and 0.16 rad, respectively, from top to bottom.
4.3 Discussion
The experiments involving QPI of a microbead and single cells demonstrated the promising performance of our phase retrieval algorithm for ZPM. To assess the effectiveness of regularization in the algorithm, we compared the phase-retrieved results obtained using different regularization methods. Figure 6 shows results with the TV and the $\ell _1$ norm, with the TV only, with the $\ell _1$ norm only, and without the TV or the $\ell _1$ norm, respectively. As shown in these results, the TV contributed to reconstruct morphological structures of the cells, and the $\ell _1$ norm suppressed background noise. The TV and the $\ell _1$ norm cooperatively functioned in the phase retrieval process. The RMSEs within the intracellular regions compared to the DH image in Fig. 6 are 0.18 rad, 0.47 rad, 0.43 rad, and 1.11 rad, respectively.
Fig. 6. Comparison of PR-ZPM images with different conditions in the regularization. The first left image is the result with both the TV and the $\ell _1$ norm, which is shown in the first row of Fig. 5. The second and third images are the results with the TV only and with the $\ell _1$ norm only, respectively. The fourth image is the result without the TV or the $\ell _1$ norm.
We also discuss the possible cause of the residual phase discrepancies between PR-ZPM and DH images shown in Fig. 5. A mismatch in spatial resolution between the two systems may partially contribute to the residual, even though we adjusted the pixel pitch of the PR-ZPM images to match that of the DH images. Another factor could be the spatial phase noise of the DH measurements, primarily arising from the coherent noise due to laser illumination. We evaluated the standard deviations of the spatial phase noise in an area where no sample exists in the DH images and found values ranging from 0.02 to 0.04 rad, which are lower than the observed residuals. Therefore, the DH images serve as reliable references for evaluating the errors in the PR-ZPM images. We attribute the dominant cause of the residual to model error, which could be reduced through more precise determination of the system parameters.
In the single-cell QPI experiment, the computational time for the phase retrieval process in the ZPM system was approximately 250 seconds. This computational process was conducted on an NVIDIA A100 SXM GPU and an AMD EPYC 7763 64-Core CPU.
5. Conclusion
In summary, we have developed a single-image phase retrieval algorithm that allows quantitative ZPM in an unambiguous phase range of $\pi$ rad. The proposed algorithm was validated through both numerical simulations and experimental demonstrations. In the experiments, QPI of a microbead and single cells was conducted using a standard off-the-shelf ZPM system, where the unambiguous phase range for the phase retrieval was from −0.25 rad to 2.89 rad. The microbead, with a phase delay of approximately 0.5 rad, exhibited a reconstruction RMSE of 0.02 rad. For single cells, the maximal phase values were around 2 rad, with reconstruction RMSEs ranging from 0.18 rad to 0.25 rad. These experimental results demonstrate the practical effectiveness of our algorithm.
To achieve more accurate phase retrieval, reducing model errors is crucial. For instance, further investigation into accurately measuring physical parameters, such as those of the condenser annulus and the phase ring, is important. Another approach to enhance reconstruction performance in the optimization process involves scheduling the tuning parameters, such as cosine annealing and decoupled weight decay regularization [25,26]. Our method is versatile and holds promise for various biomedical imaging applications, including profilometry of red blood cells and real-time morphological imaging of live cells, such as cardiomyocytes [27–29]. Since our algorithm can be implemented to any existing off-the-shelf ZPM systems without hardware modification, it could open the door to the widespread use of QPI.
Funding
Japan Society for the Promotion of Science (JP20H00125, JP20H02657, JP20H05890, JP20K05361, JP23H00273, JP23H01874); Asahi Glass Foundation; Research Foundation for Opto-Science and Technology; Nakatani Foundation for Advancement of Measuring Technologies in Biomedical Engineering; UTEC-UTokyo FSI Research Grant; UTokyo IXT Project Support Program.
Disclosures
The authors have filed a patent for the work in this paper.
Data availability
Data may be obtained from the authors upon reasonable request.
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