Quantitative phase imaging (QPI), which provides optical thickness information of transparent objects, has found its utility in numerous applications [1,2]. The most common QPI methods are based on interferometry with coherent illumination, which makes them expensive and sensitive to misalignments, vibrations, and speckle noise [3,4]. To overcome these limitations, noninterferometric or reference-free QPI methods using partially coherent illuminations have been developed, including the transport-of-intensity equation [1], Fourier ptychographic microscopy [5], and differential phase-contrast (DPC) microscopy [6]. Among these methods, DPC microscopy achieves quantitative phase recovery by utilizing a number of intensity images under asymmetric illuminations and reconstructs the complex information of objects. However, DPC operates on a linear imaging model that is applicable only to weakly scattering objects with phase delays <≈0.5 rad [7,8]. Furthermore, system aberrations can degrade the DPC reconstruction quality. Recently, a computational method for aberration-corrected DPC imaging has been demonstrated [8]; however, it is based on a linear model suitable for imaging thin phase objects and requires a complicated iterative algorithm.

Deep neural networks (DNNs) have emerged as powerful tools for phase retrieval, offering high-accuracy phase reconstruction in various platforms [9]. By leveraging large dataset and end-to-end training, DNNs can solve nonlinear inverse problems and have demonstrated a great promise for data-driven frameworks in quantitative phase microscopy [10–12]. However, the performance of these networks relies heavily on the consistency between the training and experimental settings because they are sensitive to changes in object features, instrumentation, and acquisition parameters. In contrast, an untrained neural network (UNN), inspired by the deep image prior (DIP) [13], has the potential to address these issues by integrating a physical model with DNNs. The physics-informed UNN has been proposed to solve various inverse problems (e.g., phase retrieval [14,15] and Fourier ptychography [16,17]). Motivated by these DIP and related studies, we propose a UNN-based DPC microscope, termed a UNN-DPC microscope, which enables pupil and complex object reconstruction by integrating a nonlinear image formation model without ground-truth.

Figure 1 outlines our UNN-DPC framework. We employ a U-Net-based algorithm to reconstruct complex object information by incorporating multiscale and skip connections with minimal training parameters. The output information from the network is then numerically imaged using a nonlinear forward-imaging model that incorporates pupil aberrations. The aberration is generated by a fully connected network that performs the weighted sum of Zernike functions. We evaluate and minimize the difference between the measured and estimated intensity images to obtain a joint estimate of the complex object and pupil information. The optimization problem is set as

 

Fig. 1. UNN-DPC imaging principle. Intensity measurements acquired with three different illumination patterns are fed into the UNN to generate complex object information. The physical model combined with system aberration then simulates the image formation process and generates estimated intensity measurement. The optimization problem is solved iteratively using a gradient-based procedure to minimize the difference between the measured $(I )$ and estimated $({\hat{I}} )$ intensity images.

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We first performed numerical simulations to validate the feasibility of the UNN-DPC microscopy. An LED-based microscope with system parameters the same as our experimental setup (10×/0.25 NA, $\lambda = 527$ nm) was considered, and an object composed of 3 × 3 circular step targets with various absorption and phase values was imaged (Fig. 2). We imposed pupil aberration as a weighted sum of the Zernike polynomials, with weights randomly assigned in the range of −1 to 1. The imposed pupil aberration and corresponding Zernike weights are shown in Fig. 2(a). Three DPC intensity measurements were simulated using the discrete illumination patterns in Fig. 1 and then fed into the UNN. The reconstructed absorption ($\mu $) and phase ($\phi $) images, along with the pupil aberration, are shown in Fig. 2(b). We used the learning rate and number of iterations determined in our ablation study, which is discussed next. Note the significant correspondences of the reconstructed absorption, phase, and pupil aberration compared with the ground-truth information. For the 3 × 3 circular step targets, mean absolute errors (MAEs) for the absorption, phase, and pupil aberration were determined to be 9.89 × 10⁻⁴ rad, 1.02 × 10⁻⁴ rad, and 3.70 × 10⁻³ rad, respectively. The estimated Zernike coefficients yielded an average error of 1.48%, with a maximum pupil error of 0.16 rad. Specifically, for a target of $\mu = 0.32$ and $\phi = 1.96$ rad, the UNN-DPC enabled accurate estimation of absorption and phase, with errors of 0.9% and 3.5%, respectively. We also performed UNN-DPC imaging of a lenslet array and biological tissue, which represented slowly varying (Fig. S3 in Supplement 1) and complicated phase samples (Fig. S4 in Supplement 1), respectively. Our UNN-DPC could also reconstruct object and pupil aberrations of the samples with high accuracy.

 

Fig. 2. UNN-DPC-based joint estimation of complex object information and aberration for 3 × 3 circular step target. (a), (b) Comparison of absorption and phase of phantom, pupil aberration, and weights of Zernike polynomials of (a) ground-truth and (b) UNN-DPC results. (c) Loss curves of UNN-DPC model during joint estimation process with RMSprop optimizer. lr, learning rate.

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Figure 2(c) shows the loss curves as a function of the iteration numbers at various learning rates. When using a learning rate of 1 × 10⁻², the model exhibited rapid initial convergence but suffered from instability and oscillation during the iteration. Conversely, the model with a learning rate of 1 × 10⁻⁴ resulted in slow convergence, requiring a significantly larger number of iterations to achieve comparable performance. Based on this observation, we used a learning rate of 1 × 10⁻³ to achieve the balance between convergence speed and accuracy. After 3000 iterations, the optimization was stopped to mitigate the risk of potential overfitting, as suggested by Bostan et al. [14]. This took ≈5 min on a computer [Intel Xeon Gold 6226R CPU, NVIDIA RTX A6000 graphics processing unit (GPU)].

We then experimentally validated UNN-DPC microscopy using an LED-based microscope (10×/0.25 NA objective, f = 400 mm tube lens). We employed a custom-built LED array with high-power LEDs (Shenzhen LED color, APA102-202 Super LED) as the light source, and images were acquired using a scientific CMOS (sCMOS) camera (PCO.edge, 4.2 MP format, 6.5 µm pixel size). Conventional DPC phase reconstruction was also performed using four half circle illuminations (i.e., top, bottom, left, and right half-circles) and a Tikhonov deconvolution derived from the linearized imaging model. As noted elsewhere [7,8,20,21], this linear approximation is valid for most transparent objects. However, if the object under investigation exhibits a large phase delay, the first-order approximation is no longer valid, leading to an inaccurate phase reconstruction. To demonstrate the extended phase imaging range of UNN-DPC, we conducted a series of experiments using a phase target that featured a number of Siemens stars with heights ranging from 50 nm to 350 nm (Benchmark Technologies). Figures 3(a) and 3(b) show representative UNN-DPC phase images of the phase target and profiles along the yellow dashed lines in Fig. 3(a), respectively. Phase profiles measured with a conventional DPC microscope and expected phase delays from the target specification are also presented for comparison. UNN-DPC microscopy could produce accurate phase images, whereas images from conventional DPC microscopy exhibited large errors, particularly for objects with large phase delays. Specifically, for a phase target of 1.92 rad, conventional DPC microscopy resulted in an error of 38.60%, whereas UNN-DPC microscopy enabled accurate phase estimation, with a significantly lower error of 1.72%. We performed experiments on Siemens stars with various phase values. The results are summarized in Fig. 3(c). At a phase delay of 2.17 rad, the reconstruction error with conventional DPC microscopy was 44.70%, whereas the reconstructed error with UNN-DPC was only 3.82%.

 

Fig. 3. (a) UNN-DPC imaging results of Siemens phase target with various heights (scale bar, 10 µm). (b) UNN-DPC and conventional DPC phase profiles along the yellow dashed lines in (a). Black dashed lines indicate expected phase delays from target specification. (c) Reconstructed phase with respect to height of phase target.

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To demonstrate the self-calibrating capability of the UNN-DPC platform, we then experimentally imposed a defocus aberration on the microscope and evaluated its auto-focusing performance. A Siemens star phase target was mounted on a motorized translation stage and scanned along the optical axis in steps of 5 µm in the range of −20 µm to 20 µm. On acquisition of a number of intensity images at each axial location, the phase images were reconstructed using the conventional DPC and UNN-DPC methods. The reconstructed phase and pupil aberrations for various defocus distances are shown in Fig. 4(a). One can note that the phase images from conventional DPC microscopy degraded rapidly for a defocused phase target. In contrast, UNN-DPC enabled high-resolution phase imaging over much larger defocus distances by correcting for the aberration. The images from the UNN-DPC microscopy also degraded at large defocus distances (≥20 µm), but the images maintained a high contrast, compared with those from conventional DPC. As shown in the inset of Fig. 4(a), the contrast of a conventional DPC image at the focal plane was 0.66 but dropped to 0.57 at a defocus distance of 20 µm. In comparison, the UNN-DPC images maintained a high contrast of >0.74 over the considered depth ranges (−20 µm to 20 µm). The average contrast value of the UNN-DPC images was found to be 0.76. As expected, the pupil aberrations estimated by the UNN-DPC microscopy were characterized by quadratic forms, with their curvatures increasing for large defocus distances. We extracted the defocus parameters from the UNN-DPC-estimated Zernike coefficient (i.e., the defocus component $Z_2^0$) and compared them with the experimentally measured values [Fig. 4(b)]. A significant correspondence was observed between the measured defocus parameters and those estimated from the Zernike coefficients (R²= 0.992). We further compared the imaging performance of UNN-DPC microscopy against the iterative pupil recovery algorithm from Zuo et al. [9]. For a phase object of 1.7 rad, the iterative algorithm was unable to correct for aberration, resulting in the images becoming rapidly degraded for defocused phase targets (Fig. S5 in Supplement 1). We attribute this to imaging of the phase target with a relatively large phase delay (1.7 rad), which violates the weak phase object assumption. In contrast, UNN-DPC could achieve high-accuracy phase imaging over much larger defocus distances by correcting for the aberration.

 

Fig. 4. (a) Conventional DPC and UNN-DPC imaging results of Siemens phase target at various defocus distances. Recovered pupil aberrations are also presented. (b) Estimated defocus parameters compared against experimentally measured values (R²= 0.992).

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UNN-DPC microscopy represents a novel form of learning-based DPC microscopy. Unlike conventional supervised learning, in UNN-DPC microscopy, the network output (i.e., complex object information) is propagated by a complete physical model that represents the process of nonlinear image formation with aberration to produce estimated DPC images. The images acquired with top- and left-half circle LED patterns provide DPC images along both x and y directions, which helps absorption and phase image reconstruction via an image formation model. Conversely, the image recorded with a center LED was found to be effective in aberration recovery, as it corresponds to coherent illumination with a uniform frequency response inside the objective NA [9]. The network weight and bias factors are then optimized to minimize the difference between the estimated and measured DPC images, eventually resulting in a feasible solution that satisfies the imposed physical constraints. UNN-DPC microscopy demonstrated high-accuracy imaging capability for a phase object with larger phase delays, which could not be achieved using conventional DPC microscopes. Although the method is iterative, as in DNNs, its processing can be facilitated using high-speed GPUs.

The utility of the UNN in various phase imaging platforms has been explored previously [14–17]. To the best of our knowledge, however, this study reports the first demonstration of UNN-based DPC microscopy, which is specifically aimed at addressing the limitations of conventional DPC microscopy; namely, (1) use of a linearized imaging model, which limits the range of objects for imaging, and (2) susceptibility to aberration. Our simulation and experimental results clearly validate UNN-DPC microscopy in overcoming such limitations.

The proposed scheme can be exploited in various functional derivatives of DPC microscopy. For instance, Hur et al. [21] presented a polarization-sensitive DPC (PS-DPC) microscope for birefringence imaging in the platform of DPC; however, polarization-dependent aberrations, which might be present in polarization optics, were not considered and corrected. The application of a UNN with a polarization-dependent forward-imaging model would generate polarization-dependent aberrations and phase images, which altogether would improve the measurement accuracy of birefringence information. On a final note, optimization of illumination patterns in our UNN-DPC microscope was not rigorously examined in this study. Optimization of illumination patterns, by incorporating well-characterized system physics and the nonlinear forward model, is under way to further improve imaging speed and measurement accuracy for a range of objects.