Learning a model-based neural network for quantitative phase imaging based on the transport of intensity

Computational approaches in 3D quantitative phase Imaging: Current and future trends in cellular imaging.

A computational phase retrieval method is presented that utilizes the transport of intensity equation (TIE) and its solution as a physics-based prior in an iterative reconstruction process. TIE is a non-interferometric quantitative phase imaging (QPI) method that is experimentally simple to implement, needing a minimum of two through focus intensity measurements. However, the accuracy of TIE is severely affected by certain theoretical and practical limitations such as the requirement of uniform intensity, prior knowledge of boundary conditions. Moreover, the finite-difference approximation for intensity derivatives produces blurring for large defocus distances. We propose an iterative phase recovery algorithm that updates the required phase by minimizing the difference of axial intensity derivatives obtained using experiments and expected from angular spectrum propagation theory while integrating a TIE phase-compensation solution in the iterative steps for a fast and accurate phase recovery. We show the validity of the proposed algorithm with simulations for different cases of non-uniform intensity, noise, boundary conditions, and defocus distance. We experimentally verify the working of the proposed method for large defocus distances and compare the results with standard TIE solutions and algorithms to verify its robustness.

Non-interferometric quantitative phase imaging based on Transport of Intensity Equation (TIE) has been widely used in bio-medical imaging. However, analytic TIE phase retrieval is prone to low-spatial frequency noise amplification, which is caused by the illposedness of inversion at the origin of the spectrum. There are also retrieval ambiguities resulting from the lack of sensitivity to the curl component of the Poynting vector occurring with strong absorption. Here, we establish a physics-informed neural network (PINN) to address these issues, by integrating the forward and inverse physics models into a cascaded deep neural network. We demonstrate that the proposed PINN is efficiently trained using a small set of sample data, enabling the conversion of noise-corrupted 2-shot TIE phase retrievals to high quality phase images under partially coherent LED illumination. The efficacy of the proposed approach is demonstrated by both simulation using a standard image database and experiment using human buccal epitehlial cells. In particular, high image quality (SSIM = 0.919) is achieved experimentally using a reduced size of labeled data (140 image pairs). We discuss the robustness of the proposed approach against insufficient training data, and demonstrate that the parallel architecture of PINN is efficient for transfer learning.

The transport of intensity equation (TIE) has been widely applied to phase imaging. A variety of methods for solving the TIE have been proposed. One of the most popular methods is an FFT-based method, which is simple and fast. However, this method has a strict restriction that the intensity of the object is assumed to be uniform. Otherwise, the accuracy of phase retrieval results may drop significantly. The transport of phase equation (TPE) is an equation coupled with the TIE, and both are derived from the Helmholtz equation. Few works have studied the role of the TPE in 3D imaging. In this work, a non-iterative FFT-based TIE with TPE correction is proposed. The phases at the object plane and four defocused planes are first calculated by the TIE. Ideally, the object intensity and computed phases are supposed to satisfy both the TIE and TPE. But the non-uniformity of object intensity, as well as the use of finite differences of intensities in the calculation of longitudinal derivatives for the FFT-based TIE method, introduce errors in phase retrieval result. We show that by using the TPE, the local refractive index during propagation can be updated and used as a correction in the TIE. The TIE is solved once again using the updated refractive index, and shows reduction of errors. This proposed technique can be extended to amplitude and phase imaging. It also offers the advantage of yielding the unwrapped phase with good accuracy performance and can be potentially applied to medical high-definition 3D imaging, for cells, micro bubbles, etc.

Recent advances in the field of digital sensors and computational facilities have turned digital holography (DH) into a powerful tool for many applications such as quantitative phase imaging, optical metrology, evaluation of cell parameters, optical cryptography, and optical pattern recognition among others. The transport of intensity equation (TIE) is one of the deterministic phase-retrieval methods, which does not use interferometric geometry. In this method, the phase is calculated directly from a set of intensities rather than iteratively approximating a solution. Recently, use of the TIE in the DH reconstruction process has been reported as a phase-retrieval method. Combining both DH and TIE provides better solution to the phase-retrieval. Refocusing property of the DH helps exploit the translation issue of digital sensor along the optical axis while capturing the intensity distributions at different depths. The issue of phase unwrapping is solved through the use of TIE. Hence, both the methods complement each other. In this paper, we review briefly the applicability of the combination of DH with TIE.

The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green's function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

Transport of Intensity equation(TIE) is a non-interferometric method used for quantitative phase imaging. By reformulating the TIE using Contrast Transfer Function, it can be determined that the spatial resolution of the phase retrieved using TIE is limited by the product of imaging system transfer function and a sinc function. In this work, we apply the principles of structured illumination fluorescent microscopy to develop a TIE based super resolved phase imaging technique. The sinusoidal intensity pattern down modulates high frequency spectrum of the phase into the system pass band thereby providing a convenient approach to synthetically enlarge the numerical aperture of the system. Resolution enhancement by two folds is demonstrated using simulations.

Differential phase contrast (DPC) is a non-interferometric quantitative phase imaging method achieved by using an asymmetric imaging procedure. We report a pupil modulation differential phase contrast (PMDPC) imaging method by filtering a sample's Fourier domain with half-circle pupils. A phase gradient image is captured with each half-circle pupil, and a quantitative high resolution phase image is obtained after a deconvolution process with a minimum of two phase gradient images. Here, we introduce PMDPC quantitative phase image reconstruction algorithm and realize it experimentally in a 4f system with an SLM placed at the pupil plane. In our current experimental setup with the numerical aperture of 0.36, we obtain a quantitative phase image with a resolution of 1.73μm after computationally removing system aberrations and refocusing. We also extend the depth of field digitally by 20 times to ±50μm with a resolution of 1.76μm.

Quantitative phase imaging (QPI) has advanced by accurately quantifying phase shifts caused by weakly absorbing biological and artificial structures. Despite extensive research, the diffraction limits of QPI have not been established and examined. Hence, it remains unclear whether diffraction-affected QPI provides reliable quantification or merely visualizes phase objects, similar to phase contrast methods. Here, we develop a general diffraction phase imaging theory and show that it is intrinsically connected with Rayleigh’s resolution theory. Our approach reveals the entanglement of phases under restoration, imposing diffraction bounds on spatial phase resolution and, unexpectedly, on phase accuracy. We prove that the phase accuracy depends on the size, shape, and absorption of objects forming the sample and significantly declines if the object size approaches the Rayleigh limit (a relative phase error of −16% for an Airy disk-sized object with low phase shift). We show that the phase accuracy limits can be enhanced at the cost of deteriorated phase resolution by attenuating the sample background light. The QPI diffraction limits are thoroughly examined in experiments with certified phase targets and biological cells. The study’s relevance is underscored by results showing that the phase accuracy of some structures is lost (a relative phase error of −40%) even though they are spatially resolved (a phase visibility of 0.5). A reliable procedure is used to estimate phase errors in given experimental conditions, opening the way to mitigate errors’ impact through data post-processing. Finally, the phase accuracy enhancement in super-resolution QPI is discovered, which has not been previously reported.

Phase retrieval imaging is widely significant in the scientific, optical engineering, and medical fields. The transport of intensity equation (TIE) has been extensively applied to phase retrieval imaging. In this paper, we investigate a technique for phase retrieval based on the TIE and TPE equations employing electro-optic materials such as liquid crystals to create optical path length difference through application of a bias voltage across the cell.

Fourier Ptychographic Microscopy (FPM) and Differential Phase Contrast (DPC) are quantitative phase imaging (QPI) methods that recover the complex transmittance function of a sample through coded illumination measurements and phase retrieval optimization. The successes of these methods rely upon acquiring several or possibly hundreds of illumination-encoded measurements. The multi-shot nature of such methods limits their temporal resolution. Similar to motion-induced blur during a long photographic exposure, motion occurring during these acquisitions causes spatial distortion and errors in the reconstructed phase, which inhibits these methods' ability to image fast moving live samples. Here we present a novel approach to correct for motion during QPI capture that relies on motion navigation to register measurements together prior to phase retrieval. The different illumination patterns required for QPI cause the measurements to have a different contrasts. This makes it difficult to use standard registration approaches to estimate complex sample motion directly from the measurements. Instead, we use a color-multiplexed navigator signal (red) that is comprised of a constant illumination pattern and leverage a color camera to separate it from the primary QPI information (green). The reliable motion estimate allows measurements to be shared across time points through image registration. This enables a full set of measurements for a phase retrieval problem to be solved at each time point. We demonstrate proof-of-concept experimental results in which blurring due to live sample motion (swimming Zebra fish, cell motion, and organelle movement) is reduced.

Microscopy combined with the transport of intensity equation is capable of retrieving both intensity and phase distributions of samples from both in-focus and defocus intensities. However, during measurements, the focal plane is often decided artificially and the improper choice may induce errors in quantitative intensity and phase retrieval. In order to obtain accurate in-focus information, quantitative intensity and phase imaging with the numerical focusing transport of intensity equation method combined with cellular duty ratio criterion and numerical wavefront propagation is introduced in this paper. Both numerical simulations and experimental measurements are provided proving this designed method can increase both retrieved in-focus intensity and phase accuracy and reduce dependence of focal plane determination in transport of intensity equation measurements. It is believed that the proposed method can be potentially applied in various fields as in-focus compensation for quantitative phase imaging and automatic focal plane determination, etc.

Dielectric metasurfaces have recently attracted much attention due to their ability to efficiently control light propagation, that is, its amplitude, phase and polarization, on micro- and macro- levels. Such metasurfaces also naturally allow for integration in compact optical setups due to their small sizes. An important issue of energy losses is also addressed by employing all-dielectric platform. Advancements in nanofabrication have enabled realization of metasurface-based high-resolution wavefront engineering devices, which have been employed in different imaging applications, for example, as metalenses, polarization filters, metaholograms, etc. Quantitative phase imaging is a powerful tool for the optical inspection of transparent samples in connection with biological and technical applications. By measuring thickness and/or refractive index profiles, phase imaging facilitates, for example, the observation of dynamic events in unstained cells. As such, it has become indispensable in biological imaging, wavefront correction and metrology. However, conventional cameras and photodetectors are inherently not sensitive to the phase of an incident lightwave. As a consequence, direct phase detection is challenging. Only by multiple intensity measurements or application of specialized interferometric schemes, the phase profile of an object can be in principle reconstructed. In this paper, we report on the ability of metasurfaces to contribute to quantitative phase imaging by independent control of the phase profiles in two orthogonal polarization states of an incident beam.

Blood testing has been used as an essential tool to diagnose diseases for decades. Recently, there has been a rapid developing trend in using Quantitative Phase Imaging (QPI) methods for blood cell screening. Compared to traditional blood testing techniques, QPI has the advantage of avoiding dyeing or staining the specimen, which may cause damage to the cells. However, most existing systems are bulky and costly, requiring experienced personnel to operate. This work demonstrates the integration of one QPI method onto a smartphone platform and the application of imaging red blood cells. The adopted QPI method is based on solving the Intensity Transport Equation (ITE) from two de-focused pupil images taken in one shot by the smartphone camera. The device demonstrates a system resolution of about 1 μm, and is ready to be used for 3D morphological study of red blood cells.

Quantitative differential phase-contrast (DPC) imaging is one of the commonly used methods for phase retrieval. However, quantitative DPC imaging requires several pairwise intensity measurements, which makes it difficult to monitor living cells in real-time. In this study, we present a single-shot quantitative DPC imaging method based on the combination of deep learning (DL) and color-encoded illumination. Our goal is to train a model that can generate an isotropic quantitative phase image (i.e., target) directly from a single-shot intensity measurement (i.e., input). The target phase image was reconstructed using a linear-gradient pupil with two-axis measurements, and the model input was the measured color intensities obtained from a radially asymmetric color-encoded illumination pattern. The DL-based model was trained, validated, and tested using thirteen different cell lines. The total number of training, validation, and testing images was 264 (10 cells), 10 (1 cell), and 40 (2 cells), respectively. Our results show that the DL-based phase images are visually similar to the ground-truth phase images and have a high structural similarity index (>0.98). Moreover, the phase difference between the ground-truth and DL-based phase images was smaller than 13%. Our study shows the feasibility of using DL to generate quantitative phase imaging from a single-shot intensity measurement.