Transport of intensity equation-based computational phase retrieval for robust quantitative phase imaging.

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.

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

Comparison of common-path off-axis digital holography and transport of intensity equation in quantitative phase measurement

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.

Multifilter phase imaging with partially coherent light (MFPI-PC) and phase optical transfer function recovery (POTFR) are two viable defocus-based, two-dimensional quantitative phase imaging (QPI) methods. While both methods use transfer function inversion, MFPI-PC is based on the in-focus intensity derivative, while POTFR is based on the intensity difference between symmetrically defocused images. This paper compares and contrasts MFPI-PC and POTFR. Six disadvantages (five in MFPI-PC and one in POTFR) are identified. Improvement strategies to overcome each of the six shortcomings are identified and implemented, and both methods are shown to be clearly improved. The revised MFPI-PC is shown to be more accurate than the original MFPI-PC and generally more accurate than the revised POTFR. The revised POTFR is shown to be inherently faster than the original POTFR and also slightly faster than the revised MFPI-PC.

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.

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.

Digital holographic microscopy (DHM), which provides quantitative phase imaging (QPI), has been widely applied in material and biological applications. The performance of DHM technologies relies heavily on computational reconstruction methods to provide accurate phase measurements. For example, non-telecentric DHM systems should compensate for the spherical wavefront associated with a non-telecentric configuration. The size of the ±1 diffraction orders in the hologram spectrum depends inversely on the radius of the curvature of the spherical wavefront introduced by the non-telecentric DHM system. Therefore, one can estimate the radius of curvature of the spherical wavefront by analyzing the hologram spectrum. Here, we outline the steps for the automatic reconstruction of phase images without distortions and with minimum user input from a hologram recorded in a non-telecentric DHM system. The proposed reconstruction approach can be divided into six main steps. The first step automatically selects the +1 diffraction order in the hologram spectrum. Secondly, the spherical wavefront parameters and the interference angle are estimated by analyzing the size and position of the selected +1 order. The third and fourth steps are the spatial filtering of the +1 order and the compensation of the interference angle, respectively. The next step involves the estimation of the center of the spherical wavefront. Finally, there is a fine-tuning step to optimize the estimated parameters and provide a phase image with minimum phase distortions. We have identified the relevant metrics in each step, compared multiple approaches, and selected the one with the higher performance for all our experimental holograms.

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) provides enhanced contrast for weakly absorbing specimens such as biological tissues under optical light and soft materials under X-ray. In this work, we develop a model-based phase retrieval framework by integrating the physics principles of phase imaging with the deep learning-based approach. Both measurements and the forward model are used as the inputs for a model-based neural network. The features of the object and the regularization weight of the established priors are learned by minimizing the difference between the network output to the ground truth during the training process. This method is tested on phase imaging of handwriting digital patterns and biological cells in a simulation of propagation-based TIE (transport of intensity equation) phase retrieval. We achieve enhanced accuracy for the phase retrieval compared to non-model based end-to-end neural networks and reduce the computation cost compared to traditional model-based iterative reconstruction algorithms.

Phase recovery from a stack of through-focus intensity images is an effective non-interference quantitative phase imaging strategy. Nevertheless, the implementations of these methods are expensive and time-consuming because the distance between each through-focus plane has to be guaranteed by precision mechanical moving devices, and the multiple images must be acquired sequentially. In this article, we propose a single-shot through-focus intensity image stack acquisition strategy without any precision movement. Isolated LED units are used to illuminate the sample in different colors from different angles. Due to the chromatic aberration characteristics of the objective, the color-channel defocus images on the theoretical imaging plane are mutually laterally shifted. By calculating the shift amount of each sub-image area in each color channel, the distances between each through-focus image can be obtained, which is a critical parameter in transport of intensity equation (TIE) and alternating projection (AP). Lastly, AP is used to recover the phase distribution and realize the 3D localization of different defocus distances of the sample under test as an example. Both simulation and experiments are conducted to verify the feasibility of the proposed method.

2D phase-unwrapping algorithms (PUAs) are commonly used to obtain a continuous phase map from the sawtooth-shaped phase map. However, implementing PUAs can be time consuming, and the accuracy of those algorithms may be low if there is heavy noise. In this paper, we develop a simple and robust PUA based on the transport of intensity equation (TIE). In our method, the TIE was solved using the fast cosine transform, and a phase correction operation was introduced after the TIE was solved. Because of the phase correction operation, the proposed method can obtain a satisfactory unwrapping result even in a notably hash noise condition. The simulation and experimental results are presented to validate the effectiveness of the proposed method. The detailed software package can be found at https://ww2.mathworks.cn/matlabcentral/fileexchange/68493-robust-2d-phase-unwrapping-algorithm.

Direct observation of curvature of the wave surface in transmission electron microscope using transport intensity equation

When cells are observed with a microscope, the phase information for the sample is usually obtained from the acquired intensity images using the transport of intensity equation (TIE). However, to solve the TIE, it is necessary to obtain an in-focus image. The accuracy and speed of focus positioning affect the accuracy and time of phase retrieval. We propose a fast phase retrieval method based on the TIE and the threshold duty ratio. We first perform pixel reduction on a series of acquired intensity images, and these images are segmented using the optimal threshold to calculate the threshold duty ratios. The original image corresponding to the minimum threshold duty ratio is the optimal focus image. Finally, the optimal in-focus image and related defocused images are used to solve for the phase of the sample using the TIE under the boundary conditions. The correlation coefficient for the results of the simulation experiment reached 0.9317. In a microlens array experiment, the relative error between the results from our algorithm and the actual value was only 7.1%, thus, proving the correctness and effectiveness of the proposed method.