Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation

Image reconstruction for low cost spatial light interference microscopy with fixed and arbitrary phase modulation

During the past decade, spatial light interference microscopy (SLIM) has undergone rapid development, evidenced by its broadening applications in biology and medicine. However, the need for an expensive spatial light modulator (SLM) may limit its adoption, and the requirement for multiple images per plane limits its speed in volumetric imaging. Here we propose to address these issues by replacing the SLM with a mask fabricated from a low cost optical density (OD) filter, and recover high contrast images computationally rather than through phase-shifting. This is done using a specially constructed Wiener filter to recover the object scattering potential. A crucial part of the Wiener filter is estimating the arbitrary phase introduced by the OD filter. Our results demonstrate that not only were we able to estimate the OD filter's phase modulation in situ, but also the contrast of the reconstructed images is greatly improved. Comparisons with other related methods are also performed, with the conclusion that the combination of an inexpensive OD mask and modified Wiener filtering leads to results that are closest to the traditional SLIM setup. Thus, we have demonstrated the feasibility of a low cost, high speed SLIM system utilizing computational phase reconstruction, paving the way for wider adoption of high resolution phase microscopy.

Physics-informed neural network for phase imaging based on transport of intensity equation

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.

Phase extraction neural network (PhENN) with coherent modulation imaging (CMI) for phase retrieval at low photon counts

Imaging with low-dose light is of importance in various fields, especially when minimizing radiation-induced damage onto samples is desirable. The raw image captured at the detector plane is then predominantly a Poisson random process with Gaussian noise added due to the quantum nature of photo-electric conversion. Under such noisy conditions, highly ill-posed problems such as phase retrieval from raw intensity measurements become prone to strong artifacts in the reconstructions; a situation that deep neural networks (DNNs) have already been shown to be useful at improving. Here, we demonstrate that random phase modulation on the optical field, also known as coherent modulation imaging (CMI), in conjunction with the phase extraction neural network (PhENN) and a Gerchberg-Saxton-Fienup (GSF) approximant, further improves resilience to noise of the phase-from-intensity imaging problem. We offer design guidelines for implementing the CMI hardware with the proposed computational reconstruction scheme and quantify reconstruction improvement as function of photon count.

On a universal solution to the transport-of-intensity equation

The transport-of-intensity equation (TIE) is one of the most well-known approaches for phase retrieval and quantitative phase imaging. It directly recovers the quantitative phase distribution of an optical field by through-focus intensity measurements in a non-interferometric, deterministic manner. Nevertheless, the accuracy and validity of state-of-the-art TIE solvers depend on restrictive pre-knowledge or assumptions, including appropriate boundary conditions, a well-defined closed region, and quasi-uniform in-focus intensity distribution, which, however, cannot be strictly satisfied simultaneously under practical experimental conditions. In this Letter, we propose a universal solution to TIE with the advantages of high accuracy, convergence guarantee, applicability to arbitrarily shaped regions, and simplified implementation and computation. With the "maximum intensity assumption," we first simplify TIE as a standard Poisson equation to get an initial guess of the solution. Then the initial solution is further refined iteratively by solving the same Poisson equation, and thus the instability associated with the division by zero/small intensity values and large intensity variations can be effectively bypassed. Simulations and experiments with arbitrary phase, arbitrary aperture shapes, and nonuniform intensity distributions verify the effectiveness and universality of the proposed method.

Learning to synthesize: robust phase retrieval at low photon counts

The quality of inverse problem solutions obtained through deep learning is limited by the nature of the priors learned from examples presented during the training phase. Particularly in the case of quantitative phase retrieval, spatial frequencies that are underrepresented in the training database, most often at the high band, tend to be suppressed in the reconstruction. Ad hoc solutions have been proposed, such as pre-amplifying the high spatial frequencies in the examples; however, while that strategy improves the resolution, it also leads to high-frequency artefacts, as well as low-frequency distortions in the reconstructions. Here, we present a new approach that learns separately how to handle the two frequency bands, low and high, and learns how to synthesize these two bands into full-band reconstructions. We show that this "learning to synthesize" (LS) method yields phase reconstructions of high spatial resolution and without artefacts and that it is resilient to high-noise conditions, e.g., in the case of very low photon flux. In addition to the problem of quantitative phase retrieval, the LS method is applicable, in principle, to any inverse problem where the forward operator treats different frequency bands unevenly, i.e., is ill-posed.

Detection method for the complex amplitude of a signal beam with intensity and phase modulation using the transport of intensity equation for holographic data storage

Holographic data storage (HDS), in which both the amplitude and the phase of a signal beam are modulated, has been extensively studied with the goal of increasing its storage capacity. To detect such modulation during data retrieval, it is necessary to acquire the complex amplitude of the signal beam. In this study, we focus on the transport of intensity equation (TIE) method, which allows us to detect the phase distribution of the light wave quantitatively without using interferometry, contributing to miniaturization of the optical system and improvement of the vibration tolerance of HDS. We discuss the conditions of the modulation phase distribution of the signal beam required for accurate phase detection and propose a method to estimate and eliminate the noise that frequently appears in the phase distribution detected by the TIE method.

Spectral pre-modulation of training examples enhances the spatial resolution of the phase extraction neural network (PhENN)

The phase extraction neural network (PhENN) [Optica 4, 1117 (2017)] is a computational architecture, based on deep machine learning, for lens-less quantitative phase retrieval from raw intensity data. PhENN is a deep convolutional neural network trained through examples consisting of pairs of true phase objects and their corresponding intensity diffraction patterns; thereafter, given a test raw intensity pattern, PhENN is capable of reconstructing the original phase object robustly, in many cases even for objects outside the database where the training examples were drawn from. Here, we show that the spatial frequency content of the training examples is an important factor limiting PhENN's spatial frequency response. For example, if the training database is relatively sparse in high spatial frequencies, as most natural scenes are, PhENN's ability to resolve fine spatial features in test patterns will be correspondingly limited. To combat this issue, we propose "flattening" the power spectral density of the training examples before presenting them to PhENN. For phase objects following the statistics of natural scenes, we demonstrate experimentally that the spectral pre-modulation method enhances the spatial resolution of PhENN by a factor of 2.

Transport-of-intensity computational ghost imaging

Computational ghost imaging (CGI) allows us to obtain a sample image under a low signal-to-noise-ratio (SNR) condition. However, phase information cannot be obtained by CGI; therefore, observation of transparent objects such as living cells is difficult. Although interferometry has been applied to CGI for phase retrieval, its optical setup is cumbersome. In this paper, an alternative approach, which is based on a transport-of-intensity equation, is proposed. Compared with conventional interferometric methods, the optical setup of the proposed method is robust because it does not require additional optical elements. The proposed method achieves phase retrieval by slight modification of the standard CGI setup. Numerical and optical experiments with low SNR confirm the effectiveness of the proposed method.

Fourier-based solving approach for the transport-of-intensity equation with reduced restrictions

The transport-of-intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, is based on a Fourier transform (FT). However, several assumptions have to be made to solve the TIE by these methods. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the parallelism of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.

Field of view scanning based quantitative interferometric microscopic cytometers for cellular imaging and analysis

Microimaging is of great significance in the biological and medical fields, since it can realize observations acting as important references for cellular research and disease diagnosis. However, traditional microscopy only offers qualitative sample contours; moreover, it is difficult to reach large-amount sample observations limited by the fixed field of view (FoV). To realize massive cellular measurements quantitatively, three designed quantitative interferometric microscopic cytometers based on the FoV scanning are introduced and compared in details in this article. These devices not only retrieve the quantitative sample phase distributions in the extended FoV, but also provide the detailed information of massive cells, such as cellular volume, area, and roundness. Considering their capabilities as quantitative imaging and large-amount sampling, it is believed that these quantitative interferometric microscopic cytometers (QIMCs) can be potentially adopted in high-throughput cell imaging and statistical analysis for both the biological and medical applications.

Phase retrieval based on transport of intensity and digital holography

We propose a technique in which intensity images are reconstructed from a digital hologram to provide inputs for the transport-of-intensity equation for unwrapped phase recovery. By doing this, we avoid shifting of the sample or the camera in the experiment, a method commonly employed while using the method of transport-of-intensity equation for phase retrieval. Computer simulations as well as experimental results have been demonstrated to verify the effectiveness of the proposed idea. The underlying numerical technique can also be viewed as an alternative to existing phase-unwrapping algorithms.

Quantitative phase imaging by optimized asymmetric illumination

We have presented a simple approach for quantitative phase imaging by optimizing asymmetric illumination of a conventional microscope. With this illumination, the light intensity modulation accompanying refraction at the surface profile of phase objects occurs, and "phase-gradient information" can be derived by detecting it. Two images with phase-gradient information on different axes are converted into the two-dimensional phase distribution of the specimen by introducing the phase-gradient transfer function, which is the intensity change due to refraction by the phase-gradient of a specimen. We experimentally confirm accurate and repeatable performance of our method and demonstrate phase imaging of live cells.

Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method

In order to realize high contrast imaging with portable devices for potential mobile healthcare, we demonstrate a hand-held smartphone based quantitative phase microscope using the transport of intensity equation method. With a cost-effective illumination source and compact microscope system, multi-focal images of samples can be captured by the smartphone's camera via manual focusing. Phase retrieval is performed using a self-developed Android application, which calculates sample phases from multi-plane intensities via solving the Poisson equation. We test the portable microscope using a random phase plate with known phases, and to further demonstrate its performance, a red blood cell smear, a Pap smear and monocot root and broad bean epidermis sections are also successfully imaged. Considering its advantages as an accurate, high-contrast, cost-effective and field-portable device, the smartphone based hand-held quantitative phase microscope is a promising tool which can be adopted in the future in remote healthcare and medical diagnosis.

Real-time quantitative phase imaging based on transport of intensity equation with dual simultaneously recorded field of view

Since quantitative phase distribution reflects both cellular shapes and conditions from another view, compared to traditional intensity observation, different quantitative phase microscopic methods are proposed for cellular detections. However, the transport of intensity equation-based approach not only presents phase, but also intensity, which attracts much attention. While classical transport of intensity equation needs multi-focal images which often cannot realize simultaneous phase measurement, in this Letter, to break through the limitation, a real-time quantitative phase imaging method using transport of intensity equation is proposed. Two identical CCD cameras are set at the binocular tubes to capture the same field of view but at different focal planes. With a double-frame algorithm assuming that the on-focal image is the average of over- and under-focal information, the proposed method is capable of calculating quantitative phase distributions of samples accurately and simultaneously indicating its potentialities in cellular real-time monitoring.

Recovering low spatial frequencies in wavefront sensing based on intensity measurements

The transport of intensity equation (TIE) offers a convenient method to retrieve the phase of a wave function from maps of the irradiance (images) recorded at different planes along the optic axis of an optical system. However, being a second-order partial differential equation, even for noise-free data a unique solution of the TIE requires boundary conditions to be specified which are generally not accessible experimentally, jeopardizing retrieval of the low-frequency information in particular. Here we introduce an iterative algorithm which forgoes the need for explicit boundary conditions and combines the well-known reciprocal space solution of the TIE with the charge-flipping algorithm that has originally been developed to solve the crystallographic phase problem in X-ray diffraction. Application of this algorithm to experimental data and comparison with conventionally used algorithms demonstrates an improved retrieval of the low spatial frequencies of the phase.

Phase imaging for absorptive phase objects using hybrid uniform and structured illumination transport of intensity equation

Transport of intensity equation (TIE) has been a popular and convenient phase imaging method that retrieves phase profile from the measurement of intensity differentials. Conventional 2-shot uniform illumination TIE can give reliable inversion of the phase from intensity in many situations of practical interest; however, it has a null space consisting of fields with non-zero circulation of the Poynting vector. Here, we propose the hybrid illumination TIE method to disambiguate such objects. By comparing the diffraction signals using uniform and structured (sinusoidal) illumination patterns, we obtain a modulation-induced signal that depends solely on the phase gradient. In this way, we also increase signal sensitivity in the low spatial frequency region.