Transport Of Intensity Equation Research Articles

3D fluorescence imaging through scattering medium using transport of intensity equation and Fresnel ping-pong algorithm

3D fluorescence imaging through scattering medium using transport of intensity equation and Fresnel ping-pong algorithm

Monitoring the dynamics of cultured tobacco cells division by TIE-based 3D fluorescence microscopy with two-synchronized image sensors

The dynamic activity of the microtubule cytoskeleton is critical to cellular organization, including structural support, intracellular transport, and engagement in processes such as cell division and cell motility. In this work, we monitor the volumetric cell division process of tobacco BY-2 suspension cells including the dynamics of microtubule in a single-shot by using a 3D epi-fluorescence microscope based on the transport of intensity equation (3DFM-TIE) with two synchronized image sensors. The technique makes it possible to recognize and track cell mobility by examining the fluorescently tagged proteins and studying their behavior and dynamics during the cell division process. In light of experimental findings, the 3DFM-TIE system could be a favored option for live cell imaging and analysis because of its real-time 3D imaging capability and simple and compact optical design, and it has the potential to be employed in both basic and clinical investigations.

Quantitative phase imaging with a compact meta-microscope

Quantitative phase imaging (QPI) based on the transport-of-intensity equation (TIE) is a powerful technique in label-free microscopy. The image stack required for a successful TIE-QPI is traditionally obtained by translating the object or image plane, and the optical elements used in the conventional TIE-QPI systems are usually bulky and cumbersome. Stable and compact TIE-QPI methods capable of non-motion optical zooming can significantly facilitate applications that demand portability. Here, we propose a non-motion TIE-QPI method based on a dispersive metalens. The dispersive nature of the metalens is utilized to provide a spectral focal tuning. With fixed object and image planes, seven through-focus intensity images are captured by changing the illumination wavelength. The QPI performance is validated by retrieving the surface phase profiles of a microlens array and a phase resolution target, showing a high phase detection accuracy (deviation less than 0.03 wavelength). Subsequently, we established a compact meta-microscope by integrating the metalens with a commercially available CMOS image sensor, which shows good performance in microscopic imaging of unstained bio-samples. Our approach, based on the large-dispersive metalens, facilitates a compact and robust QPI system for optical metrology and label-free microscopy.

3D fluorescence imaging through scattering medium using transport of intensity equation and iterative phase retrieval.

In this paper, we have proposed a method of three-dimensional (3D) fluorescence imaging through a scattering medium. The proposed method combines the numerical digital phase conjugation propagation after measurement of the complex amplitude distribution of scattered light waves by the transport of intensity equation (TIE) with followed iterative phase retrieval to achieve 3D fluorescence imaging through a scattering medium. In the experiment, we present the quantitative evaluation of the depth position of fluorescent beads. In addition, for time-lapse measurement, cell division of tobacco-cultured cells was observed. Numerical results presented the effective range of the phase amount in the scattering medium. From these results, the proposed method is capable of recovering images degraded by a thin scattering phase object beyond a small phase change approximation.

Real-time phase retrieval in division of aperture microscopy with the transport of intensity equation.

The transport of intensity equation (TIE) allows to recover the phase of a microscopy sample from differently focused intensity measures along the axial direction of its optical field. In the present work, we propose a cost-effective technique for snapshot phase retrieval with TIE. The optics of a commercially available camera is replaced with a doublet system consisting of a microscope objective and a lenslet array with an extra lens mask attached to it. The system allows to obtain, in real-time and with no mechanical shift of either the sample or the sensor, the in-focus as well as a defocused image of the sample. From these two sub-aperture images, the intensity derivative term in TIE can then be approximated after image rectification. Phase is then retrieved for static as well as dynamic samples over the common view area. Validation experiments are presented.

TIE-GANs: single-shot quantitative phase imaging using transport of intensity equation with integration of GANs.

Artificial intelligence (AI) has become a prominent technology in computational imaging over the past decade. The expeditious and label-free characteristics of quantitative phase imaging (QPI) render it a promising contender for AI investigation. Though interferometric methodologies exhibit potential efficacy, their implementation involves complex experimental platforms and computationally intensive reconstruction procedures. Hence, non-interferometric methods, such as transport of intensity equation (TIE), are preferred over interferometric methods. TIE method, despite its effectiveness, is tedious as it requires the acquisition of many images at varying defocus planes. The proposed methodology holds the ability to generate a phase image utilizing a single intensity image using generative adversarial networks (GANs). We present a method called TIE-GANs to overcome the multi-shot scheme of conventional TIE. The present investigation employs the TIE as a QPI methodology, which necessitates reduced experimental and computational efforts. TIE is being used for the dataset preparation as well. The proposed method captures images from different defocus planes for training. Our approach uses an image-to-image translation technique to produce phase maps and is based on GANs. The main contribution of this work is the introduction of GANs with TIE (TIE:GANs) that can give better phase reconstruction results with shorter computation times. This is the first time the GANs is proposed for TIE phase retrieval. The characterization of the system was carried out with microbeads of size and structural similarity index (SSIM) for microbeads was found to be 0.98. We demonstrated the application of the proposed method with oral cells, which yielded a maximum SSIM value of 0.95. The key characteristics include mean squared error and peak-signal-to-noise ratio values of 140 and 26.42dB for oral cells and 100 and 28.10dB for microbeads. The proposed methodology holds the ability to generate a phase image utilizing a single intensity image. Our method is feasible for digital cytology because of its reported high value of SSIM. Our approach can handle defocused images in such a way that it can take intensity image from any defocus plane within the provided range and able to generate phase map.

Enhancing Contrast of Spatial Details in X-ray Phase-Contrast Imaging through Modified Fourier Filtering

In-line X-ray phase contrast imaging, which is simple to experiment with, provides significantly higher sensitivity, compared to conventional X-ray absorption imaging. The inversion of the relationship between recorded Fresnel diffraction intensity and the phase shift induced by the object is called phase retrieval. The transport of intensity equation (TIE), a simple method of phase retrieval, which is solved by the fast Fourier transform algorithm proposed by Paganin et al., has been widely adopted. However, the existing method suffers from excessive suppression of high-frequency information, resulting in loss of image details after phase retrieval, or insufficient detail contrast, leading to blurry images. Here, we present a straightforward extension of the two-distance FFT-TIE method by modifying the Fourier filter through the use of a five-point approximation to calculate the inverse Laplacian in a discrete manner. Additionally, we utilize a combination of continuous Fourier transform and a four-point approximation to compute the gradient operator. The method is evaluated by simulating samples with a shape similar to the resolution test map and by using a photograph of a dog for further evaluation. The algorithm that incorporates the modified gradient operator and the algorithm that solely utilizes the continuous Fourier transform for gradient computation were compared with the results obtained using the two-distance FFT-TIE method. The comparisons were conducted using the results obtained from two distances from the sample to the detector. The results show that this method improves the contrast of spatial details and reduces the suppression of high spatial frequencies compared to the two-distance FFT-TIE method. Furthermore, in the low-frequency domain, our algorithm does not lose much information compared to the original method, yielding consistent results. Furthermore, we conducted our experiments using carbon rods. The results show that both our method and the FFT-TIE method exhibit low-frequency distortion due to the requirement of close proximity between the absorption maps and the detector. However, upon closer inspection, our proposed method demonstrates superior accuracy in reproducing the finer details of the carbon rod fibers.

Flexible dynamic quantitative phase imaging based on division of focal plane polarization imaging technique.

This paper proposes a flexible and accurate dynamic quantitative phase imaging (QPI) method using single-shot transport of intensity equation (TIE) phase retrieval achieved by division of focal plane (DoFP) polarization imaging technique. By exploiting the polarization property of the liquid crystal spatial light modulator (LC-SLM), two intensity images of different defocus distances contained in orthogonal polarization directions can be generated simultaneously. Then, with the help of the DoFP polarization imaging, these images can be captured with single exposure, enabling accurate dynamic QPI by solving the TIE. In addition, our approach gains great flexibility in defocus distance adjustment by adjusting the pattern loaded on the LC-SLM. Experiments on microlens array, phase plate, and living human gastric cancer cells demonstrate the accuracy, flexibility, and dynamic measurement performance for various objects. The proposed method provides a simple, flexible, and accurate approach for real-time QPI without sacrificing the field of view.

Multi-particles detection in transformer oil based on lens-free digital holography and transport of intensity equation

Transformer oil quality plays an important role in health condition monitoring of a transformer. In order to better monitor the quality of transformer oil, a hybrid method combining lens-free digital holography and transport of intensity equation is proposed. We take a single-shot hologram of sample oil with a large field of view, and then reconstruct intensity images with different distances, which are necessary inputs of transport of intensity equation. After that, we can get relatively high-resolution image of oil. Via experiments, we compare the results from basic lens-free digital holography reconstruction and hybrid method reconstruction. The results show that, the hybrid method can effectively improve the image reconstruction accuracy and achieve a resolution of 2.67 μm across a large field-of-view of 60.35 mm2, which means that we can get high-resolution zoom-in oil image with more details and quantitative parameters in real time, which is a breakthrough for quick transformer oil testing.

Target-surface multiplexed quantitative dynamic phase microscopic imaging based on the transport-of-intensity equation.

Microscopic phase digital imaging based on the transport of intensity equation, known as TIE, is widely used in optical measurement and biomedical imaging since it can dispense with the dependence of traditional phase imaging systems on mechanical rotational scanning and interferometry devices. In this work, we provide a single exposure target-surface multiplexed phase reconstruction (SETMPR) structure based on TIE, which is remarkably easy to construct since it directly combines a conventional bright-field inverted microscope with a special image plane transmission structure that is capable of wavefront shaping and amplification. In practice, the SETMPR is able to achieve dynamic, non-interferometric, quantitative refractive index distribution of both static optical samples and dynamic biological samples in only one shot, meaning that the only limitation of measuring frequency is the frame rate. By comparing the measurement results of a microlens array and a grating with a standard instrument, the quantitative measurement capability and accuracy are demonstrated. Subsequently, both in situ static and long-term dynamic quantitative imaging of HT22 cells were performed, while automatic image segmentation was completed by introducing machine learning methods, which verified the application prospect of this work in dynamic observation of cellular in the biomedical field.

Common-path quantitative phase imaging by propagation through a sinusoidal intensity mask

We present a new quantitative approach for phase retrieval that is derived from the analysis of light intensity propagation through a phase object and a sinusoidal intensity mask positioned into the light path. Unlike the Transport of Intensity Equation (TIE), we deduce a new algorithm that results in a simpler expression since it does not include the Laplacian of the phase. In practice, the method is implemented in a common path optics where each optical component of the assembly remains fixed and phase unwrapping is not required. We introduce the analytical expressions for this algorithm and show experimental results including the phase retrieval of a phase object and of cheek cell samples.

Noise robust Zernike phase retrieval via learning based algorithm only with 2-step phase shift measurements.

We present a noise robust deep learning based aberration analysis method using 2-step phase shift measurement data. We first propose a realistic aberration pattern generation method to synthesize a sufficient amount of real-world-like aberration patterns for training a deep neural network by exploiting the asymptotic statistical distribution parameters of the real-world Zernike coefficients extracted from a finite number of experimentally measured real-world aberration patterns. As a result, we generate a real-world-like synthetic dataset of 200,000 different aberrations from 15 sets of real-world aberration patterns obtained by a Michelson interferometer under a variety of measurement conditions using the 4-step derivative fitting method together with the exploitation of the Gaussian density estimation. We then train the deep neural network with the real-world-like synthetic dataset, using two types of network architectures, GoogLeNet and ResNet101. By applying the proposed learning based 2-step aberration analysis method to the analysis of numerically generated aberrations formed under 100 different conditions, we verify that the proposed 2-step method can clearly outperform the existing 4-step iterative methods based on 4-step measurements, including the derivative fitting, transport of intensity equation (TIE), and robust TIE methods, in terms of noise robustness, root mean square error (RMSE), and inference time. By applying the proposed 2-step method to the analysis of the real-world aberrations experimentally obtained under a variety of measurement conditions, we also verify that the proposed 2-step method achieves compatible performance in terms of the RMSE between the reconstructed and measured aberration patterns, and also exhibits qualitative superiority in terms of reconstructing more realistic fringe patterns and phase distributions compared to the existing 4-step iterative methods. Since the proposed 2-step method can be extended to an even more general analysis of aberrations of any higher order, we expect that it will be able to provide a practical way for comprehensive aberration analysis and that further studies will extend its usefulness and improve its operational performance in terms of algorithm compactness, noise robustness, and computational speed.

Accurate dynamic quantitative phase imaging using multi-wavelength multiplexing

We present a novel, accurate, full-filed, dynamic quantitative phase imaging (QPI) technique by using multi-wavelength multiplexing and multi-plane iterative phase retrieval algorithm. In our method, a liquid crystal spatial light modulator is employed to flexibly generate multiple defocus intensity images at once, using its adjustable phase modulation characteristics of different wavelengths. Then these images contained at different wavelengths are captured by two color cameras with single exposure. To achieve accurate QPI, a multi-plane iterative phase reconstruction algorithm is also proposed based on transport of intensity equation (TIE). Finally, with these multiple defocus images, an accurate dynamic phase result can be provided by our approach. In addition, the errors caused by color coupling of color camera and chromatic aberration of the optical system are both analyzed and effectively compensated. Experiments conducted on the phase plate, living human colorectal cancer cells and human red blood cells well demonstrate the accuracy, dynamic measurement ability and flexibility of our method.

Quantum enhanced non-interferometric quantitative phase imaging

Quantum entanglement and squeezing have significantly improved phase estimation and imaging in interferometric settings beyond the classical limits. However, for a wide class of non-interferometric phase imaging/retrieval methods vastly used in the classical domain, e.g., ptychography and diffractive imaging, a demonstration of quantum advantage is still missing. Here, we fill this gap by exploiting entanglement to enhance imaging of a pure phase object in a non-interferometric setting, only measuring the phase effect on the free-propagating field. This method, based on the so-called “transport of intensity equation", is quantitative since it provides the absolute value of the phase without prior knowledge of the object and operates in wide-field mode, so it does not need time-consuming raster scanning. Moreover, it does not require spatial and temporal coherence of the incident light. Besides a general improvement of the image quality at a fixed number of photons irradiated through the object, resulting in better discrimination of small details, we demonstrate a clear reduction of the uncertainty in the quantitative phase estimation. Although we provide an experimental demonstration of a specific scheme in the visible spectrum, this research also paves the way for applications at different wavelengths, e.g., X-ray imaging, where reducing the photon dose is of utmost importance.

High accuracy quantitative phase imaging based on transport-of-intensity equation

For quantitative phase imaging (QPI) based on transport-of-intensity equation (TIE), the phase obtained by TIE has suffered a lot from the higher-order intensity derivatives caused by large defocus distance and the phase discrepancy caused by Teague assumption. To address these problems and obtain the higher accuracy phase without increasing the number of defocus images, we propose a fast compensation algorithm that requires only a few iterations. We theoretically prove the convergence of our algorithm and the efficiency is verified by experiments. We believe that this method has the characteristics of fast and higher accuracy, which will be conducive to application of QPI.

Video-rate quantitative phase imaging with dynamic acousto-optic defocusing

Various quantitative phase imaging techniques exist capable of characterizing transparent and low-contrast samples without the addition of dyes or fluorescent probes. Among them, the transport of intensity equation (TIE) allows phase retrieval by capturing information from different focal planes without complex inteferometric setups. However, current implementations can be limited in speed or accuracy by the lack of optical systems suitable for fast, reliable, and customizable focal plane selection. Here, we report how combining acousto-optics with pulsed illumination enables accurate and on-demand electronic defocus control suitable for TIE phase imaging at speeds only limited by the camera frame rate. The system exhibits diffraction-limited spatial resolution and high reconstruction fidelity, undistinguishable from traditional mechanical defocusing. We demonstrate its feasibility by measuring different dynamic events at rates as high as 100 phase maps per second. The tunability and ease of implementation of our system can pave the way to democratizing quantitative phase imaging in histopathology, fluid dynamics, and other fields involving thin transparent samples.

Complete transport of intensity equation for phase retrieval of optical vortex beams

We derive the complete form of the transport of intensity equation (TIE) for an arbitrary wave field, in which additional terms describing the effect of optical vortices are included. The Poynting vector of optical vortices goes in a spiral around the vortex core through space. Consequently, the intensity variation is not observable in the manner of wave propagation, thereby precluding the possibility of phase retrieval via near-field intensity measurements. This difficulty is elegantly overcome by introducing a hard-edged aperture, as a common procedure to produce the required Neumann boundary signal, to break the azimuthal symmetry of the propagation, and visualize the rotation of the beam’s local Poynting vector around the aperture boundary. The basic principle, numerical solution, and practical implementation of how to use this new version of TIE for phase retrieval of arbitrary wave fields including optical vortices are developed. Based on the new approach, we present the first experimental demonstration of non-interferometric phase retrieval of Laguerre–Gaussian vortex beams of different orders using only two near-field intensity measurements. Our results consummate the basic theoretical framework of TIE and represent an important step towards phase retrieval of arbitrary wave fields carrying phase singularities.

Multiplexed superresolution phase microscopy with transport of intensity equation

Resolving power in imaging systems, optical microscopy in particular, is limited by several factors being diffraction probably the most important one. Digital holographic microscopy (DHM) allows diffraction-limited quantitative phase imaging (QPI) with high accuracy by retrieving complex fields using interferometric recording, and a bunch of superresolution (SR) techniques have been widely reported through the past. However, there are additional ways to retrieve the complex amplitude distribution (such as the transport of intensity equation – TIE), having additional advantages over interferometric recording. We present how to increase the resolution limit in low/medium numerical aperture (NA) microscope objectives by using time and angular multiplexing while providing quantitative phase imaging (QPI) retrieved by TIE algorithm. This is, to the best of our knowledge, the first time that the resolution limit is experimentally improved in combination with TIE for phase retrieval in QPI. The proposed approach is validated in a real microscope embodiment (Olympus IX81) using resolution targets for quantitative resolution analysis and verifying a resolution gain factor of 2 from an experimental point of view.

Quantitative phase imaging comparison of digital holographic microscopy and transport of intensity equation phase through simultaneous measurements of live cells

Quantitative phase imaging (QPI) of live cells allows label-free assessment of cell morphology and quantitative estimation of cell thickness, and therefore volume. Of the techniques available for QPI, digital holographic microscopy (DHM) is an accurate and reliable high-resolution phase imaging technique, but assembly and operation of a DHM capable setup can require significant optical engineering experience and specialized equipment that may not be practical for many labs. Transport of intensity equation (TIE) imaging is a QPI method that is compatible with conventional bright field microscopy, has a lower implementation cost without the need for specialized optics or alignment, and has significantly reduced spatial and temporal coherence requirements when compared to DHM. However, TIE is susceptible to different forms of noise than DHM and its accuracy in live cell imaging has not been well studied. Here, we present a microscope capable of performing simultaneous DHM and TIE phase reconstructions to compare both methods and validate TIE as an accurate phase retrieval method both for a static sample array and for live cells.

Recent Advances in Coherent Optical Communications for Short-Reach: Phase Retrieval Methods

Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.