Experimental robustness of Fourier ptychography phase retrieval algorithms

Classifying breast cancer and fibroadenoma tissue biopsies from paraffined stain-free slides by fractal biomarkers in Fourier Ptychographic Microscopy

Breast cancer is one of the most spread and monitored pathologies in high-income countries. After breast biopsy, histological tissue is stored in paraffin, sectioned and mounted. Conventional inspection of tissue slides under benchtop light microscopes involves paraffin removal and staining, typically with H&E. Then, expert pathologists are called to judge the stained slides. However, paraffin removal and staining are operator-dependent, time and resources consuming processes that can generate ambiguities due to non-uniform staining. Here we propose a novel method that can work directly on paraffined stain-free slides. We use Fourier Ptychography as a quantitative phase-contrast microscopy method, which allows accessing a very wide field of view (i.e., mm2) in one single image while guaranteeing high lateral resolution (i.e., 0.5 µm). This imaging method is multi-scale, since it enables looking at the big picture, i.e. the complex tissue structure and connections, with the possibility to zoom-in up to the single-cell level. To handle this informative image content, we introduce elements of fractal geometry as multi-scale analysis method. We show the effectiveness of fractal features in describing and classifying fibroadenoma and breast cancer tissue slides from ten patients with very high accuracy. We reach 94.0 ± 4.2% test accuracy in classifying single images. Above all, we show that combining the decisions of the single images, each patient's slide can be classified with no error. Besides, fractal geometry returns a guide map to help pathologist to judge the different tissue portions based on the likelihood these can be associated to a breast cancer or fibroadenoma biomarker. The proposed automatic method could significantly simplify the steps of tissue analysis and make it independent from the sample preparation, the skills of the lab operator and the pathologist.

Influence of loss function and electron dose on ptychography of 2D materials using the Wirtinger flow

Iterative phase retrieval is based on minimising a loss function as a measure of the consistency of an initial guess and underlying experimental data. Under ideal experimental conditions, real data contains Poissonian noise due to counting statistics. In this work, we use the Wirtinger Flow concept in combination with four common loss functions, being the L1 loss, the mean-squared error (MSE), the amplitude loss and the Poisson loss. Since only the latter reflects the counting statistics as an asymmetric Poisson distribution correctly, our simulation study focuses on two main cases. Firstly, high-dose momentum-resolved scanning transmission electron microscopy (STEM) of an MoS2 monolayer is considered for phase retrieval. In this case, it is found that the four losses perform differently with respect to chemical sensitivity and frequency transfer, which we interprete in terms of the substantially different signal level in the bright and dark field part of diffraction patterns. Remedies are discussed using further simulations, addressing the use of virtual ring detectors for the dark field, or restricting loss calculation to the bright field. Secondly, a dose series is presented down to 100 electrons per diffraction pattern. It is found that all losses yield qualitatively reasonable structural data in the phase, whereas only MSE and Poisson loss range at the correct amplitude level. Chemical contrast is, in general, reliably obtained using the Poisson concept, which also provides the most continuous spatial frequency transfer as to the reconstructed object transmission function.

Enhanced tissue slide imaging in the complex domain via cross-explainable GAN for Fourier ptychographic microscopy

Achieving microscopy with large space-bandwidth products plays a key role in diagnostic imaging and is widely significant in the overall field of clinical practice. Among quantitative microscopy techniques, Fourier Ptychography (FP) provides a wide field of view and high-resolution images, suitable to the histopathological field, but onerous in computational terms. Artificial intelligence can be a solution in this sense. In particular, this research delves into the application of Generative Adversarial Networks (GAN) for the dual-channel complex FP image enhancement of human kidney samples. The study underscores the GANs' efficacy in promoting biological architectures in FP domain, thereby still guaranteeing high resolution and visibility of detailed microscopic structures. We demonstrate successful GAN-based enhanced reconstruction through two strategies: cross-explainability and expert survey. The cross-explainability is evaluated through the comparison of explanation maps for both real and imaginary components underlining its robustness. This comparison further shows that their interplay is pivotal for accurate reconstruction without hallucinations. Secondly, the enhanced reconstruction accuracy and effectiveness in a clinical workflow are confirmed through a two-step survey conducted with nephrologists.

High-performance 4-nm-resolution X-ray tomography using burst ptychography

Advances in science, medicine and engineering rely on breakthroughs in imaging, particularly for obtaining multiscale, three-dimensional information from functional systems such as integrated circuits or mammalian brains. Achieving this goal often requires combining electron- and photon-based approaches. Whereas electron microscopy provides nanometre resolution through serial, destructive imaging of surface layers1, ptychographic X-ray computed tomography2 offers non-destructive imaging and has recently achieved resolutions down to seven nanometres for a small volume3. Here we implement burst ptychography, which overcomes experimental instabilities and enables much higher performance, with 4-nanometre resolution at a 170-times faster acquisition rate, namely, 14,000 resolution elements per second. Another key innovation is tomographic back-propagation reconstruction4, allowing us to image samples up to ten times larger than the conventional depth of field. By combining the two innovations, we successfully imaged a state-of-the-art (seven-nanometre node) commercial integrated circuit, featuring nanostructures made of low- and high-density materials such as silicon and metals, which offer good radiation stability and contrast at the selected X-ray wavelength. These capabilities enabled a detailed study of the chip's design and manufacturing, down to the level of individual transistors. We anticipate that the combination of nanometre resolution and higher X-ray flux at next-generation X-ray sources will have a revolutionary impact in fields ranging from electronics to electrochemistry and neuroscience.

High-resolution, large field-of-view label-free imaging via aberration-corrected, closed-form complex field reconstruction

Computational imaging methods empower modern microscopes to produce high-resolution, large field-of-view, aberration-free images. Fourier ptychographic microscopy can increase the space-bandwidth product of conventional microscopy, but its iterative reconstruction methods are prone to parameter selection and tend to fail under excessive aberrations. Spatial Kramers-Kronig methods can analytically reconstruct complex fields, but is limited by aberration or providing extended resolution enhancement. Here, we present APIC, a closed-form method that weds the strengths of both methods while using only NA-matching and darkfield measurements. We establish an analytical phase retrieval framework which demonstrates the feasibility of analytically reconstructing the complex field associated with darkfield measurements. APIC can retrieve complex aberrations of an imaging system with no additional hardware and avoids iterative algorithms, requiring no human-designed convergence metrics while always obtaining a closed-form complex field solution. We experimentally demonstrate that APIC gives correct reconstruction results where Fourier ptychographic microscopy fails when constrained to the same number of measurements. APIC achieves 2.8 times faster computation using image tile size of 256 (length-wise), is robust against aberrations compared to Fourier ptychographic microscopy, and capable of addressing aberrations whose maximal phase difference exceeds 3.8π when using a NA 0.25 objective in experiment.

Res-U2Net: untrained deep learning for phase retrieval and image reconstruction

Conventional deep learning-based image reconstruction methods require a large amount of training data, which can be hard to obtain in practice. Untrained deep learning methods overcome this limitation by training a network to invert a physical model of the image formation process. Here we present a novel, to our knowledge, untrained Res-U2Net model for phase retrieval. We use the extracted phase information to determine changes in an object's surface and generate a mesh representation of its 3D structure. We compare the performance of Res-U2Net phase retrieval against UNet and U2Net using images from the GDXRAY dataset.

Dual-wavelength Fourier ptychographic microscopy for topographic measurement

Topographic measurements of micro- or nanostructures are essential in cutting-edge scientific disciplines such as optical communications, metrology, and structural biology. Despite the advances in surface metrology, measuring micron-scale steps with wide field of view (FOV) and high-resolution remains difficult. This study demonstrates a dual-wavelength Fourier ptychographic microscopy for high-resolution topographic measurement across a wide FOV using an aperture scanning structure. This structure enables the capture of a three-dimensional (3D) sample's scattered field with two different wavelength lasers, thus allowing the axial measurement range growing from nano- to micro-scale with enhanced lateral resolution. To suppress the unavoidable noises and artifacts caused by temporal coherence, system vibration, etc., a total variation (TV) regularization algorithm is introduced for phase retrieval. A blazed grating with micron-scale steps is used as the sample to validate the performance of our method. The agreement between the high-resolution reconstructed topography with our method and that with atomic force microscopy verified the effectiveness. Meanwhile, numerical simulations suggest that the method has the potential to characterize samples with high aspect-ratio steps.

Fourier Ptychographic Microscopy 10 Years on: A Review

Fourier ptychographic microscopy (FPM) emerged as a prominent imaging technique in 2013, attracting significant interest due to its remarkable features such as precise phase retrieval, expansive field of view (FOV), and superior resolution. Over the past decade, FPM has become an essential tool in microscopy, with applications in metrology, scientific research, biomedicine, and inspection. This achievement arises from its ability to effectively address the persistent challenge of achieving a trade-off between FOV and resolution in imaging systems. It has a wide range of applications, including label-free imaging, drug screening, and digital pathology. In this comprehensive review, we present a concise overview of the fundamental principles of FPM and compare it with similar imaging techniques. In addition, we present a study on achieving colorization of restored photographs and enhancing the speed of FPM. Subsequently, we showcase several FPM applications utilizing the previously described technologies, with a specific focus on digital pathology, drug screening, and three-dimensional imaging. We thoroughly examine the benefits and challenges associated with integrating deep learning and FPM. To summarize, we express our own viewpoints on the technological progress of FPM and explore prospective avenues for its future developments.

Redundant information model for Fourier ptychographic microscopy

Fourier ptychographic microscopy (FPM) is a computational optical imaging technique that overcomes the traditional trade-off between resolution and field of view (FOV) by exploiting abundant redundant information in both spatial and frequency domains for high-quality image reconstruction. However, the redundant information in FPM remains ambiguous or abstract, which presents challenges to further enhance imaging capabilities and deepen our understanding of the FPM technique. Inspired by Shannon's information theory and extensive experimental experience in FPM, we defined the specimen complexity and reconstruction algorithm utilization rate and reported a model of redundant information for FPM to predict reconstruction results and guide the optimization of imaging parameters. The model has been validated through extensive simulations and experiments. In addition, it provides a useful tool to evaluate different algorithms, revealing a utilization rate of 24%±1% for the Gauss-Newton algorithm, LED Multiplexing, Wavelength Multiplexing, EPRY-FPM, and GS. In contrast, mPIE exhibits a lower utilization rate of 19%±1%.

Maximum-likelihood estimation in ptychography in the presence of Poisson-Gaussian noise statistics

Optical measurements often exhibit mixed Poisson-Gaussian noise statistics, which hampers the image quality, particularly under low signal-to-noise ratio (SNR) conditions. Computational imaging falls short in such situations when solely Poissonian noise statistics are assumed. In response to this challenge, we define a loss function that explicitly incorporates this mixed noise nature. By using a maximum-likelihood estimation, we devise a practical method to account for a camera readout noise in gradient-based ptychography optimization. Our results, based on both experimental and numerical data, demonstrate that this approach outperforms the conventional one, enabling enhanced image reconstruction quality under challenging noise conditions through a straightforward methodological adjustment.

Fast and robust Fourier ptychographic microscopy with position misalignment correction

Significance

Fourier ptychographic microscopy (FPM) is a new, developing computational imaging technology. It can realize the quantitative phase imaging of a wide field of view and high-resolution (HR) simultaneously by means of multi-angle illumination via a light emitting diode (LED) array, combined with a phase recovery algorithm and the synthetic aperture principle. However, in the FPM reconstruction process, LED position misalignment affects the quality of the reconstructed image, and the reconstruction efficiency of the existing LED position correction algorithms needs to be improved.

Aim

This study aims to improve the FPM correction method based on simulated annealing (SA) and proposes a position misalignment correction method (AA-C algorithm) using an improved phase recovery strategy.

Approach

The spectrum function update strategy was optimized by adding an adaptive control factor, and the reconstruction efficiency of the algorithm was improved.

Results

The experimental results show that the proposed method is effective and robust for position misalignment correction of LED arrays in FPM, and the convergence speed can be improved by 21.2% and 54.9% compared with SC-FPM and PC-FPM, respectively.

Conclusions

These results can reduce the requirement of the FPM system for LED array accuracy and improve robustness.

LED array microscopy system correction method with comprehensive error parameters optimized by phase smoothing criterion

LED array microscopy is a novel computational imaging technique that can achieve two-dimensional (2D) phase imaging and three-dimensional (3D) refractive index imaging with both high resolution and a large field of view. Although its experimental setup is simple, the errors caused by LED array position and light source central wavelength obviously decrease the quality of reconstructed results. To solve this problem, comprehensive error parameters optimized by the phase smoothing criterion are put forward in this paper. The central wavelength error and 3D misalignment model with six freedom degree errors of LED array are considered as the comprehensive error parameters when the spatial positional and optical features of arbitrarily placed LED array are unknown. Phase smoothing criterion is also introduced to the cost function for optimizing comprehensive error parameters to improve the convergence results. Compared with current system correction methods, the simulation and experimental results show that the proposed method in this paper has the best reconstruction accuracy, which can be well applied to an LED array microscope system with unknown positional and optical features of the LED array.

Hybrid full-pose parameter calibration of a freeform illuminator for Fourier ptychographic microscopy

As a typical computational method, Fourier ptychographic microscopy (FPM) can realize high spatial resolution and quantitative phase imaging while preserving the large field of view with a low numerical aperture (NA) objective. A programmable light-emitting diode (LED) array is used as a typical illuminator in an FPM system, and the illumination parameters of each LED element are crucial to the success of the FPM reconstruction algorithm. Compared with LED arrays arranged in rectangular arrays, LED arrays with special structures such as domes or rings can effectively improve FPM imaging results and imaging efficiency. As a trade-off, their calibration difficulty is greatly increased due to the lack of geometric constraints of rectangular arrays. In this paper, we propose an effective hybrid full-pose parameter calibration method for freeform LED array illuminators, combining stereoscopic 3D imaging techniques and the geometric constraints of the microscopic platform. First, a stereovision system is used to obtain the accurate 3D position of each LED element of the freeform illuminator and to construct a rigid 3D coordinate LED array system. Then, calibration between the coordinate system of the LED array and that of the optical imaging component is realized according to the geometric features of the brightfield-to-darkfield edges. Finally, we verify the feasibility and effectiveness of the proposed method through full-pose parameter calibration of LED arrays with different arrangement rules.

Spatial- and Fourier-domain ptychography for high-throughput bio-imaging

First envisioned for determining crystalline structures, ptychography has become a useful imaging tool for microscopists. However, ptychography remains underused by biomedical researchers due to its limited resolution and throughput in the visible light regime. Recent developments of spatial- and Fourier-domain ptychography have successfully addressed these issues and now offer the potential for high-resolution, high-throughput optical imaging with minimal hardware modifications to existing microscopy setups, often providing an excellent trade-off between resolution and field of view inherent to conventional imaging systems, giving biomedical researchers the best of both worlds. Here, we provide extensive information to enable the implementation of ptychography by biomedical researchers in the visible light regime. We first discuss the intrinsic connections between spatial-domain coded ptychography and Fourier ptychography. A step-by-step guide then provides the user instructions for developing both systems with practical examples. In the spatial-domain implementation, we explain how a large-scale, high-performance blood-cell lens can be made at negligible expense. In the Fourier-domain implementation, we explain how adding a low-cost light source to a regular microscope can improve the resolution beyond the limit of the objective lens. The turnkey operation of these setups is suitable for use by professional research laboratories, as well as citizen scientists. Users with basic experience in optics and programming can build the setups within a week. The do-it-yourself nature of the setups also allows these procedures to be implemented in laboratory courses related to Fourier optics, biomedical instrumentation, digital image processing, robotics and capstone projects.

PtyLab.m/py/jl: a cross-platform, open-source inverse modeling toolbox for conventional and Fourier ptychography

Conventional (CP) and Fourier (FP) ptychography have emerged as versatile quantitative phase imaging techniques. While the main application cases for each technique are different, namely lens-less short wavelength imaging for CP and lens-based visible light imaging for FP, both methods share a common algorithmic ground. CP and FP have in part independently evolved to include experimentally robust forward models and inversion techniques. This separation has resulted in a plethora of algorithmic extensions, some of which have not crossed the boundary from one modality to the other. Here, we present an open source, cross-platform software, called PtyLab, enabling both CP and FP data analysis in a unified framework. With this framework, we aim to facilitate and accelerate cross-pollination between the two techniques. Moreover, the availability in Matlab, Python, and Julia will set a low barrier to enter each field.

Undersampled Fourier ptychography for reflective-based long range imaging

Fourier ptychography (FP) can be a promising technique for long-range and high-resolution imaging. In this work, we explore reconstructions with undersampled data for meter-scale reflective based Fourier ptychographic imaging. To reconstruct with under-sampling captures, we propose a novel cost function for FP phase retrieval and design a new optimization algorithm based on gradient descent. To verify the proposed methods, we perform the high-fidelity reconstruction of the targets with sampling parameter less than one. Compared to the state-of-the-art alternative-projectionbased FP algorithm, the proposed one can achieve the same performance but with much less data.

Multi-angle lensless ptychographic imaging via adaptive correction and the Nesterov method

Lensless systems based on ptychographic imaging can simultaneously achieve a large field of view and high resolution while having the advantages of small size, portability, and low cost compared to traditional lensed imaging. However, lensless imaging systems are susceptible to environmental noise and have a lower resolution of individual images than lens-based imaging systems, which means that they require a longer time to obtain a good result. Therefore, in this paper, to improve the convergence rate and robustness of noise in lensless ptychographic imaging, we propose an adaptive correction method, in which we add an adaptive error term and noise correction term in lensless ptychographic algorithms to reach convergence faster and create a better suppression effect on both Gaussian noise and Poisson noise. The Wirtinger flow and the Nesterov algorithms are used in our method to reduce computational complexity and improve the convergence rate. We applied the method to phase reconstruction for lensless imaging and demonstrated the effectiveness of the method by simulation and experiment. The method can be easily applied to other ptychographic iterative algorithms.

Fourier ptychographic topography

Topography measurement is essential for surface characterization, semiconductor metrology, and inspection applications. To date, performing high-throughput and accurate topography remains challenging due to the trade-off between field-of-view (FOV) and spatial resolution. Here we demonstrate a novel topography technique based on the reflection-mode Fourier ptychographic microscopy, termed Fourier ptychograhpic topography (FPT). We show that FPT provides both a wide FOV and high resolution, and achieves nanoscale height reconstruction accuracy. Our FPT prototype is based on a custom-built computational microscope consisting of programmable brightfield and darkfield LED arrays. The topography reconstruction is performed by a sequential Gauss-Newton-based Fourier ptychographic phase retrieval algorithm augmented with total variation regularization. We achieve a synthetic numerical aperture (NA) of 0.84 and a diffraction-limited resolution of 750 nm, increasing the native objective NA (0.28) by 3×, across a 1.2 × 1.2 mm² FOV. We experimentally demonstrate the FPT on a variety of reflective samples with different patterned structures. The reconstructed resolution is validated on both amplitude and phase resolution test features. The accuracy of the reconstructed surface profile is benchmarked against high-resolution optical profilometry measurements. In addition, we show that the FPT provides robust surface profile reconstructions even on complex patterns with fine features that cannot be reliably measured by the standard optical profilometer. The spatial and temporal noise of our FPT system is characterized to be 0.529 nm and 0.027 nm, respectively.

Beyond conventional microscopy: Observing kidney tissues by means of fourier ptychography

Kidney microscopy is a mainstay in studying the morphological structure, physiology and pathology of kidney tissues, as histology provides important results for a reliable diagnosis. A microscopy modality providing at same time high-resolution images and a wide field of view could be very useful for analyzing the whole architecture and the functioning of the renal tissue. Recently, Fourier Ptychography (FP) has been proofed to yield images of biology samples such as tissues and in vitro cells while providing high resolution and large field of view, thus making it a unique and attractive opportunity for histopathology. Moreover, FP offers tissue imaging with high contrast assuring visualization of small desirable features, although with a stain-free mode that avoids any chemical process in histopathology. Here we report an experimental measuring campaign for creating the first comprehensive and extensive collection of images of kidney tissues captured by this FP microscope. We show that FP microscopy unlocks a new opportunity for the physicians to observe and judge renal tissue slides through the novel FP quantitative phase-contrast microscopy. Phase-contrast images of kidney tissue are analyzed by comparing them with the corresponding renal images taken under a conventional bright-field microscope both for stained and unstained tissue samples of different thicknesses. In depth discussion on the advantages and limitations of this new stain-free microscopy modality is reported, showing its usefulness over the classical light microscopy and opening a potential route for using FP in clinical practice for histopathology of kidney.

Optical ptychography for biomedical imaging: recent progress and future directions [Invited]

Ptychography is an enabling microscopy technique for both fundamental and applied sciences. In the past decade, it has become an indispensable imaging tool in most X-ray synchrotrons and national laboratories worldwide. However, ptychography's limited resolution and throughput in the visible light regime have prevented its wide adoption in biomedical research. Recent developments in this technique have resolved these issues and offer turnkey solutions for high-throughput optical imaging with minimum hardware modifications. The demonstrated imaging throughput is now greater than that of a high-end whole slide scanner. In this review, we discuss the basic principle of ptychography and summarize the main milestones of its development. Different ptychographic implementations are categorized into four groups based on their lensless/lens-based configurations and coded-illumination/coded-detection operations. We also highlight the related biomedical applications, including digital pathology, drug screening, urinalysis, blood analysis, cytometric analysis, rare cell screening, cell culture monitoring, cell and tissue imaging in 2D and 3D, polarimetric analysis, among others. Ptychography for high-throughput optical imaging, currently in its early stages, will continue to improve in performance and expand in its applications. We conclude this review article by pointing out several directions for its future development.

Learning-based single-shot long-range synthetic aperture Fourier ptychographic imaging with a camera array

In this Letter, we report a new long-range synthetic aperture Fourier ptychographic imaging technique, termed learning-based single-shot synthetic aperture imaging (LSS-SAI). LSS-SAI uses a camera array to record low-resolution intensity images corresponding to different non-overlapping spectral regions in parallel, which are synthesized to reconstruct a super-resolved high-quality image based on a physical model-based dual-regression deep neural network. Compared with conventional macroscopic Fourier ptychographic imaging, LSS-SAI overcomes the stringent requirement on a large amount of raw data with a high spectral overlapping ratio for high-resolution, high signal-to-noise imaging of reflective objects with diffuse surfaces, making single-shot long-range synthetic aperture imaging possible. Experimental results on rough reflective samples show that our approach can improve the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) by 10.56 dB and 0.26, respectively. We also demonstrate the single-shot ptychography capability of the proposed approach by the synthetic aperture imaging of a dynamic scene at a camera-limited speed (30 fps). To the best of our knowledge, this is the first demonstration of macroscopic Fourier ptychography to single-shot synthetic aperture imaging of dynamic events.

Learning to Sense for Coded Diffraction Imaging

In this paper, we present a framework to learn illumination patterns to improve the quality of signal recovery for coded diffraction imaging. We use an alternating minimization-based phase retrieval method with a fixed number of iterations as the iterative method. We represent the iterative phase retrieval method as an unrolled network with a fixed number of layers where each layer of the network corresponds to a single step of iteration, and we minimize the recovery error by optimizing over the illumination patterns. Since the number of iterations/layers is fixed, the recovery has a fixed computational cost. Extensive experimental results on a variety of datasets demonstrate that our proposed method significantly improves the quality of image reconstruction at a fixed computational cost with illumination patterns learned only using a small number of training images.

High-resolution polarization-sensitive Fourier ptychography microscopy using a high numerical aperture dome illuminator

Polarization-sensitive Fourier-ptychography microscopy (pFPM) allows for high resolution imaging while maintaining a large field of view, and without mechanical movements of optical-setup components. In contrast to ordinary light microscopes, pFPM provides quantitative absorption and phase information, for complex and birefringent specimens, with high resolution across a wide field of view. Using a semi-spherical home-built LED illumination array, a single polarizer, and a 10x /0.28NA objective, we experimentally demonstrate high performance pFPM with a synthesized NA of 1.1. Applying the standard quantitative method, a measured half-pitch resolution of 244 nm is achieved for the 1951 USAF resolution test target. As application examples, the polarimetric properties of a herbaceous flowering plant and the metastatic carcinoma of human liver cells are analyzed and quantitatively imaged.

Batch-based alternating direction methods of multipliers for Fourier ptychography

Fourier ptychography (FP) has been developed as a general imaging tool for various applications. However, the redundancy data has to be enforced to get a stable recovery, leading to a large dataset and a high computational cost. Based on the additive property of the optical pupils in FP recovery, we report batch-based alternating direction methods of multipliers (ADMM) for FP reconstruction. The reported scheme is performed by implementing partial updates in sub-problems of the standard ADMM. We validate the reconstruction performance using both simulated and experimental measurements. Compared with the embedded pupil function recovery (EPRY) algorithm, the proposed algorithms can converge faster and produce higher-quality images.

High-fidelity intensity diffraction tomography with a non-paraxial multiple-scattering model

We propose a novel intensity diffraction tomography (IDT) reconstruction algorithm based on the split-step non-paraxial (SSNP) model for recovering the 3D refractive index (RI) distribution of multiple-scattering biological samples. High-quality IDT reconstruction requires high-angle illumination to encode both low- and high- spatial frequency information of the 3D biological sample. We show that our SSNP model can more accurately compute multiple scattering from high-angle illumination compared to paraxial approximation-based multiple-scattering models. We apply this SSNP model to both sequential and multiplexed IDT techniques. We develop a unified reconstruction algorithm for both IDT modalities that is highly computationally efficient and is implemented by a modular automatic differentiation framework. We demonstrate the capability of our reconstruction algorithm on both weakly scattering buccal epithelial cells and strongly scattering live C. elegans worms and live C. elegans embryos.

High-speed multi-objective Fourier ptychographic microscopy

The ability of a microscope to rapidly acquire wide-field, high-resolution images is limited by both the optical performance of the microscope objective and the bandwidth of the detector. The use of multiple detectors can increase electronic-acquisition bandwidth, but the use of multiple parallel objectives is problematic since phase coherence is required across the multiple apertures. We report a new synthetic-aperture microscopy technique based on Fourier ptychography, where both the illumination and image-space numerical apertures are synthesized, using a spherical array of low-power microscope objectives that focus images onto mutually incoherent detectors. Phase coherence across apertures is achieved by capturing diffracted fields during angular illumination and using ptychographic reconstruction to synthesize wide-field, high-resolution, amplitude and phase images. Compared to conventional Fourier ptychography, the use of multiple objectives reduces image acquisition times by increasing the area for sampling the diffracted field. We demonstrate the proposed scaleable architecture with a nine-objective microscope that generates an 89-megapixel, 1.1 µm resolution image nine-times faster than can be achieved with a single-objective Fourier-ptychographic microscope. New calibration procedures and reconstruction algorithms enable the use of low-cost 3D-printed components for longitudinal biological sample imaging. Our technique offers a route to high-speed, gigapixel microscopy, for example, imaging the dynamics of large numbers of cells at scales ranging from sub-micron to centimetre, with an enhanced possibility to capture rare phenomena.

Integration of Fourier ptychography with machine learning: an alternative scheme

As the core task of the reconstruction in conventional ptychography (CP) and Fourier ptychographic microscopy (FPM), the meticulous design of ptychographical iterative engine (PIE) largely affects the performance of reconstruction algorithms. Compared to traditional PIE algorithms, the paradigm of combining with machine learning to cross a local optimum has recently achieved significant progress. Nevertheless, existing designed engines still suffer drawbacks such as excessive hyper-parameters, heavy tuning work and lack of compatibility, which greatly limit their practical applications. In this work, we present a complete set of alternative schemes comprised of a kind of new perspective, a uniform design template, and a fusion framework, to naturally integrate Fourier ptychography (FP) with machine learning concepts. The new perspective, Dynamic Physics, is taken as the preferred tool to analyze a path (algorithm) at the physical level; the uniform design template, T-FP, clarifies the physical significance and optimization part in a path; the fusion framework follows two workable guidelines that are specially designed to keep convergence and make later localized modification for a new path, and further establishes a link between FP iterations and the gradient update in machine learning. Our scheme is compatible with both traditional FP paths and machine learning concepts. By combining ideas in both fields, we offer two design examples, MaFP and AdamFP. Results for both simulations and experiments show that designed algorithms following our scheme obtain better, faster (converge at the early stage after a few iterations) and more stable recovery with only minimal tuning hyper-parameters, demonstrating the effectiveness and superiority of our scheme.

Morphological variations to a ptychographic algorithm

Ptychography is a technique widely used in microscopy for achieving high-resolution imaging. This method relies on computational processing of images gathered from diffraction patterns produced by several partial illuminations of a sample. We numerically studied the effect of using different shapes for illuminating the aforementioned sample: convex shapes, such as circles and regular polygons, and unconnected shapes that resemble a QR code. Our results suggest that the use of unconnected shapes seems to outperform convex shapes in terms of convergence and, in some cases, accuracy.

Diffractive optical system design by cascaded propagation

Modern design of complex optical systems relies heavily on computational tools. These frequently use geometrical optics as well as Fourier optics. Fourier optics is typically used for designing thin diffractive elements, placed in the system's aperture, generating a shift-invariant Point Spread Function (PSF). A major bottleneck in applying Fourier Optics in many cases of interest, e.g. when dealing with multiple, or out-of-aperture elements, comes from numerical complexity. In this work, we propose and implement an efficient and differentiable propagation model based on the Collins integral, which enables the optimization of diffractive optical systems with unprecedented design freedom using backpropagation. We demonstrate the applicability of our method, numerically and experimentally, by engineering shift-variant PSFs via thin plate elements placed in arbitrary planes inside complex imaging systems, performing cascaded optimization of multiple planes, and designing optimal machine-vision systems by deep learning.

Pixel Super-Resolution Phase Retrieval for Lensless On-Chip Microscopy via Accelerated Wirtinger Flow

Empowered by pixel super-resolution (PSR) and phase retrieval techniques, lensless on-chip microscopy opens up new possibilities for high-throughput biomedical imaging. However, the current PSR phase retrieval approaches are time consuming in terms of both the measurement and reconstruction procedures. In this work, we present a novel computational framework for PSR phase retrieval to address these concerns. Specifically, a sparsity-promoting regularizer is introduced to enhance the well posedness of the nonconvex problem under limited measurements, and Nesterov’s momentum is used to accelerate the iterations. The resulting algorithm, termed accelerated Wirtinger flow (AWF), achieves at least an order of magnitude faster rate of convergence and allows a twofold reduction in the measurement number while maintaining competitive reconstruction quality. Furthermore, we provide general guidance for step size selection based on theoretical analyses, facilitating simple implementation without the need for complicated parameter tuning. The proposed AWF algorithm is compatible with most of the existing lensless on-chip microscopes and could help achieve label-free rapid whole slide imaging of dynamic biological activities at subpixel resolution.

Addressing phase-curvature in Fourier ptychography

In Fourier ptychography, multiple low resolution images are captured and subsequently combined computationally into a high-resolution, large-field of view micrograph. A theoretical image-formation model based on the assumption of plane-wave illumination from various directions is commonly used, to stitch together the captured information into a high synthetic aperture. The underlying far-field (Fraunhofer) diffraction assumption connects the source, sample, and pupil planes by Fourier transforms. While computationally simple, this assumption neglects phase-curvature due to non-planar illumination from point sources as well as phase-curvature from finite-conjugate microscopes (e.g., using a single-lens for image-formation). We describe a simple, efficient, and accurate extension of Fourier ptychography by embedding the effect of phase-curvature into the underlying forward model. With the improved forward model proposed here, quantitative phase reconstruction is possible even for wide fields-of-views and without the need of image segmentation. Lastly, the proposed method is computationally efficient, requiring only two multiplications: prior and following the reconstruction.

Perceptually motivated loss functions for computer generated holographic displays

Understanding and improving the perceived quality of reconstructed images is key to developing computer-generated holography algorithms for high-fidelity holographic displays. However, current algorithms are typically optimized using mean squared error, which is widely criticized for its poor correlation with perceptual quality. In our work, we present a comprehensive analysis of employing contemporary image quality metrics (IQM) as loss functions in the hologram optimization process. Extensive objective and subjective assessment of experimentally reconstructed images reveal the relative performance of IQM losses for hologram optimization. Our results reveal that the perceived image quality improves considerably when the appropriate IQM loss function is used, highlighting the value of developing perceptually-motivated loss functions for hologram optimization.

Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers

Fourier ptychographic microscopy (FPM) has risen as a promising computational imaging technique that breaks the trade-off between high resolution and large field of view (FOV). Its reconstruction is normally formulated as a blind phase retrieval problem, where both the object and probe have to be recovered from phaseless measured data. However, the stability and reconstruction quality may dramatically deteriorate in the presence of noise interference. Herein, we utilized the concept of alternating direction method of multipliers (ADMM) to solve this problem (termed ADMM-FPM) by breaking it into multiple subproblems, each of which may be easier to deal with. We compared its performance against existing algorithms in both simulated and practical FPM platform. It is found that ADMM-FPM method belongs to a global optimization algorithm with a high degree of parallelism and thus results in a more stable and robust phase recovery under noisy conditions. We anticipate that ADMM will rekindle interest in FPM as more modifications and innovations are implemented in the future.

High-resolution 3D Fourier ptychographic reconstruction using a hemispherical illumination source with multiplexed-coded strategy

Fourier ptychography is a promising and flexible imaging technique that can achieve 2D quantitative reconstruction with higher resolution beyond the limitation of the system. Meanwhile, by using different imaging models, the same platform can be applied to achieve 3D refractive index reconstruction. To improve the illumination NA as much as possible while reducing the intensity attenuation problem caused by the LED board used in the traditional FP platform, we apply a hemispherical lighting structure and design a new LED arrangement according to 3D Fourier diffraction theory. Therefore, we could obtain the illumination of 0.98NA using 187 LEDs and achieve imaging half-pitch resolutions of ∼174 nm and ∼524 nm for the lateral and axial directions respectively, using a 40×/0.6NA objective lens. Furthermore, to reduce the number of captured images required and realize real-time data collection, we apply the multiplexed-coded illumination strategy and compare several coded patterns through simulation and experiment. Through comparison, we determined a radial-coded illumination pattern that could achieve more similar results as sequential scanning and increase the acquisition speed to above 1 Hz. Therefore, this paper provides the possibility of this technique in real-time 3D observation of in vitro live samples.

Snapshot ptychography on array cameras

We use convolutional neural networks to recover images optically down-sampled by 6.7 × using coherent aperture synthesis over a 16 camera array. Where conventional ptychography relies on scanning and oversampling, here we apply decompressive neural estimation to recover full resolution image from a single snapshot, although as shown in simulation multiple snapshots can be used to improve signal-to-noise ratio (SNR). In place training on experimental measurements eliminates the need to directly calibrate the measurement system. We also present simulations of diverse array camera sampling strategies to explore how snapshot compressive systems might be optimized.

Accurate and stable two-step LED position calibration method for Fourier ptychographic microscopy

SIGNIFICANCE

Fourier ptychography microscopy (FPM) is a computational optical imaging technology that employs angularly varying illuminations and a phase retrieval algorithm to achieve a wide field of view and high-resolution imaging simultaneously. In the FPM, LED position error will reduce the quality of the reconstructed high-resolution image. To correct the LED positions, current methods consider each of the LED positions as independent and use an optimization algorithm to get each of the positions. When the positional misalignment is large or the search position falls into a local optimal value, the current methods may lack stability and accuracy.

AIM

We improve the model of the LED position and propose an accurate and stable two-step correction scheme (tcFPM) to calibrate the LED position error.

APPROACH

The improved LED positions model combines the overall offset, which represents the relative deviation of the LED array and the optical axis, with the slight deviation of each LED's independent position. In the tcFPM, the overall offset of the LED array is corrected at first, which obtains an approximate value of the overall offset of the LED array. Then the position of each LED is precisely adjusted, which obtains the slight offset of each LED.

RESULTS

This LED position error model is more in line with the actual situation. The simulation and experimental results show that the method has high accuracy in correcting the LED position. Furthermore, the reconstruction process of tcFPM is more stable and significantly improves the quality of the reconstruction results, which is compared with some LED position error correction methods.

CONCLUSIONS

An LED position error correction technology is proposed, which has a stable iterative process and improves the reconstruction accuracy of complex amplitude.

Generalized optimization framework for pixel super-resolution imaging in digital holography

The imaging quality of in-line digital holography is challenged by the twin-image and aliasing effects because sensors only respond to intensity and pixels are of finite size. As a result, phase retrieval and pixel super-resolution techniques serve as two essential ingredients for high-fidelity and high-resolution holographic imaging. In this work, we combine the two as a unified optimization problem and propose a generalized algorithmic framework for pixel-super-resolved phase retrieval. In particular, we introduce the iterative projection algorithms and gradient descent algorithms for solving this problem. The basic building blocks, namely the projection operator and the Wirtinger gradient, are derived and analyzed. As an example, the Wirtinger gradient descent algorithm for pixel-super-resolved phase retrieval, termed as Wirtinger-PSR, is proposed and compared with the classical error-reduction algorithm. The Wirtinger-PSR algorithm is verified with both simulated and experimental data. The proposed framework generalizes well to various physical settings and helps bridging the gap between empirical studies and theoretical analyses.

Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained "blind" ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

Parameter retrieval of small particles in dark-field Fourier ptychography and a rectangle in real-space ptychography

We present a parameter retrieval method which incorporates prior knowledge about the object into ptychography. The proposed method is applied to two applications: (1) parameter retrieval of small particles from Fourier ptychographic dark field measurements; (2) parameter retrieval of a rectangular structure with real-space ptychography. The influence of Poisson noise is discussed in the second part of the paper. The Cramér Rao Lower Bound in both applications is computed and Monte Carlo analysis is used to verify the calculated lower bound. With the computation results we report the lower bound for various noise levels and analyze the correlation of particles in application 1. For application 2 the correlation of parameters of the rectangular structure is discussed.

Double-flow convolutional neural network for rapid large field of view Fourier ptychographic reconstruction

Fourier ptychographic microscopy is a promising imaging technique which can circumvent the space-bandwidth product of the system and achieve a reconstruction result with wide field-of-view (FOV), high-resolution and quantitative phase information. However, traditional iterative-based methods typically require multiple times to get convergence, and due to the wave vector deviation in different areas, the millimeter-level full-FOV cannot be well reconstructed once and typically required to be separated into several portions with sufficient overlaps and reconstructed separately, which makes traditional methods suffer from long reconstruction time for a large-FOV (of the order of minutes) and limits the application in real-time large-FOV monitoring of live sample in vitro. Here we propose a novel deep-learning based method called DFNN which can be used in place of traditional iterative-based methods to increase the quality of single large-FOV reconstruction and reducing the processing time from 167.5 to 0.1125 second. In addition, we demonstrate that by training based on the simulation dataset with high-entropy property (Opt. Express 28, 24 152 [2020]), DFNN could has fine generalizability and little dependence on the morphological features of samples. The superior robustness of DFNN against noise is also demonstrated in both simulation and experiment. Furthermore, our model shows more robustness against the wave vector deviation. Therefore, we could achieve better results at the edge areas of a single large-FOV reconstruction. Our method demonstrates a promising way to perform real-time single large-FOV reconstructions and provides further possibilities for real-time large-FOV monitoring of live samples with sub-cellular resolution.

Accelerating ptychographic reconstructions using spectral initializations

Ptychography is a promising phase retrieval technique for label-free quantitative phase imaging. Recent advances in phase retrieval algorithms witnessed the development of spectral methods to accelerate gradient descent algorithms. Using spectral initializations on experimental data, for the first time, we report three times faster ptychographic reconstructions than with a standard gradient descent algorithm and improved resilience to noise. Coming at no additional computational cost compared to gradient-descent-based algorithms, spectral methods have the potential to be implemented in large-scale iterative ptychographic algorithms.

Neural network model assisted Fourier ptychography with Zernike aberration recovery and total variation constraint

SIGNIFICANCE

Fourier ptychography (FP) is a computational imaging approach that achieves high-resolution reconstruction. Inspired by neural networks, many deep-learning-based methods are proposed to solve FP problems. However, the performance of FP still suffers from optical aberration, which needs to be considered.

AIM

We present a neural network model for FP reconstructions that can make proper estimation toward aberration and achieve artifact-free reconstruction.

APPROACH

Inspired by the iterative reconstruction of FP, we design a neural network model that mimics the forward imaging process of FP via TensorFlow. The sample and aberration are considered as learnable weights and optimized through back-propagation. Especially, we employ the Zernike terms instead of aberration to decrease the optimization freedom of pupil recovery and perform a high-accuracy estimation. Owing to the auto-differentiation capabilities of the neural network, we additionally utilize total variation regularization to improve the visual quality.

RESULTS

We validate the performance of the reported method via both simulation and experiment. Our method exhibits higher robustness against sophisticated optical aberrations and achieves better image quality by reducing artifacts.

CONCLUSIONS

The forward neural network model can jointly recover the high-resolution sample and optical aberration in iterative FP reconstruction. We hope our method that can provide a neural-network perspective to solve iterative-based coherent or incoherent imaging problems.

Structure-dependent amplification for denoising and background correction in Fourier ptychographic microscopy

Fourier Ptychographic Microscopy (FPM) allows high resolution imaging using iterative phase retrieval to recover an estimate of the complex object from a series of images captured under oblique illumination. FPM is particularly sensitive to noise and uncorrected background signals as it relies on combining information from brightfield and noisy darkfield (DF) images. In this article we consider the impact of different noise sources in FPM and show that inadequate removal of the DF background signal and associated noise are the predominant cause of artefacts in reconstructed images. We propose a simple solution to FPM background correction and denoising that outperforms existing methods in terms of image quality, speed and simplicity, whilst maintaining high spatial resolution and sharpness of the reconstructed image. Our method takes advantage of the data redundancy in real space within the acquired dataset to boost the signal-to-background ratio in the captured DF images, before optimally suppressing background signal. By incorporating differentially denoised images within the classic FPM iterative phase retrieval algorithm, we show that it is possible to achieve efficient removal of background artefacts without suppression of high frequency information. The method is tested using simulated data and experimental images of thin blood films, bone marrow and liver tissue sections. Our approach is non-parametric, requires no prior knowledge of the noise distribution and can be directly applied to other hardware platforms and reconstruction algorithms making it widely applicable in FPM.

High-resolution and large field-of-view Fourier ptychographic microscopy and its applications in biomedicine

Fourier ptychographic microscopy (FPM) is a promising and fast-growing computational imaging technique with high resolution, wide field-of-view (FOV) and quantitative phase recovery, which effectively tackles the problems of phase loss, aberration-introduced artifacts, narrow depth-of-field and the trade-off between resolution and FOV in conventional microscopy simultaneously. In this review, we provide a comprehensive roadmap of microscopy, the fundamental principles, advantages, and drawbacks of existing imaging techniques, and the significant roles that FPM plays in the development of science. Since FPM is an optimization problem in nature, we discuss the framework and related work. We also reveal the connection of Euler's formula between FPM and structured illumination microscopy. We review recent advances in FPM, including the implementation of high-precision quantitative phase imaging, high-throughput imaging, high-speed imaging, three-dimensional imaging, mixed-state decoupling, and introduce the prosperous biomedical applications. We conclude by discussing the challenging problems and future applications. FPM can be extended to a kind of framework to tackle the phase loss and system limits in the imaging system. This insight can be used easily in speckle imaging, incoherent imaging for retina imaging, large-FOV fluorescence imaging, etc.

Adaptive high-dynamic-range Fourier ptychography microscopy data acquisition with a red-green-blue camera

Fourier ptychography microscopy is a powerful tool for wide-field and high-spatial-resolution imaging. It can achieve a large field of view and high-spatial-resolution imaging with a low numerical aperture objective by capturing a series of low-resolution (LR) images that contain the information of different spatial frequencies and then stitching them together in the Fourier domain. Furthermore, the phase information of the object can also be recovered simultaneously. In this Letter, we propose a method to realize adaptive high-dynamic-range (HDR) LR image acquisition with a red, green, and blue camera, which needs only single exposure for each light-emitting diode (LED) illumination. According to the imaging principle of a color camera, the filter of each color channel still allows additional light within a certain wavelength range to pass through, but with a much smaller transmittance. By illuminating the sample with monochromatic light and combining the raw data from three color channels together, an HDR image corresponding to each LED illumination is obtained. The feasibility and good performance of our method are demonstrated by the experimental results.

Acoustofluidic Scanning Nanoscope with High Resolution and Large Field of View

Optical imaging with nanoscale resolution and a large field of view is highly desirable in many research areas. Unfortunately, it is challenging to achieve these two features simultaneously while using a conventional microscope. An objective lens with a low numerical aperture (NA) has a large field of view but poor resolution. In contrast, a high NA objective lens will have a higher resolution but reduced field of view. In an effort to close the gap between these trade-offs, we introduce an acoustofluidic scanning nanoscope (AS-nanoscope) that can simultaneously achieve high resolution with a large field of view. The AS-nanoscope relies on acoustofluidic-assisted scanning of multiple microsized particles. A scanned 2D image is then compiled by processing the microparticle images using an automated big-data image algorithm. The AS-nanoscope has the potential to be integrated into a conventional microscope or could serve as a stand-alone instrument for a wide range of applications where both high resolution and large field of view are required.

Diffraction tomography with a deep image prior

We present a tomographic imaging technique, termed Deep Prior Diffraction Tomography (DP-DT), to reconstruct the 3D refractive index (RI) of thick biological samples at high resolution from a sequence of low-resolution images collected under angularly varying illumination. DP-DT processes the multi-angle data using a phase retrieval algorithm that is extended by a deep image prior (DIP), which reparameterizes the 3D sample reconstruction with an untrained, deep generative 3D convolutional neural network (CNN). We show that DP-DT effectively addresses the missing cone problem, which otherwise degrades the resolution and quality of standard 3D reconstruction algorithms. As DP-DT does not require pre-captured data or pre-training, it is not biased towards any particular dataset. Hence, it is a general technique that can be applied to a wide variety of 3D samples, including scenarios in which large datasets for supervised training would be infeasible or expensive. We applied DP-DT to obtain 3D RI maps of bead phantoms and complex biological specimens, both in simulation and experiment, and show that DP-DT produces higher-quality results than standard regularization techniques. We further demonstrate the generality of DP-DT, using two different scattering models, the first Born and multi-slice models. Our results point to the potential benefits of DP-DT for other 3D imaging modalities, including X-ray computed tomography, magnetic resonance imaging, and electron microscopy.

VIPR: vectorial implementation of phase retrieval for fast and accurate microscopic pixel-wise pupil estimation

In microscopy, proper modeling of the image formation has a substantial effect on the precision and accuracy in localization experiments and facilitates the correction of aberrations in adaptive optics experiments. The observed images are subject to polarization effects, refractive index variations, and system specific constraints. Previously reported techniques have addressed these challenges by using complicated calibration samples, computationally heavy numerical algorithms, and various mathematical simplifications. In this work, we present a phase retrieval approach based on an analytical derivation of the vectorial diffraction model. Our method produces an accurate estimate of the system's phase information, without any prior knowledge about the aberrations, in under a minute.

Fourier ptychography: current applications and future promises

Traditional imaging systems exhibit a well-known trade-off between the resolution and the field of view of their captured images. Typical cameras and microscopes can either "zoom in" and image at high-resolution, or they can "zoom out" to see a larger area at lower resolution, but can rarely achieve both effects simultaneously. In this review, we present details about a relatively new procedure termed Fourier ptychography (FP), which addresses the above trade-off to produce gigapixel-scale images without requiring any moving parts. To accomplish this, FP captures multiple low-resolution, large field-of-view images and computationally combines them in the Fourier domain into a high-resolution, large field-of-view result. Here, we present details about the various implementations of FP and highlight its demonstrated advantages to date, such as aberration recovery, phase imaging, and 3D tomographic reconstruction, to name a few. After providing some basics about FP, we list important details for successful experimental implementation, discuss its relationship with other computational imaging techniques, and point to the latest advances in the field while highlighting persisting challenges.