Fourier Ptychography

Reference: Guoan Zheng, Cheng Shen, Shaowei Jiang, Pengming Song, and Changhuei Yang, “Concept, implementations and applications of Fourier ptychography,” Nature Reviews Physics, 3, 207-223 (2021). (free-to-read version of the paper). Supplementary note on how to set up a Fourier ptychographic imaging platform. The experimental data and code can be downloaded here: data and code download. 

3. Implementations of FP

Table 1 categorizes different FP implementations into 4 groups. Figure 2 shows several representative schematics of these implementations. 

3.1 Strategies for performance improvement

The first group of implementations is related to the performance improvement of the original LED-array microscope setup. In the following, we will discuss strategies for improving the resolution, acquisition speed, device portability, and phase imaging capability. The resolution d of the complex amplitude in an FP system is determined by both the NA of the objective lens28. It is clear that higher FP resolution can be achieved by either increasing the objective NA or using an LED array with larger incident angles24,29-32. Ou et al. demonstrated the first high-NA FP system that used a 40× 0.75-NA objective, and a maximum illumination NA of 0.7 to synthesize a system NA of 1.4529. Similarly, Sun et al. demonstrated a 1.6-NA FP system using a 10× 0.4-NA objective and a 1.2-NA oil-immersion condenser lens24.

The acquisition speed of the original FP setup is limited by a large number of image acquisitions and the relatively long exposure time for acquiring the darkfield images. To reduce the number of acquisitions, better sampling strategies can be employed by selecting or arranging the LED illumination wavevectors in the Fourier space. One example is content-adaptive illumination33-35, where a subset of illumination angles is selected based on the spectral energy distribution in the Fourier space. This strategy is most effective for specimens that have a highly structured Fourier spectrum, such as the USAF resolution target. It may be ineffective for biological samples whose frequency contents are often uniformly expanded in the Fourier space. Another strategy for reducing the number of acquisitions is to perform non-uniform sampling in the Fourier domain (left panel of Fig. 2a)36. As most signal energy concentrates at the central region of the Fourier spectrum, a higher density of sampling points ( s can be employed for the central region. For darkfield images with large incident angles, the aperture overlap between adjacent acquisitions can be reduced to 20% or less. With the same Fourier bandwidth coverage, this strategy can substantially reduce the number of acquisitions from 137 in the original FP setup to 6836. We also note that, the sampling points ( s in the original FP setup form a quasi-periodic grid in the Fourier domain. Such a periodic scan pattern is susceptible to image artifacts known as raster scan pathology13. Non-uniform sampling strategy can also address this issue by breaking the translational symmetry of the sampling points in the Fourier domain. The third example for reducing the number of acquisitions is to perform multiplexed sampling in the Fourier domain. In this case, multiple LEDs with different incident angles38,40 or different colours37,39,40 can be turned on at the same time for sample illumination (Fig. 2b). Tian et al. demonstrated a high-speed in vitro FP implementation41 using this strategy (right panel of Fig. 2b). In this implementation, an initial guess of the complex object was generated via differential-phase-contrast imaging132, which was performed by turning on half of the LED array for image acquisitions. With a good initial guess, it is possible to recover the high-resolution complex object with multiplexed darkfield measurements. 

The use of the LED array in the original FP setup leads to a relatively long exposure time for acquiring the darkfield images. To improve the light delivering efficiency, the LED elements can also be angled towards the sample or arranged on a hemisphere dome (right panel of Fig. 2a)30-32. Another strategy to shorten the exposure time is to use a steerable laser beam, in place of the LED array, for sample illumination43-45. The use of laser allows the intensity to be substantially increased beyond the limitation of LEDs. To this end, Chung et al. demonstrated the use of a guided laser beam for angle-varied illumination44. The scheme is shown in the left panel of Fig. 2c, where a mirror array and a 2D Galvo scanner are used to provide planewave illuminations with centimeter beam size at diverse incidence angles. Similarly, a digital mirror device can be coded to generate angle-varied illumination at high speed (right panel of Fig. 2c)43,45. In this case, the beam size and the FOV is limited by the projection lens. For low-light applications, single-photon detection can also be employed in FP to reduce the exposure time and improve the signal-to-noise ratio133.     

The FP approach can be implemented in portable and turnkey systems. For example, low-cost lenses can be used to develop high-resolution imaging systems. Dong et al. demonstrated a 3D-printed, field-portable FP microscope using a cellphone lens47. More recently, Aidukas et al. demonstrated the use of Raspberry Pi and its low-cost camera module for building an FP microscope with sub-micron resolution49. The FOV of most FP implementations discussed so far is limited by the size of the image sensor. A robotic scanning system can be integrated with FP for expanding the FOV. To this end, Guo et al. demonstrated a low-cost whole slide scanner using an LED array for sample illumination50. It can be used for both regular incoherent microscopy and coherent FP imaging. 

Phase retrieval process generally requires a good initial guess. In FP, this initial guess can be created by using the captured low-resolution images with the phase set to a constant1. Over the past few years, there have been innovations on improving the initial guess. Tian et al.41 reported an improved initialization that uses the differential-phase-contrast imaging132 to generate the initial phase. Another strategy is through annular illumination40,51. In this approach, the illumination angle is close to the maximum acceptance angle of the objective, i.e., images are captured near the brightfield-to-darkfield transition region. The phase information of the specimen, especially for the low-frequency phase components, can be effectively converted into the intensity variations for detection51.

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Table 1| Summary of different FP implementations

Improving the performance of the original FP implementation

▪   Improving the resolution: larger incident angles29-32, oil-immersion condenser lens24.

▪   Improving the acquisition speed: content-adaptive illumination33-35, nonuniform sampling strategy in the Fourier plane36, spectral and angular multiplexing37-40, multiplexed source coding41, symmetrical illumination42, rapid laser beam scanning43-45.

▪   Improving the system portability: beam steering via a low-cost liquid crystal display46, a cellphone-lens system47, a moldless-lens system48, a Raspberry Pi system49, and a low-cost whole slide scanner with LED array illumination50.

▪   Improving phase imaging capability: better initial guess of phase via different-phase-contrast imaging41, annular illumination40,51.

Reconstruction approaches and algorithms

▪   System corrections: LED intensity correction52,53, LED alignment calibration and incident angle correction53-58, correcting under- and over-exposed pixels7, joint sample-pupil recovery59-62, sample motion correction63, stray light detection64, vignetting effect65.

▪   Phase retrieval algorithms: stochastic gradient descent59, Gauss-Newton method with amplitude-based cost function57, adaptive step size for gradient descent66, Wirtinger gradient67-71, convex relaxation72, regularization via sparsity73,74, noise modelling and denoising68,69,75,76, stochastic proximal gradient descent77, low-rank recovery78.

▪   Neural network and related approaches: high-resolution image recovery from low-resolution FP measurements via deep neural networks79-84, incorporating physical model for designing the coded illumination pattern85-89, neural network model for the optimization process (also known as automatic differentiation)90-93, virtual brightfield / fluorescence staining and other unsupervised image style transfer of the FP reconstructions94.

New hardware implementations

▪   Aperture-scanning FP: mechanical scanning of a confined aperture61,95,96 and a diffuser97, digital scanning via a spatial light modulator60,98,99, dual-wavelength implementation for phase unwrapping96, and single-shot implementation100.

▪   Long-range super-resolution imaging: camera-scanning FP95,101,102.

▪   Reflective implementations: angle-varied illumination103-107, aperture scanning61,96.

▪   Multi cameras: 6108 and 96 cameras109 for parallel acquisition, Scheimpflug arrangement110.

▪   Single-shot implementations: diffraction grating100,111, lens array112, colour-multiplexing40.

▪   Translated speckle illumination for super-resolution microscopy and long-range imaging: fluorescence113-115, coherent imaging115,116, far-field speckle scanning116,117.

▪   X-ray imaging: pinhole scanning118, lens and detector translation118-120, Bragg configuration121.  

Addressing the model limitation: 2D wave-object multiplication

▪   Better forward imaging model: multi-slice modelling9,122,123, diffraction tomography124-129.

▪   Modulating the exit wave at the detection path: aperture-scanning FP61,95-98, diffuser modulation for coherent super-resolution microscopy130,131.

3.2 Reconstruction approaches and algorithms

A successful FP reconstruction relies on an accurate imaging model of the optical system. 3 common system imperfections of an FP setup can be corrected in the iterative reconstruction process: 1) LED intensity variations, 2) incident angles of the LED elements, and 3) pupil aberrations. 

For correcting LED intensity variations, Bian et al. demonstrated the use of an image-quality metric to update the LED intensity during the iterative reconstruction process52. The same metric can also be used to recover the Zernike modes of pupil aberration. For correcting incident angles from different LED elements, simple alignment of the LED array can be performed following the procedure discussed in Box 2. This alignment procedure is often sufficient for LED arrays that have a well-defined pitch. For other illuminators without a well-defined pitch, further refinement of the illumination wavevector is needed in the iterative recovery process53-58. This wavevector refinement process is similar to the positional correction process in real-space ptychography23. For pupil aberration correction, one can directly measure the spatially varying aberrations using a calibration target134. Another solution is to jointly recover both the object and the pupil in the iterative process59. Field-dependent nature of the optical pupil can also be leveraged for a more robust full-field aberration recovery62.  

Different phase retrieval algorithms have also been implemented and tested for the FP platform. Table 1 summarizes some of these developments. Despite its simplicity, alternating projection remains an effective algorithm for phase retrieval. More advanced optimization strategies for ptychographic phase retrieval have been discussed by Chao and Marchesini et al.135 in a wide-ranging survey covering conjugate gradient, Newton-type second-order optimization, set projection approaches, and the relaxed average alternating reflections method. Yeh et al.57 has performed a comprehensive review of first- and second-order optimization approaches for FP datasets. It was shown that the second-order Gauss-Newton method with amplitude-based cost function gave the best results in general. In practice, given a carefully conducted FP experiment with reasonably clean data8, the phase retrieval process is typically well conditioned and amenable to solution by any optimization technique – from alternating projection to nonlinear heavyweights. 

Another trend of algorithm development related to FP is the employment of data-driven approaches or neural networks in the imaging and reconstruction process. These developments can be categorized into 4 groups. The first group focus on inferring the high-resolution intensity and/or the phase images from the low-resolution FP measurements79-84. For example, Nguyen et al.79 and Zhang et al.84 demonstrated the use of deep neural networks to produce high-resolution images from FP raw measurements. More recently, Xue et al. demonstrated the use of a Bayesian network to output both the high-resolution phase images and the uncertainty maps that quantifies the uncertainty of the predictions82. 

The second group focuses on incorporating the physical model into the design of the network85-89. The training of the network can jointly optimize the physical parameters used in the acquisition, such as the coded illumination pattern of the LED array. Horstmeyer et al. demonstrated the use of a neural network to jointly optimize the LED array illumination to highlight important sample features for classification task88. Kellman et al. demonstrated a framework to create interpretable context-specific LED source patterns for FP illumination89. 

The third group of developments is to model the forward FP imaging problem using neural networks and perform optimization via a network training process90-93. This process is also termed automatic differentiation or algorithmic differentiation, where derivatives can be efficiently evaluated based on the forward model136. For FP, the complex object can be treated as the learnable weights of a convolutional layer. The output of the network can be modeled as the loss function one aims to minimize. Different solvers and acceleration strategies from machine-learning libraries can then be adopted in the optimization process. 

Lastly, the recovered image of FP can be passed through a neural network for further improvement. For example, virtual brightfield and fluorescence staining can be performed on FP recovered images without paired training data94. Coherent artifacts of FP reconstructions can also be reduced in this unsupervised image translation process.

3.3 New hardware implementations

LED-array microscope is not the only approach for implementing FP3. Various hardware-based FP schemes have been reported in the past years. We will discuss 7 representative examples in the following. The first example is aperture-scanning FP, where the specimen is illuminated by a fixed planewave and an aperture is placed at the Fourier plane of the optical system (Fig. 2d). By translating the aperture to different lateral positions in the Fourier plane, images can be captured and synthesized in the Fourier domain to perform FP reconstruction. The key innovation of this scheme is that the recovered image depends upon how the complex wavefront exits the sample instead of entering it. Thus, the sample thickness becomes irrelevant during reconstruction. One can back-propagate the exit wave to any plane along the optical axis for 3D imaging. Dong et al. first demonstrated this scheme by using a mechanical stage to translate the aperture (left panel of Fig. 2d)95. Ou et al. later adopted a digital scanning scheme using a spatial light modulator (middle panel of Fig. 2d)98. Other related development includes scanning a diffuser at the Fourier plane (right panel of Fig. 2d)97, dual-wavelength implementation for phase unwrapping96, and single-shot implementation using a grating100. 

The second example is camera-scanning FP for long-range super-resolution imaging95,101,102. In this approach (Fig. 2e), the specimen is placed at the far-field and a photographic camera is used to capture images. Far-field propagation of the light waves corresponds to the operation of a Fourier transform. As such, the lens aperture placed at the far-field serves as a confined aperture constraint in the Fourier domain. By scanning the entire camera to different positions, one can capture multiple images corresponding to different circular apertures at the Fourier plane. These images can then be synthesized to form a super-resolution exit wavefront of the object. The final resolution is not determined by the lens aperture. Instead, it is determined by how far one can translate the camera. In the original demonstration, Dong et al. first experimented with a transmission mode (left panel of Fig. 2e)95. Holloway et al. later demonstrated it in a reflection mode for imaging diffused objects (right panel of Fig. 2e)101.

The third example is reflective implementations61,96,103-107. Angle-varied illumination in the original FP setup can be implemented in the epi-illumination arm in a reflection mode. A LED ring can be mounted outside the objective lens for sample illumination with large incident angles103,106. Other related developments include an off-axis configuration with angle-varied illumination107, and the implementation of aperture-scanning FP in a reflection mode61. 

The fourth example is multi-camera implementation. Chan et al. demonstrated a high-throughput screening system using 96 repeating low-cost microscope units (left panel of Fig. 2g)109. This platform can image the entire 96-well plate at a rate of 0.7 frames per second. Another configuration was reported by Konda et al., where a camera array was arranged according to the Scheimpflug principle (right panel of Fig. 2g)110. Using multiple cameras in an FP platform has the potential to substantially increase the imaging throughput. 

The fifth example is the single-shot FP implementation shown in Fig. 2h. Lee et al. demonstrated a single-shot FP scheme based on a lens array and the angular multiplexing strategy (left panel of Fig. 2h)112. By placing the lens array at the Fourier plane of the objective lens, multiple intensity profiles can be collected in a single shot. Similarly, single-shot implementations can also be achieved via a diffraction grating100,111  (right panel of Fig. 2h) and colour-multiplexed annular illumination40. These different implementations essentially trade off the imaging FOV to achieve the single-shot capability.  

The sixth example is the speckle illumination scheme shown in Fig. 2i. Here the speckle pattern on the specimen can be considered as an amalgamation of localized phase gradients, each approximating a plane wave incident at a different angle. A lateral translation of the speckle or the object causes a given region of the specimen to be illuminated by a different phase gradient, similar to the original angle-varied illumination concept in FP. One key distinction is that the translated speckle illumination scheme can be used for both coherent and incoherent imaging. When it is implemented for incoherent fluorescence imaging, it is conceptually similar to structured illumination microscopy137. The final archivable resolution is jointly determined by the NA of the objective lens and the feature size of the speckle pattern. Dong et al. first demonstrated this scheme for fluorescence imaging beyond the diffraction limit of the employed objective lens113. Guo et al. further demonstrated the use of a high-NA condenser to generate speckle patterns and a low-NA objective lens to acquire fluorescence images114. A 4-fold resolution gain was achieved for a fluorescence microscope setup and a 13-fold resolution gain was achieved for imaging through a low-pass diffusing layer. More recently, this concept has been extended to coherent imaging systems115,116. Zhang et al. used a 0.12-NA objective lens for image acquisition and achieve a synthetic NA of 0.85 for the reconstructed images (left panel of Fig. 2i)116. Interestingly, the translated speckle scheme can also be used for long-range super-resolution imaging. As shown in the right panel of Fig. 2i, a translated speckle pattern is projected on the remote objects and a regular photographic lens is used for image acquisition. Based on the modulation process of the projected speckle pattern, the otherwise inaccessible high-resolution object information can now be encoded into the captured images. This scheme has been validated for both coherent116 and incoherent long-range imaging settings116,117. Up to 7-fold resolution improvement has been experimentally demonstrated116. 

Lastly, the FP method can be implemented in coherent X-ray microscopy for super-resolution nano-scale imaging118-121. Figure 2j shows two representative schemes of X-ray FP. The first one employed a scanning pinhole to select different illumination angles from a condenser’s different subfields118. The resolution of this scheme remains limited by the combined NA of the condenser and objective. The second scheme scans both the objective lens and detector through the diffraction beam118-120. It is conceptually similar to the camera-scanning implementation in Fig. 2e. For X-ray imaging, scanning the lens and the detector avoids the complexity of keeping the illumination constant while changing the incident angle. The achievable resolution is limited by the scan range, which extends the NA of the objective, in effect forming a synthetic lens and exceeding the NA of any available X-ray optics. The X-ray FP experiment also exhibits unexpected high dose efficiency compared to the regular transmission X-ray microscopy118. More recently, a Fourier synthesis strategy similar to FP has also been implemented in a Bragg coherent X-ray diffraction imaging experiment121. Resolution improvement in 3D has been demonstrated for crystalline materials.    

3.4 Addressing the model limitation

In the forward imaging model of FP discussed in Box 1, the complex object is multiplied with the incident planewave before passing through the low-pass microscope system. This multiplication process assumes the object section is infinitesimally thin. For any practical object, tilting the illumination planewave would change the spectrum of the object rather than just shifting it in the Fourier space. In a wave optics picture, tilting the illumination would rock the Ewald sphere around the origin. The objective lens lets through a solid angle of scattered beams that subsequently form an image at the detector plane. As such, each captured image corresponds to a spherical cap of the Ewald sphere in the Fourier space. These spherical caps do not span a single 2D plane in the Fourier domain. There are two strategies to address this problem. 

The first strategy is to adopt a better forward imaging model by considering the 3D nature of the object. Multi-slice modelling, originally proposed for electron microscopy138, is one example of this strategy and it has shown great success in real-space ptychography22,139. In this approach, the 3D object is represented by a series of 2D thin slices separated by a certain distance. The incident wave from the light source multiplies with the first layer to form an exit wave. The exit wave is then propagated to the second layer, where it forms a new incident wave. This process goes through the whole specimen. Tian et al.122 and Li et al.9 have demonstrated the use of multi-slice modelling to image 3D specimens using an FP microscope setup. Its advantage is that it does not make any weak or single-scattering approximations. The serial scattering from each slice accounts for multiple scattering effect in the forward direction. Backscattered light has not been considered in current implementations. 

Another approach to account for the 3D nature of the object is diffraction tomography. In this approach, the object is represented by 3D scattering potential. Under the first-order Born or Rytov approximations, the captured images in an FP microscope can be used to update the corresponding spherical cap region of the 3D scattering potential in the Fourier domain. Horstmeyer et al. reported the first implementation of diffraction tomography with FP124. Zuo et al. further employ darkfield measurements with large illumination NA to demonstrate a high-throughput FP diffraction tomography platform. 

Despite these successes, isotropic resolution remains elusive for these two methods due to the missing cone problem. Specifically, the 3D coherent transfer function of an imaging system under single or weak scattering approximations has a donut shape due to the limited aperture. The missing frequency near the origin along the kz direction in the Fourier domain results in a loss of spatial resolution along the z axis. It has been shown that multiple scattering can mitigate the missing cone problem, hence the multi-slice model may perform better in principle123. Computationally filling in the missing cone via some priors is also a possible strategy for both approaches. Similarly, a better forward imaging model can bring in reconstruction with higher fidelity. Thus, future developments include a multi-slice model considering backward scattering and a diffraction tomography model considering multiple scattering. A challenge for both approaches is, perhaps, the high computational demand in the reconstruction process, which often can take hours or days to recover the full 3D volume. 

The second strategy for addressing the 2D modelling problem in FP is to keep the illumination angle fixed and perform modulation at the detection path. In this case, the recovered complex image represents the exit wavefront of the object. The thickness of the object is irrelevant in the modelling. The aperture-scanning and camera-scanning FP are two examples of this strategy (Fig. 2d and 2e). For aperture-scanning FP, however, the achievable resolution is limited by the NA of the first objective lens. A recent development to address this resolution limitation is to translate a thin diffuser in-between the specimen and the objective lens, as shown in Fig. 2k130,131. In this approach, the diffuser serves as a thin scattering lens for lightwave modulation. The multiplication process between the object and the tilted planewave in the original FP setup now becomes the multiplication between the object exit wavefront and the thin diffuser profile. This multiplication process allows the otherwise inaccessible high-resolution object information to be encoded into the captured images. To this end, Song et al. demonstrated a 4.5-fold resolution improvement over the diffraction limit of the objective lens using this approach130. It was also shown that a 4-fold resolution gain can be achieved with as few as ~30 raw images. 

5. Outlook 

Real-space ptychography has become an indispensable imaging modality in most X-ray synchrotron laboratories worldwide. We anticipate that FP, currently in its early stage, will continue to grow and expand in applications. We identified the following areas for potential further developments of the FP approach. 

Algorithms for blind FP reconstruction. Current challenging cases for blind FP reconstruction include specimens with many phase wraps (e.g. rough reflected surfaces), optical systems with severe unknown aberrations, light sources with low spatial and temporal coherence, measurements with interference fringe or other artefacts, and combinations of the above conditions. There have been attempts to convert the nonconvex phase retrieval problem to a convex optimization problem. One prominent example is PhaseLift145, which has been implemented in an FP setting72. In this approach, convex relaxation is needed to replace the rank constraint by a norm. For certain phase retrieval algorithms, it has been shown that the conditions guarantee the correspondence between the nonconvex problem and its convex relaxation are also sufficient to guarantee that the nonconvex problem does not have local minimums146. Therefore, recasting to convex relaxations may not necessarily give a better result compared to the nonconvex ones. The developments of memory-efficient nonconvex tools for handling these challenging cases are highly desired for the development of both FP and real-space ptychography. 

Implementations of FP in the EUV regime. The adoption of 13.5 nm EUV lithography has become a new standard for manufacturing new generation of silicon chips. Identifying the defect in EUV mask is a challenging task and the related tools are among the most expensive ones in semiconductor fabrication facility. It is also important to perform the inspection at the same 13.5-nm wavelength since defects are often wavelength-sensitive. Early development has demonstrated the use of angle-varied illumination to improve the resolution107. We envision the lens translation strategy in X-ray FP can also be implemented for EUV light. 

FP for chemical imaging. Infrared wavelengths contain vibration modes of many molecular bonds. The higher wavenumber region is associated with stretching vibrations such as S-H, C-H, N-H, and O-H, whereas the lower wavenumber regions typically correspond to bending and carbon skeleton fingerprint vibrations. Implementing FP in this region can find important applications in chemical imaging. A recent theoretical analysis has also shown the potential of applying FP for coherent anti-Stokes Raman scattering147. If successful, it can be an efficient approach for imaging thin tissue sections on specific Raman vibration transitions. 

FP for electron microscopy. Current tilted-series method in electron microscopy acquires images under angle-varied illuminations. However, it performs image reconstruction using linear restoring filters26,27. The iterative phase retrieval process in FP can be employed in tilted series reconstruction for 2D thin specimens. Fourier ptychographic diffraction tomography124,127 or multi-slice modelling9,122,123 can be used to handle 3D specimens. 

FP for education. The concept of FP can help students understand the basics of Fourier optics, including coherent transfer function, NA, pupil aberration, Ewald sphere, etc. The FP approach has become a sub-chapter in Goodman’s textbook “Introduction to Fourier Optics (4th edition)”. We envision the development of an FP microscope to be part of an optics lab course. 

Other applications. Many FP applications remain to be seen. To name a few, the aberration metrology capability59,62,99 can be used to characterize the lens profile and the related optical transfer function. The high-throughput 96-camera platform109 can be used for drug screening. The QPI capability in 2D and 3D can be used to detect bloodborne and waterborne pathogens. Sample rotation can be integrated with the transmission X-ray FP for 3D nanoscopy. Diffraction tomography can be integrated with angle-varied illumination for imaging diffused reflective objects.

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