Transport of intensity equation: a tutorial

Fourier ptychography (FP) utilizes illumination control and computational post-processing to increase the resolution of bright-field microscopes. In effect, FP extends the fixed numerical aperture (NA) of an objective lens to form a larger synthetic system NA. Here, we build an FP microscope (FPM) using a 40X 0.75NA objective lens to synthesize a system NA of 1.45. This system achieved a two-slit resolution of 335 nm at a wavelength of 632 nm. This resolution closely adheres to theoretical prediction and is comparable to the measured resolution (315 nm) associated with a standard, commercially available 1.25 NA oil immersion microscope. Our work indicates that Fourier ptychography is an attractive method to improve the resolution-versus-NA performance, increase the working distance, and enlarge the field-of-view of high-resolution bright-field microscopes by employing lower NA objectives.

High numerical aperture Fourier ptychography: principle, implementation and characterization.

We report an imaging scheme, termed aperture-scanning Fourier ptychography, for 3D refocusing and super-resolution macroscopic imaging The reported scheme scans an aperture at the Fourier plane of an optical system and acquires the corresponding intensity images of the object The acquired images are then synthesized in the frequency domain to recover a high-resolution complex sample wavefront; no phase information is needed in the recovery process We demonstrate two applications of the reported scheme In the first example, we use an aperture-scanning Fourier ptychography platform to recover the complex hologram of extended objects The recovered hologram is then digitally propagated into different planes along the optical axis to examine the 3D structure of the object We also demonstrate a reconstruction resolution better than the detector pixel limit (ie, pixel super-resolution) In the second example, we develop a camera-scanning Fourier ptychography platform for super-resolution macroscopic imaging By simply scanning the camera over different positions, we bypass the diffraction limit of the photographic lens and recover a super-resolution image of an object placed at the far field This platform’s maximum achievable resolution is ultimately determined by the camera’s traveling range, not the aperture size of the lens The FP scheme reported in this work may find applications in 3D object tracking, synthetic aperture imaging, remote sensing, and optical/electron/X-ray microscopy

Aperture-scanning Fourier ptychography for 3D refocusing and super-resolution macroscopic imaging

The incremental gradient approaches, such as PIE and ePIE, are widely used in the field of ptychographic imaging due to their great flexibility and computational efficiency. Nevertheless, their stability and reconstruction quality may be significantly degraded when non-negligible noise is present in the image. Though this problem is often attributed to the non-convex nature of phase retrieval, we found the reason for this is more closely related to the choice of the step-size, which needs to be gradually diminishing for convergence even in the convex case. To this end, we introduce an adaptive step-size strategy that decreases the step-size whenever sufficient progress is not made. The synthetic and real experiments on Fourier ptychographic microscopy show that the adaptive step-size strategy significantly improves the stability and robustness of the reconstruction towards noise yet retains the fast initial convergence speed of PIE and ePIE. More importantly, the proposed approach is simple, nonparametric, and does not require any preknowledge about the noise statistics. The great performance and limited computational complexity make it a very attractive and promising technique for robust Fourier ptychographic microscopy under noisy conditions.

Adaptive step-size strategy for noise-robust Fourier ptychographic microscopy.

Fluorescence microscopy plays a vital role in modern biological research and clinical diagnosis. Here, we report an imaging approach, termed pattern-illuminated Fourier ptychography (FP), for fluorescence imaging beyond the diffraction limit of the employed optics. This approach iteratively recovers a high-resolution fluorescence image from many pattern-illuminated low-resolution intensity measurements. The recovery process starts with one low-resolution measurement as the initial guess. This initial guess is then sequentially updated by other measurements, both in the spatial and Fourier domains. In the spatial domain, we use the pattern-illuminated low-resolution images as intensity constraints for the sample estimate. In the Fourier domain, we use the incoherent optical-transfer-function of the objective lens as the object support constraint for the solution. The sequential updating process is then repeated until the sample estimate converges, typically for 5-20 times. Different from the conventional structured illumination microscopy, any unknown pattern can be used for sample illumination in the reported framework. In particular, we are able to recover both the high-resolution sample image and the unknown illumination pattern at the same time. As a demonstration, we improved the resolution of a conventional fluorescence microscope beyond the diffraction limit of the employed optics. The reported approach may provide an alternative solution for structure illumination microscopy and find applications in wide-field, high-resolution fluorescence imaging.

High-resolution fluorescence imaging via pattern-illuminated Fourier ptychography

Information multiplexing is important for biomedical imaging and chemical sensing. In this paper, we report a microscopy imaging technique, termed state-multiplexed Fourier ptychography (FP), for information multiplexing and coherent-state decomposition. Similar to a typical Fourier ptychographic setting, we use an array of light sources to illuminate the sample from different incident angles and acquire corresponding low-resolution images using a monochromatic camera. In the reported technique, however, multiple light sources are lit up simultaneously for information multiplexing, and the acquired images thus represent incoherent summations of the sample transmission profiles corresponding to different coherent states. We show that, by using the state-multiplexed FP recovery routine, we can decompose the incoherent mixture of the FP acquisitions to recover a high-resolution sample image. We also show that, color-multiplexed imaging can be performed by simultaneously turning on R/G/B LEDs for data acquisition. The reported technique may provide a solution for handling the partially coherent effect of light sources used in Fourier ptychographic imaging platforms. It can also be used to replace spectral filter, gratings or other optical components for spectral multiplexing and demultiplexing. With the availability of cost-effective broadband LEDs, the reported technique may open up exciting opportunities for computational multispectral imaging.

Spectral multiplexing and coherent-state decomposition in Fourier ptychographic imaging

Fourier ptychography (FP) is a recently developed imaging approach that facilitates high-resolution imaging beyond the cutoff frequency of the employed optics. In the original FP approach, a periodic LED array is used for sample illumination, and therefore, the scanning pattern is a uniform grid in the Fourier space. Such a uniform sampling scheme leads to 3 major problems for FP, namely: 1) it requires a large number of raw images, 2) it introduces the raster grid artefacts in the reconstruction process, and 3) it requires a high-dynamic-range detector. Here, we investigate scanning sequences and sampling patterns to optimize the FP approach. For most biological samples, signal energy is concentrated at low-frequency region, and as such, we can perform non-uniform Fourier sampling in FP by considering the signal structure. In contrast, conventional ptychography perform uniform sampling over the entire real space. To implement the non-uniform Fourier sampling scheme in FP, we have designed and built an illuminator using LEDs mounted on a 3D-printed plastic case. The advantages of this illuminator are threefold in that: 1) it reduces the number of image acquisitions by at least 50% (68 raw images versus 137 in the original FP setup), 2) it departs from the translational symmetry of sampling to solve the raster grid artifact problem, and 3) it reduces the dynamic range of the captured images 6 fold. The results reported in this paper significantly shortened acquisition time and improved quality of FP reconstructions. It may provide new insights for developing Fourier ptychographic imaging platforms and find important applications in digital pathology.

Optimization of sampling pattern and the design of Fourier ptychographic illuminator.

The large consumer market has made cellphone lens modules available at low-cost and in high-quality. In a conventional cellphone camera, the lens module is used to demagnify the scene onto the image plane of the camera, where image sensor is located. In this work, we report a 3D-printed high-resolution Fourier ptychographic microscope, termed FPscope, which uses a cellphone lens in a reverse manner. In our platform, we replace the image sensor with sample specimens, and use the cellphone lens to project the magnified image to the detector. To supersede the diffraction limit of the lens module, we use an LED array to illuminate the sample from different incident angles and synthesize the acquired images using the Fourier ptychographic algorithm. As a demonstration, we use the reported platform to acquire high-resolution images of resolution target and biological specimens, with a maximum synthetic numerical aperture (NA) of 0.5. We also show that, the depth-of-focus of the reported platform is about 0.1 mm, orders of magnitude longer than that of a conventional microscope objective with a similar NA. The reported platform may enable healthcare accesses in low-resource settings. It can also be used to demonstrate the concept of computational optics for educational purposes.

Pariksheet Nanda

Papers

Spectral multiplexing and coherent-state decomposition in Fourier ptychographic imaging

High-resolution fluorescence imaging via pattern-illuminated Fourier ptychography

Optimization of sampling pattern and the design of Fourier ptychographic illuminator.

FPscope: a field-portable high-resolution microscope using a cellphone lens

Spectrum multiplexing and coherent-state decomposition in Fourier ptychographic imaging