Phase retrieval is an important tool to unveil wavefront of light, especially in high performance microscopy such as Fourier ptychographic microscopy (FPM). In general phase-retrieval methods, the resolution and the number of measurements are in a trade-off relationship. Inspired by FPM, we devise what we believe is a novel microscopic phase-retrieval method, termed single-shot FPM (SSFPM). In our approach, the imaging performance exceeds the trade-off relationship in that it enables phase retrieval for high resolution with a single measurement. By placing the lens array at the Fourier plane of the objective lens, multiple intensity profiles required for the FPM algorithm are collected in a single shot. To achieve enough redundancy of data for satisfying convergence condition of FPM, the specimen is simultaneously illuminated by multiple light-emitting diodes. SSFPM reconstructs quantitative phase profile and enhances the resolution sacrificed by applying lens-array imaging. We demonstrate the performance of SSFPM with numerical simulation and experiments. The prototype achieves lateral resolution of 3.10 μm over a field of view of . Without an interferometer or scanning devices, SSFPM can reconstruct high resolution of a complex profile with a single shot.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Phase retrieval is a prominent method to discover three-dimensional (3D) distribution of nonabsorbing samples such as biological cells, tissues, and micro-elements [1–3]. While only the intensity profile is measured in general microscopic imaging systems, there is research to retrieve the phase of transmitted or reflected light, which can be categorized as multiple-shot imaging [4–16] and single-shot imaging methods [17–29]. Multiple-shot imaging techniques, including on-axis digital holography (DH), transport of intensity equation (TIE) imaging, and ptychography of coherent diffraction imaging (CDI), are representative methods that recover the complex wavefront by measuring multiple intensity patterns. Although all of these techniques quantitatively retrieve phase while keeping the resolution and field of view (FOV) of an image without degradation, they require electrical controllable elements (e.g., a piezoelectric actuator), or mechanical scanning parts in their system. Moreover, the requirement of multiple measurements makes them sensitive to small noise during the acquisition process, and the system is restricted to capturing the static objects only. On the other hand, the single-shot phase imaging is advantageous in dynamic imaging with its high frame rate and tolerance for signal fluctuation during the scanning process. However, in order to retrieve a phase profile with a single measurement, conventional methods have limitations so that reconstructed results undergo loss of spatial resolution [17–19], FOV [20,21], or constraint of high-frequency components of the sample [22,23]. Although there has been research to overcome the trade-off relationship between the number of measurements and space-bandwidth product (SBP), the systems require a specific specimen’s sparsity [26,27], additional object support , massive deep-learning complex data , or interferometry with many times wider receiver bandwidth than the cutoff frequency of the signal [30–32]. Here, we present a phase-retrieval microscopic method that provides the spatial resolution and FOV at the level of a conventional microscope using a single shot. This method, termed single-shot Fourier ptychographic microscopy (SSFPM), is devised by combination of microscopic imaging methods: Fourier ptychographic microscopy (FPM) [33–42], lens-array imaging [43–47], and multiplexed illumination [36–39]. FPM is a recently developed holographic imaging technique that reconstructs the Fourier spectrum by extending bandwidth beyond the numerical aperture (NA) of an objective lens. This technique sequentially illuminates specimen using an LED array and collects intensity patterns corresponding to different regions of spatial frequency. By stitching the intensity patterns in the Fourier domain, high-resolution of the phase as well as the intensity profile are quantitatively achieved . Nevertheless, the requirements for scanning are the most challenging problem for FPM. There are prior studies that effectively reduce the number of measurements required in FPM using multiplexed illumination [36–38]. Although these previous studies have shown great quality results, taking multiple shots limits their application, including the high frame rate of dynamic imaging.
The main novelty of our research is that we only take a single-shot image for quantitative high-resolution imaging. Also, the significance of SSFPM is to overcome not only the trade-off relationship between SBP and the number of measurements, but also the requirements of the scanning process in FPM. Using the lens-array imaging method, the system collects various angular intensity profiles with a single acquisition. LEDs generate quasi-monochromatic plane waves rather than diffusive incoherent light, so that subimages formed by the lens array correspond to spatial frequency components for the case of thin samples. In addition, LEDs have advantages of competitive price, compact optical systems, and the capability of multiplexed illumination. Since FPM requires data redundancy (overlapping in the spectral domain), we simultaneously turn LEDs on to make a single-shot image retain sufficient data for satisfying the convergence condition of FPM. As the subimages correspond to low-resolution images of conventional FPM, the system can adopt a multiplexed FPM algorithm and reconstruct wide bandwidth of a complex spectrum whose cutoff frequency is equal to the sum of the NA of the objective lens and the illumination angle of the marginal LED. Thus, even if the lens-array imaging method degrades resolution of each subimage, high resolution of the complex profile is retrieved through FPM reconstruction.
This paper analyzes the constraints for applying FPM with a single-shot system and demonstrates the performance of SSFPM using numerical simulation and experiments. By imaging the 1951 USAF resolution target, we confirm that the prototype represents resolution of 3.10 μm with a wide FOV of , which is comparable in performance to coherent imaging using a conventional microscope. In addition, a biological specimen experiment is carried out in order to show that SSFPM can be utilized for quantitative phase imaging.
A. Imaging System for Single-Shot Fourier Ptychography
In typical FPM, an LED array replaces a light source of a conventional microscope, and an illuminated light from each LED is considered as a plane wave with a unique spatial frequency . While scanning the LED array, FPM measures intensity patterns of the object field that corresponds to48] or the Fienup method . FPM reconstructs the high resolution of intensity and phase images when the convergence condition is satisfied, i.e., the intensity profiles should overlap each other at least 40% in the Fourier domain .
Figures 1(a) and 1(b) show the experimental setup and schematic diagram of SSFPM, respectively. Following the basic assumption of the FPM, an LED illuminates a thin specimen in the form of a coherent plane wave within its coherency length range. As the scattered light passes through the objective lens, the objective lens performs Fourier transform by multiplying its pupil function. Thus, the object field passing through the Fourier plane is
Our purpose is to place a lens array at the position of the Fourier plane to obtain multiple intensity images corresponding to different regions of spatial frequencies. However, it is difficult to directly place the lens array at the back focal plane because the Fourier plane is generally formed inside the objective lens. Thus, we implement a imaging system that consists of two lenses with focal lengths of and . In addition, there are several advantages of using the system. First, the beam width of the spectral region, , can be adjusted by determining the ratio of the focal lengths. Here, and are the focal length and the NA of the objective lens, respectively. The beam width can be appropriately adopted to match with the size of the lens array and the imaging area of the sensor. Second, a spatial filter can be placed at the first imaging plane [IP in Fig. 1(b)]. Without the filter, each subimage appears in circular form due to the shape of the exit pupil, and subimages are arranged periodically following the structure of the lens array. We used a long f-number of lens array to make subimages overlap and remove the overlapped parts using the square filter. This process allows most of the pixels in the sensor to be efficiently used. In addition, a combination of a long f-number lens array and a square filter can alleviate the f-number matching problem occurring in lens-array imaging [43–47,50]. The vignetting effect is also mitigated by magnifying the object to the level of a conventional microscope.
The lens array is located at the relayed Fourier plane, and the CCD sensor measures the intensity pattern at the back focal plane of the lens array. Each lens performs a Fourier transform, so that the detected th subintensity pattern in sensor using th LED is given by1) and (3), the intensities measured in SSFPM correspond to those measured in FPM. Thus, the detected subimages can be applied to a conventional FPM algorithm.
B. Multiplexed Illumination
Using a lens array, we obtain multiple intensity images corresponding to different regions of the spectrum in a single exposure [see Fig. 2(a)]. Although the images represent different spatial frequency components like conventional FPM, we cannot directly apply these to the FPM algorithm, since they do not satisfy the overlapping condition for convergence of the FPM. The structure of the lens array consists of physically separated lenses (the gap between adjacent lenses is represented as in the Fourier domain). As each lens performs Fourier transform in its structure, the detected intensities do not have overlapping information in the spectral region. This problem cannot be solved by shifting the position of the lens array, because it only adds an additional phase term to the wavefront, which disappears when measuring the intensity pattern at the back focal plane of the lens array. To achieve enough information in a single shot, several LEDs are turned on simultaneously, as shown in Fig. 2(b1). Even though the light from an LED is considered as a coherent wave in its coherence length, the lights illuminated from different LEDs are mutually incoherent. Thus, the th subimage using multiple illumination appears as the sum of each intensity pattern:2(b)]. It has been presented that the superposed information could be decomposed during the FPM reconstruction process [36,37]. Thus, we decompose multiplexed subimages to subimages satisfying the overlapping condition. The details are explained in Section 2.C. In our SSFPM, the LED matrix (index of ) and lens array (index of ) have a periodic regular rectangular arrangement, requiring optimal selection of to efficiently collect data in a single shot. For example, if , the decomposed spectrum sets of red and blue circles in Fig. 2(b4) correspond to the same area in the six parts. In this case, they neither constrain each other nor do they provide sufficient object support to recover phase information. Therefore, we take the that is out of the lens array’s period in order that the decomposed intensities are overlapped while providing object support, as shown in Fig. 2(b4). In a fashion similar to previous studies, it is difficult to analytically find and indicate the optimal with satisfying the convergence condition while providing enough object support due to the nonlinear property of the FPM algorithm. Instead, we find a suitable range for choosing the value that minimizes errors between the reconstructed phase and ground truth using numerical simulation. The detailed process is explained in Supplement 1.
C. Reconstruction Algorithm
FPM uses the iterative algorithm, in which the intensity corresponding to each illumination is updated with the measured intensity data. For multiplexing FPM imaging, a step of decomposing the summated intensity is added in the conventional FPM reconstruction process. The key idea is that a set of intensities () is updated to multiplying the measured intensity by the ratio between the sum of set of intensity and each intensity . We extend the multiplexing FPM algorithm and develop the SSFPM reconstruction algorithm to apply the lens-array imaging system. The SSFPM reconstruction algorithm is summarized as shown in algorithm 1. The computational cost per each iteration represents using big notation. denotes the total number of a subimage’s pixels. The computational cost is linearly proportional to the (multiplexed number of LEDs).
A. Numerical Simulation
The performance of SSFPM is verified using numerical simulation. In this simulation, the system is numerically designed as shown in Fig. 1, with the parameters considering experimental reproduction. The system consists of an objective lens ( magnification, ), two lenses for system (), and a lens array (pitch of 1 mm, ). We simulated the case in which LEDs () illuminate an object simultaneously with a wavelength of 630 nm, and the object has complex transmittance, as shown in Figs. 3(a) and 3(b). The LEDs are placed 145 mm apart from the sample, and the interval between adjacent LEDs is 12 mm. Since the back focal plane diameter of the objective lens is 9.3 mm, the central lens array () is used to satisfy be nearly one. Then, the scattered light is sequentially propagated from the object through the objective lens, system, lens array, and to the imaging sensor. The calculated intensity pattern is displayed in Fig. 3(c). Multiplexed illumination makes subimages have multiregional frequency information, which can be confirmed through the appearance of the DC bright-field component in the central peripheral images. Next, the intensity pattern is divided into subimages. Figure 3(d) represents a central subimage of the captured image, marked in white. After generating subimages, the high resolution of the object function is reconstructed using the SSFPM algorithm, as shown in Figs. 3(e) and 3(f). From the simulation results, SSFPM shows the validity of retrieving a phase profile using only a single intensity pattern. Also, the sacrificed spatial resolution due to using a lens array is recovered up to resolution of the input complex profiles.
B. Experimental Demonstrations
Next, SSFPM is experimentally demonstrated using an optical system, shown in Fig. 1. The system adopts an objective lens ( magnification, , ) and a CCD sensor (Pointgrey Grasshopper3) with a pixel pitch of 3.69 μm and spatial resolution of . A system is implemented using two aberration compensated lenses (Canon, F1.8, focal length of 50 mm). An LED array (Adafruit, 6 mm spacing) is placed at 145 mm apart from the sample and LEDs (wavelength of 630 nm, bandwidth of 20 nm) are turned on simultaneously. The lateral gap of LED is 12 mm, and the marginal LEDs have a lateral illumination of 0.0825, so that the maximum received NA into the objective lens is . Before SSFPM experiments, a calibration process is implemented by placing a diffuser below the specimen. The diffuser leads the measured subimages to represent not the spatial frequency components but the perspective views, which is a concept similar to integral floating microscopy [45,47]. The system is optically and computationally calibrated to make all the subimages appear at the same position using test samples (1951 USAF resolution target and pinhole).
After the calibration process, 1951 USAF resolution target imaging is carried out to demonstrate the performance of SSFPM, such as resolution enhancement and the convergence of the algorithm. The measured intensity pattern consists of 49 subimages whose resolution is low because each lens constrains maximum spatial frequency and the number of pixels, as shown in Fig. 4(a). The white dashed figure shows that element 2 of group 6 (71.8 lp/mm, resolution of 13.92 μm) is barely distinguished in the centered bright subimage. The overall magnification ratio is calculated by . The measured intensity patterns share the FOV of the sensor, so that the effective FOV of an image is divided by the lateral number of subimages. Thus, the system has FOV of , which corresponds to the microscope (multiplying the overall magnification ratio with the lateral number). The measured intensity pattern is cropped to generate a set of subimages and apply the SSFPM algorithm to them. The reconstruction results show that the high-resolution intensity as well as phase map is recovered, as shown in Figs. 4(c) and 4(d). It is possible to distinguish element 3 of group 8 (287.4 lp/mm, resolution of 3.10 μm), as shown in Fig. 4(f). To compare the imaging performance with a conventional microscope, an additional experiment is implemented, as shown in Fig. 4(b). The resolution target is illuminated by a single LED (wavelength of 630 nm) for coherent imaging. In this experiment, we use the objective lens (NA of 0.25) combined with a commercialized microscope (Olympus BX53F), and an even number of pixels with the single-shot image () are used. The Abbe limit for coherent illumination is given by . From Figs. 4(g), element 4 of group 8 (362.0 lp/mm, resolution of 2.76 μm) can be resolved, which accords with the theoretical value of 2.52 μm. Since SSFPM can restore sacrificed resolution owing to the lens array from 13.92 μm to 3.1 μm (1.23 times for the diffraction limit and 1.12 times for the experiment result), it is confirmed that SSFPM can provide performance comparable to that of a conventional microscope.
Next, a biological specimen experiment is carried out using allium cepa (onion) epidermal cells to show SSFPM can be implemented for 3D microscopic imaging. The specimen is illuminated by green LEDs (wavelength of 532 nm), and other experimental conditions are the same as the above experiments. Using the SSFPM method, intensity as well as phase map of the allium cepa epidermal cells is reconstructed with FOV of , as shown in Figs. 5(a) and 5(b), respectively. The conventional coherent microscope images taken with objective are shown for comparison in Figs. 5(c3)–5(e3). In our experiment, sacrificed resolution due to using a lens array is recovered up to that of a conventional microscope, as shown in Figs. 5(c1)–5(e1), and the phase profile is retrieved, as shown in Figs. 5(c2)–5(e2).
Figure 6(a) indicates the 3D distribution of the chosen FOV using the phase profile retrieved by SSFPM. The distribution obviously indicates that the allium cepa epidermal cell consists of cell walls and nuclei with a periodical structure. To verify SSFPM provides the quantitative phase imaging, phase-imaging experiments using phase-shifting DH is carried out for comparison [Fig. 6(b)]. In the DH experiment, a total of five images is taken, one for amplitude information, and the other four for restoring phase information (). For a more accurate comparative analysis, we also implement phase imaging using the TIE method. The detailed explanation about DH and TIE with experimental results are discussed in Supplement 1. By comparing the phase distributions retrieved by these methods, it is asserted that SSFPM can provide quantitative phase profile and has competitiveness over conventional phase retrieval methods.
We present a new phase-retrieval method with single-intensity measurement as combining FPM, lens-array imaging, and multiplexed illumination. The method overcomes the challenges of the conventional phase-retrieval method, which indicates a trade-off relation between SBP and the number of measurements. Using the intensity multiplexing strategy, enough information to recover the lost phase is obtained with single measurement. The main advantage of SSFPM is that the resolution loss or narrow FOV that normally appear with other single-shot phase retrieval is diminished. In addition, collecting intensity data in a space domain leads to high signal-to-noise ratio data in both bright- and dark-field images, and allows using partial coherence light by cropping an image to small patches considering coherence length. SSFPM consists of a similar optical system in integral floating microscopy or light field microscopy (LFM) in the sense that a lens array is used to acquire multiple images in a single shot [45,47]. In LFM, however, the perspective views of a 3D thick sample rather than spatial frequency components appear based on ray optics using an incoherent light source (e.g., a halogen lamp). Although there is analysis of LFM based on wave optics, it does not recover the sacrificed resolution that is the fundamental limit of LFM . SSFPM is a method based on wave optics that collects the spatial frequency components of a thin sample as data, and the monochromatic relation between spatial and frequency domain (Fourier transform) can be applied. In other words, the data are not just used to generate perspective views, but synthesizes complex Fourier spectrum based on an FPM algorithm, so that SSFPM recovers the resolution of image up to the NA of a conventional microscope with a 3D profile of a specimen. From another perspective, SSFPM can be considered extended Fourier analysis of coherent LFM with thin samples. FPM also can be applied to restore the refractive index of thick samples by considering the volumetric coherent transfer function . Since it requires more information and redundancy of data than 2D FPM reconstruction, SSFPM may have the advantage of applying 3D FPM with multiplexing strategy. For instance, our prototype obtains intensity profiles in a single exposure using a lens array. Also, we believe SSFPM generates more accurate complex profiles with fewer computational costs by optimizing the multiplexing pattern and the structure of the lens array. Finally, since our method is based on FPM algorithm, SSFPM can be applied in various applications of FPM imaging such as pupil function recovery , reflection FPM , fluorescence FPM , and full-color imaging .
This paper presented a single-shot phase-imaging microscopy, termed SSFPM. Using lens array and multiplexing strategy, multiple intensity images satisfying the convergence condition of FPM were acquired in a single measurement. The sacrificed resolution due to lens-array imaging was recovered via FPM, so that high resolution of phase as well as intensity profiles was acquired. We demonstrated that the performance of SSFPM, resolution, and FOV was comparable with a conventional microscope by numerical simulations and imaging the 1951 USAF resolution target. Furthermore, SSFPM provided the quantitative phase profile of a biological specimen, showing the feasibility of 3D imaging. SSFPM has the potential to improve image quality with aberration correction and to be applied in subsequent research of FPM and lens-array imaging. We believe SSFPM can provide new possibilities in phase-retrieval microscopy in that high resolution of quantitative phase imaging is acquired with a single measurement.
Korean National Police Agency (KNPA) (PA-H000001).
This research was supported by Projects for Research and Development of Police Science and Technology under the Center for Research and Development of Police Science and Technology and KNPA.
See Supplement 1 for supporting content.
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