Wide-field high-resolution 3D microscopy with Fourier ptychographic diffraction tomography

We report a computational 3D microscopy technique, termed Fourier ptychographic diffraction tomography (FPDT), that iteratively stitches together numerous variably illuminated, low-resolution images acquired with a low-numerical aperture (NA) objective in 3D Fourier space to create a wide field-of-view (FOV), high-resolution, depth-resolved complex refractive index (RI) image across large volumes. Unlike conventional optical diffraction tomography (ODT) approaches that rely on controlled bright-field illumination, holographic phase measurement, and high-NA objective detection, FPDT employs tomographic RI reconstruction from low-NA intensity-only measurements. In addition, FPDT incorporates high-angle dark-field illuminations beyond the NA of the objective, significantly expanding the accessible object frequency. With FPDT, we present the highest-throughput ODT results with 390nm lateral resolution and 899nm axial resolution across a 10X FOV of 1.77mm2 and a depth of focus of ~20{\mu}m. Billion-voxel 3D tomographic imaging results of biological samples establish FPDT as a powerful non-invasive and label-free tool for high-throughput 3D microscopy applications.


Introduction
In optical microscopy, there has been a continued need towards increasing resolution and contrast for visualizing 3D subcellular features of transparent biological samples over a large FOV and an extended period of time. Confocal microscopy 1, 2 , as the paradigmatic tool of 3D microscopy, is able to collect serial optical sections from thick fluorescence specimens with both high-resolution and high-specificity. However, the focused, high-intensity laser irradiation is detrimental to living cells and tissues 3 . Two-photon 4 and light-sheet microscopy 5 are superior alternatives with advantages of deeper penetration, efficient light detection, and reduced photobleaching. More recently, super-resolution fluorescence microscopy has opened new view towards nanoscale subcellular structure [6][7][8] . However, all these techniques require fluorescent proteins as biomarkers, and are thus ill-suited for samples that are non-fluorescent or cannot be fluorescently tagged. Besides, the photobleaching and phototoxicity of fluorescent agents prevent live cells imaging over extended periods of time 9 . Overcoming these limitations becomes extremely challenging, as acquiring data over multi-dimension prolongs exposure of the specimen to the excitation light, reducing its viability. Furthermore, the throughput of these imaging modalities is limited by the space-bandwidth product (SBP) of the optical system 10 . In order to achieve sub-cellular imaging resolution and depth sectioning, a high-magnification objective has to be used, resulting in a proportionally smaller FOV. For example, a standard 60× 0.9 NA objective has a lateral resolution of ∼0.38µm but a very limited FOV of ∼200µm in diameter.
The refractive index (RI) distribution serves as an important endogenous contrast agent, which indeed enables the visualization of intracellular structures of biological samples without the need for specific staining or fluorescent tagging. In 1969, Wolf 11 proposed optical diffraction tomography (ODT) as a solution to infer the 3D RI distribution by combining the X-ray tomography principle with optical holography. Different from conventional holography 12,13 and other quantitative phase imaging (QPI) techniques [14][15][16][17][18][19][20][21][22] , which only capture 2D integral phase shifts introduced by the sample, the ODT is a true 3D imaging technique in the sense that volumetric information inside the sample can be accessible 23 . In a typical ODT system, the sample is illuminated from various directions [24][25][26][27] or axially scanned at different depths [28][29][30][31] , and the resulting complex diffraction patterns in the far field is measured based on QPI techniques. These measurements may then be synthesized in Fourier space with Fourier slice theorem 25,32 or Fourier diffraction theorem 11,24 into a 3D tomographic reconstruction.
Due to synthetic aperture principles 33 , ODT naturally comes with the additional benefit of improving the imaging aperture larger than (maximum 2 times for transmissive configuration) the physical aperture set by the microscope 23,34 . Nevertheless, as a technique that generally requires both angular/depth scanning and multiple phase measurements, ODT has been primarily imple-mented in well-controlled, customized optical setups, prohibiting their widespread use in biological and medical science. Furthermore, in order to alleviate the "missing cone" problem due to the limited angle of acceptance of the imaging system, ODT typically employs a high-magnification objective at the cost of significant FOV reduction 35,36 , bringing additional challenges for their applications to several important problems such as rare cell imaging or optical phenotyping of model organisms, where large-scale high-throughput microscopy is highly desired.
Here, we present a computational ODT platform that is capable of providing 1.3 NA resolution (390nm lateral resolution and 899nm axial resolution) across a 10× FOV of ∼1.77mm 2 . As a result, our ODT platform merges high spatial resolution in 3D with a significantly large imaging volume, offering a 3D SBP that is unmatched by existing ODT approaches. Our technique, termed Fourier ptychographic diffraction tomography (FPDT), uses a low-NA objective to acquire a sequence of intensity images corresponding to different illumination angles scanned sequentially with a programmable light-emitting-diode (LED) array. Then, it gradually combines these intensity images into the 3D spectrum of the object using an ODT-based ptychographic reconstruction algorithm. After convergence of the algorithm, an inverse Fourier transform is performed to get the sample's 3D RI distribution. Probing a large volume with a decent 3D spatial resolution, FPDT could provide a powerful tool for high-throughput imaging applications in, e.g., cell and developmental biology.
Fourier diffraction theorem in finite-aperture optical systems As illustrated in Fig. 1a, the physical quantity describing a thick 3D sample is termed as scattering potential f (x) , which is the function of the 3D distribution of complex RI n (x) of the object: where k 0 = 2π/λ is the wave-number in free space, λ is the illumination wavelength, n m is the RI of the surrounding medium, and x ≡ (x, y, z) ≡ (x T , z) is the short-hand notation for the 3D spatial coordinate (transverse coordinate x T plus z). When the 3D sample is illuminated by a plane wave U in (x) , the resultant total field U (x) is the superposition of the incident field, U in (x) , and the scattered field, U s (x) , that is, . In order to solve the inverse scattering problem, we introduce a new quantity U s1 (x) , which represents the first order scattered field. Under (first) Born 11 or Rytov approximations 24 , U s1 (x) is connected with U in (x) and U s (x) through the following equations (see Supplementary Section A): Based on Green's function method 37 , the linearized relation between the first order scattered field and the scattering potential of the object can be established, and the corresponding form in the frequency domain is well-known as the Fourier diffraction theorem 11 : where j is the imaginary unit, k m is the wave-number in the medium, k i is the 3D wave vector of the incident plane wave, the exponential term in Fig. 3 accounts for the coordinate shift in the z direction and will automatically vanishes if the measurement is performed at the nominal 'in-focus' plane (z D = 0),f (k) andÛ s1 (k T ; z = z D ) are the 3D and 2D Fourier transforms of f (x) and U s1 (x T ; z = z D ) , respectively (we use the "hat" to denote the signal spectrum in the 2D/3D Fourier domain). Because the 3D frequency vector, k = (k T , k z ) , is laid on the 2D surface of the so-called Ewald sphere under the constraint k z = k 2 m − |k T | 2 , the information defined byÛ s1 (k T ) , is directly related to a particular semi-spherical surface with a radius of k m in 3D Fourier space that is displaced by −k i (see Fig. 1b). Thus, the planar 2D Fourier spectrum is projected onto a semi-spherical surface, as depicted in Fig. 1c. However, for a practical microscopic system, only forward propagating waves falling within the system aperture can contribute to the image formation, as illustrated in Fig. 1d. In 2D imaging, the effect of the lens aperture is usually described by the 2D complex pupil function [i.e., coherent transfer function (CTF)] P (k T ), which ideally is a circ-function with a radius of k 0 N A obj , determined by the NA of the objective. For 3D imaging, the complex pupil function should be projected onto the spherical surface, i.e. P (k) = P (k T ) δ k z − k 2 m − |k T | 2 , resulting a subsection of the Ewald sphere called the generalized aperture (i.e. 3D CTF) 38 (see Fig. 1e). Limited by the generalized aperture, the forward scattered field by sub-wavelength delta-like features (subtending an angle of ±90 • ) completely captured by the microscope. As shown in Fig. 1e,f, the aperture angle θ denotes the largest cone of wave-vectors that can pass from the sample into the imaging lens, so only when θ = 90 • , the generalized aperture can cover the half-sphere.
The Fourier diffraction theorem (Fig. 3) suggests that for each illumination angle, only partial spherical cap bounded by the generalized aperture can be probed. Illuminating the object at different angles will shift different regions of the object's frequency spectrum into a fixed microscope objective lens, enlarging the accessible object frequency domain. In conventional ODT systems, the complex amplitude (both amplitude and phase) of the total field, U ( x T ), is measured by interferometric or holographic approaches, and complex function of incident plane wave illumination,U in (x), can be known beforehand or determined through proper calibration. So U s1 (x T ) can be calculated by Fig. 2 with either first Born approximation or Rytov approximation, and then mapped on the particular Eward sphere according to Fig. 3. With a series of angledependent interferometric complex amplitude measurements, certain region of the 3D Fourier spectrum of the object can be completed, which will allow us to reconstruct the scattering potential of the 3D sample. It should be mentioned that the Rytov approximation is valid as long as the phase gradient in the sample is small and is independent of the sample size and the total phase shift, so it was considered less restrictive and shown to demonstrate superior reconstruction performance than the Born approximation for thick biological samples 24 .

FPDT forward imaging model and reconstruction algorithm
Since all computational imaging approaches rely on inversion algorithms to retrieve the parameters of interest of the sample, they require an accurate modelling of the link between the measured data and the sample parameters. For a thin sample characterized by its 2D amplitude and phase, each angled illumination shifts the object spectrum around in 2D Fourier space, with the objective aperture selecting out different sections 20 . The recorded images are intensities of resultant complex fields corresponding to different segments of the object spectrum. For a thick sample characterized by its 3D scattering potential, the object spectrum is shifted by the incident wave-vector in 3D Fourier space in a similar way, and only the frequencies covered by the 3D generalized aperture can pass through the imaging system and contribute the image formation. However, U s1 (x) is not the field we detected in the image plane. Instead, it is related to our measurements by Fig. 2 with either Born or Rytov approximations. So a forward imaging model linking measured intensities with the sample' scattering potential should be established.
As illustrated in Fig. 2a,b, for bright-field imaging where the incident illumination falls within the objective pupil, the field measured is the total field U (x) (the sum of U in (x) and U s (x)), which is similar to the case of conventional ODT but only the intensity is recorded. For dark-field imaging illustrated in Fig. 2c, the 3D pupil function do not intersect the zero order of the object spectrum at k = 0, and the incident illumination (zero diffraction order), which falls out of the objective, is absent in the image. Hence, only the scattered component contributes to the image formation.
In the following text, we introduce three new quantities, U sn (x T ) , U n (x T ) and U s1n (x T ) , which are normalized versions of the first order scattered field, total field, and scattered field with respect to the incident field, that is, . Without loss of generality, the incident illumination is considered as an angled plane wave with a unit amplitude, so the normalization process will not modify the intensities of the fields which we detect. With these new notations, Fig. 2 can be simplified as: The inverse relation, which connects the measured field to the first order scattered field (scattered for Born approximation, and for Rytov approximation. Based on these imaging models, we develop an iterative reconstruction algorithm, mirroring that from FPM 20 , to "fill in" the high-resolution k-space scattering potential computationally with data from a set of N captured low resolution intensity images, I i c (x T ) , with their corresponding illumination wavevector k i in , with i = 1, 2, ..., N . Here we use the superscript i to denote that this image is formed by the ith LED illumination. Figure 3a displays the general process of the FPDT reconstruction algorithm, which alternates between the spatial and Fourier domain according to the following steps: 1. Making an initial guess of the high-resolution k-space scattering potential,f (k) . It has been found that the initial value forf (k) is not critical to the final reconstruction result, so we simply initializef (k) with all zeros in this work.
2. From the first illumination angle, select values off (k) taken along its associated shell corresponding to k − k i and bounded by the 3D generalized aperture, P (k) (radius k 0 , maximum width 2k 0 N A obj ). The sub-region of the 3D spectrum is projected along the axial frequency coordinate to obtain a low-resolution 2D Fourier sub-spectrumf i (k T ) , which is directly , and inverse Fourier transform the resultant spectrumÛ s1 (k T − k i inT ) to the spatial domain.
This is equivalent to the actual physical scenario of using our current high-resolution k-space scattering potential estimate to simulate the formation of the low-resolution normalized first 4. Convert the estimated U i s1n (x T ) to the measured field and enforce the amplitude constraint.
Note that the update formula depends both on the area which the illumination vector belonging to (bright-field or dark-field) and the approximation assumed for ODT (Born or Rytvo).
For Born approximation, the update formula is derived based on the relation given by Eqs. 5 and 6 For Rytov approximation, the update formula is derived based on the relation given in Eqs. 5 and 7 Unless otherwise noted, Rytov approximation is employed for FPDT reconstruction due to its higher RI reconstruction accuracy (see Supplementary Section E for detailed comparison).
5. Fourier transform this amplitude-constrained estimate,Ū i s1n (x T ) , to the frequency domain.
The resultant spectrum is further shifted back to its originally position with a translation vec- . This forms an update of the first order scattered field,Ū i s1 (k T ) . Based on the values off i (k T ) = − kz πjŪ i s1 (k T ) , we locally update the corresponding subregion of the 3D spectrumf (k) enclosed in 3D generalized aperture, which is just the spherical cap we extracted in Step 2. This completes one sub-iteration of the FPDT algorithm.
Next, we move to the next illumination angle, which corresponds to a new spectrum region. Steps 2-5 are then repeated until scanning all N images and the whole iteration scheme is repeated over M cycles to achieve a self-consistent solution. To accelerate and stabilize the convergence of the algorithm, we use adaptive relaxation strategy to gradually diminish the weight of updating as the iteration accumulates 39 . At the end of this iterative recovery process, the converged solution in Fourier space will typically cover a significantly extended support. With our current configuration, we could get an image within illumination angles of up to ±64.2 • (see FPDT platform and characterization). As a result, the entire region of frequency space still cannot be filled. Thus, in the final step of the reconstruction algorithm, we use an iterative constraint algorithm 24,36 to computationally fill this missing information, which mitigates the elongation of the reconstructed shape along the optical axis and generates more accurate estimation of RI (see Methods).

Frequency coverage and resolution analysis
In FPDT, the support for accessible object frequencies can be used to assess both lateral and axial resolution limit theoretically, as illustrated in Fig. 3b,c for different illumination configurations.
Without loss of generality, here we consider a microscope in air with a rotationally symmetric aperture, which allows us to limit our attenuation to only one cross-section (y-z slice) of the 3D CTF.
For a coherent microscope with a fixed orthogonal plane wave illumination, the accessible object frequency just corresponds to the support of the generalized aperture, with a half-side lateral frequency extent of N A obj /λ and half-side axial extent of 1 − 1 − N A 2 obj λ , as illustrated in Fig. 3b. The 3D CTF is only a fraction of the Eward sphere with no axial support, resulting in a very limited discrimination along the optical axis 34,40,41 . ODT techniques circumvent this issue by sequentially illuminating a 3D object from various directions to enlarge the accessible 3D Fourier domain. In a conventional ODT system, the maximum illumination angle allowed is limited to the maximum collection NA, forming a torus-shaped structure in 3D frequency space 23,40 (Fig. 3c).
The half-side lateral and axial extensions of the frequency supports are given by Note the frequency support of a conventional ODT system is the same as the one obtained in a conventional incoherent illuminated microscope, with a doubled lateral resolution compared to the coherent diffraction limit 23,41 . Furthermore, the sequential synthetic aperture procedure also allows to transmit all spatial frequencies within the support without any attenuation (the transfer function of ODT is always 1 within its support), whereas the optical transfer function (OTF) of a conventional incoherent microscope strongly dims high spatial frequencies 42 . However, the limited angular coverage of the incident beam leads to a very restricted frequency support, which yields a relatively low axial resolution especially when a low NA objective is used. As the case shown in Fig. 3c, when the objective N A obj = 0.4 and λ = 507nm, the depth resolution 1/∆u z ≈ 6.07µm, which is almost one order of magnitude lower than the lateral resolution 1/∆u x,y ≈ 634nm.
If the synthetic aperture process can further incorporate high-angle dark-field illuminations that may be much greater than that allowed by the NA of the objective, the accessible object frequency will be improved significantly, as shown in Fig. 3c. By processing these dark-field data and inverting it into an image, we can in principle obtain a synthetic transfer function which can be many times larger than the usable lens transfer function. As illustrated in Fig. 3d, the half-side lateral and axial extensions of the resultant frequency supports are It can be calculated that by employing high-angle illumination with N A ill = 0.9 which is much larger than N A obj = 0.4 , the lateral and axial resolution can be increased to 390nm and 899nm respectively, suggesting a remarkable resolution enhancement in both lateral and axial directions.
Besides, the accessible frequency support (the volume of the non-zero 3D spectral region) was expanded to 9.2 times its original volume, resulting in a significant increase in the SBP. However, the dark-field scattered light is generally very weak and creates severe speckle noise in the case of coherent illumination, preventing reliable interferometric measurement of its phase component.
This challenge can be effectively bypassed by our FPDT method since it only utilizes a standard microscope to detect image intensities under different angled illuminations without requiring direct phase measurement.

FPDT platform and characterization
As depicted in Fig. 4a We demonstrate imaging performance of our FPDT platform experimentally by measuring a 1951 USAF resolution target (Ready Optics Company, USA). First, the resolution target was placed on the sample stage of the microscope horizontally. Figure 4c shows the raw full-FOV image of the object under axial illumination from the central LED, and red-boxed region (64 × 64 pixels), which contains the smallest groups of features (Groups 10 and 11), is extracted and shown in Fig. 4d. It can be seen that under coherent illumination, the final resolution is limited by the pixel size of the camera (650nm at the object plane), instead of the coherent diffraction limit (1.27µm, Group 9, Element 5), creating significant aliasing effects (pixelization). Since the flat USAF target can be regarded as a thin 2D object, we first used conventional FPM algorithm to recover the 2D super-resolved images of this target and meanwhile calibrated the spatial position of each LED element numerically 43 . Figure 4d2,3 presents the images recovered by FPM without or with darkfield raw images, revealing that the highest resolution achieved is enhanced from 616nm (Group 10, Element 5) to 388nm (Group 11, Element 3), which agrees well with the theoretical resolution limit of 634nm (N A syn = 0.8 ) and 390nm (N A syn = 1.3 ), respectively.
Next, the same data set was used to implement 3D tomography based on FPDT. When only bright-field images were used for the reconstruction, the theoretical lateral and axial resolution limits can be determined as 634nm and 6.07µm, according to the lateral and axial sections of the recovered high-resolution 3D absorption spectrum shown in Fig. 4e1,2. Since the recovered information occupies only a very limited portion of the whole spectrum, the missing information can hardly be compensated by the iterative algorithm, leading to significant grid artifacts (Fig. 4f1,2).
When all raw images, including dark-field images, were used for 3D tomographic reconstruction, the accessible frequency support was significantly expanded, as illustrated in Fig. 4e3,4. Besides, the missing cone problem in the 3D Fourier spectrum was significantly allievated by the iterative algorithm, as shown in Fig. 4f3,4. Figure 4g and  Figure   S3, we further determine the depth-of-focus (DOF) of the proposed FPDT platform to be ∼20µm without significantly compromising the lateral resolution, which is 5-fold longer than the 4µm natural DOF associated with the 10× 0.4NA objective used in the experiment, and ∼40-fold longer than that of a conventional microscope objective with the same 1.3 NA (501 nm).

3D tomographic imaging of Pandorina
In this Section, the proposed FPDT technique was applied to a bleached paraffin section of Pandorina morum algae (P. morum). Figure 5 displays the 3D rendering results of a 16-celled P. morum using Rytov approximation based FPDT algorithm (with and without using dark-field imaging) in x-y, y-z and x-z directions, respectively. Besides, the recovered through-slice RI stacks of the P. morum as well as the corresponding 3D rendered RI images are animated in Supplementary Movie 3. In these results, the inner architecture of the algae is clearly visualized, and we can see clearly how different cells are held together with respect to each other in 3D space (indicated by white arrows in Fig. 5b). When dark-field images were used in the FPDT technique, more subcellular details inside the P. morum, e.g. the periphery of pyrenoids, can be revealed (indicated by the green arrows in Fig. 5b), as illustrated in the x-y slices of RI distribution images at 5 different depths (Fig. 5b). Besides, the vertical cellular structure became more clear and compact, as can be observed in the x-z (Fig. 5c1,c2) and y-z slices (Fig. 5d1,d2). The line profiles shown in Fig. 5e1-e4 confirm that more fine structures within the algae can be resolved by incorporating the dark-field measurements. In Supplementary Section F and Supplementary Figure S6, we further demonstrate the tomographic imaging results of another algae, Pinnularia sp. diatom.
The recovered through-slice RI stacks and corresponding 3D rendered RI images are animated in Supplementary Movie 5. Once again, our FPDT clearly reveals its structural features such as the central nodule and densely spaced striae when high-NA dark-field measurements are incorporated.

Wide-field tomographic imaging of unstained HeLa cells
Finally, to demonstrate the performance of the FPDT approach for wide-FOV and high-resolution 3D imaging in life sciences, we imaged a large population of adherent HeLa cells. To preserve the cell morphology, HeLa cells were fixed in PBS buffer (n m = 1.34) on a microscope slide. Figure 5a shows the full-FOV RI slice of the fixed HeLa cells slide at z = 10.4µm reconstructed using the FPDT algorithm (with dark-field measurements). To illustrate the subcellular structures inside individual HeLa cells, the RI slices of three zoomed areas (62.4 × 62.4 µm) enclosing three typical cells within the entire FOV are selected and shown in Fig. 5b1-b3. Their corresponding rendered x-y, y-z, x-z projections are displayed in Fig. 5c1-c3 ∼17.2 billion voxels of quantitative 3D RI data with ∼20,000 blood cells over 1.77mm 2 FOV, revealing the potential of our FPDT approach and platform for high-content quantitative analysis of a large population of cells, which is of utmost importance for many applications, such as cancer screening, stem cell research, and drug development.

Conclusion and discussion
In this study, we have demonstrated FPDT, which is based on intensity-only measurements captured by an off-the-shelf microscope with a simple LED array source add-on. By synthesizing a set of variably illuminated, low-resolution intensity images acquired with a low-NA objective in 3D Fourier space, a wide-field, high-resolution depth-resolved complex RI image across large volumes can be reconstructed. There are two distinct features of FPDT: First, without resorting to holographic phase measurement, FPDT achieves phase retrieval, synthetic aperture, and tomographic reconstruction from low-NA intensity-only measurements simultaneously. Second, it incorporates high-angle dark-field illuminations (up to 0.9NA) to significantly expand the accessible object frequency. As a result, the FPDT platform offers the highest-throughput ODT results with 390nm lateral resolution together with an axial resolution of 899nm across a 10× FOV of 1.77mm 2 and a DOF of ∼20µm, creating an effective voxel size of 0.0043µm 3 across a sample volume of 0.074mm 3 . The tomographic imaging performance of FPDT has been quantified by imaging a USAF resolution target and demonstrated by quantitative measurement of the complex RI through several types of thick specimens. The experimental results suggest that the proposed FPDT is an effective and promising tool for large-scale high throughput 3D bio-imaging in a labelfree fashion.
Being a recording and post-processing technique, the processing speed of FPDT has not yet been fully optimized. The large data requirement (∼12.6 gigapixels per whole FOV dataset) poses severe burden on both storage and processing. We have implemented the FPDT reconstruction in MATLAB. The time required for reconstructing the entire tomogram is ∼23 hours on our workstation. The processing speed may be significantly improved to several minutes by using graphics processing units (GPUs) with larger graphics memory, since each smaller sub-tomogram can be independently and parallelly processed by the FPDT recovery procedure. Furthermore, the FPDT method requires much higher redundancy compared with the FPM method 46 since it requires a sufficient overlap of the generalized aperture in 3D Fourier specturm to guarantee fast and stable convergence. Therefore, the overall size of the data set is significantly larger than that of the FPM.
Besides, in our current system, the distribution of LED elements in the array suggests that the high-frequency dark-field region of the 3D object spectrum has a larger overlapping ratio than that of the central bright-field region. So it is expected that reconstructions can still be successful with much fewer intensity measurements than we currently used in this work by properly redesigning the LED illuminator 44,45 or carefully selecting the active LEDs in the array 46 . Finally, it would be worth exploring the multiplexed illumination strategy 47,48 and recently emerged deep learning reconstruction approaches 49,50 to further reduce the data size and speed up the acquisition procedure.
By merging these approaches, it may be possible to increase the image acquisition speed from 91 seconds per 3D frame to less than 10 seconds per 3D frame without incurring data management issue, enabling capturing 3D videos of subcellular dynamical phenomena over a wide FOV for time lapse in vitro microscopy applications.

Methods
Iterative constraint algorithm To minimize the artifact introduced by the missing angle space, we applied an iterative constraint algorithm based on the prior knowledge that the object RI is higher than that of the surrounding medium and the imaginary part of the complex IR should always be no more than zero (no negative absorption). We first take inverse 3D Fourier transform of the originally retrieved spectral data with zero values for the missing space (Fig. 3a). In the spatial domain, there are pixels whose index values are smaller than RI of the medium, we force these to be the same as the index of the medium 24,36 . Besides, for pixels whose absorption coefficients are larger than zero, we also set these to zero to prevent negative absorption. Then we take 3D Fourier transform and get the updated spectral data in which the missing space are no longer zero. In this way, we obtain an approximate solution for the missing angles. At the same time, the data within the original frequency support is replaced with the initial retrieved data. The rest remains unchanged. This constitutes a single iteration of the iterative constraint algorithm. We iterate this procedure until the reconstructed object function converges. Usually, up to 50 iterations are required to guarantee the convergence. Finally, the negative bias in the RI and absorption is removed and the reconstructed object function becomes more accurate. which, however, did not support this work. Figure 1: Fourier diffraction theorem in finite-aperture optical systems. a, A thick 3D sample is illuminated by plane waves from different directions. b, Particular semi-spherical surface in 3D Fourier space corresponding to each illumination direction. c, 2D Fourier spectrum projecting onto a semi-spherical surface. d, The forward scatted wave is limited by the aperture of the objective. e, f, 3D and 2D illustrations of the finite aperture effect on the Fourier diffraction theorem.  Step 1

Experimental setup
Step 2 Impose amplitude constraint Extract 3D sub-spectrum Step 2 Step 3 Step 4 Step 4 Step 5 Step 5 The light source of an off-the-shelf inverted microscope (IX71, Olympus, Japan) is replaced by an LED array. b, A photograph of the FPDT platform. c, Bright-field low-resolution raw image acquired with a full-FOV of a 10× objective. d1, Magnified raw image corresponding to the red-boxed central part of the USAF target. d2, High-resolution recovery result using ordinary FPM method without using dark-field images. d3, High-resolution recovery result of the same sub-region using ordinary FPM method with all of the raw images, including dark-field images. e1-e2, Lateral and axial sections of the recovered high-resolution 3D absorption spectrum without using dark-field images. e3-e4, Lateral and axial sections section of the recovered high-resolution 3D absorption spectrum using all of the raw images. f1-f2, Lateral and axial sections of the recovered high-resolution 3D absorption spectrum without using dark-field images after iterative constraint algorithm. f3-f4, Lateral and axial sections section of the recovered high-resolution 3D absorption spectrum using all of the raw images after iterative constraint algorithm. g1-g2, Volume-rendered image of the USAF target reconstructed by FPDT using and without using dark-field images. Three pairs absorption slices taken at depths of -0.52, 0, and 2.99µm. h1-h2, Corresponding line profiles shown to compare the imaging resolution (indicated by the yellow lines in g1-g2 respectively). x-z slice y-z slice x-y slices  Figure 5: 3D quantitative RI reconstruction of a P. morum using FPDT. a1-a2, 3D rendering results of the RI of the P. morum using FPDT algorithm (without and with using dark-field imaging) in xy, y-z and x-z directions. b, Five 2D slices of RI distribution images at different depths recovered using FPDT without and with using dark-field imaging. c1-c2, d1-d2, Two RI slices along the x-z and y-z directions recovered using FPDT without and with using dark-field imaging. e1-e4, Corresponding line profiles shown to compare the imaging resolution (indicated by the yellow lines in b-d respectively). x-y x-z x-y x-z x-y x-z y-z  Figure 6: Wide-FOV and high-resolution 3D imaging of a large population of HeLa cells using FPDT. a, Full-FOV RI slice of the fixed HeLa cells slide at z = 10.4µm reconstructed using FPDT algorithm. b1-b3, RI slices of three zoomed areas enclosing three typical cells within the entire FOV at different depths. c1-c3, Rendered x-y, y-z, x-z RI projections corresponding to the selected 3 HeLa cells.