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

Article

https://doi.org/10.1038/s41467-024-49126-y

High-resolution, large

fi

eld-of-view label-free

imaging via aberration-corrected, closed-

form complex

fi

eld reconstruction

Ruizhi Cao

1,2

, Cheng Shen

1,2

& Changhuei Yang

Computational imaging methods empower modern microscopes to produce

high-resolution, large

fi

eld-of-view, aberration-f

ree images. Fourier ptycho-

graphic microscopy can increase the s

pace-bandwidth pr

oduct of conven-

tional microscopy, but its iterative

reconstruction methods are prone to

parameter selection and tend to fail u

nder excessive aberrations. Spatial

Kramers

–

Kronig methods can analytically reconstruct complex

fi

elds, but is

limited by aberration or providing extended resolution enhancement. Here,

we present APIC, a closed-form meth

od that weds the strengths of both

methods while using only NA-matching and dark

fi

eld measurements. We

establish an analytical phase retrie

val framework which demonstrates the

feasibility of analytically

reconstructi

ng the complex

fi

eld associated with

dark

fi

eld measurements. APIC can retrieve complex aberrations of an imaging

system with no additional hardware and a

voids iterative algorithms, requiring

no human-designed convergence metrics while always obtaining a closed-

form complex

fi

eld solution. We experimentally demonstrate that APIC gives

correct reconstruction results where F

ourier ptychographic microscopy fails

when constrained to the same number of measurements. APIC achieves 2.8

times faster computation using image ti

le size of 256 (length-wise), is robust

against aberrations compared to Four

ier ptychographic microscopy, and

capable of addressing aber

rations whose maximal phase difference exceeds

3.8

π

when using a NA 0.25 objective in experiment.

The pursuit of microscopy techniques that can simultaneously provide

high-resolution and large

fi

eld-of-view (FOV) can improve digital

pathology and be broadly applied in other high-throughput imaging

applications. Computational imaging, a keystone of modern micro-

scopy, plays a crucial role in achieving such goals. Over the past few

decades, remarkable progresses have been made in both

fl

uorescence

and label-free imaging

fi

eld

–

. One such representative label-free

technique, Fourier ptychographic microscopy (FPM), leverages the

power of computation to provide high-resolution and aberration

correction abilities to low numerical aperture (NA) objectives

.FPM

operates by collecting a series of low-resolution images under tilted

illumination and applies a core iterative phase retrieval algorithm to

reconstructs sample

’

s high spatial frequency features and optical

aberration, resulting in high-resolution aberration-free imaging that

preserves the inherently large FOV associated with the low numerical

aperture objectives. It greatly increases the spatial bandwidth product

of standard microscopy in a simple but surprisingly effective way. Due

to these attractive traits, FPM has found diverse applications in

quantitative phase imaging, aberration metrology, digital pathology,

and other

fi

elds

Although FPM is an important advancement in label-free micro-

scopy, its essential iterative reconstruction algorithm poses several

Received: 3 October 2023

Accepted: 20 May 2024

Check for updates

Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, USA.

These authors contributed equally: Ruizhi Cao, Cheng Shen.

e-mail:

rcao@caltech.edu

Nature Communications

| (2024) 15:4713

1234567890():,;

challenges. First and foremost, the iterative reconstruction of FPM is a

non-convex optimization process, which means that it is not guaran-

teed to converge onto the actual solution

–

. In practice, the

algorithm executes alternating projections between real space and

spatial frequency space until certain conditions are met, such as its loss

function decreasing rate reaches a lower bound, the execution reaches

the allowed maximum iteration number, or the algorithm satis

fi

other pre-de

fi

ned metric thresholds

–

.Asaresult,FPMdoesnot

guarantee that the global optimal solution is ever reached. This is

problematic for exacting applications, such as digital pathology, where

even small errors in the image are not tolerable. Furthermore, the joint

optimization of aberration and sample spectrum can fail when the

system

’

s aberrations are suf

fi

ciently severe

—

leading to poor

reconstructions

. The iterative nature of FPM reconstruction algo-

rithm has prompted researchers to adapt machine learning concepts

to its implementation, in pursuit of computational load reduction,

artifact abatement, and aberration correction

–

. These, in turn, lead

to other problems, such as contextual sensitivity and potentially

greater drift away from the global optimal solution. It is worth con-

sidering at this juncture whether it is possible to develop a closed-form

solution to this class of computational imaging problems, so that all

these challenges can be more effectively addressed.

Recent studies have shown that the complex

fi

eld can be non-

iteratively reconstructed in one speci

fi

c varied illumination micro-

scopy scenario by matching the illumination angle to the objective

’

maximal acceptance angle (the NA-matching angle) and exploiting the

signal analyticity, for example, through spatial-domain

Kramers

–

Kronig imaging

–

.These

fi

ndings are important and

impactful as they eliminate the need for an iterative reconstruction

framework and do not require a human-engineered convergence cri-

terion. However, it is worth noting that this approach does not possess

the capability to correct hybrid aberrations nor provide great resolu-

tion enhancement beyond the diffraction limit of the objective NA. As

such, FPM remains a more appealing choice in various scenarios.

In this study, we present an analytical method, termed Angular

Ptychographic Imaging with Closed-form method (APIC), that weds

the strengths of both methods. APIC builds on complex

fi

eld recon-

struction using Kramers

–

Kronig relations and employs analytical

techniques to retrieve aberration and reconstruct the dark

fi

eld asso-

ciated high spatial frequency spectrum. By using NA-matching and

dark

fi

eld measurements, APIC is capable of retrieving high-resolution,

aberration-free complex

fi

elds when a low magni

fi

cation, large FOV

objective is used for data acquisition. From both simulations and

experiments, APIC demonstrates unprecedented robustness against

aberrations, while FPM drastically fails. Due to its analytical nature,

APIC is inherently insensitive to optimization parameters and offers a

guaranteed analytical complex

fi

eld solution. We additionally show

that APIC performs better than FPM when subjected to the same

constraint on input data size, as it does not require an overly large data

redundancy needed by FPM for a good

convergence. By incorporating

dark

fi

eld measurements, APIC effectively achieves the same theore-

tical resolution enhancement as FPM. We believe APIC represents an

impactful step forward in the

fi

eld of computational imaging.

Results

Principle

APIC collects both NA-matching and dark

fi

eld intensity measurements

for high-resolution reconstruction

. Its reconstruction process begins

by analytically solving for the sample

’

s spatial frequency spectrum and

aberration with the NA-matching measurements. Then, the dark

fi

eld

measurements are used to extend the sample

’

s spatial frequency

spectrum to greatly enhance the resolution of a NA-limited imaging

system. The system setup, data acquisition process, and its recon-

struction

fl

owchart are illustrated in Fig.

.InAPIC

’

s data acquisition

step, the LEDs whose illumination angles match up with the maximal

acceptance angle of the imaging system are sequentially lit. The

measurements under these NA-matching angle illuminations con-

stitute the NA-matching measurements of APIC. LEDs whose illumi-

nation angles are greater than the acceptance angle are then

successively lit for acquiring dark

fi

eld measurements. In the following

sections, we use the word

“

spectrum

”

as shorthand for spatial fre-

quency spectrum (the Fourier transform of the sample

’

scomplex

fi

eld). We note that the spectrum is different from the Fourier trans-

form of an acquired image, which is the Fourier transform of a pure

intensity measurement.

APIC operates by

fi

rst reconstructing the complex

fi

eld corre-

sponding to NA-matching measurements using Kramers

–

Kronig rela-

tions. These measurements are taken with LED illumination angles that

matchwiththeobjective

’

s maximal receiving angle. For a realistic

imaging system, aberrations are inevitably superimposed on the

spectrums

’

phase. To extract the objective

’

s aberrations, we focus on

the overlapping region in their spectrums (the overlap of two trans-

lated CTFs, where the translation is shown on the left side of Fig.

). As

the sample dependent phases are identical in the overlapped region of

the two spectrums, subtracting their phases cancels out the sample

dependent phase term, leaving only the phase differences between

different parts of the pupil function (see Fig. S16 for more informa-

tion). Consequently, the overlapping regions give us a linear equation

with respect to the aberration term. By solving this linear equation, the

aberrationoftheimagingsystemcanbeextracted,whichcanthen,in

turn, be used to correct the original reconstructed spectrums. The

corrected spectrums are then stitched together to obtain an aberra-

tion-free, two-fold resolution-enhanced sample image.

We can then extend the spectrum by using the dark

fi

eld mea-

surements. In this step, the reconstruction spectrum and the aberra-

tion obtained in the

fi

rst step serve as the a priori knowledge. The step-

by-step reconstruction operates in the following way. We choose a

measurement whose spectrum is closest to the known spectrum (say,

the

th measurement) and crop out the known spectrum based on what

is sampled in this measurement, as shown in Fig.

.Thiscropped

spectrum, however, only contains part of the information of the

measurement. Our goal is to recover the unknown part of the spec-

trum so that it can be

fi

lled in for spectrum spanning.

We can see that the Fourier transform of our

th intensity mea-

surement consists of cross-correlation of the known and unknown

spectrum and their autocorrelations. In the following, we show that by

using the known spectrum, we can construct a linear equation with

respect to the unknown spectrum, which can be analytically solved.

First, the autocorrelation of the known part is calculated and

subtracted from the Fourier transform of the measurement. After

subtraction, the autocorrelation of the unknown part and the cross-

correlations are left. One important observation is that these parts are

not fully coincided in the spatial frequency domain (Fig.

). As such, we

can focus on the non-overlap region where the cross-correlation solely

contributes to the signal.

We can then construct a linear equation with respect to the

unknown spectrum. When calculating the cross-correlation, one of the

signals is shifted and multiplied with another signal. The correlation

coef

fi

cient is the summation of this product. Assuming one of the two

signals is known, we essentially use the known signal as the weights

and calculate the summation of a weighted version of the other sig-

nal to

fi

nd this coef

fi

cient. This is a linear operation. Thus, applying the

known spectrum, we can construct a linear operator that takes the

unknown spectrum and produces this cross-correlation. By extracting

the non-overlapping part of the cross-correlation term, we can form

and analytically solve a linear equation with respect to the unknown

spectrum. That is, we obtain the closed-form solution of the unknown

spectrum by solving this equation.

For a practical imaging system, we need to consider its aber-

ration in the imaging process. To match up with the measurement,

Article

https://doi.org/10.1038/s41467-024-49126-y

Nature Communications

| (2024) 15:4713

the recovered aberration is initially introduced to the cropped out

known spectrum. After recovering the unknown spectrum, the

aberration gets corrected, and this corrected spectrum is

fi

lled back

into the reconstructed spectrum. This process stops when all dark-

fi

eld measurement has been reconstructed. The detailed derivation

of the aforementioned analytical complex

fi

eld reconstruction and

aberration extraction methods can be found in Section 12 in the

supplementary note.

Once the above steps are completed, we obtain a high-resolution

and aberration-free sample image. The theoretical optical resolution of

APIC is determined by the sum of the illumination NA and the objective

NA, which is identical to FPM

’

s NA-resolution formulae

.Wenotethat

FPM requires an iterative process to recover the spectrum and is

sensitive to the choice of optimization parameters. On the other hand,

APIC analytically recovers the actual spectrum. This direct and ef

fi

cient

approach sets APIC apart from FPM, offering a more straightforward

and robust spectrum recovery process. In the following section, we will

report on our experimental demonstration that APIC is computation-

ally ef

fi

cient and achieves consistent, high-quality complex

fi

eld

reconstructions even under large aberrations, whereas FPM struggles

due to the increased complexity in its optimization problem.

Experiment results

We used a low magni

fi

cation objective (10× magni

fi

cation, NA 0.25,

Olympus) for data acquisition. A LED ring (Neopixel ring 16, Adafruit)

glued onto a LED array (32 × 32 RGB LED Matrix, Adafruit) served as the

illumination unit. The two LED clusters were mounted on a motorized

stage for position and height adjustment, and they were individually

controlled by two Arduino boards (Arduino Uno, Arduino). In the

acquisition process, we lit up one LED at a time, and simultaneously

triggered the camera (Prosilica GT6400, Allied Vision) to capture an

image when the LED was on. This process continued until all desired

LEDs were lit once. We then performed reconstruction using both APIC

and FPM. The calibration of the system can be found in Section 1 in the

supplementary note.

In our

fi

rst experiment, we imaged a Siemens star target and chose

to acquire a small dataset to perform reconstruction using APIC and

FPM. The dataset acquired in this experiment consisted of 9 bright

fi

eld

measurements, 8 NA-matching measurements, and 28 dark

fi

eld mea-

surements. We note that there are works that apply multiplexed illu-

mination scheme to reduce the number of the measurements in

FPM

, these methods are not as reliable as the conventional FPM data

acquisition scheme. Thus, we only focus on the more reliable

Optimize spectrum

and CTF

Convergence

Evaluation

Converged

Otherwise

Initialize/Update

spectrum and CTF

...

Darkfield Brightfield

Extend spectrum

using dark-field

Stitch spectrum

CTF extraction

Image-wise field

reconstruction

Angular Ptychographic Imaging with Closed-form method (APIC)

Setup and data acquisition in Angular Ptychographic Imaging with Closed-form method (APIC)

Fourier Ptychographic Microscopy (FPM)

NA-Matching

Darkfield

Reconstruction

Pupil

Reconstruction

Pupil

-π

Moderate aberration

Large aberration

Reconstruction

Pupil

Reconstruction

Pupil

-π

Moderate aberration

Large aberration

Fig. 1 | Concept of angular ptychographic imaging with closed-form method

(APIC) and comparison between the reconstruction process of APIC and

Fourier ptychographic microscopy (FPM). a

Setup of APIC. The LEDs whose

illumination angle matches up with the numerical aperture (NA) of the objective are

lit sequentially to obtain the NA-matching measurements. Then, the LEDs whose

illumination angle is larger than the objective

’

s receiving angle are successively lit

for the dark

fi

eld measurements.

Reconstruction process of APIC. Once the

aberration is extracted, it is used to correct aberration in the image-wise

fi

eld

reconstruction. The aberration-corrected spectrums are then stitched together and

serves as a prior knowledge in the spectrum extension. Using dark

fi

eld measure-

ments, the spectrum is furtherly extended to obtain a high-resolution, aberration-

free reconstruction.

Reconstruction process of Fourier ptychographic micro-

scopy (FPM). FPM iteratively updates the spectrum and the aberration to minimize

the differences in the measurement and reconstruction output. This iterative

process is terminated upon convergence to obtain the spectrum and coherent

transfer function (CTF) estimate. Pupil in the

fi

gure denotes the reconstructed

aberrations.

Article

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Nature Communications

| (2024) 15:4713

acquisition scheme in this study. The nominal scanning pupil overlap

rate was approximately 65%. In our experiments, the second-order

Gauss-Newton FPM reconstruction algorithm was applied for recon-

struction as it was found to be the most robust FPM reconstruction

algorithm

, and the sum of all measurements was used for initializa-

tion. We also note that we used 6 sets of parameters in the recon-

struction of FPM and chose the best result, as the reconstruction

quality of FPM heavily depends on its parameters. Some representative

FPM results are also shown in Fig.

, which con

fi

rms such parameter

dependency. On the contrary, the faithfulness and correctness are

guaranteed in APIC, bene

fi

ting from its analytical phase retrieval fra-

mework. We found that APIC was able to render the correct complex

fi

eld while FPM failed, as shown in Fig.

. As shown by the result, the

reconstructed

fi

ner spokes were distorted in all reconstruction results

of FPM. Moreover, noticeable wavy reconstruction artifacts existed in

the phases reconstructed by FPM. When the measurements were given

to APIC, the reconstructed phases and amplitudes were less wavy. The

reconstructed amplitude is also closer to the ground truth, which is

5μm

-π

APIC

FPM, 1

FPM, 2

FPM, 3

5μm

Pupil

10x

Amplitude

Phase

FPM, trial 1

FPM, trial 2

FPM, trial 3

APIC

40x (GT)

5μm

-π

Fig. 3 | Reconstruction result of APIC and FPM using a small number of mea-

surements.

For comparison, we also acquired a ground truth (GT) image, which

was imaged under a high-NA, 40× objective. The Kohler illuminated image under

the same 10× objective used in APIC

’

s data acquisition is shown on the upper left.

The reconstructed pupils are shown on the right side of the

fi

gure. For FPM

reconstructions, we selected three representative results from all 6 parameter sets

we used in FPM, and trial 1 is the best result we got. We note that when the ground

truth is unknown, all FPM reconstruction results might be falsely treated as the

correct solution as they all possess good contrast and

fi

ne details. However,

apparent discrepancies are noticeable when comparing these results with the

ground truth image. APIC, as an analytical method, is not prone to parameter

selection.

Fig. 2 | Reconstruction pipeline for APIC.

By changing the illuminating angle, we

effectively shift the CTF to different positions in the spatial frequency domain, and

samples different regions of sample

’

s spectrum. For measurements under NA-

matching angle illumination, we

fi

rst use Kramers

–

Kronig relation to recover the

corresponding spectrums. The phase differences of two spectrums with overlaps in

their sampled spectrum are used to extract the imaging system

’

s aberration. Then,

the image-wise reconstructed spectrums are corrected for aberration and get

stitched, which forms our prior knowledge in the reconstruction process involving

dark

fi

eld measurements. To extend the spectrum using dark

fi

eld measurement,

the known spectrum in the

th measurement is used to isolate cross-correlation

from other autocorrelation terms. By solving a linear equation involving the iso-

lated cross-correlation, the unknown spectrum can be analytically obtained. We

then use the newly reconstructed spectrum to extend the recovered spectrum. The

extended spectrum then serves as the prior for the

(i+1)

th measurement.

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| (2024) 15:4713

sampled using a high-NA objective. We stress that when the ground

truth is not given, all three FPM results shown in Fig.

may be per-

ceived as a good reconstruction in practice as they preserve good

contrast and are detail-rich. However, these reconstructions are of low

fi

delity as they all deviate from the ground truth we acquired. We can

fi

nd some of the spokes of the Siemens star target were missing in the

FPM

’

s reconstruction, which indicates the failure of FPM. This experi-

ment showcased the ability of APIC to better retrieve a high-resolution

complex

fi

eld when the raw data size is constrained because it is an

analytical method and does not rely as heavily on pupil overlap

redundancy for solution convergence that FPM requires.

It is also worth noting, for input image tile of length 256 pixels on

both sides, APIC reconstruction took 9 seconds on a personal com-

puter (CPU: Intel Core i9-10940X with 64 GB RAM), while FPM required

25 seconds to

fi

nish the reconstruction. The relative computational

fi

ciency of APIC can again be attributed to the analytical nature of its

approach in contrast to FPM. We note that this computational ef

fi

ciency is image tile size dependent

—

the smaller the tile is, the more

fi

cient APIC can be (see section 6 in the supplementary note for more

information). As it is generally preferred to divide a large image into

smaller tiles in parallel computing, APIC

’

s computational ef

fi

ciency for

smaller tiles aligns well with practical computation considerations.

In the next experiment, we studied the robustness of APIC and

FPM in addressing optical aberrations. For this experiment, we

acquired a total of 316 images, which consisted of 52 normal bright

fi

eld

measurements, 16 NA-matching measurements, and 248 dark

fi

eld

measurements. The nominal scanning pupil overlap ratio of our

dataset was ~87%, and the

fi

nal theoretical synthetic NA was equal to

0.75 when all dark

fi

eld measurements were used. We note that this

large degree of spectrum overlap was chosen to provide suf

fi

cient data

redundancy for the best performance of FPM. APIC does not require

such a large dataset (examples can be found in Fig.

and Fig. S3 of our

supplementary note). In our reconstruction, APIC only used the NA-

matching and dark

fi

eld measurements, whereas FPM used the entire

dataset, including these additional 52 bright

fi

eld measurements cor-

responding to illumination angles that were below the objective

’

acceptance angle.

We deliberately defocused a Siemens star target to assess how the

two methods perform under different aberration levels. In this

experiment, the sample was defocused to different levels, and the

defocus information was hidden from both methods. The results of

FPM and APIC are shown in Fig.

a. Clearly, for large aberrations whose

phase standard deviation exceeded 1.1

π

(the case when Siemens star

target was defocused by 32

μ

m, and the maximal phase difference is

~3.8

π

), FPM failed to

fi

nd the correct solution and the reconstructed

images were considerably different from the ground truth, even when

the algorithm indicated its convergence criterion was reached. At a

lower aberration level, the amplitude reconstructions of FPM appeared

to be close to the ideal case. However, the reconstructed phases were

substantially different from the result when no defocus was intro-

duced. In contrast, APIC was highly robust to different levels of aber-

rations. Although the contrast of APIC

’

s reconstruction dropped under

larger aberrations, it retrieved the correct aberrations and gave high-

resolution complex

fi

eld reconstructions that matched with the in-

-2

FPM

APIC

Zernike

decomposition

Coefficients

Zernike mode

-π

Defocus 0 μm

Defocus 16 μm

Defocus 32 μm

-π

Amplitude

Phase

Pupil

-π

Amplitude

Phase

Pupil

-π

Amplitude

Phase

Pupil

5μm

FPM

Sum

APIC

20 μm

Pupil, FPM

Pupil, APIC

-π

20 μm

Amplitude, FPM

Amplitude, APIC

Ground truth

Sum of measurements

Phase, APIC

Phase, FPM

Fig. 4 | Reconstruction under different levels of aberrations. a

Reconstructed

complex

fi

elds and aberrations with different defocus distances. For the recon-

struction of APIC and FPM, the defocus distance is labeled on top of each group. In

our reconstruction, the actual defocus distance is hidden from both algorithms.

APIC reconstructed amplitude and phase are shown on the upper right of each

group and highlighted by the dashed magenta line. FPM reconstructed amplitude

and phase are shown on the lower left of each group and highlighted by the dashed

cyan line. The reconstructed pupils of APIC and FPM are also color-coded by

magenta and cyan, respectively.

Reconstruction of a human thyroid carcinoma

cell sample using APIC and FPM. Sum of measurements denotes the summation of

all 316 images we acquired, which can be treated as the incoherent image we would

get under the same objective. The ground truth was acquired using an objective

whose magni

fi

cation power is 40, and NA equals 0.75. The NA of this 40× objective

equals the theoretical synthetic NA of APIC and FPM. We calculated the square root

of the summed image and the 40× image to match up with the amplitude recon-

struction. The zoomed images of the highlighted boxes are shown on the lower

right of

. The Zernike decompositions of retrieved aberrations using FPM and

APIC are shown on the far-right side of

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| (2024) 15:4713

focus result. The measured resolution for both FPM and APIC is ~

870 nm when the in-focus measurements were used, which is close to

the 840 nm theoretical resolution (Fig. S4).

To test the two methods under more complex aberrations, we

used an obsolete Olympus objective (10× magni

fi

cation, NA 0.25) that

was designed to work with another type of tube lens for image mea-

surement in this particular experiment. A human thyroid adenocarci-

noma cell sample was imaged to see their performance. As the

standard deviation of the phase of the imaging system

’

s aberration was

close to 2

π

/5, FPM failed to reconstruct a high-resolution image. From

Fig.

b, the reconstructed amplitude of FPM was heavily distorted by

the reconstruction artifacts. APIC recovered all the

fi

ner details that

were in good correspondence with the image we acquired using a 0.75

NA objective.

We then conducted an experiment using a hematoxylin and eosin

(H&E) stained breast cancer cell sample. We used red, green, and blue

LEDs to acquire datasets for these three different channels and then

applied APIC for the reconstruction. In this experiment, the sample

was placed at a

fi

xed height in the data acquisition process. As a result,

we see different levels of defocus in different channels lying on top of

the chromatic aberrations of the objective (Fig. S5). To acquire the

ground truth image, we switched to a 40× objective and manually

focused each channel. We calibrated the illumination angles for the

central patch (side length: 512 pixels) and then calculated the angles

for off-axis patches using geometry. These calibrated illumination

angles were used as the input parameter in our reconstruction. The

fi

nal reconstructed region is a square of side length of 1.2 mm in this

experiment.

The reconstructed color image is shown in Fig.

.Thecomparison

of all three channels can be found in our supplementary note (Fig. S5).

From the zoomed images in Fig.

, we can see that the reconstructions

of FPM were noisy for the blue channel. We found that FPM did not

work well with this weakly absorptive sample under a relatively high

aberration level. It failed to extract the aberration of the imaging sys-

tem. As such, the color image generated by FPM appeared grainy, and

the high spatial frequency information was only partially recovered.

We also see that the color reconstruction of APIC retained all high

spatial frequency features that were closely matched up with the

ground truth we acquired. This demonstrates that the aberration and

complex

fi

eld reconstruction of APIC is considerably more accurate

compared with FPM.

Discussion

We showed that APIC can extract large aberrations and synthesize

large FOV and high-resolution images using low NA objectives. APIC

empowers computational label-free microscopy with high robustness

against aberration. Under the same high-aberration conditions, FPM

fails to recover the aberration, and its reconstruction result largely

inherits such aberration and thus cannot produce aberration-free,

high-resolution reconstructions.

Moreover, some of the fundamental problems in the conventional

phase retrieval algorithm, such as being prone to optimization para-

meters and getting stuck in local minimum, are solved in APIC. Previous

results demonstrated that reconstruction artifacts appear in FPM with-

out properly selected parameters or loss functions

,whichisin

consistent with our experiment results shown in Fig.

. Without a

properly engineered metric, the selection of parameters becomes highly

subjective. This again indicates that it is often unclear on whether FPM is

even converging close to the real complex

fi

eld solution. APIC is robust

against this problem as it does not require an iterative algorithm for

reconstruction. It circumvents th

e need to choose optimization para-

meters or designing metric for convergence. However, as an analytical

method, the knowledge about the position of the LEDs, as well as the

alignment of the NA-matching angle illuminations, is important in APIC.

When large calibration errors show up, the solution of APIC will be

negatively impacted (Fig. S14). Thus, a good calibration is required in

APIC to get the correct solution. In addition, we note that the amplitude

of the CTF is assumed to be unit in our prototype as the aperture of our

experiment system has negligible amplitude variation. For an aperture

with intrinsic amplitude variation,

we anticipate that this can be cor-

rected using a similar approach applied for the aberration correction.

Instead of subtracting the phase, o

ne would calculate the ratio of the

spectrum

’

s amplitude for the overlapping part and then use this ratio to

correct for the unevenness in CTF

’

samplitude.

As APIC directly solves the complex

fi

eld, it avoids the potentially

time-consuming iterative process. When a reasonable image patch size

is chosen, APIC is computationally more ef

fi

cient compared with FPM.

As such, APIC alleviates the lengthy processing in FPM, making it a

more appealing method (Fig. S7 in supplementary note).

While this work demonstrates a working APIC prototype, we note

that a key aspect of the prototype would need further design

improvements if a larger

fi

eld of view is desired. Speci

fi

cally, in this

prototype, we treat the LED illumination as a plane wave at the sample

200 μm

FPM

APIC

FPM

-π

FPM

APIC

π/4

-π/4

10x

FPM

APIC

40x (GT)

5 μm

Blue channel

Amplitude

Color

Phase

Pupil

APIC reconstruction, color image

Fig. 5 | Reconstructed high-resolution image of hematoxylin and eosin (H&E)

stained breast cancer cells.

APIC reconstructed aberration corrected, high-

resolution color image is shown on the left. The zoomed image of the highlighted

region in the color image is shown on the right. The image with the label

“

10×

”

denotes the image acquired using the same 10× magni

fi

cation objective, which was

used for data acquisition and thus showed poorer resolution due to its limited NA.

The color image with label

“

40× (GT)

”

denotes the ground truth we acquired using a

40× objective whose NA equals the theoretical synthetic NA of APIC. We note that

we manually focused the image under red, green, and blue LED illumination when

acquiring the ground truth as the best focal planes for them are different, while no

tuning was applied in APIC

’

s data acquisition. We picked a blue channel as an

example in this illustration, and the complex

fi

eld reconstructions and retrieved

aberrations are shown at the bottom of the rounded box. From the zoomed images,

APIC shows good correspondence with the ground truth, while FPM is much

noisier.

Article

https://doi.org/10.1038/s41467-024-49126-y

Nature Communications

| (2024) 15:4713

plane. However, for a large

fi

eld of views, the illumination angle can be

quite different for different patches in the entire FOV. This may lead to

noisy reconstruction of NA-matching measurements, as a previous

study has indicated

. We anticipate that this problem can be mitigated

by increasing the distance between the LED and the sample. It can also

be solved by designing better LED illumination systems. Additionally,

the number of LED illuminations can be reduced in future systems by

decreasing the overlap of two measured spectrums.

In conclusion, we demonstrate that APIC can provide high-

resolution and large FOV label-free imaging with unprecedented

robustness to aberrations. As an analytical method, APIC is insensitive

to parameter selections and can compute the correct imaging

fi

eld

without getting trapped in local minimums. APIC

’

sanalyticityispar-

ticularly important in a range of exacting applications, such as digital

pathology, where even minor errors are not tolerable. APIC guarantees

the correct solution, while FPM-like iterative methods cannot. Addi-

tionally, APIC brings new possibilities to label-free computational

microscopy as it affords greater freedom in the use of engineered

pupils for various imaging purposes. We anticipate the APIC concept

can be fruitfully adopted for other methods, such as the aberration

found by APIC, which can potentially be used to correct incoherent

imaging. The idea of using the known spectrum to reconstruct the

unknown spectrum can be readily adapted for use in other scenarios.

Reporting summary

Further information on research design is available in the Nature

Portfolio Reporting Summary linked to this article.

Data availability

Part of the data that supports this study is available on GitHub (

https://

github.com/rzcao/APIC-analytical-complex-

fi

eld-reconstruction

). The

complete data that support the plots within this paper and other

fi

ndings of this study are available from the corresponding authors

upon request. Source data are provided with this paper.

Code availability

The reconstruction code that supports the plots within this paper and

other

fi

ndings of this study is available on Supplemetary Code 1 as well

as GitHub (

https://github.com/rzcao/APIC-analytical-complex-

fi

eld-

reconstruction

) and the zenodo repository

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Acknowledgements

This research is supported by Heritage Medical Research Institute (HMRI)

(Award number HMRI-15-09-01). The authors thank Dr. Jerome Mertz for

the insightful discussion of this work.

Author contributions

R.C. conceived the idea. R.C. conduc

ted the theoretical analysis, con-

ducted the simulations and wrote the reconstruction algorithm. C.S.

built the experiment setup, wrote t

he hardware controlling code and

performed the calibration of the system. R.C. and C.S. conducted the

experiments. C.Y. supervised this project. All authors contributed to the

preparation of the manuscript.

Competing interests

The authors (R.C., C.S., and C.Y.) declare the following competing

interests: On 30 March 2023, the California Institute of Technology

led

a provisional patent application for APIC, which covered the concept

and implementation of the APIC system described here.

Article

https://doi.org/10.1038/s41467-024-49126-y

Nature Communications

| (2024) 15:4713

Additional information

Supplementary information

The online version contains

supplementary material available at

https://doi.org/10.1038/s41467-024-49126-y

Correspondence

and requests for materials should be addressed to

Ruizhi Cao.

Peer review information

Nature Communications

thanks Pavan Konda

and the other, anonymous, reviewer(s)

for their contribution to the peer

review of this work. A peer review

le is available.

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is available at

http://www.nature.com/reprints

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