11institutetext: PAII Inc., Bethesda, MD, USA 22institutetext: University of
Maryland, College Park, MD, USA 33institutetext: Chang Gung Memorial Hospital,
Linkou, Taiwan, ROC 44institutetext: Ping An Technology, Shenzhen, China

# Semi-Supervised Learning for Bone Mineral Density Estimation in Hip X-ray
Images

Kang Zheng 11 Yirui Wang 11 Xiao-Yun Zhou 11 Fakai Wang 1122 Le Lu 11
Chihung Lin 33 Lingyun Huang 44 Guotong Xie 44 Jing Xiao 44 Chang-Fu Kuo 33
Shun Miao 11

###### Abstract

Bone mineral density (BMD) is a clinically critical indicator of osteoporosis,
usually measured by dual-energy X-ray absorptiometry (DEXA). Due to the
limited accessibility of DEXA machines and examinations, osteoporosis is often
under-diagnosed and under-treated, leading to increased fragility fracture
risks. Thus it is highly desirable to obtain BMDs with alternative cost-
effective and more accessible medical imaging examinations such as X-ray plain
films. In this work, we formulate the BMD estimation from plain hip X-ray
images as a regression problem. Specifically, we propose a new semi-supervised
self-training algorithm to train the BMD regression model using images coupled
with DEXA measured BMDs and unlabeled images with pseudo BMDs. Pseudo BMDs are
generated and refined iteratively for unlabeled images during self-training.
We also present a novel adaptive triplet loss to improve the model’s
regression accuracy. On an in-house dataset of 1,090 images (819 unique
patients), our BMD estimation method achieves a high Pearson correlation
coefficient of 0.8805 to ground-truth BMDs. It offers good feasibility to use
the more accessible and cheaper X-ray imaging for opportunistic osteoporosis
screening.

###### Keywords:

Bone Mineral Density Estimation Hip X-ray Semi-supervised Learning.

##  1 Introduction

Osteoporosis is a common skeletal disorder characterized by decreased bone
mineral density (BMD) and bone strength deterioration, leading to an increased
risk of fragility fracture. All types of fragility fractures affect the
elderly with multiple morbidities, reduced life quality, increased dependence,
and mortality. FRAX is a clinical tool for assessing bone fracture risks by
integrating clinical risk factors and BMD. While some clinical risk factors
such as age, gender, and body mass index can be obtained from electronic
medical records, the current gold standard to measure BMD is dual-energy X-ray
absorptiometry (DEXA). However, due to the limited availability of DEXA
devices, especially in developing countries, osteoporosis is often under-
diagnosed and under-treated. Therefore, alternative lower-cost BMD evaluation
protocols and methods using more accessible medical imaging examinations,
_e.g._ , X-ray plain films, are highly desirable.

Previous methods aiming to use imaging obtained for other indications to
estimate BMD or classify osteoporosis have been proposed. The clinical value
of this “opportunistic screening” approach seems apparent with low additional
cost, high speed to obtain essential patient information, and potentially
greater accuracy over traditional prediction models. For example, previous
studies [1, 8, 10, 6, 11, 2] have attempted to estimate BMD through CT scans
for osteoporosis screening. However, the undesired prediction performance,
high cost, long acquisition time, and particularly high radiation dose of CT
imaging compared to DEXA and plain film X-ray are barriers to their wide
clinical adoption. In contrast, plain film X-ray is a more accessible and
lower-cost imaging tool than CT for opportunistic screening. In this work, we
attempt to estimate BMD from plain film hip X-ray images for osteoporosis
screening.

Our work is based on the assumption that hip X-ray images contain sufficient
information on visual cues for BMD estimation. We use a convolutional neural
network (CNN) for regressing BMD from hip X-ray images. Paired hip X-ray image
and DEXA measured BMD taken within six months apart (as ground-truth) are
collected as labeled data for supervised regression learning. To improve
regression accuracy, we also propose a novel adaptive triplet loss, enabling
the model to better discriminate samples with dissimilar BMDs in the feature
space. Since it is often difficult to obtain a large amount of hip X-ray
images paired with DEXA measured BMDs, we propose to use semi-supervised
learning to exploit the usefulness of large-scale hip X-ray images without
ground-truth BMDs, which can be efficiently collected at scale. Our method is
evaluated on an in-house dataset of 1,090 images from 819 patients and
achieves a Pearson correlation coefficient of 0.8805 to ground-truth BMDs.

Our main contributions are three-fold: 1) This work is the first to estimate
bone mineral density from the hip plain films, which offers a potential
computer-aided diagnosis (CAD) application to opportunistic osteoporosis
screening with more accessibility and reduced costs. 2) We propose a novel
adaptive triplet loss to improve the model’s regression accuracy, applicable
to other regression tasks as well. 3) We present a new semi-supervised self-
training algorithm to boost the BMD estimation accuracy via exploiting
unlabeled hip X-ray images.

##  2 Method

Given a hip X-ray image, we first crop a region-of-interest (ROI) around the
femoral neck111This automated ROI localization module can be achieved by re-
implementing the deep adaptive graph (DAG) network [9]. as input to the
convolutional neural network. Denoting the hip ROI as $I$ and its associated
ground-truth (GT) BMD as $y$, our goal is to provide an estimation
$y^{\prime}$ close to $y$. We formulate the problem as a regression problem
since BMD values are continuous.

Figure 1: An overview of our proposed framework, consisting of (a) supervised
pre-training stage and (b) semi-supervised self-training stage. In the
supervised pre-training stage, we train the model on labeled images using the
MSE loss and (c) a novel adaptive triplet loss, which encourages the distance
between feature embeddings of samples correlated to their BMD difference. In
the self-training stage, we fine-tune the model on labeled data and pseudo-
labeled data. We update the pseudo labels when the model achieves higher
performance on the validation set.

Figure 1 illustrates the overall framework of our proposed method, which
consists of two stages: supervised pre-training and semi-supervised self-
training. In the first stage, we pre-train a model on a small amount of
labeled images. We adopt VGG-11 [12] as the backbone with its last two fully-
connected (FC) layers replaced by a global average pooling (GAP) layer to
obtain a 512-dimensional embedding feature vector. Then, an FC layer is
applied to regress BMD from the feature vector. We train the model using Mean
Square Error (MSE) loss and a novel adaptive triplet loss, as shown in Figure
1(a). The MSE loss is defined between the predicted BMD $y^{\prime}$ and the
GT BMD $y$, written as:

$\mathcal{L}_{mse}=\left(y-y^{\prime}\right)^{2}.$ (1)

With the proposed adaptive triplet loss, the model learns more discriminative
feature embeddings for images with different BMDs, thus improving its
regression accuracy. We elaborate on the adaptive triplet loss in Section 2.1.
In the second stage, we fine-tune the model with a novel semi-supervised self-
training algorithm using both labeled and unlabeled images. We iteratively
predict the pseudo BMDs for unlabeled images and combine them with labeled
ones to fine-tune the model. The self-training algorithm is detailed in
Section 2.2.

###  2.1 Adaptive Triplet Loss

As BMD is a continuous value, hip ROIs’ embeddings are expected to be
continuous in the feature space as well. In particular, the distance between
embeddings of two samples in the feature space should be correlated with their
BMD discrepancy. Motivated by this, we propose a novel adaptive triplet loss
encouraging the model to discriminate samples with different BMDs in feature
space, as shown in Figure 1(c). For a given triplet, one sample is selected as
anchor. Among the other two samples, the one with a BMD closer to the anchor
is regarded as near sample, and the other as far sample. We want the embedding
of the anchor to be closer to near sample than far sample in the feature space
by a margin, _i.e._ ,

$||F_{a}-F_{n}||^{2}_{2}+m<||F_{a}-F_{f}||^{2}_{2},$ (2)

where $m$ is the margin. $F_{a}$, $F_{n}$, and $F_{f}$ are embeddings of the
anchor, near, and far samples, respectively. The margin that separates the
near sample from the far sample should account for their relative BMD
differences. Therefore, we define the adaptive triplet loss as follows:

$\mathcal{L}_{triplet}=\left[||F_{a}-F_{n}||^{2}_{2}-||F_{a}-F_{f}||^{2}_{2}+\alpha
m\right]_{+}.$ (3)

$\alpha$ is the adaptive coefficient based on the BMD differences, defined as:

$\alpha=||y_{a}-y_{f}||^{2}_{2}-||y_{a}-y_{n}||^{2}_{2},$ (4)

where $y_{a}$, $y_{n}$, and $y_{f}$ are the GT BMD values of the anchor, near,
and far samples, respectively. For network training, we combine the MSE loss
with adaptive triplet loss as follows:

$\mathcal{L}=\mathcal{L}_{mse}+\lambda\mathcal{L}_{triplet},$ (5)

where $\lambda$ is the weight for adaptive triplet loss. We set $\lambda$ to
0.5 for all experiments.

###  2.2 Semi-Supervised Self-Training

Given limited images coupled with GT BMDs, the model can easily overfit the
training data and yield poor performance on unseen test data. To overcome this
issue, we propose a semi-supervised self-training algorithm to leverage both
labeled and unlabeled data. An overview of the proposed self-training
algorithm is illustrated in Figure 1(b). We first use the pre-trained model to
predict pseudo GT BMDs on unlabeled images to obtain additional supervisions.
The unlabeled images with pseudo GT BMDs are subsequently combined with
labeled images to fine-tune the model. To improve the quality of estimated
pseudo GT BMDs, we propose to refine them using the better fine-tuned models
during self-training. We assume that if a fine-tuned model achieves higher
performance on the validation set, it can also produce more accurate/reliable
pseudo GT BMDs for unlabeled images. Specifically, after each self-training
epoch, we evaluate the model performance on the validation set using Pearson
correlation coefficient and mean square error. If the model indeed achieves
higher correlation coefficient and lower mean square error than previous
models at the same time, we use it to re-generate pseudo GT BMDs for the
unlabeled images for self-training. This process is repeated until the total
self-training epoch is reached. We summarize this semi-supervised self-
training algorithm in Algorithm 1.

In addition, we augment each image twice and employ consistency constraints
between their features and between predicted BMDs, which regularize the model
to avoid being misled by inaccurate pseudo labels. Denoting two augmentations
of the same image as $I_{1}$ and $I_{2}$, their features $F_{1}$ and $F_{2}$,
and their predicted BMDs $y_{1}$ and $y_{2}$, we define the consistency loss
as:

$\mathcal{L}_{c}=||F_{1}-F_{2}||^{2}_{2}+||y_{1}-y_{2}||^{2}_{2}.$ (6)

For self-training, the total loss is:

$\mathcal{L}=\mathcal{L}_{mse}+\lambda\mathcal{L}_{triplet}+\lambda^{\prime}\mathcal{L}_{c},$
(7)

where $\lambda^{\prime}$ is the consistency loss weight set to 1.0.

Algorithm 1 Semi-supervised self-training algorithm

1:Initialize the best R-value $\hat{\eta}\coloneqq 0$ and MSE
$\hat{\epsilon}\coloneqq\infty$

2:Initialize training epoch $e\coloneqq 0$ and set total training epoch $E$

3:Initialize model with pre-trained weights

4:while $e<E$ do

5: Evaluate model performance R-value $\eta$ and MSE $\epsilon$ on the
validation set

6: if $\eta>\hat{\eta}$ and $\epsilon<\hat{\epsilon}$ then

7: $\hat{\eta}\coloneqq\eta$

8: $\hat{\epsilon}\coloneqq\epsilon$

9: Generate pseudo BMDs for unlabeled images

10: Fine-tune model on labeled images and unlabeled images with pseudo BMDs

11: $e\coloneqq e+1$

##  3 Experiment Results

Dataset and evaluation metrics. We collected 1,090 hip X-ray images with
associated DEXA-measured BMD values from 819 patients. The X-ray images are
taken within six months of the BMD measurement. We randomly split the images
into training, validation, and test sets of 440, 150, and 500 images based on
patient identities. For semi-supervised learning, 8,219 unlabeled hip X-ray
images are collected. To extract hip ROIs around the femoral neck, we train a
localization model with the deep adaptive graph (DAG) network (re-implemented
from [9]) using about 100 images with manually annotated anatomical landmarks.
We adopt two metrics, Pearson correlation coefficient (R-value) and Root Mean
Square Error (RMSE), to evaluate the proposed method and all compared methods.

Implementation details. The ROIs are resized to $512\times 512$ pixels as
model input. Random affine transformations, color jittering, and horizontal
flipping are applied to each ROI during training. We adopt VGG-11 with batch
normalization [5] and squeeze-and-excitation (SE) layer [4] as the backbone,
since it outperforms other VGG networks [12] and ResNets [3] as demonstrated
later. The margin for adaptive triplet loss is set to 0.5. We use Adam
optimizer with a learning rate of $10^{-4}$ and weight decay of $4\times
10^{-4}$ to train the network on labeled images for 200 epochs. The learning
rate is decayed to $10^{-5}$ after 100 epochs. For semi-supervised learning,
we fix the learning rate at $10^{-5}$ to fine-tune for another 100 epochs.
After each training/fine-tuning epoch, we evaluate the model on the validation
set and select the one with the highest Pearson correlation coefficient for
testing. All models are implemented using PyTorch 1.7.1 and trained on a
workstation with an Intel(R) Xeon(R) CPU, 128G RAM, and a 12G NVIDIA Titan V
GPU. We set the batch size to 16. The adaptive triplet loss is calculated for
triplets randomly constructed within each batch. Specifically, each sample is
used as an anchor with two other samples randomly selected as the near/far
samples depending on their BMD distances to the anchor.

Backbone selection. Here we study how different backbones affect the baseline
performance without adaptive triplet loss or self-training. The compared
backbones include VGG-11, VGG-13, VGG-16, ResNet-18, ResNet-34, and ResNet-50.
As shown in Table 1, VGG-11 achieves the best R-value of 0.8520 and RMSE of
0.0831. The lower performance of other VGG networks and ResNets may be
attributed to overfitting from more learnable parameters. In light of this, we
conjecture there might be a more appropriate network backbone for BMD
regression, which is outside this paper’s scope. Therefore, we adopt VGG-11
with batch normalization and SE layer as our network backbone.

Table 1: Comparison of baseline methods using different backbones. Backbone | VGG-11 | VGG-13 | VGG-16 | ResNet-18 | ResNet-34 | ResNet-50
---|---|---|---|---|---|---
R-value | 0.8520 | 0.8501 | 0.8335 | 0.8398 | 0.8445 | 0.8448
RMSE | 0.0831 | 0.0855 | 0.1158 | 0.0883 | 0.0946 | 0.1047

Quantitative comparison results. We compare our method with three existing
semi-supervised learning (SSL) methods: $\rm{\Pi}$-model [7], temporal
ensembling [7], and mean teacher [13]. Regression MSE loss between predicted
and GT BMDs is used on labeled images for all semi-supervised learning
methods. $\rm{\Pi}$-model is trained to encourage consistent network output
between two augmentations of the same input image. Temporal ensembling
produces pseudo labels via calculating the exponential moving average of
predictions after every training epoch. Pseudo labels are then combined with
labeled images to train the model. Instead of directly ensembling predictions,
mean teacher uses an exponential moving average of model weights to produce
pseudo labels for unlabeled images. For the proposed method and compared SSL
methods, all models are fine-tuned from the weights pre-trained on labeled
images.

Table 2: Comparison with semi-supervised learning methods. (Temp. Ensemble: temporal ensembling) Method |  $\rm{\Pi}$-model | Temp. Ensemble | Mean Teacher | Proposed
---|---|---|---|---
R-value | 0.8637 | 0.8722 | 0.8600 | 0.8805
RMSE | 0.0828 | 0.0832 | 0.0817 | 0.0758

As shown in Table 2, the proposed method achieves the best R-value of 0.8805
and RMSE of 0.0758. $\rm{\Pi}$-model outperforms the baseline by enforcing
output consistency as a regularization. While both temporal ensembling and
mean teacher obtain improvements with the additional pseudo label supervision,
averaging labels or weights can accumulate more errors over time. In contrast,
our method only updates pseudo labels when the model performs better on the
validation set, proving to be an effective strategy. Figure 2 shows the
predicted BMDs of our model against GT BMDs. Overall, the model has a larger
prediction error for lower or higher BMD cases. This situation is expected
because lower/higher BMD cases are less common, and the model tends to predict
moderate values.

Figure 2: Predicted BMDs against ground-truth BMDs on the test set.

Ablation study. To assess the importance of various components in our proposed
method, we conduct experiments to remove some components or vary some hyper-
parameters while keeping others fixed. Firstly, we analyze the proposed
adaptive triplet loss (ATL) by comparing it to its counterpart without the
adaptive coefficient in Eq. 4 and varying the preset margin. Results in Table
3 show that the non-adaptive counterpart deteriorates the model’s regression
accuracy, which proves the necessity of adaptive coefficient. Since BMD
differences vary for different triplets, it is unreasonable to use a fixed
margin to uniformly separate samples with dissimilar BMDs. We also observe
that using ATL achieves higher R-values than the baseline regardless of the
margin value. Specifically, $m=0.5$ produces the best R-value of 0.8670 and
RMSE of 0.0806. It demonstrates the effectiveness of our proposed adaptive
triplet loss.

Table 3: Ablation study of adaptive triplet loss. Margin | Triplet Loss | ATL
---|---|---
R-value | RMSE | R-value | RMSE
0.1 | 0.8515 | 0.0917 | 0.8563 | 0.1013
0.3 | 0.8459 | 0.1000 | 0.8657 | 0.0814
0.5 | 0.8538 | 0.0866 | 0.8670 | 0.0806
0.7 | 0.8549 | 0.0823 | 0.8565 | 0.0836
1.0 | 0.8522 | 0.0829 | 0.8524 | 0.1215

To better understand the benefits of our proposed semi-supervised self-
training algorithm, we compare it with a straightforward semi-supervised
learning (SSL) approach. We use the pre-trained model to predict pseudo BMDs
for unlabeled images and combine them with labeled ones for further fine-
tuning. We apply this approach to fine-tune models pre-trained using only MSE
loss and the combination of MSE and ATL losses, denoted as SSL and SSL+ATL,
respectively. Finally, we also analyze the contribution of the consistency
loss in Eq. 6 by removing it during the self-training stage. The comparison
results are shown in Table 4.

Table 4: Ablation study of adaptive triplet loss and self-training algorithm. Method | R-value | RMSE
---|---|---
Baseline | 0.8520 | 0.0831
Baseline + ATL | 0.8670 | 0.0806
SSL | 0.8605 | 0.0809
SSL + ATL | 0.8772 | 0.0767
Proposed w/o Consistency | 0.8776 | 0.0761
Proposed | 0.8805 | 0.0758

Employing the straightforward SSL strategy is effective as expected. It
increases the baseline R-value to 0.8605 and reduces the RMSE to 0.0809. The
model pre-trained using MSE and ATL losses is also improved. The results prove
the effectiveness of using pseudo labels of unlabeled images. Our proposed
self-training algorithm can obtain further improvement by updating pseudo
labels during fine-tuning. However, the results show that the improvement
becomes marginal from 0.8772 to 0.8776 in R-value without the consistency
loss, even if pseudo labels are updated a few more times. The consistency loss
can regularize model training by encouraging consistent output and features.
Without it, the model is prone to overfitting to inaccurate pseudo labels and
deteriorates. Combining ATL, self-training algorithm, and consistency loss,
our proposed method achieves the best R-value of 0.8805 and RMSE of 0.0758.
Compared to the baseline, the R-value is improved by 3.35% and the RMSE is
reduced by 8.78%.

##  4 Conclusion

In this work, we investigate the feasibility of obtaining bone mineral density
(BMD) from hip X-ray images instead of DEXA measurement. It is highly
desirable for various practical reasons. We employ a convolutional neural
network to estimate BMDs from preprocessed hip ROIs. Besides the MSE loss, a
novel adaptive triplet loss is proposed to train the network on hip X-ray
images with paired ground-truth BMDs. We further present a semi-supervised
self-training algorithm to improve our model using large-scale unlabeled hip
X-ray images. Experiment results on an in-house dataset of 1,090 images from
819 patients show that our method effectively achieves a high Pearson
correlation coefficient of 0.8805. It implies the strong feasibility of X-ray
based BMD estimation, which can provide potential opportunistic osteoporosis
screening with more accessibility and at reduced cost.

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# Direct 3D imaging through spatial coherence of light

Gianlorenzo Massaro  1,2,* Barbara Barile  3 Giuliano Scarcelli  4 Francesco
V. Pepe  1,2 Grazia Paola Nicchia  3,5,6,$\dagger$ and Milena
D’Angelo1,2,$\dagger$ 1Dipartimento Interateneo di Fisica, Università degli
Studi di Bari Aldo Moro, Bari, 70125,Italy
2Sezione di Bari, Istituto Nazionale di Fisica Nucleare, Bari, 70125,Italy
3Dipartimento di Bioscienze, Biotecnologie e Ambiente,Università degli Studi
di Bari Aldo Moro, Bari, 70125,Italy
4Fischell Department of Bioengineering, University of Maryland, College Park
MD 20742 USA
5Institute for Organic Synthesis and Photoreactivity, National Research
Council of Italy, Bologna, 40129, Italy
6Dominik P.Purpura Department of Neuroscience, Albert Einstein College of
Medicine, New York, 10461, NY, USA
$\dagger$The authors contributed equally to this work.
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Wide-field imaging is widely adopted due to its fast acquisition, cost-
effectiveness and ease of use. Its extension to direct volumetric
applications, however, is burdened by the trade-off between resolution and
depth of field (DOF), dictated by the numerical aperture of the system. We
demonstrate that such trade-off is not intrinsic to wide-field imaging, but
stems from the spatial incoherence of light: images obtained through spatially
coherent illumination are shown to have resolution and DOF independent of the
numerical aperture. This fundamental discovery enabled us to demonstrate an
optimal combination of coherent resolution-DOF enhancement and incoherent
tomographic sectioning for scanning-free, wide-field 3D microscopy on a
multicolor histological section.

##  1 Introduction

Wide-field imaging is amongst the most common imaging modalities for the
observation and characterization of absorbing specimens, as done, for
instance, in bright-field microscopy [1]. Some of the reasons behind its
widespread use across many diverse applications are its ease of use, cost-
effectiveness, fast acquisition, and its direct imaging capability (namely,
the availability of the output image in real time, with no need for inverse
computation techniques on the collected intensity). Although conventional
devices work extremely well with 2D samples, having negligible thickness along
the optical axis ($z$), their use with 3D samples is significantly complicated
by the well-known dependence of both resolution and depth of field (DOF) on
the numerical aperture (NA) of the imaging device: this dependence results in
a strong trade-off between image resolution and DOF, and imposes the need to
z-scan the whole sample in order to collect the complete volumetric profile.
The operation of z-scanning requires that either the imaging device or the
sample itself are mechanically shifted along the optical axis, so as to change
the plane at focus and perform multiple acquisitions of different transverse
planes [2, 3]. The intrinsically long acquisition required by moving
components implies limited in vivo applicability and comes with further
disadvantages, such as the need for precise stabilization, requiring large and
heavy devices, costly mechanical parts with the required precision, as well as
high maintenance costs, which preclude the use of scanning microscopes in low-
budget applications. The limitations of axial scanning become particularly
relevant in large-NA devices, where the higher resolution comes at the expense
of a narrower DOF. This has detrimental effects on the number of axial
measurements necessary to characterize the entire sample, so that a common
option to keep the measurement time low is to under-sample along the optical
axis, with a consequent loss of information. In 3D imaging, resolution, axial
sampling and acquisition speed are thus in direct conflict.

Several apporaches have been proposed in the literature to address this
problem; optical coherence tomography (OCT) is one of the most noticeable
examples [4]. However, in all cases, the limitations imposed by the NA of the
imaging system persist; in OCT, for example, small NA are required, at the
expenses of resolution and signal-to-noise ratio (SNR), for addressing the
loss of intensity implied by large-NA optics [5].

A recent and rapidly developing approach to scanning-free wide-field 3D
microscopy is light-field (LF) imaging, where direct images of thick samples
containing heavily defocused planes are acquired and then refocused, in post-
processing [6, 7, 8, 9, 10]. Directional information about light from the
sample is in fact acquired by a microlens array and employed, in post-
processing, to perform software z-scans with similar features to the typical
mechanical scans. LF devices thus enable scanning-free single-shot acquisition
of a 3D sample, but its fast acquisition comes at the expenses of a dramatic
loss of resolution, well beyond the diffraction limit [11]. In fact, due to
its geometric-optics-based working principle, the maximum achievable DOF is
defined by the circle of confusion (CoC), namely, by the projection of the
lens aperture over the acquired defocused planes [12]. The resolution of the
refocused images is thus not determined by the Airy disk, as is typically the
case in microscopy, but is rather dominated by the geometrical effects of
defocusing, as typically occurring in photography. In addition, the lenslets
introduce an even stronger trade-off between resolution and DOF, consisting in
the loss of resolution at focus with the improvement of the volumetric
performance [13].

In this work, we demonstrate that the resolution versus DOF trade-off of
defocused images is governed by the spatial coherence properties of light, and
is naturally relieved when the sample is illuminated with spatially coherent
light (i.e., the coherence area on the sample is either comparable or larger
than the sample details, as explained in the Methods and Results). Coherent
imaging is thus found to entail a much slower image degradation with
defocusing, a result that leads to discover the direct 3D imaging potential of
spatially coherent light: by combining the extremely large DOF of coherent
imaging with the strong localization capability of incoherent imaging, we
design a direct, scanning-free, wide-field 3D microscope and demonstrate its
working principle by means of both test and histological samples. In
particular, we characterize the properties of direct coherent wide-field
imaging and show 3D reconstruction compatible with absorbing non-fluorescent
dyes routinely used for histochemistry [14].

An important aspect to remark is that the required coherence exclusively
relates with the transverse coherence of the field illuminating the sample,
disregarding both the temporal and the spatial coherence of the source [15]:
although our findings apply to both temporally and spatially coherent sources
such as lasers, as well as to collimated beams, none of these properties are
necessary to our scopes, and our results are thus not confined to these
scenarios. On the contrary, NA-independent resolution and DOF are shown to be
obtained with virtually any source of spatially and temporally incoherent
light, such as a LED, since the required spatial coherence can always be
acquired through propagation (Van Cittert-Zernike theorem [16]).

Our 3D microscope is, in fact, based on a conventional bright-field imaging
device, integrated with an array of LEDs for implementing a dedicated coherent
illumination strategy. Conversely, typical bright-field illumination is
obtained with extended sources, shining spatially incoherent light on the
sample [17, 1]. Spatial coherence, on the other hand, is used in a plethora of
non real-time imaging modalities relying on post-processing of the acquired
data aimed at recovering rich phase information about the sample; techniques
such as holography [18, 19] and ptychography [20], for example, can achieve
super-resolution, wavefront reconstruction, and correction of optical
aberrations. Most notably, techniques based on computational illumination from
LED arrays [21, 22] have demonstrated high-resolution 3D amplitude and phase
reconstruction [23] by exploiting sequential multi-angle plane-wave
illumination and recursive phase-retrieval algorithms. However, all such
coherent imaging techniques are indirect, due to the time-consuming algorithms
they require for data analysis, and are thus not suitable for real-time
imaging [24, 25].

In this work, we show that the spatial coherence of light can be exploited in
direct wide-field imaging to obtain a breakthrough improvement of the image
resolution over large DOF. This result is supported by the discovery of the
completely different physical mechanisms regulating resolution loss in
defocused images obtained through spatially coherent and spatially incoherent
illumination. In fact, while the peculiarities of focused images, whether
coherent or incoherent, are well known [26, 27], the properties of coherent
defocused images have been so far mostly unexplored, with the only exception
of the very special case of collimated light illumination [28, 23, 27]. In
this work, the introduction of a dedicated formalism and an unbiased image
quantifier enables to study the properties of coherent images and to compare
them with the ones of conventional incoherent imaging. One of the main results
we shall present is that neither the NA nor the design of the imaging system
affect the quality of defocused coherent images; in fact, the NA-dependent
trade-off between resolution and DOF defined, in incoherent defocused images,
by the CoC, naturally disappears when illuminating the sample with spatially
coherent light. We shall profit from this effect to perform the typical
tomographic reconstruction of LF imaging and retain its multicolor capability,
but with enhanced resolution both at focus (where we recover Rayleigh-limited
resolution) and in refocused planes. No phase retrieval and time-consuming
post processing of the acquired images are required in our approach, paving
the way toward 3D real-time imaging.

##  2 Methods

As mentioned in the Introduction, we refer to coherent imaging whenever the
coherence area of the illumination [29], on the sample, is larger than the
spatial features of the sample one wishes to resolve [15]. According to the
size of the details composing a given object, an imaging system might thus
behave coherently for object details smaller than the coherence area, and
incoherently for larger details. In this respect, we should highlight that the
size of the coherence area with respect to the whole field of view (FOV) of
the image does not play any role. The transition from one regime to the other
will be discussed in details later in the paper. For the sake of simplicity,
we shall now disregard the effects of partial coherence, and only consider
coherent systems as having a coherence area larger than any object detail, and
incoherent systems as having a negligible (point-like) coherence area on the
sample.

###  2.1 Resolution and DOF in coherent imaging

Figure 1:  Comparison of resolution versus DOF in incoherent and coherent
imaging. Panel a). Incoherent imaging setup (top) and corresponding focused
and defocused images (bottom). In the setup, a transmissive sample (a double-
slit mask) is illuminated by the incoherent wavefronts coming from an extended
source and is imaged by a conventional imaging system. The focused image
(bottom left) is obtained by placing the sample at the working distance of the
imaging system; the defocused images (bottom center and right) are obtained by
$z$-scanning of the sample. Panel b). Coherent imaging setup (top) and
corresponding focused and defocused images (bottom). In the setup, the same
transmissive sample as in panel a) is illuminated by spatially coherent light
from a small incoherent source, and is imaged by the same conventional imaging
system. The focused (bottom left) and defocused (bottom center and right)
coherent images are obtained by placing the sample at exactly the same
positions as in panel a). Panel c). Axial section of the 3D cube obtained by
z-scanning the sample in the setups of panels (a) and (b), namely, in the case
of incoherent (left panel) and coherent (right panel) illumination,
demonstrating the extremely larger DOF of coherent imaging.

The upper part of panel b) suggests one of many possible ways for obtaining
coherent illumination from an incoherent source: since the coherence area on
the sample scales proportionally with the ratio between the source diameter
and the source distance, the desired coherence can easily be obtained by
reducing the source size [16]. An obvious alternative would be to employ laser
light illumination, but the presented results are not limited to this
scenario. In Figs. 1 a) and b), we report two typical examples of incoherent
and coherent imaging, respectively, as obtained by changing the illumination
in the same exact imaging system. The opposite situation of incoherent
illumination is typically achieved by placing extended natural sources at a
small distance from the sample, as reported in the upper part of panel a). In
the lower part of panels a) and b), we report the corresponding incoherent and
coherent images, both focused (left panels) and defocused (central and right
panels), of a two-dimensional sample (a double-slit mask). Defocused images
have very distinctive features depending on the spatial incoherence or
coherence of the light on the sample: while incoherent images tend to quickly
blur upon defocusing, coherent images do not blur. This is even more apparent
in panel c), reporting a section, in the $(x,z)$ plane, of the 3D cubes
obtained by mechanically z-scanning the two-slit mask, in both cases of
incoherent (left panel) and coherent (right panel) imaging. Whereas, in
incoherent imaging, z-scanning quickly gives rise to flat intensity
distributions as the object is moved out of focus, in coherent imaging, the
transmissive details contain rich spatial modulations and stay well separated
from each other over a much longer axial range compared to the corresponding
incoherent image. Transmissive details thus appear “resolved” at a much larger
distance from the plane at focus, before being completely altered by
diffraction. A much longer DOF (or, equivalently, higher resolution of
defocused images) is thus expected in coherent imaging, with image degradation
not due to blurring. Upon quantitatively describing these effects, we shall
find that resolution and DOF of defocused coherent images are actually
completely independent of the NA of the imaging system.

The differences between coherent and incoherent systems can be traced back to
the different underlying image formation processes, as formally expressed by
the intensity distributions describing the images[27]:

$\displaystyle
I_{\text{inc}}(\bm{x})=\left\lvert\mathcal{A}(\bm{x})\right\rvert^{2}*\left\lvert\mathcal{P}\left(\bm{x}\right)\right\rvert^{2}\
\ \ \ \ \ \ \
I_{\text{coh}}(\bm{x})=\left\lvert\mathcal{A}(\bm{x})*\mathcal{P}\left(\bm{x}\right)\right\rvert^{2},$
(1)

where $\mathcal{A}(\bm{x})$ is the complex transmission function of the
object, $\mathcal{P}(\bm{x})$ is the Green’s function describing the field
propagation through the optical system, and $f*g$ denotes the convolution
between two complex-valued functions, $f$ and $g$. Unlike the incoherent image
formation process, which is linear in the optical intensity, coherent imaging
is non-linear with respect to the object $\mathcal{A}(\bm{x})$. Therefore,
although the same quantities are involved in both intensity distributions of
Eq. (1), those contributing to the incoherent image formation are real and
positive, whereas coherent imaging is sensitive to both the amplitude and
phase of complex functions describing both the field distribution within the
sample and its propagation through the imaging system [19, 20, 21, 23]. In
fact, upon neglecting optical aberrations, the coherent (i.e. complex) PSF
$\mathcal{P}(\bm{x})$ of Eq. (1) can be decomposed into two contributions:

$\mathcal{P}(\bm{x})=\mathcal{D}_{z-\delta}(\bm{x})*\mathcal{P}_{0}(\bm{x}),$
(2)

where $\mathcal{P}_{0}$ is the complex PSF describing the focused coherent
image and determining the well-known Airy disk [30], and
$\mathcal{D}_{z-\delta}$ represents the field propagation over a distance
$z-\delta$, with $\delta$ and $z$ the axial coordinates of the object point
and of the plane at focus, respectively. Depending on the placement of the
sample and the numerical aperture of the device, the quality of the output
image can thus be dominated either by the effects of out-of-focus propagation
or by the Airy disk, with the two effects blending into each other only when
the object is placed close to (but not perfectly on) focus. The corresponding
transition between the focused and defocused image is well known in incoherent
imaging: at focus, both the resolution ($\lambda/$NA) and the DOF
($\lambda/$NA2) are determined by wave optics (Airy disk), with $\lambda$ the
illumination wavelenght. However, as the object is moved outside of the
natural DOF of the focused device, the PSF $\mathcal{P}$ is dominated by
geometrical optics effects and reduces to the circle of confusion (namely, the
projection of the lens aperture onto the defocused image plane) [12], which
induces a typically circular blurring with a radius proportional to both the
defocusing $|z-\delta|$ and the effective lens radius.

The different physics regulating coherent and incoherent imaging helps
developing an intuition about the different behaviour observed in Fig. 1, but
does not suffice to quantitatively compare the resolution and DOF of coherent
and incoherent imaging. In fact, image quality estimators typically used for
characterizing imaging performance, from two-point resolution criteria, such
as Rayleigh’s and Abbe’s [31, 26], to more advanced ones, such as modulation
transfer functions [32], all rely on the linearity of the (incoherent) image
formation and the positiveness of the PSF, and thus fail in assessing the
performance of coherent imaging. For instance, the definition of a Rayleigh
criterion prescribes that, because of the broadening effect of the incoherent
PSF, the image of two ”points” is the superposition of two disks (Airy disks,
at focus, CoC, out of focus). The resolution is then easily defined by
arbitrarily setting an acceptable threshold to when the two disks are
perceived as separated. But these methods cannot be applied as effectively to
a non-linear process such as the coherent image formation, since coherence
induces the appearance of spurious spatial frequency components.Therefore,
neither an approach based on modulation transfer functions, that require the
harmonic content to be unaltered, nor the two-point visibility, which requires
a relative minimum separation between the images of two points, can be used.

To quantify the performance of coherent and incoherent imaging systems, we
thus introduce a general-purpose quality estimator: the functional $F_{A}$,
which we shall refer to as image fidelity, defined as a positive quantity
$F_{A}[I(\bm{x})]$ that compares the intensity distribution $I(\bm{x})$ of the
image produced by an imaging system directly with the original intensity
profile of the object $A=\lvert\mathcal{A}\rvert^{2}$, namely,

$F_{A}[I]=\int\sqrt{A\left(\frac{\bm{x}}{M}\right)I(\bm{x})}\,d\bm{x},$ (3)

where $M$ is the magnification of the imaging system in its plane at focus.
Both $A$ and $I$ are normalized quantities for the definition of the fidelity
to be consistent and to saturate to unity in the ideal case of perfect imaging
($I=A$). Being completely independent of any detail of the image formation
process, the fidelity enables performing image quality evaluation through any
imaging device, as long as the shape of the known reference object is known:
resolution and DOF shall thus be defined as the minimum object size and the
maximum axial range producing a “faithful” image, as identified by a threshold
set to the fidelity. Both these definitions apply equally well to focused and
defocused images, thus enabling to study how resolution changes with
defocusing. Since incoherent imaging is only sensitive to the intensity
transmitted by the sample, our study will now be restricted to non-diffusive
objects and will disregard phase information, namely, we shall consider field
transmission profiles with $\mathrm{arg}\left(\mathcal{A}\right)=0$ uniformly
in the sample, so that $\mathcal{A}=\left\lvert\mathcal{A}\right\rvert\geq 0$.

Figure 2:  Image fidelity in coherent and incoherent imaging. The colored
areas (blue for coherent illumination, orange for incoherent illumination)
highlight the regions in which an $s$-sized object placed at an axial
coordinate $z$ can be imaged with a fidelity larger than $95\%$, with $z=0$
identifying the position of the object at focus. The black dashed line is the
fidelity equivalent of the Rayleigh criterion, intended as the minimum object
size that can be imaged faithfully. The orange dashed curve is the NA-
dependent geometrical circle of confusion at $95\%$ fidelity, obtained by
evaluating the fidelity on the geometrical optics approximation. The blue
dashed line represents the curve of $95\%$ fidelity obtained in the wave
optics regime (namely, by considering the free space coherent propagation of
the field) in the case of infinite NA of the imaging system. The imaging
system is a $20\times$ microscope with NA$=0.5$, illuminated with
monochromatic light with wavelength $500\,$nm. The object is a mask with a
single transmissive detail with Gaussian shape of width $s$.

###  2.2 Resolution limits

The plot reported in Fig. 2 employs the fidelity to offer a quantitative
interpretation of the coherent and incoherent $z$-scans reported in Fig. 1c).
The colored areas in Fig. 2 highlight how far from the plane at focus
(abscissas) an $s$-sized object (ordinates) can be placed to produce an image
with fidelity higher than $95\%$. The orange area refers to spatially
incoherent illumination, whereas the blue area refers to the coherent case.
The physical regimes leading to the dashed curves that delimit the high-
fidelity regions associated with coherent and incoherent imaging offer a clear
perspective on the physical mechanisms regulating the two image formation
processes and enable quantifying the resolution versus DOF trade-off in the
two cases. Such boundaries can be interpreted as resolution limit curves,
giving the functional dependence of the resolution on the displacement of the
sample from focus, at the threshold of the image fidelity above which an image
is considered resolved. These curves are obtained from the analytical
expression of the image fidelity, written in terms of the parameters on which
the image depends, which in our case are: the dimension $s$ of the features of
the sample, the axial coordinate $\delta$ where the sample is located, and the
axial location $z$ of the plane focused by the imaging system. The image
fidelity associated with $I(\bm{x})=I(\bm{x};\delta-z,s)$ is thus a two-
variable function: $F_{A}[I](\delta-z,s)$. Since the quality of the image,
upon mechanical z-scanning, only depends on the relative distance between the
object and the focused plane, we shall set for simplicity $\delta=0$ and
interpret $z$ as the relative defocusing distance.

By studying the analytical expression of $F_{A}[I](z,s)$, exact expressions of
relevant image quantifiers can be extracted. For instance, $F_{A}[I](0,s)$
gives the image fidelity in the plane with Rayleigh-limited resolution, as a
function of the object size. By inversion, one obtains, both for coherent and
incoherent imaging,

$s_{\text{foc}}=\frac{\lambda}{\text{NA}}\,f_{\text{foc}}(c),$ (4)

where $f_{\text{foc}}(c)$ is a coefficient depending of the threshold image
fidelity, amounting to $0.157$ for $c=0.95$ (Fig. 2). Apart from the
multiplying constant, which only depends on the arbitrary choice of a
threshold on the fidelity, the equation corresponds to the well-known
diffraction-limited resolution of focused imaging systems (dashed black line
in Fig. 2), as determined by the Airy disk. Therefore, the analysis in terms
of fidelity recovers the well-known fact that the optical performance of
focused coherent and incoherent systems is analogous.

The differences between the two illumination strategies emerge when
investigating defocused images in two different physical regimes. The
geometrical optics regime is explored by considering the fidelity in the limit
$\lambda\rightarrow 0$, namely,

$F_{\text{geom}}[I](z,s)=\lim_{\lambda\rightarrow 0}F_{A}[I].$ (5)

If this physical regime is investigated in the incoherent imaging case, the
implicit curves $F_{\text{geom}}[I_{\text{inc}}]=c$, in the $(z,s)$ plane,
have an explicit expression, which, unsurprisingly, prescribes the well-known
circle of confusion of geometrical optics:

$s_{\text{geom}}(z)=\text{NA}\,\lvert z\rvert\,f_{\text{geom}}(c),$ (6)

with $f_{\text{geom}}(0.95)=1.97$. As shown in Fig. 2, the CoC-defined trend
perfectly traces the boundary of the fidelity area. Hence, the fidelity
analysis confirms that wave optics has negligible effects on the optical
performance of an incoherent system when the sample is moved away from perfect
focus. By exploring the same physical limit in the case of coherent imaging,
the obtained analytical expression does not describe any physically relevant
situation and does not have a counterpart in the shape of the fidelity region.

Instead, interesting results are obtained, in the coherent case, by
investigating the opposite regime, namely by neglecting geometrical effects.
This is done by considering the radius of the limiting aperture
$l\rightarrow\infty$, so as to completely ignore the influence of the imaging
device. This condition is equivalent to considering an imaging system where
the image formation process is solely governed by diffraction, from the object
plane up to the plane at focus; in fact, in Eq. (2),
$\mathcal{P}(\bm{x})\rightarrow\mathcal{D}_{z}(\bm{x})$, indicating that no
CoC exists in this case. Upon setting a threshold $c$ to the fidelity of a
coherent system with infinite NA

$F_{\text{diff}}[I_{\text{coh}}]=\lim_{l\rightarrow\infty}F_{A}[I_{\text{coh}}],$
(7)

we obtain the resolution limit curves

$s_{\text{diff}}(z)=\sqrt{\lambda\lvert z\rvert}\,f_{\text{diff}}(c),$ (8)

with $f_{\text{diff}}(0.95)=0.396$. Rather surprisingly, this square-root
scaling of the resolution with defocusing perfectly reproduces the boundary of
coherent imaging out of the plane at focus, as reported by the blue dashed
line in Fig. 2. As in the previous case, exploring the same physical limit in
the case of incoherent illumination yields no interesting conclusion. The
optical performance of coherent and incoherent imaging are thus defined by two
entirely different processes: the geometrical CoC (hence, the system NA) is
basically the only factor limiting the resolution of defocused incoherent
imaging; on the contrary, the aperture size and optical design of the imaging
system play no role in coherent imaging, where the sole responsible for image
degradation is diffraction and free-space space propagation from the object to
the observation plane. The different physical phenomena governing image
degradation (geometric optics, as opposed to diffraction and wave propagation)
have surprising effects on the image quality. Resolution and DOF of coherent
imaging are found to be independent of the NA of the imaging system, and their
trade-off is extremely relieved with respect to incoherent imaging, as defined
by the square-root law (dashed blue line) compared to the linear dependence
(dashed red line).

###  2.3 Coherent 3D imaging with incoherent sectioning capability

The newly discovered properties of direct coherent images can be integrated
with the strong axial localization capability of incoherent imaging to achieve
scanning-free 3D wide-field imaging of absorbing samples with enhanced
volumetric resolution. In fact, spatially coherent illumination will enable a
(NA-independent) square-root scaling of transverse resolution, thus offering
high lateral resolution over a long DOF; at the same time, the axial
sectioning typical of spatially incoherent illumination entails, in the wide
DOF accessed because through coherence, a precise sectioning capability, as
enabled by large-NA tomographic systems [33]. We should clarify that, in this
context, the concepts of DOF and axial resolution are rather distinct: while
the DOF represents the axial length of the volume where object details of a
given size can be faithfully imaged, the axial resolution represents the axial
sectioning capabilities, that is, how finely transverse planes within the DOF
can be isolated along the axis.

The underlying principle for achieving high-resolution 3D imaging within a
direct wide-field coherent system is similar to LF imaging: in both cases,
information about the propagation direction of light enables scanning-free
volumetric reconstruction. However, while LF imaging acquires the required
directional information by means of the microlens array, our proposal
prescribes to do it with spatially coherent illumination of the sample from
different locations. Our approach will be shown to come with two major
advantages: a much larger DOF, as defined by the square-root (as opposed to
linear) scaling of the resolution with defocusing, and Rayleigh-limited images
at focus.

In the proposed scheme, 3D information about the sample is acquired by
accessing the 4D function $I(\bm{x}_{0},\bm{x})$, where $\bm{x}_{0}$ is the
transverse coordinate of a point-like emitter enabling spatially coherent
illumination of the sample, and $\bm{x}$ is the transverse coordinate of the
collected image. Sampling of the complete 4D function is performed by
sequentially sweeping an illumination plane made of point-like emitters
centered in $\bm{x}_{0}$, and collecting, for each coordinate $\bm{x}_{0}$,
the resulting intensity

$I(\bm{x}_{0},\bm{x})=\left\lvert\left[\mathcal{L}_{\bm{x}_{0}}(\bm{x})\mathcal{A}(\bm{x})\right]*\mathcal{P}\left(\bm{x}\right)\right\rvert^{2},$
(9)

where $\mathcal{A}$ and $\mathcal{P}$ are the same object transmittance and
coherent PSF as in Eq. (1), and $\mathcal{L}_{\bm{x}_{0}}$ is the Green’s
function propagating the field from the point-like source centered in
$\bm{x}_{0}$ to the sample plane. As we shall discuss in the “Results”
section, the wide freedom in the choice of $\mathcal{L}_{\bm{x}_{0}}$ (hence,
of the illumination scheme) enables to greatly customize the optical
performances of the proposed 3D imaging system. Specifically, in order to
encode 3D information into $I(\bm{x}_{0},\bm{x})$, illuminating the sample
from many different angles is not necessary. In previous works (see, e.g.,
Ref. [23]), in fact, $\mathcal{L}_{\bm{x}_{0}}$ has always been arranged in
such a way to have an illumination distance $L$ between a source at coordinate
$\bm{x}_{0}$ and the sample, such that the latter can be considered to be
illuminated by tilted plane waves, corresponding to the choice

$\mathcal{L}_{\bm{x}_{0}}(\bm{x})=\exp\left[i\,\frac{2\pi}{\lambda}\frac{\bm{x}_{0}}{L}\cdot\bm{x}\right],$
(10)

as conventionally done in tomographic systems. However, as we shall discuss in
the “Results” section, our complete formal analysis, and the consequent
understanding of coherent imaging, enable to demonstrate that neither the
angular illumination nor the requirement of collimated light are in any way
necessary to encode 3D information into $I(\bm{x}_{0},\bm{x})$. Most
importantly, understanding the underlying physics of coherent and incoherent
imaging is the key for achieving scanning-free direct 3D imaging, with no need
for time-consuming phase retrieval algorithms.

The intensity distribution described by Eq. (9) is easily recognized as a
coherent image, as in Eq. (1), with the only difference that the object is now
replaced by the expression $\mathcal{L}_{\bm{x}_{0}}\,\mathcal{A}$,
emphasizing the role of the illumination scheme and the wide freedom in its
design. The acquired 4D intensity can thus be expected to have mostly the
features we have attributed to coherent images, such as the decoupling of the
lateral resolution and DOF. However, the large DOF entails the lack of axial
localization: thick 3D samples are imaged with high transverse resolution, but
lack any axial localization. To address the issue, we shall integrate the
proposed technique with the properties of incoherent imaging, in which the
tomographic properties are defined by the angular acceptance of the lens. The
analogy with incoherent systems can easily be understood by considering the
image resulting from the sum of the coherent images obtained from different
illumination coordinates, namely,

$R_{0}(\bm{x})=\int I(\bm{x}_{0},\bm{x})\,d\bm{x}_{0}=I_{\text{inc}}(\bm{x}).$
(11)

This equality can be analytically verified by plugging the expression of the
plane-wave illumination into Eq. (9) and integrating the result; however, any
other illumination schemes presented in this work yields the same result. In
fact, from a more intuitive standpoint, we can recognize that integrating over
the entire illumination plane is equivalent to shining uniform incoherent
light onto the sample, which is exactly the typically sought-after
experimental condition of uniform illumination in conventional systems (e.g.
Kohler illumination). It is thus not surprising that the integration must
yield exactly the same results as conventional (incoherent) imaging: Rayleigh-
limited resolution at focus, CoC blurring out of focus, and dependence on NA,
with the only difference that the uniform illumination is achieved in post-
processing. However, the mere integration reported in Eq. (11) is a rather
poor way of employing the much larger amount of information contained within
$I(\bm{x}_{0},\bm{x})$: due to the shallow DOF of incoherent imaging, the
features of the sample are rapidly lost as it is moved away from perfect
focus, more so for large NA. On the contrary, a Radon transformation of
$I(\bm{x}_{0},\bm{x})$ (here expressed as a line integral):

$R_{z}(\bm{x}^{\prime})=\int_{\gamma(\bm{x}^{\prime})}I(\bm{x}_{0},\bm{x})\,dl,$
(12)

enables localizing the object within the much larger DOF characterizing
coherent imaging. In Eq. (12), $\gamma(\bm{x}^{\prime})$ are two lines of
equations $\sin{\theta(z)}\bm{x}_{0}+\cos{\theta(x)}\bm{x}=\bm{x}^{\prime}$
defined in the spaces $(x_{0},x)$ and $(y_{0},y)$. In fact, the Radon
transform $R_{z}(\bm{x}^{\prime})$ isolates a specific axial coordinate $z$ by
integrating over the whole dataset $I(x_{0},x)$ at a $z$-dependent angle
$\theta(z)$; this allows one to perform, in post-processing, a software
$z$-scanning similar to the hardware scan done by manually moving the focus of
a conventional (incoherent imaging) device.

The relation between the integration angle and the reconstructed axial plane
can be understood by considering that the object point at coordinate
$\bm{x}^{\prime}$, once illuminated by the source lit at transverse coordinate
$\bm{x}_{0}$, is mapped onto the detector coordinate $\bm{x}$, which depends
on both $\bm{x}_{0}$ and $\bm{x}^{\prime}$. As anticipated, the geometrical
locus of the points of the sensor $\bm{x}$ corresponding to the same object
coordinate $\bm{x}^{\prime}$ is a line in the $(x_{0},x)$ space with equation

$s_{x^{\prime}}:\alpha(\delta)\,x_{0}+\beta(\delta)\,x+x^{\prime}=0,$ (13)

where $\alpha$ and $\beta$ are two functions depending on both the defocusing
distance $\delta$ and the particular illumination scheme; as in conventional
LF imaging, they are obtained through ray tracing in a geometrical optics
context. The same holds, with the same coefficients $\alpha$ and $\beta$, for
the other two coordinates $(y_{0},y)$. Therefore, for an object imaged at an
axial displacement $\delta$ from the focused plane, the most accurate
reconstructed image is $R_{z=\delta}$, as obtained by performing the
integration in Eq. (12) along lines with a tilting
$\theta=\arctan(-\alpha/\beta)$.

As we shall discuss in the “Results” section, the 3D performance of the
proposed coherent 3D imaging technique can be characterized in terms of the
image fidelity $F_{A}[R_{z}]$. However, volumetric reconstruction shall
require the image fidelity to be evaluated onto a three-parameter space: since
the focus of the system is fixed at a given coordinate, both the relative
position $\delta$ of the $s$-sized object and the reconstruction coordinate
$z$ can be moved independently with respect to the plane at focus.

##  3 Results

Figure 3:  Experimental comparison of resolution and DOF in incoherent and
coherent imaging. Comparison of the resolution versus DOF trade-off in two
cases of incoherent imaging (red and blue points), corresponding to two
different illumination NA, and in the case of coherent imaging (orange
points), as obtained by single LED illumination. For both coherent and
incoherent imaging, an image is considered to be resolved when the
corresponding fidelity is at least $75\%$. The areas of $75\%$ fidelity
(colored regions) are identified by the smallest slit width imaged with
fidelity over $75\%$ (circles) and the largest slit width imaged with fidelity
lower than $75\%$ (bullets). All data are taken with a conventional $20\times$
microscope, whose details are reported in the Supplementary document. The
analysis is based on triple-slit masks from a USAF test target, having slit
width $s$ and center-to-center slit distance $d=2s$. The four images, reported
as a reference quality of the targeted fidelity, are taken at the same axial
distance of $z=250\,\mu$m and correspond to the 4 different reported values of
the slit width.

The introduction of the image fidelity has enabled to directly compare the
performance of coherent and incoherent systems and to discover that, in
coherent imaging, the degradation of the image resolution with defocusing is
not related with geometrical blurring mechanisms such as the CoC. On the
contrary, the degradation of the image quality is governed almost entirely by
diffraction from the object plane to the imaged plane; this leads to a square-
root law scaling $\sqrt{|z|}$ of the resolution with the distance from focus
$z$, as opposed to the linear scaling characterizing the CoC. Thus, in
coherent imaging, the axial range in which the object is resolved thus scales
quadratically with the object size, rather than linearly. Therefore, in
addition to being independent of the NA of the imaging device, coherent
imaging yields a quantitative DOF advantage over conventional (incoherent)
imaging.

In Fig. 3, we report the experimental demonstration of this prediction and
show that coherent illumination enables a $4$ times larger DOF at $2\,\mu$m
resolution and an almost $20$ times larger DOF at $10\,\mu$m resolution, with
respect to incoherent illumination. The experimental images shown in the
figure are obtained by illuminating the masks with an array of green LEDs,
placed $110\,$mm apart from the plane at focus of a $20\times$ magnification
conventional microscope. Although it is a well-known fact in microscopy, and
in bright-field imaging in general, that source collimation indeed implies DOF
augmentation [30], we should highlight that the square-root trend we
experimentally demonstrate is the result of an entirely different physical
phenomenon and cannot be understood in terms of source collimation, but only
in terms of its spatial coherence. The conventional (incoherent imaging)
explanation of the DOF improvement through source collimation, in fact, is
related to the divergence of the illuminating beam becoming smaller than the
acceptance angle of the optical devices, so that the optical properties of the
imaging device are no longer dictated by the NA of imaging device, but rather
by the effective NA defined by the illumination itself. However, this effect
is profoundly different from the DOF extension enabled by spatial coherence,
where collimation is by no means a requirement. The presented DOF advantage,
in fact, is by all means maintained even with a quasi-infinite illumination
NA, as one could get by bringing the illumination stage in extreme proximity
to the sample and employing smaller sources, such as quantum dots and single-
molecule LEDs.

Figure 4: Effect of partial coherence in the transition from coherent to
incoherent imaging. Log-log plot of the $95\%$-fidelity (black continuous
line) evaluated in the case of partial coherence due to the finite transverse
size of the light source. The dashed red and blue curves are the
$95\%$-fidelities evaluated, respectively, in the geometric optics
approximation and in wave optics but with infinte NA. All curves are obtained
by considering a microscope with NA=$0.5$, illuminated by a green light
emitter placed at $11\,$cm from its plane at focus and having a Gaussian
intensity profile of width $w=0.3\,$mm; the object is a Gaussian slit of width
$s$. The two dashed orange lines identify, for each defocusing $z$, the values
of the slit width $s$ corresponding to $1/2$ (lower curve) to $2$ (upper
curve) times the coherence area; the orange colored area identify the values
of $s$ for which partial coherence enters into play giving rise to the
transition from incoherent to coherent imaging.

The incoherent effects of DOF improvement, however, indeed exist in the regime
in which the illumination collimation is such to define the effective NA of
the system, but the coherence area on the sample is not wide enough for the
system to behave in a coherent manner. To gain more insight about the role
played by the NA of the illumination system, we shall now study the transition
from incoherent to coherent imaging and consider the general case of a finite-
sized source, matching the experimental conditions of Fig. 3. In Fig. 4, we
plot the $95\%$-fidelity curve (solid black line) of the image of a
transmissive mask (an $s$-sized slit) as a function of its distance from a
$w$-sized incoherent emitter; strictly speaking, $z$ is the distance of the
object from the plane at focus, but its variation naturally changes the
object-to-source distance as well. When the object is so close to the source
that the spatial coherence acquired through propagation towards the sample is
smaller than $s$, the resolution versus DOF trade-off is determined by the
numerical aperture of the illumination, as expected in a conventional system;
this is demonstrated by the overlap of the evaluated fidelity (black line)
with the one obtained in the geometrical optics approximation (red dashed
line), for large values of $z$. In this regime, imaging is thus incoherent.
However, as the object is moved farther away from the source, the coherence
area on the object becomes proportionally larger till coherence effects become
dominant, and the fidelity trend (black line) detaches from the geometrical
optics prediction and overlaps on the coherent trend (dashed blue line),
completely NA-independent. The yellow region highlights the transition from
the incoherent to the coherent imaging, and shows that coherent effects enter
into play, as predicted, when the coherence area becomes comparable to the
details one wishes to resolve. Although this result is quite intuitive, its
implications are very relevant. First, it demonstrates that the maximum DOF
that direct imaging can achieve is ultimately limited, at least in the realms
of classical optics, by the spatial coherence of the illumination on the
sample; this is in contrast with the approximately infinite DOF one might
incorrectly expect by interpreting the case of perfectly collimated
illumination in terms of conventional incoherent imaging, along the line of
what discussed above. Second, it leads to better appreciate the implications
of the approach employed for obtaining the results of Fig. 3, namely, the
reduction of the illumination stage area for showing, on the one hand, the
effect of the NA on the CoC of incoherent imaging (blue and red points), on
the other hand, the transition from incoherent to coherent imaging (orange
points). The trend of the blue and red points indicate that the decrease of
the illumination NA by a factor of $2$ increases the DOF by the same amount,
as expected for incoherent imaging; however, in both cases, the illumination
area (i.e., the radius of the area of lit LEDs) is not yet small enough for
the coherence area to be comparable with the desired resolution. By further
shrinking the illumination, a point is reached where the linear trend is lost
and the of NA-independent square-root trend emerge, as a consequence of the
larger coherence acquired by the illumination. The transition is explained in
terms of spatial coherence.

Figure 5:  3D imaging capability of a high-NA microscope exploiting spatial
coherence. Panel a). Schematic representation of three possible illumination
schemes for performing 3D imaging through coherent illumination. The plots on
the right hand side report the corresponding scaling of the resolution as a
function of the defocusing $\delta$, as prescribed by the coherent scaling
$\sqrt{d(\lambda)}$ (blue curves); the different scaling obtained within a
given scheme for varying values of the object-to-illumination distance
($z_{0}$ in the first case, $\delta_{0}$ in the third case) are also reported
(dashed blue lines). Panel b). Axial section of a stack of reconstructed
images $R_{z}(\bm{x})$ of a double slit mask placed at $\delta=1\,\mu$m
(left), showing the capability of localizing the sample. The two images
reconstructed at the focused distance $z=0$ and at the correct sample
locations $z=\delta=1\,\mu$m are reported in the top right panels, and
compared with the corresponding two reference images reported in the bottom
right panels: the defocused images collected by the same imaging device in the
cases of incoherent (left) and coherent (right) illumination (same images
reported in Fig. 1). The illumination scheme adopted for this simulations is
the upper one of panel a), with an illumination distance $Z_{0}=110\,$mm.
Panel c). Characterization of the resolution as a function of the defocusing
$z$ for five different object positions $\delta$, showing the sectioning
capability and the overall resolution versus DOF performance of the proposed
approach. The yellow and blue area are a reference for the performance of
incoherent and coherent imaging, respectively; the last representing the
maximum achievable DOF of the proposed technique. The five “V-” shaped areas
show the sectioning capability enabled by the software z-scanning and the
characterize the performance of the reconstructions, for the five different
object placements; the software z-scan for a focused object (yellow) is shown
to give the same resolution versus DOF performance as incoherent imaging.

Let us now show why localized coherent illumination is so convenient for
performing 3D scanning-free imaging. As detailed in the “Methods” section, 3D
information about the sample is gathered by measuring the 4D function
$I(\bm{x}_{0},\bm{x})$, obtained by sequentially illuminating the object from
different point-source on the illumination plane. The image dataset
$I(\bm{x}_{0},\bm{x})$ can then be Radon-transformed to isolate axial planes
of the samples, as prescribed by Eq. (12). In particular, an extremely
interesting result is obtained by applying the Radon reconstruction to the
plane $\delta$ on which an object with transmission function $\mathcal{A}$ is
placed, namely:

$R_{\delta}(\bm{x})=\left\lvert\mathcal{A}(\bm{x})*\tilde{\mathcal{P}}(\bm{x})\right\rvert^{2},$
(14)

with

$\tilde{\mathcal{P}}(\bm{x})=\mathcal{D}_{d(\delta)}(\bm{x})*\mathcal{P}_{0}(\bm{x}),$
(15)

where $\mathcal{P}_{0}$ is the coherent PSF of the imaging system in its
focus, and $\mathcal{D}_{z}$ is propagation in vacuum by a distance $z$, as in
Eq. (2). The properties of the reconstructed image are easily understood by
noticing that Eq. (14) is exactly the expression of a coherent image (see Eqs.
(1) and (2)), as observed by the same imaging device, but affected by an
equivalent defocusing $d(\delta)$. More specifically, coherent illumination
gives rise to images having a resolution at focus defined by the Rayleigh
limit and a resolution out-of-focus scaling with the square root of the
defocusing $\sqrt{\delta}$, whereas the images reconstructed by our method
have the same resolution at focus, but an out-of-focus resolution scaling with
$\sqrt{d(\delta)}$. As reported in Fig. 5a), different illumination schemes
are thus possible, each one characterized by a specific scaling of the
resolution with the defocusing; the optical performance of the device can thus
be greatly enlarged, offering a wide flexibility in view of a variety of
different applications. In fact, the plots indicate that the scaling of the
lateral resolution as a function of the defocusing is a pure square-root only
in the case of plane-wave illumination, namely, when $Z_{0}\rightarrow\infty$,
in the first scheme, or when the middle scheme is adopted. In the other cases,
the scaling remains defined by a square-root law, but the actual dependence
also involves the illumination distance ($Z_{0}$ in the first scheme,
$\delta_{0}$ in the third one).

Fig. 5b) demonstrates that the great (NA-independent) DOF extension typical of
coherent imaging is integrated with very accurate (NA-dependent) axial
localization, due to the incoherent imaging properties brought in by the
reconstruction process. The reported software $z$-scan shows that a double-
slit object placed outside of the native DOF of the microscope can actually be
localized extremely well around the plane where the most accurate
reconstruction happens. Also, at a glance, one immediately recognizes that the
depth of the reconstruction is not what one would expect by coherent imaging,
but rather exactly the native (NA-defined) incoherent DOF of the device. As
shown on the right hand side of the axial scanning, however, the reconstructed
image is exactly the same image that coherent imaging would give, with an
object displaced by an equivalent defocusing $d(\delta)$.

All the aforementioned properties are summarized by Fig. 5 c). The blue and
yellow areas identify the resolution performance of coherent and incoherent
illumination, respectively, with their characteristic linear (CoC) and square-
root trends. The colored “V”-shaped regions, instead, show the optical
properties of the images reconstructed, through software z-scanning, by the
proposed 3D imaging modality, and correspond to five different axial positions
of the object. As expected from our findings, the depth of the
reconstructions, which represents the axial resolution of the 3D imaging
technique, coincide with the DOF of incoherent imaging, namely, it is the same
one would obtain by focusing the equivalent incoherent imaging system (i.e.,
be lighting on the whole array of small sources at once) on the correct object
plane; the only difference with respect to the image obtained by mechanical
z-scanning within an equivalent incoherent imaging system is in the minimum
resolution, which in our approach lies on the square-root curve defined by
coherent imaging. On the other hand, when the object is placed at focus
($\delta=0$), the image reconstructed with our method is exactly the same as
in incoherent imaging, both in the minimum Rayleigh-limited resolution and in
the CoC-defined axial localization.

We shall now employ these results to experimentally demonstrate the high-
resolution volumetric multicolor capability of the proposed technique (Fig.
6). Coherent illumination from localized emitters is obtained through an array
of commercial RGB LEDs, placed far enough from the sample plane for the
coherence area on the sample plane to be comparable with the details of
interest. The sample is a $10\,\mu$m-thick mouse brain section, where cell
nuclei and cytoplasm have been labeled, respectively, by hematoxylin and
eosin. The acquired 3D information enables to clearly compensate for the sub-
optimal placement of the microscope slide, whose closest part to focus is
$10\,\mu$m away from the focused plane, and mounted with a tilting of about
$10$ degrees. Unlike the color-independent CoC, the square-root scaling of the
resolution of coherent imaging has a weak dependence on wavelength
($\propto\sqrt{\lambda}$), thus giving rise to images characterized by
negligible chromatic aberration.

Figure 6:  Multicolor 3D reconstruction of a histological mouse brain section
exploiting spatially coherent illumination The resolution versus DOF advantage
granted by spatially coherent illumination is used to reconstruct the true-
color wide-field image a histological mouse brain section marked at two
wavelengths. Due to both a slight tilting of the sample holder and a axial
misplacement, conventional incoherent illumination yields an unfocused image.
By exploiting sequential coherent illumination from an array of RGB LEDs,
volumetric information is obtained and employed, through the software axial
scans, for reconstructing the different portions of the sample at different
$z$ coordinates. The 3D information is then used to compensate for the tilting
of about $10$ degrees and obtain a single wide-field image at focus. The
optical microscope is a conventional $20\times$, $0.75$ NA wide-field device.

##  4 Discussion

We have found that the lateral resolution and DOF of defocused images obtained
through spatially coherent illumination are decoupled from the numerical
aperture of the imaging system. Such independence is particularly convenient
for designing 3D imaging devices exploiting transverse coherence of light: the
resulting overall DOF of the technique becomes independent of the optical
components used for the image acquisition, and is instead entirely defined by
the coherent illumination scheme, as shown in Fig. 5. Despite being based on
the same image reconstruction principle as LF imaging, our resolution scaling
is much more convenient, with the additional benefit of retaining Rayleigh-
limited resolution at focus. In fact, compared to conventional LF devices,
which achieve DOF extension at the expense of the lateral resolution, 3D
imaging systems based on spatially coherent illumination have a DOF that
scales quadratically with the desired resolution, thus always yielding an
advantage over the linear scaling typical of LF [34]. Furthermore, since a
large NA has no effect on the resolution and DOF of the system, large
apertures can be used to obtain optimal sectioning capability upon refocusing,
enabling a strong suppression of the background neighbouring planes, as in
high-NA tomographic systems.

Since our proposal only requires transverse spatial coherence, these systems
work with temporally incoherent sources, which induce negligible to modest
radiation damage, as required by in vivo biological applications. Although
image reconstruction through Radon transform does not recover the phase
content, as opposed to computational techniques based on coherence [23, 21,
19], it carries the enormous advantage of being performed in real-time with
current GPU architectures and FPGAs [35], or through the use of holographic
screens [36]. The proposed 3D wide-field imaging technique can thus be used
both for direct and real-time imaging.

The extreme simplicity and low cost of the optical design, also compared to LF
imaging, has high potentials to open up the possibility of employing 3D
imaging in new scenarios, low-budget applications as well as for public
healthcare in developing countries.

Funding All Authors acknowledge funding from Università degli Studi di Bari
through the Horizon Europe Seeds program, project INTERGLIO (S081). M.D. and
F.V.P. are supported by PNRR MUR project PE0000023-NQSTI. M.D., G.M. and
F.V.P. are supported by INFN project QUISS. MD, GM and FVP acknowledge funding
under project ADEQUADE: this project has received funding from the European
Defence Fund (EDF) under grant agreement EDF-2021-DIS-RDIS-ADEQUADE
(n°101103417). G.P.N. is supported by: 1) AstroDyn (FA9550-19-1-0370),
AstroColl (FA9550-21-1-00352) and Stochastic Biophysical Interactions within
Aquaporin-4 Assemblies (FA9550-20-1-0324) funded by AFOSR; 2) Marie
Skłodowska-Curie Actions -ITN-2020 ASTROTECH (GA956325) funded by the European
Commission; 3) NEXTGENERATIONEU (NGEU) funded by the Ministry of University
and Research (MUR), National Recovery and Resilience Plan (NRRP), project
MNESYS (PE0000006) – A Multiscale integrated approach to the study of the
nervous system in health and disease (DD 1553, 11.10.2022); 4)
NEXTGENERATIONEU (NGEU) funded by the Ministry of University and Research
(MUR), National Recovery and Resilience Plan (NRRP), project CN00000041 -
National Center for Gene Therapy and Drugs based on RNA Technology (DD n.1035,
17.06.2022).
Note. Funded by the European Union. Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect those of the
European Union or the European Commission. Neither the European Union nor the
granting authority can be held responsible for them.

Author contributions G.M. conceptualized the idea, developed theory and
simulations, designed and built the experimental setup, developed the
software, performed data analysis, and wrote the original draft. G.M. and B.B.
performed the experiments. B.B. prepared the histological sample and
contributed to write the biological part in the Supplemental document. F.V.P.
supervised the theoretical work. G.P.N. supervised the biological part of the
work. M.D. supervised the overall physical part of the work. M.D. and G.S.
contributed to the organization and writing of the manuscript M.D. and G.P.N.
were responsible for fundings. All authors read and edited the manuscript.

Disclosures The authors declare no competing interests.

Data AvailabilityAll the data leading to the results discussed in the paper
are available upon request to the corresponding author.

Supplemental document See Supplementary text for supporting content.

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