## Abstract

Propagation-based X-ray phase-contrast imaging (PBI) is a powerful nondestructive imaging technique that can reveal the internal detailed structures in weakly absorbing samples. Extending PBI to CT (PBCT) enables high-resolution and high-contrast 3D visualization of microvasculature, which can be used for the understanding, diagnosis and therapy of diseases involving vasculopathy, such as cardiovascular disease, stroke and tumor. However, the long scan time for PBCT impedes its wider use in biomedical and preclinical microvascular studies. To address this issue, a novel CT reconstruction algorithm for PBCT is presented that aims at shortening the scan time for microvascular samples by reducing the number of projections while maintaining the high quality of reconstructed images. The proposed algorithm combines the filtered backprojection method into the iterative reconstruction framework, and a weighted guided image filtering approach (WGIF) is utilized to optimize the intermediate reconstructed images. Notably, the homogeneity assumption on the microvasculature sample is adopted as prior knowledge, and therefore, a prior image of microvasculature structures can be acquired by a k-means clustering approach. Then, the prior image is used as the guided image in the WGIF procedure to effectively suppress streaking artifacts and preserve microvasculature structures. To evaluate the effectiveness and capability of the proposed algorithm, simulation experiments on 3D microvasculature numerical phantom and real experiments with CT reconstruction on the microvasculature sample are performed. The results demonstrate that the proposed algorithm can, under noise-free and noisy conditions, significantly reduce the artifacts and effectively preserve the microvasculature structures on the reconstructed images and thus enables it to be used for clear and accurate 3D visualization of microvasculature from few-projection data. Therefore, for 3D visualization of microvasculature, the proposed algorithm can be considered an effective approach for reducing the scan time required by PBCT.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microvasculature usually refers to the network of small blood vessels with less than 100 µm diameter, including capillaries, venules and arterioles. Microvasculature plays an important role in microcirculation and can be used for delivering and exchanging blood, metabolic substances and hormones. The development and variation in many diseases are associated with alterations of microvascular structure and morphology (e.g., changes in vessel diameter, vascular distortion and vascular network complexity), such as cardiovascular disease, stroke and tumor. Therefore, the qualitative and quantitative analyses of microvascular networks contribute to full understanding, effective diagnosis and accurate evaluation of the therapy or prognosis of diseases involving vasculopathy [1,2]. Currently, microscopic imaging of vessels reliant on histopathological sections remains the gold standard for observing microvascular changes [3]. However, the preparation of histopathological material requires tissue dissection, formalin fixation, paraffin embedding, tissue slicing and staining. This procedure may cause the deformation of anatomy and limit further analysis with different methods [4]. Additionally, histopathological sections provide two-dimensional (2D) information from a given cross section of the microvasculature architecture only, which often yields discontinuous and incomplete information. Analyzing the 3D characteristics of microvasculature architecture and morphology is important because it allows one to evaluate, at the micrometer scale, the relationship between structure and function for advancing the understanding of the role of vasculature that cannot be fully analyzed in 2D representative sections. Thus, a nondestructive imaging technique as a possible adjunct to histopathology, which allows 3D visualization and quantification of complicated microvascular networks while retaining a precise anatomical context, is required.

CT angiography (CTA) is a traditional imaging technique for 3D visualization of the vasculature, which can enhance the vasculature by the absorption contrast between the contrast medium and the surrounding tissue. However, due to insufficient spatial resolution and low contrast resolution, conventional CTA has difficulty delineating tiny vessels with a diameter of 200 µm or less, and therefore, it cannot provide microvascular changes at the micron or submicron levels [5]. Along with the rapid development of synchrotron radiation (SR) X-ray sources and the great improvement of detector technology, SR-based micro-CT angiography (SR-µCTA) can achieve high-resolution 3D visualization of microvascular networks, where the diameter of the smallest vascular structures that can be distinguished is approximately 7.4 µm [6]. Generally, SR-µCTA provides the microangiography network, but it is difficult to observe the surrounding soft tissues simultaneously, which makes it difficult to study the relationship between the microangiography network and the soft-tissue morphology. In addition, with the advanced development of some diseases or tumors, the vessels become leaky, and the contrast agents can penetrate into the surrounding tissue through the damaged parts of the vessel walls, which might pose a challenge for accurate microvascular research. Unlike traditional absorption-based X-ray imaging techniques, phase-contrast imaging (PCI) is a phase-based X-ray imaging technique; namely, it is based on the collection of phase shift in the beam that arises from the refraction of X-ray in the detected specimen [7]. For hard X-ray imaging of low-Z materials (e.g., biological soft tissues), the alterations of phase shifts incurred to X-rays as they pass through matter are approximately 3 orders of magnitude larger than those of the attenuation coefficients [8], which allows imaging biological soft tissues at a lower absorbed dose to achieve higher image contrast compared with absorption-based X-ray imaging mechanisms. Over the past two decades, a variety of PCI techniques have been developed to transform phase shifts into measurable intensity variations on the detector, such as crystal interferometry imaging, analyzer-based imaging, propagation-based X-ray phase-contrast imaging (PBI), edge-illumination imaging and grating-based imaging. Among the abovementioned techniques, PBI is the simplest to implement in reality because it only requires that the incident X-ray beams are sufficiently spatially coherent, and the sample-to-detector distance (SDD) can be adjusted, and no additional optical elements or multiexposures are needed [9]. In general, by extending PBI to CT (PBCT), PBCT can be used for high-resolution 3D visualization of the internal detailed structures in weakly absorbing objects. Compared with SR-µCTA, SR-based PBCT (SR-PBCT) might be a promising novel approach to enable 3D visualization and precise quantification of microvascular networks without the need for contrast agent injection [10]. To date, SR-PBCT has been widely used in biomedical and preclinical experimental studies to reconstruct the microvasculature of tissues and organs, including nerve tissue [11], alveoli [12], kidney [13], liver [14], spinal cord [15], and brain [16].

The standard filtered backprojection (FBP) is an analytical CT
reconstruction method based on the Fourier-slice theorem, and it typically
consists of ramp filtering and backprojection procedures, where ramp
filtering is usually used to avoid the blur effect on the reconstructed
image that arises from the backprojection procedure. Due to its simplicity
and high computational efficiency, the FBP algorithm has been commonly
utilized to reconstruct CT images in SR-PBCT. According to the
Nyquist-Shannon sampling theory, the minimal number of tomographic
projections required for high-quality FBP reconstruction is approximately
π*d* /(2*r*), where
*d* is the maximum thickness of the sample and
*r* is the detector pixel pitch. Under the condition of the
minimum projection (CMP), when using FBP reconstruction in SR-PBCT, the
high-resolution 3D visualization of microvascular networks typically
requires a very long scan time (from tens of minutes to a few hours) for
the acquisition of sufficient projections. However, the long scan time
might impede the wider use of SR-PBCT in microvascular imaging for
biomedical and preclinical experimental studies. First, the long scan time
will result in high radiation doses, which not only poses health risks to
subjects for *in vivo* imaging but also has the risk of
causing damage to the structures and biological properties of the samples
when used for *ex vivo* imaging [17]. Second, the long scan time will cause motion
artifacts, which arise from not only the heartbeat and breathing of living
subjects during the scan but also the deformation of fresh tissue without
chemical fixatives that occurs when the scan time is more than
approximately 15 minutes [4,18]. To shorten the scan time, one
possible solution is to reduce the number of projections required for CT
reconstruction. However, few-projection sampling will make the projection
data nonuniform, and this can introduce streaking artifacts into the CT
image. The streaking artifacts can severely degrade the image quality,
which may affect subsequent 3D visualization and quantification of the
microvasculature. In fact, ramp filtering in the FBP algorithm, regardless
of whether other window functions are added, typically has a limited
capability to reduce the streaking artifacts.

Recently, many CT reconstruction methods for the few-projection data have
been developed. These methods can be mainly divided into two categories.
The first category is the deep-learning (DL) based reconstruction methods.
For instance, Han *et al*. [19] proposed a deep residual learning network based reconstruction
method via the persistent homology analysis; Jin *et al*.
[20] developed a deep convolutional
neural network based reconstruction method that combined the FBP and
U-net; Zhang *et al*. [21] presented a reconstruction method based on combination of
DenseNet and Deconvolution. The more information can refer to a review on
the subject of applying DL [22].
The DL based reconstruction methods can perform better than many
conventional CT approaches, however, they generally rely on the
high-quality big data and meticulos training, and this might be difficult
to achieve in some special fields, such as the PCI field. The second
category is the iterative reconstruction (IR) methods. The IR methods can
utilize prior knowledge to reduce noise or artifacts. According to the
employed prior knowledge, the IR methods can be grouped into two types:
empirical-knowledge and prior-image knowledge based methods. The
empirical-knowledge based methods can utilize the known information (e.g.,
the boundary, shape, density range and the sparsity) of an object to
formulate the reconstruction model, such as the total variation (TV) based
reconstruction algorithm [23],
tensor based reconstruction algorithm [24], low-rank based reconstruction algorithm [25], and so on. The prior-image knowledge
based methods explore both image sparsity and similarity by utilizing the
high-quality prior images, e.g., the dictionary learning based
reconstruction algorithm [26], the
prior image constrained compressed sensing (PICCS) algorithm [27], and the guided image filtering (GIF)
[28] based simultaneous algebraic
reconstruction technique algorithm [29]. The IR methods have the excellent performance for the
few-projection CT reconstruction, however, they often require large memory
space for very large computational tasks, and they also take a long
execution time (typically costing tens of minutes and even a few hours for
one CT). Moreover, they commonly contain many parameters; these parameters
may be difficult to adjust, and in many cases, the optimal parameter
setting for one reconstructed slice cannot show good robustness for all
slices of one sample. In addition, for the prior-image knowledge based
methods, they typically need the aid of previously scanned high-quality CT
images, and these prior images may not be available in practice. The
abovementioned problems might impede the wider use of IR methods in
SR-PBCT. Therefore, a novel SR-PBCT reconstruction algorithm for 3D
visualization of microvasculature by means of few-projection data, which
enables combining the advantages of the FBP algorithm (simplicity and high
computational efficiency) and IR algorithms (utilize prior knowledge for
processing artifacts in iteration), is highly desirable.

The GIF is a guided image-based filtering technique that utilizes the edge
information of the guided image to preserve the edges of the processing
image, but it suffers from halo artifacts [28]. Li *et al* incorporated an edge-aware
weighting into the GIF to form a WGIF, and it can avoid halo artifacts in
the edge-preserving filtering process [30]. Due to its superior performance and fast processing speed,
the WGIF has played an important role in many applications, such as noise
reduction, detail smoothing and image matting. Inspired by the
abovementioned work, we incorporated FBP reconstruction into the iterative
reconstruction framework and adopted the WGIF to optimize the intermediate
reconstructed images. It is noteworthy that one approximation on
biological soft tissue containing microvasculature was utilized, i.e.,
biological soft tissue can be approximated as a single-material object.
Based on this approximation, a binarized image of microvasculature,
namely, the prior image of microvasculature structures, can be accurately
obtained by a k-means clustering method [31] and then used as the guided image in the WGIF procedure to
effectively suppress streaking artifacts and preserve microvasculature
structures, in which no previous high-quality prior images are needed. To
assess the effectiveness and capability of the proposed algorithm,
simulation experiments on the 3D microvasculature numerical (3D-MV)
phantom and SR-PBCT reconstruction experiments on a real microvasculature
sample were performed.

## 2. Methods

#### 2.1. Phase retrieval for PBI

An object illuminated by spatially coherent X-ray can be described with its 3D complex refractive index distribution:

where*δ*(

*x*,

*y*,

*z*) represents the phase information, and

*β*(

*x*,

*y*,

*z*) represents the absorption information; (

*x*,

*y*) denotes the coordinates of a projection plane orthogonal to the propagation direction, and

*z*denotes the propagation axis.

In PBI, as the spatially coherent X-ray beams propagate from an illuminated object, the density variations of the object cause phase shifts of X-rays. Owing to Fresnel diffraction, the phase shifts are converted into detectable intensity modulations and subsequently recorded by the detector set at a specific distance. In general, the raw projections from PBI consist of both absorption information and the Laplacian of phase information. To obtain the phase-information distribution of the illuminated object, we first needed to extract the phase-shift information from the raw projections, and thus, phase retrieval was required [32]. To acquire the accurate phase shift for each tomographic projection, phase retrieval generally needs at least two phase-contrast projections taken at different SDDs. However, the multiple-SDD phase retrieval methods are usually difficult to arrange experimentally due to the complicated registration problem and the long execution time. In this work, we assumed that the microvasculature sample could be approximated as a single-material object; namely, the object was ‘homogeneous’, and thus, a so-called ‘homogeneous’ transport of intensity Equation (TIE-Hom) phase retrieval method [33] that required just a single-SDD projection was used to accurately extract the phase-shift information. The TIE-Hom method can be described by the following formula:

*D*; ${\cal F}$ and ${{\cal F}^{ - 1}}$ are the 2D FFT and its inverse operators, respectively; $\gamma$ represents the ratio between

*δ*(

*x*,

*y*,

*z*) and

*β*(

*x*,

*y*,

*z*), and

*λ*denotes the wavelength of the X-ray; $\xi$ and $\eta$ are the coordinates of

*x*and

*y*in the Fourier domain, respectively.

#### 2.2. The proposed CT reconstruction algorithm for SR-PBCT

### 2.2.1. The forward projection model of phase shift in SR-PBCT

In the near-field regime of PBI, the refraction and scattering effects between the X-ray and illuminated object are weak, and hence, the X-ray beam propagation path inside the illuminated object can be assumed to be straight. Therefore, the phase-shift function of the sample can be written as follows:

where ${L_\theta }$ denotes the linear propagation path inside the sample at the projection angle $\theta$, and $\delta (x, y, z)$ denotes the 3D phase-information distribution of the sample.Generally, the SR X-ray source has a monochromatic property, and
thus, the forward projection model of phase shift in SR-PBCT,
namely, the phase-shift set belonging to the same slice
(*y *= *constant*)
that consists of ${\varphi _\theta
}(x)$ ranging from 0° to
180°, can be approximately expressed as the following
discrete linear system:

*A*is the system matrix used for modeling the parallel-beam X-ray forward projections [34], and

*P*denotes $(\lambda /2\pi )$ times the phase-shift set.

### 2.2.2. The iterative reconstruction strategy based on the FBP algorithm

Following phase retrieval, according to Eq. (2) and Eq. (3), the 2D phase-information distribution of the sample, i.e., the SR-PBCT images of the sample, can be acquired by CT reconstruction. Here, the solution for the reconstruction problem of the 2D phase-information distribution of the sample using FBP was formulated as follows:

For simplicity, a so-called FBP operator $\Re$ was utilized, and Eq. (5) can be rewritten as following:

where $\delta$ is an abbreviation for $\delta (x, z)$.In this study, an iterative reconstruction strategy based on the FBP algorithm [35] was used to solve the inverse problem in Eq. (4), and the solution can be described as follows:

where $\alpha$ is the step size, and the superscripts*t*and

*t*+ 1 represent the

*t-*th iteration and (

*t*+ 1)th iteration of the recursive scheme, respectively.

### 2.2.3. The WGIF algorithm based on a prior image

In practice, the solution expressed in Eq. (7) would lead to poor performance for the few-projection CT reconstruction problem because it neither builds an effective artifacts model nor utilizes prior knowledge for artifacts suppression in the reconstruction strategy. Generally, the biological soft tissue surrounding the microvasculature can be considered a single-material object, i.e., the biological soft tissue is ‘homogeneous’. Moreover, owing to the phase-based contrast generation mechanism, the interfaces between the biological soft tissue and microvasculature typically show a high image contrast. Therefore, a binarized image of microvasculature, where the flat regions represent prior knowledge of the homogeneous biological soft tissue and the background (typically the air) for smoothing and, additionally, the interfaces between the highlight and shadow regions represent prior knowledge of the microvascular structures for preserving, can be accurately generated by the k-means clustering approach. In this study, the binarized image of the microvasculature was considered the prior image.

The GIF is a local filtering-based edge-preserving smoothing
technique, and it is assumed to be a local linear model between
the guided image *G* and the filtering output $\mathop \delta
\limits^ \wedge $; namely, the abovementioned prior
image can be used as the guided image of the microvasculature
image. In principle, the local linear model between
*G* and $\mathop \delta
\limits^ \wedge $ can ensure that the output $\mathop \delta
\limits^ \wedge $ has an edge only if the guided
image *G* has an edge, and it can be formulated as
follows:

*k*, and

*l*is the pixel index.

To determine the linear coefficients ${a_k}$ and ${b_k}$, a constraint is added to the filtering input $\delta$ and the filtering output $\mathop \delta \limits^ \wedge $; the constraint is expressed as follows:

where the output $\mathop \delta \limits^ \wedge $ represents homogeneous areas with sharp edges, and ${e_l}$ represents unwanted components, such as noise or artifacts.Based on the constraint in Eq. (9), the values of ${a_k}$ and ${b_k}$ can be determined by minimizing a cost function $E({a_k},{b_k})$ in the window ${\omega _k}$, which is defined as follows:

In general, the cost function $E({a_k},{b_k})$ can be utilized to minimize the
difference between the filtering input $\delta$ and the filtering output $\mathop \delta
\limits^ \wedge $ while maintaining the linear
model in Eq. (8). However, as a local optimization-based function, $E({a_k},{b_k})$ cannot preserve sharp edges, and
it usually produces halo artifacts near some edges. To overcome
this deficiency, the WGIF was presented that incorporated an
edge-aware weighting into the GIF [30]. The WGIF computed the edge-aware weightings in the
guided image *G*, and then, these weightings were
used for the better preservation of edges in the filtering input $\delta$. Here, the edge-aware weighting $W_k^G$ is defined by utilizing local
variances of 4×4 windows of all pixels as follows:

*M*represents the total number of pixels in the guided image

*G*, and ${\varepsilon _0}$ represents a small number used to avoid singularities. In this study, ${\varepsilon _0}$ is set to 1.0×10

^{−6}; ${(\sigma _k^{G,1})^2}$ defines the variance in the guided image

*G*in the 4×4 window centered at the pixel

*k*, and ${(\sigma _l^{G,1})^2}$ represents the variance in the guided image

*G*in the 4×4 window centered at pixel

*l*.

According to Eq.(10) and Eq.(11), the cost function $\mathop E\limits^ \wedge ({a_k},{b_k})$ of the WGIF can be written as follows:

*G*in window ${\omega _k}$; $|{\omega _k}|$ represents the number of pixels in window ${\omega _k}$, and additionally, ${\mathop \delta \nolimits^ - _k} = \frac{1}{{|{\omega _k}|}}\sum\nolimits_{l \in {\omega _k}} {{\delta _l}}$ is the mean value of $\delta$ in window ${\omega _k}$.

Following that ${a_k}$ and ${b_k}$ were computed for all windows ${\omega _k}$ in the filtering input $\delta$, the final filtering output $\mathop \delta \limits^ \wedge $ can be obtained as follows:

*l*.

### 2.2.4. Steps of the proposed CT reconstruction algorithm

The proposed algorithm mainly consists of two steps per iteration: the FBP step and the WGIF step. Before the FBP step, the two images were first acquired: one was the CT reconstruction image using FBP from the inpainted projection data that was obtained by interpolation between undersampled angles based on CMP, and the other was the few-projection CT reconstruction image using FBP that was processed by the WGIF. In the filtering process of the WGIF, the binarized image of the few-projection CT reconstruction image generated by the k-means clustering method was used as the guided image. Considering that the guided image was not updated, it was named a static guided image. By fusing the abovementioned two images, a good initial guess was obtained, and this was important for the convergence of the proposed algorithm. Then, the FBP step was used to acquire few-projection CT reconstruction images with artifacts. Following the FBP step, the WGIF step was performed to reduce the artifacts in CT reconstruction images obtained by the FBP step and improve image quality. It is worth mentioning that the guided image was updated per iteration in the WGIF step, which was termed a dynamic guided image. Subsequently, the resultant image from the WGIF step was reprojected based on CMP to produce the forward projection data, and then, the projection residual was modeled as the inpainted projection data subtracting the forward projection data. Finally, the projection residual flowed into the FBP step for the next iteration, and this procedure was updated based on Eq. (7). The FBP step and the WGIF step alternated during the iteration until the stop criterion was met. The steps of the proposed algorithm are better illustrated in Fig. 1.

### 2.2.5. Pseudocode for the proposed CT reconstruction algorithm

To explain the proposed algorithm more clearly, the corresponding pseudocode is presented, as shown in Algorithm 1.

#### 2.3. Quantitative assessment

In this study, the mean structural similarity index (MSSIM) [37], signal-to-noise ratio (SNR) [38], relative loss rate and relative difference (RD) were used as quantitative metrics. The MSSIM is widely utilized to evaluate the similarity between a reconstructed image and a reference image, and it is usually between 0 and 1, which increases with increasing similarity. The SNR is a traditional measure of image quality, and a larger SNR value indicates better image quality. The relative loss rate is used to measure the relative volume loss between the reconstructed microvasculature and the reference microvasculature, and a smaller value indicates more accuracy. The RD is used to assess the difference between two reconstructed images in adjacent iterations, and the smaller RD value represents the smaller difference.

- (i) MSSIM is defined as follows,$$SSIM({x_j},{y_j}) = \frac{{(2{u_{{x_j}}}{u_{{y_j}}} + {c_1})(2{\sigma _{{x_j}{y_j}}} + {c_2})}}{{(u_{{x_j}}^2 + u_{{y_j}}^2 + {c_1})(\sigma _{{x_j}}^2 + \sigma _{{y_j}}^2 + {c_2})}},$$where $x$ and $y$ are the reference image and the reconstructed image, respectively; ${x_j}$ and ${y_j}$ are the $j$th image patch of
*x*and*y*, respectively; ${u_{{x_j}}}$ and ${u_{{y_j}}}$ are the mean values of ${x_j}$ and ${y_j}$, respectively; $\sigma _{{x_j}}^2$ and $\sigma _{{y_j}}^2$ are the variances of ${x_j}$ and ${y_j}$, respectively; ${\sigma _{{x_j}{y_j}}}$ denotes the standard deviation between ${x_j}$ and ${y_j}$; ${c_1}$ and ${c_2}$ represent two constants to stabilize the division with weak denominator, i.e., ${c_1} = {({k_1}L)^2}$, ${c_2} = {({k_2}L)^2}$; $T$ is the total number of image patch. In this study, the $T$ was set to 729, $L$ was set to 255, ${k_1}$ and ${k_2}$ was set to 0.01 and 0.03, respectively; - (ii) SNR is defined as follows, where ${S_{sample}}$ denotes the mean value of the pixel values in the local sample region of the reconstructed image $y$, and ${\sigma _{background}}$ the standard deviation of the pixel values in the local background region of the reconstructed image $y$.
- (iii) Relative loss rate can be defined as follows,$$Relative \,loss \,rate = \frac{{\textrm{Vesse}{\textrm{l}_{GT}} - \textrm{Vesse}{\textrm{l}_R}}}{{\textrm{Vesse}{\textrm{l}_{GT}}}} \times 100\%$$where $\textrm{Vesse}{\textrm{l}_{GT}}$ denotes the microvasculature volume calculated from the reference images, and $\textrm{Vesse}{\textrm{l}_{\mathop{\textrm R}\nolimits} }$ represents the microvasculature volume calculated from the reconstructed images.

## 3. Simulation experiment

#### 3.1. Simulations

To assess the performance of the proposed algorithm, a 3D-MV phantom
designed via the real SR-PBCT images of the microvasculature in a
mouse-liver sample was employed, as shown in Fig. 2. The 3D-MV phantom consists of
three different types of compositions, such as air, teflcn and water,
where the air and teflcn parts located in the field of view (FOV) were
used to simulate the parts inside and outside the microvasculature
and, additionally, the water located outside the FOV was used as the
background for computing the SNR. The pixel values of air, teflcn and
water in the 3D-MV phantom were set to 0,
7.02×10^{−7}, and
3.69×10^{−7}, respectively, which represented
the corresponding phase values of the abovementioned three types of
compositions at an energy of 25 keV (the energy dependence of the
complex refractive index can be obtained for a variety of chemical
compounds under http://henke.lbl.gov/optical_constants/getdb2.html). The
volume of the 3D-MV phantom was 216×216×100
voxel^{3}, and the size of the voxel pitch was
3.25×3.25×3.25 um^{3}. The detector is modeled
as a straight-line array with 306 bins, and the size of the
reconstructed images was 216×216 pixel^{2}. The total
number of reconstructed slices was 100. According to CMP, the number
of the full projection required for FBP was approximately 340. In this
study, 68 and 34 uniformly distributed projections over acquisition
orbital extents (*θ*) ranging from 0° to
180° (full-scan range) were used as few-view projections. The
experiments with CT reconstruction were performed using the MATLAB
programming language on a workstation computer equipped with an Intel
(R) Xeon (R) E5-2640 CPU at 2.4 GHz and 128 GB of RAM. Finally, the 3D
surface-rendering images were generated from a sequential series of CT
slices using Amira 6.3.0 software (Visage Imaging, Berlin,
Germany).

#### 3.2. Experimental results

### 3.2.1. Noise-free reconstructed results

In this section, the optimal parameters for the proposed algorithm,
such as *α* and *ɛ*,
were determined by comparing both the quantitative metrics and the
visual assessment of the reconstructed images from the different
parameter values, and hence, three sets of parameters were
obtained as follows: (i) one set for 340 projections: $\alpha =
1.2,$ $\varepsilon =
0.01;$ (ii) another set for 68
projections: $\alpha =
1.0,$ $\varepsilon =
0.01;$ and (iii) the other set for 34
projections: $\alpha =
0.4,$ $\varepsilon =
0.01.$ The total iteration number ${t_{\max
}}$ was set to 5, which was utilized
as the stop criterion for the proposed algorithm.

Figure 3 shows the surface-rendering images of the 3D-MV phantom in which all slices were reconstructed using the FBP and the proposed algorithm under the noise-free 340 projections, 68 projections and 34 projections. Additionally, Fig. 3 also shows the 24th reconstructed slices using the FBP and the proposed algorithm under the noise-free 340 projections, 68 projections and 34 projections, which were used for comparing tomographic sections. Figure 3(b) and Fig. 3(d) clearly show that the visual effect of the CT image reconstructed using the proposed algorithm under the 340 projections was almost the same as that reconstructed using the FBP, and the same was true for their corresponding 3D images (see Fig. 3(b) and Fig. 3(d)). As the number of projections decreased, the streaking artifacts on the CT images reconstructed using the FBP increased (see Fig. 3(f) and Fig. 3(j)), and the false structures on the corresponding 3D images, which are typically formed by the streaking artifacts on the CT images, also increased (see Fig. 3(e) and Fig. 3(i)). Obviously, compared with the reconstructed results of the FBP under the few-projection conditions, those of the proposed algorithm showed a better visual effect, which suggests that the proposed algorithm has the ability to reduce the streaking artifacts on the CT images (see Fig. 3(h) and Fig. 3(l)) and thereby enables the suppression of the false structures on their corresponding 3D images (see Fig. 3(g) and Fig. 3(k)). To further assess the performance of the proposed algorithm, the profiles along the same position in the reconstructed images, the position labeled with the red line in Fig. 3(b), were drawn, as shown in the upper right corner of the CT images in Fig. 3. When the number of projections decreased, the profiles of the reconstructed images using the FBP began fluctuating in the homogeneous areas and near the boundaries (see Fig. 3(b, f, j)), which means that the observation for some boundaries and fine features were affected. However, as the number of projections decreased, the profiles of the reconstructed images using the proposed algorithm remained smooth in the homogeneous areas and near the boundaries (see Fig. 3(d, h, l)), which demonstrates that the proposed algorithm has the ability to suppress the streaking artifacts in the homogeneous areas and preserve the boundaries (i.e., the microvasculature structures).

### 3.2.2. Noisy reconstructed results

In this study, a combination of Poisson noise and
Gaussian noise [39], which
was used to simulate the statistical noise in the projections, was
added into the phase-shift sets for analyzing the robustness and
reliability of the proposed algorithm in the presence of
statistical noise. The parameters of the noise model were set as
follows: the photon number of Poisson noise ${I_0}$ was set to
1.0×10^{5}, while the mean value ${m_G}$ and variance $\sigma
_{_G}^2$ of the Gaussian noise were set to
0 and 5, respectively. In this experiment, the optimal values of
the parameters, such as *α* and
*ɛ*, were selected by comparing both the
quantitative metrics and the visual assessment of the
reconstructed images based on the different parameter values.
Finally, three sets of parameters were acquired as follows: (i)
one set for 340 projections: $\alpha =
1.0,$ $\varepsilon =
0.04;$ (ii) another set for 68
projections: $\alpha =
1.0,$ $\varepsilon =
0.04;$ and (iii) the other set for 34
projections: $\alpha =
0.4,$ $\varepsilon =
0.04.$ The total number of iterations ${t_{\max
}}$ for the proposed algorithm in the
presence of noise was set as 5.

Figure 4 depicts the surface-rendering images of the 3D-MV phantom where all slices were reconstructed using the FBP and the proposed algorithm under the noisy 340 projections, 68 projections and 34 projections, as shown in Fig. 4(a, e, i) and Fig. 4(c, g, k). In addition, Fig. 4 also depicts the 24th reconstructed slices using the FBP and the proposed algorithm under the noisy 340 projections, 68 projections and 34 projections, as shown in Fig. 4(b, f, j) and Fig. 4(d, h, l). Compared with the reconstructed slices using the FBP under the noisy few-projection conditions (see Fig. 4(b, f, j)), the reconstructed slices using the proposed algorithm under the noisy 340 projections, 68 projections and 34 projections (see Fig. 4(d, h, l)), had less noise and fewer artifacts, which achieves a better visual effect for observing the microvasculature structures. Obviously, it can be found that the surface-rendering images of the 3D-MV phantom where all slices were reconstructed using the FBP under the noisy few-projection conditions (see Fig. 4(a, e, i)) were severely affected by considerable false structures, which led to a poor visual effect for distinguishing the microvasculature structures. However, the surface-rendering images of the 3D-MV phantom where all slices were reconstructed using the proposed algorithm under the noisy few-projection conditions (see Fig. 4(c, g, k)) showed a superior visual effect for observing the microvasculature structures, where the false structures were effectively suppressed. Additionally, from the profiles of the reconstructed images in Fig. 4(b, f, j) and Fig. 4(d, h, l), it can be clearly seen that the profiles of the latter are much smoother in the homogeneous areas and nearby the boundaries than those of the former, which demonstrates that the proposed algorithm enables suppression of the noise and artifacts, as well as enables preservation of the boundaries.

#### 3.3. Assessments

In this section, both the image quality and the accuracy of the 3D-MV phantom where all slices were reconstructed using the FBP and the proposed algorithm were assessed via quantitative metrics, such as MSSIM, SNR and relative loss rate. The quantitative metrics for the 3D-MV phantom were measured based on all slices (100 slices), where all slices were reconstructed using the FBP and the proposed algorithm under the noise-free and noisy few-projection conditions, respectively. The measured quantitative metrics are presented as the mean ± standard deviation, as displayed in Fig. 5. Under the noise-free and noisy few-projection conditions, the MSSIM values and the SNR values from the reconstructed slices using the proposed algorithm are larger than those of the reconstructed slices using the FBP. The higher MSSIM and SNR values confirmed that the proposed algorithm effectively reduced the noise and artifacts.

Due to the existence of false structures, it is difficult to accurately measure the microvasculature volume. Therefore, both the reconstructed microvasculature without the false structures and the reference microvasculature (see Fig. 2(a)) were used to compute the relative loss rate, and the corresponding results are shown in Fig. 5(e, f). From Fig. 5(e, f), it can be observed that the proposed algorithm remains small relative loss rate values (<3%) of microvasculature structures under the noise-free and noisy few-projection conditions, which demonstrates that the proposed algorithm enables a high reconstruction accuracy for microvasculature.

#### 3.4. The influences of the step size α on the reconstructed images

In this section, to analyze the influences of the step size
*α*, as depicted in Eq. (7), the different reconstructed images
based on *α* values ranging from 0 to 1.8 were
evaluated by the MSSIM, as shown in Fig. 6. From Fig. 6, it can be observed that the noise and the
artifacts in the reconstructed images increased as the
*α* values increased, and the edges (i.e., the
microvasculature structures) were more accurate and clearer. However,
if the *α* values became too small, the imaged
edges and fine features were severely deformed, and if the
*α* values became too large, the imaged edges
and fine features were severely destroyed by the noise and the
artifacts. Therefore, the step size *α* can
serve as a balance parameter that controls the trade-off between the
structure fidelity of the edges or fine features and the smoothness of
the artifacts or noise. Since the step size *α*
plays a critical role in the CT reconstruction process, the selection
of the optimal *α* will be important. In this
study, the optimal *α* values were selected by
comparing both the MSSIM and the visual assessment of the
reconstructed images based on different *α*
values.

#### 3.5. Convergence analysis and computational time

To assess the convergence performance of the proposed algorithm under the noise-free and noisy few-projection conditions, the MSSIM-based and SNR-based convergence curves were presented, as shown in Fig. 7. The proposed algorithm converged near the 5th iteration. Therefore, in this study, the total number of iterations ${t_{\max }}$ was set to 5, and it was used as the stop criterion for the proposed algorithm. In addition, the reconstructed time of all slices from the 3D-MV phantom using both the FBP and the proposed algorithm under the noise-free and noisy few-projection conditions were computed, as presented in Table 1. The computational time of the proposed algorithm was approximately two minutes, which was much longer than that of the FBP, but this was acceptable for 3D reconstruction tasks.

## 4. Real experiment on the microvasculature using PBI

#### 4.1. Sample preparation

The animal experiments were performed in accordance with the guiding principles for the care and use of laboratory animals approved by the Research Ethics Committee of the Beijing Friendship Hospital, Capital Medical University, China. In this study, a healthy female C57BL/6 mouse (7 weeks old, weighing 18-20 g) used for SR-PBCT experiment was euthanized with a dose of 800 mg/kg of sodium pentobarbital, and then an entire liver lobe was removed from the mouse. After removal, the entire liver lobe was rinsed by phosphate buffer and perfused with saline 0.9% to clear the blood vessels, finally fixed in 10% neutral buffered formalin solution for 4 months. Before imaging, the entire liver lobe was dehydrated using an ethanol series (50%, 70%, 80%, 95%, 95%, 100% and 100%), and then it was transferred to the imaging facility in motion-proof container to avoid risk of tissue damage during shipping.

#### 4.2. PBI data acquisition

The PBI data of the entire liver lobe were collected at the BL13W1
beamline at the Shanghai Synchrotron Radiation Facility, Shanghai,
China. In this experiment, the energy of the incoming monochromatic
photons was set as 14 keV, and the SDD was adjusted to 0.2 m. A
charge-coupled device camera (Photonic Science, UK) with a 6.656 mm
(horizontal)×4.264 mm (vertical) FOV was adopted as the imaging
detector, and its effective pixel pitch was 3.25×3.25
µm^{2}. According to CMP, the reference number of the
projection required for FBP was approximately 1740. For the SR-PBCT
imaging, 1200 projections at equidistant angles from 0° to
180° were collected. The size of the projection image was
2048×1312 pixel^{2}, and the exposure time per
projection was 1.5 s. The total scan time was approximately 40
minutes. In addition, ten dark current images (without the X-ray
beams) were utilized to reduce the dark signals in projections, and
twenty flat-field images (with the X-ray beams on, but without the
sample) were utilized to correct the pixel-to-pixel nonuniformity in
projections [40]. In the
few-projection SR-PBCT experiments, 400 and 200 projections were
uniformly chosen from 1200 projections recorded by the detector. The
phase retrieval was performed using the TIE-Hom method, and the $\gamma$ was set to 200. After phase
retrieval, 1200, 400 and 200 projections were used for the
few-projection CT reconstruction. Finally, 550 slices containing the
microvasculature information were chosen from the 1312 reconstructed
slices and then used for the 3D visualization of the
microvasculature.

#### 4.3. Experimental results

In this experiment, the optimal parameters for
*α* and *ɛ* were selected
by comparing both the quantitative metrics and the visual effect of
the reconstructions from different parameter values. Finally, three
sets of parameters were acquired as follows: (i) one set for 1200
projections: $\alpha =
1.0,$ $\varepsilon =
0.10;$ (ii) another set for 400 projections: $\alpha =
1.0,$ $\varepsilon =
0.13;$ and (iii) the other set for 200
projections: $\alpha =
0.8,$ $\varepsilon =
0.14.$ The total number of iterations ${t_{\max }}$ was set to 8.

Figure 8 shows the surface-rendering images and the 318th slices of the entire liver lobe in which all slices were reconstructed using the FBP and the proposed algorithm from 1200 projections, 400 projections and 200 projections. Figure 8(a, e, i) depict the surface-rendering images of the entire liver lobe in which all slices were reconstructed using the FBP from 1200 projections, 400 projections and 200 projections, respectively. As the number of projections decreased, it can be obviously observed that the false structures on the corresponding 3D images increased, and the same was true for the streaking artifacts on the corresponding CT images (see Figs. 8(b, f, j)). The 3D reconstruction process for microvasculature in which all slices were reconstructed using the FBP approach from 200 projections can be observed in Visualization 1, where the formation process of the false structures can also be seen. Figure 8(c, g, k) depict the surface-rendering images of the entire liver lobe in which all slices were reconstructed using the proposed algorithm from 1200 projections, 400 projections and 200 projections, respectively. Compared with Figs. 8(a, e, i), Figs. 8(c, g, k) have clearer 3D microvasculature structures, where the false structures were greatly reduced, and thus enabled the subsequent structure segmentation, measurement and analysis for the 3D microvasculature. Additionally, compared with Figs. 8(b, f, j), Figs. 8(d, h, l) presented the smoother tissue areas and clearer edge details, which indicated that the proposed algorithm was capable of suppressing the streaking artifacts and preserving the microvasculature structures under the few-projection condition. The 3D reconstruction process for microvasculature in which all slices were reconstructed using the proposed approach from 200 projections can be observed in Visualization 2, where clear 3D microvasculature structures without false structures can also be observed.

#### 4.4. Result analysis

To provide a better visual effect for the result analysis, the areas in the reconstructed images, as marked by the green rectangle in Fig. 8(b), were selected as the regions of interest (ROIs) and then enlarged. The enlarged images of ROIs are presented for comparison, as shown in Figs. 9(a-f). Figures 9(a, b, c) are the enlarged ROIs of the reconstructed images using the FBP approach from 1200 projections, 400 projections and 200 projections, respectively. Figures 9(d, e, f) are the enlarged ROIs of the reconstructed images using the proposed approach from 1200 projections, 400 projections and 200 projections, respectively. Compared with Figs. 9(a, b, c), it is obvious that Figs. 9(d, e, f) provided the more superior visual effect for microvasculature, where the streaking artifacts were effectively smoothed, and the edges of microvasculature were well preserved, while some imaged blood vessels of the former were severely destructed by the streaking artifacts and the subsequent structure segmentation, measurement and analysis for the microvasculature were significantly affected. Moreover, the profiles along the same position in enlarged ROIs, the position marked with the red line in Fig. 9(a), were drawn, as presented in Figs. 9(g, h, i). Figures 9(g, h, i) represent the profiles of the reconstructed images using the FBP approach and the proposed approach from 1200 projections, 400 projections and 200 projections, respectively. From Figs. 9(g, h, i), it can be seen that the profiles from the proposed algorithm are much smoother in the tissue regions (homogeneous areas) and near the edges (boundaries) than those from the FBP approach, which demonstrates that the proposed algorithm is capable of smoothing the streaking artifacts, as well as preserving edges (i.e., the microvasculature structures).

#### 4.5. Convergence analysis and reconstructed time

In this section, the quantitative metric RD, as depicted in Eq. (20), was used to evaluate the convergence performance of the proposed algorithm. Figure 10 presents the RD-based convergence curves of the proposed algorithm using 1200 projections, 400 projections and 200 projections. It could be clearly observed that the proposed algorithm converged near the 8th iteration under different few-projection conditions. Thus, in this experiment, the total iteration number ${t_{\max }}$ was set to 8. Additionally, under different few-projection conditions, the reconstructed time for 550 slices of the entire liver lobe based on the FBP and the proposed algorithm were measured, as shown in Table 2. The reconstructed time of the proposed algorithm was approximately twenty hours, which was acceptable for 3D reconstruction tasks.

## 5. Discussion and conclusion

As the widely used FBP algorithm fails to reconstruct high-quality SR-PBCT images of microvasculature under the few-projection conditions, we developed a new few-projection SR-PBCT reconstruction algorithm for microvasculature imaging. In the experiments, the 3D-MV phantom and SR-PBI data of the mouse-liver sample were utilized to assess the accuracy and feasibility of the proposed algorithm, and the FBP algorithm was used for comparison. The simulation experiment results confirmed that the proposed algorithm was an effective method for suppressing the streaking artifacts as well as preserving the edges under the few-projection conditions and that it had robustness and reliability in the presence of statistical noise. The SR-PBCT experiment results demonstrated that the proposed algorithm had the ability to effectively suppress the streaking artifacts and well preserve the edges of microvasculature under the few-projection conditions, and the same was true for the corresponding 3D images, where the false structures could be effectively suppressed, and the 3D microvasculature structures could be clearly presented. Overall, the proposed algorithm is significant for SR-PBCT reconstruction of microvasculature from few-projection data, and it is also practical for the 3D visualization of microvasculature since the data acquisition time is significantly reduced.

Generally, SR-PBCT enables high-resolution and high-contrast 3D
visualization of microvasculature and has the potential to serve as an
adjunct to histopathological sections for observing microvascular changes.
However, the main limitation of the SR-PBCT method is that the SR source
requires an expensive synchrotron facility, which impedes its wider use in
preclinical and biomedical microvascular imaging studies. Recently,
transferring the PBCT method onto laboratory X-ray sources, such as
microfocus X-ray sources, has been an active area of research. Compared
with the SR source, the laboratory X-ray sources typically have lower
photon flux [9,41], which results in longer scan times for collecting
data (typically a few hours). For laboratory X-ray source-based PBCT
technology, few-projection PBCT reconstruction algorithms will also be
highly desirable, and they have great potential to make both fresh and
chemically unfixed biological tissues available for this technology [4]. Additionally, the laboratory X-ray
source-based PBCT technology may also be useful for the intraoperative
scanning of excised microvascular tissues and even available for
*in vivo* microvascular imaging of small animals, where the
few-projection PBCT reconstruction algorithms will have important
significance due to the long scan time and the high radiation dose. In
summary, we believe that this work will facilitate the development and
application of PBCT technology in microvasculature imaging.

In this study, the homogeneity assumption on the microvasculature sample
was utilized. Based on this approximation, accurate phase information was
obtained by the TIE-Hom method, and a binarized image acquired by the
k-means clustering method was used as the prior image of microvasculature
structures. For some multimaterial samples, the TIE-Hom method would still
be applicable if the regions of different materials could be approximated
as homogeneous, and the k-means clustering method would also be available
in this case. Thus, the proposed algorithm has the potential to
reconstruct other single-material or multimaterial samples, and this will
be investigated in our future work. Furthermore, GPU-based parallel
computing techniques will be implemented to improve the reconstructed
speed of the proposed algorithm, and the adaptivity of the parameters in
the proposed algorithm will also be researched. In addition, future
studies will also be performed to test whether the proposed algorithm
applies for laboratory X-ray source-based PBCT technology and *in
vivo* imaging data.

## Funding

National Natural Science Foundation of China (81671683, 81670545, 81371549); Natural Science Foundation of Tianjin City in China (16JCYBJC28600); The Foundation of Tianjin university of technology and education (KJ12-01, KJ17-36).

## Acknowledgements

Thanks to the BL13W1 beamline of Shanghai Synchrotron Radiation Facility.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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