New reconstruction method for few-view grating-based phase-contrast imaging via dictionary learning

Huiping Bai; Weikang Zhang; Jun Zhao; Yujie Wang; Jianqi Sun

doi:10.1364/OE.26.026566

1. Introduction

In X-ray optics, the interaction of X-rays with materials is encoded in their indices of refraction n = 1-δ + iβ, where δ and β characterize the materials’ phase shift and absorption properties of X-rays, respectively. For soft tissues, δ is approximately 1000 times bigger than β in the hard X-ray regime, enabling imaging soft tissue structures without exogenous contrast agents using X-ray phase-contrast imaging (PCI) technique which measures δ only[1]. In the past decades, several PCI imaging methods have been proposed, including interferometric methods [2], propagation-based imaging [3–6], diffraction enhanced imaging [7], and grating interferometers [8,9]. However, the requirement of high spatial coherence ordinarily requires a synchrotron X-ray source, hindering the wide implementation of PCI in medical or industrial applications.

With the demonstration of three-grating interferometry using low-brilliance tube-based X-ray sources [10], the application of grating interferometers with conventional X-ray source becomes very promising. Additionally, grating-based PCI can generate contrast not only on phase information, but also on absorption and scattering information, which can be combined in advanced image analyses to provide more complete quantitative information on samples under test. Therefore, Grating-based phase-contrast computed tomography (GPCT) has become a promising merging modality technique in recent years.

However, medical imaging applications of GPCT are limited by its long data-acquisition time and relatively high radiation doses caused by the phase stepping procedure. A natural way to conquer these problems is to reduce the exposure time or the number of projections. As a result, several iterative image reconstruction algorithms for GPCT have been proposed [11–14].

Although phase contrast is superb for soft tissues, its contrast for high absorption elements like bones are not as clear as that of the absorption image. So far, few phase-contrast methods take the full advantage of the information contained in other two modalities (i.e., absorption and scattering). Hahn D et al. [15] used the bone information segmented from the absorption image to overcome the limited dynamic range problem, so that the bone part can be shown in the phase contrast image without metal artifacts. The study used full-view projections and part of the information of absorption image in the reconstruction. In this paper, we propose a novel method using both absorption and phase information to accomplish few-view GPCT reconstruction. This study combines dual-dictionary learning and compressive sensing (CS) techniques to improve image reconstruction. Using the proposed method, one can considerably reduce the number of projection and the image acquisition time, which can reduce or avoid motion artifact during image acquisition. Therefore, this few-view GPCT has a great potential for applications in in-vivo imaging and future clinical usage.

2. Method

2.1 First dual-quality dictionary learning: from low-resolution to high-resolution

Dictionary learning (DL) is an important sparse representation method, which builds adaptive sparse transform from particular images for sparse approximation. It has been shown that adaptive sparse transform is promising in many image processing applications such as image denoising, deblurring, and demosaicing [16]. The concept of image dictionary is explained below: A dictionary consists of atoms, which are usually in the form of image patches. An atom is supposed to contain critical image features such as bone structures or tissue boundaries. In order to extract common characteristics from original image patches, K-singular value decomposition (K-SVD) algorithm[17] was adopted to construct the dictionary $D$ . The optimization problem can be formulated as:

\min_{D, {α_{j}}} {\sum_{j} ‖ D α_{j} - R_{j} x ‖}_{2}^{2} subject to {‖ α_{j} ‖}_{0} \leq ρ, \forall j,

where

R_{j} x

extracts image patches from x,

α_{j}

is the sparse representation of patch

x_{j}

under dictionary D,

|| \cdot {||}_{0}

is

l_{0}

norm, and

ρ

represents the preset target of sparsity level of α. Afterwards, the dictionary is applied to approximate other target images via sparse representation

α

:

\min_{α_{j}} {\sum_{j} ‖ D α_{j} - R_{j} x ‖}_{2}^{2} subject to {‖ α_{j} ‖}_{0} \leq ρ, \forall j .

The orthogonal MP (OMP) algorithm[18] is used to solve this problem.

Building upon previous work, researchers have optimized DL method to dual-dictionary learning (DDL) for further improvement. DDL method was initially designed for super-resolution imaging. Then this method is validated by Lu et al[19] in few-view CT reconstruction. Considering the case of our few-view CT reconstruction, the quality of corresponding absorption imaging can be improved with the DDL method.

In our study, two reconstructed series, FBP reconstructed series $x_{h i g h}$ using full view and iterative reconstructed series $x_{l o w}$ using under-sampled insufficient data, were prepared respectively. Before they were trained into dictionaries, they were concatenated in the optimization problem Eq. (1) in order to keep their correspondence. Our trained D composed of $D_{h i g h}$ and $D_{l o w}$ can be described as follows:

[\begin{array}{l} D_{h i g h} \\ D_{l o w} \end{array}] = k s v d {[\begin{array}{l} x_{h i g h} \\ x_{l o w} \end{array}]} .

In Eq. (3), ksvd {} denotes the process of patch extraction and dictionary construction. It is also proved that the high-quality image patch and its corresponding low-quality image patch share the same representation under different dictionaries[19]:

R_{j} [\begin{array}{l} x_{h i g h} \\ x_{l o w} \end{array}] = [\begin{array}{l} D_{h i g h} \\ D_{l o w} \end{array}] α_{j} .

Thus, with the $α$ obtained from Eq. (5) using orthogonal MP (OMP) algorithm[18], the high quality image can be reconstructed:

\min_{α_{j}} {\sum_{j} ‖ D_{l o w} α_{j} - R_{j} x_{l o w} ‖}_{2}^{2} subject to {‖ α_{j} ‖}_{0} \leq ρ, \forall j,

x_{h i g h} = {(\sum_{j} R_{j}^{T} R_{j})}^{- 1} \sum_{j} R_{j}^{T} D_{h i g h} α_{j} .

2.2 Second dual-modality dictionary learning: from absorption to phase image

Combining the good contrast for heavy absorption elements of the absorption image in the phase reconstruction process could achieve a more accurate restoration of few-view phase image in principle. Inspired by the relation between the two dictionaries $D_{l o w}$ and $D_{h i g h}$ above, the high quality absorption image and phase image can be combined to form the second dual dictionary: i.e., absorption versus phase dual dictionary. Similar to the dual dictionary in 2.1, the new dual dictionary of absorption and phase information can be established:

[\begin{array}{l} D_{a b s o r p t i o n} \\ D_{p h a s e} \end{array}] = k s v d {[\begin{array}{l} x_{a b s o r p t i o n} \\ x_{p h a s e} \end{array}]} .

Sparse representation is also implemented using the dual dictionary. The estimated phase image $x_{p h a s e}^{e s t}$ can then be obtained as follows:

\min_{β_{j}} {\sum_{j} ‖ D_{a b s o r p t i o n} β_{j} - R_{j} x_{a b s o r p t i o n} ‖}_{2}^{2} subject to {‖ β_{j} ‖}_{0} \leq ρ, \forall j,

x_{p h a s e}^{e s t} = {(\sum_{j} R_{j}^{T} R_{j})}^{- 1} \sum_{j} R_{j}^{T} D_{p h a s e} β_{j} .

The implementation details of DDL algorithm is listed below:

(1) Prepare a set of high quality phase images and absorption images, which are reconstructed from adequate projections using filtered back projection (FBP) algorithm;
(2) Extract image patches with the size of n × n from each of the two sets;
(3) Remove redundant image features and establish dictionaries with one-to-one correspondence between atoms by Eq. (7);
(4) Search the dictionary $D_{a b s o r p t i o n}$ for the best representation of each patch of a given image using OMP, and then replace them by their counterparts in the dictionary $D_{p h a s e}$ .

2.3 Estimated image Guided reconstruction

According to Xu et al. [11], a forward model suitable for the differential phase reconstruction can be obtained using blobs, which has been proven to be effective in the reconstruction by Kohler et al. [12]. Element in m’^th row and n^th col of forward model A for phase reconstruction is given by:

{[A]}_{m ’, n} = {[A]}_{m ’ = t \times S + s, n} = - \frac{{(2 π α)}^{1 / 2}}{I_{m} (α)} \times \frac{ξ}{a} {(\sqrt{1 - {(\frac{ξ}{a})}^{2}})}^{m - 1 / 2} \times I_{m - 1 / 2} (α \sqrt{1 - {(\frac{ξ}{a})}^{2}}),

ζ =s Δ_{d} - x_{n} \cos (t Δ_{ϑ}) - y_{n} \sin (t Δ_{ϑ}) .

In Eq. (10), S is the total number of projection data for each view, s is the integer-valued detector element index, t is the view index, $Δ_{ϑ}$ is the angular sampling interval between view angles, $Δ_{d}$ denotes the detector element dimension, $(x_{n}, y_{n})$ specifies the coordinate of the nth lattice point on a uniform Cartesian lattice, $I_{m} (α)$ denotes the modified Bessel function of order m, and a and α determine the blob’s radius and specific shape respectively. In this study, m, a and α were empirically set as 2, 2, 10.4 respectively.

With this image model, many iterative reconstruction algorithms of conventional CT can be applied for DPCT.

In the case of few-view reconstruction, a popular assumption for a medical image is the sparsity of its gradient image. Then, the minimization of the l₁ norm of the gradient image, which is well-known as total variation (TV) regularization, has become one of the most commonly adopted regulation methods [20,21].

The major issue in few-view reconstruction is to solve the equation Ax = b, for either phase or absorption. Here we rewrite it into an optimization problem as described in Eq. (12):

\min_{x} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ {‖ F x ‖}_{1} .

Here, F is a gradient operation in both x and y direction, and

|| \cdot {||}_{1}

is

l_{1}

norm.

With estimated images obtained from the two dual-dictionary learning methods, we put forward the optimization problem for few-view phase and absorption reconstruction:

\min_{x} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ_{1} {‖ F x ‖}_{1} + λ_{2} {‖ F (x - x^{e s t}) ‖}_{1} .

In Eq. (13), λ₁ is used to balance the regularization effects of TV in an image, $x$ can be the phase image or the absorption image, and $x^{est}$ is the corresponding estimated image from dual dictionary learning. The problem can be solved using Douglas–Rachford Splitting TV (DRS-TV) algorithm in [22].With this optimization method, an optimal balance between artifact suppression and spatial resolution is obtained.

We can subsequently obtain an improved phase image with more details in both soft tissue and high absorbing elements. Similarly, with this enhanced phase image, we can also obtain an improved absorption image via dual-dictionary learning procedure. The entire absorption and phase image reconstruction is described below:

Initialization:

(1) $x_{a b s o r p t i o n}$ reconstruction using Eq. (12): $x_{a b s o r p t i o n}^{l o w} = \underset{x}{\arg \min} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ {‖ F x ‖}_{1};$
(2)Dual dictionary learning from $x_{a b s o r p t i o n}^{l o w}$ to $x_{a b s o r p t i o n}^{est}$ ;
(3) $x_{a b s o r p t i o n}$ reconstruction: $x_{a b s o r p t i o n} = \underset{x}{\arg \min} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ_{1} {‖ F x ‖}_{1} + λ_{2} {‖ F (x - x_{a b s o r p t i o n}^{e s t}) ‖}_{1};$
Loop:
(4)Dual dictionary learning from absorption image $x_{a b s o r p t i o n}$ to phase image $x_{p h a s e}^{e s t}$ ;
(5) $x_{p h a s e}$ reconstruction: $x_{p h a s e} = \underset{x}{\arg \min} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ_{1} {‖ F x ‖}_{1} + λ_{2} {‖ F (x - x_{p h a s e}^{e s t}) ‖}_{1}$
(6)Dual dictionary learning from $x_{p h a s e}$ to $x_{a b s o r p t i o n}^{e s t}$ ;
(7) $x_{a b s o r p t i o n}$ reconstruction: $x_{a b s o r p t i o n} = \underset{x}{\arg \min} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ_{1} {‖ F x ‖}_{1} + λ_{2} {‖ F (x - x_{a b s o r p t i o n}^{e s t}) ‖}_{1} .$

After several iterations of dual image reconstruction, both phase and absorption images can be improved. The whole objective function can be written as:

\min_{x_{a}, x_{p}} \frac{1}{2} {‖ A_{1} x_{a} - b_{1} ‖}_{2}^{2} + λ {‖ F x_{a} ‖}_{1} + \frac{1}{2} {‖ A_{2} x_{p} - b_{2} ‖}_{2}^{2} + λ {‖ F x_{p} ‖}_{1} .

Here, subscript “a” and “p” denote the corresponding parameters in absorption and phase image reconstruction, respectively. In our experiment, divergence was not found; however, to avoid divergence, an experimental number of loops should be used. In our experiment, we did the loop twice.

3. Results and discussion

In our study, the original data is collected from Shanghai Synchrotron Radiation Facility (SSRF) [23]. The specimen was the right paw excised from a C57/Black6 mouse (weight 24g) which was fixed in formalin solution over 24 hours right after excision. The experimental parameters are listed below in Table 1:

Table 1. Experimental parameters for the tomographic scans

View Table

Projection images were captured by a CCD detector (Photonic Science, UK) with an effective pixel size of 9 µm. At the BL13W beamline of SSRF, a Si(111) double-crystal monochromator was used, which produced a quasi-monochromatic X-ray beam with a relative bandwidth smaller than 3 × 10⁻³. The X-ray source was located ~34 m away from the sample stage, and the beam size at the sample stage was about 45 (H) mm × 5 (V) mm. The integrated flux per pixel at the quoted exposure time was approximately 4.2 × 10⁴ phs. A gold π/2-phase grating with a pitch of 2.4 µm was used as G1, and a gold absorption grating with a pitch of 2.4 µm was located at a distance of 46.4 mm downstream of G1, corresponding to the first Talbot order. The transverse coherence length at the sample position was 4.66 µm. The sample was located 100 mm in front of G1, which is sufficiently far from the sample so that constraints imposed by temporal coherence do not inflict with the spatial resolution [24]. The reconstructed phase and absorption tomograms of the specimen are shown in Fig. 1.

Fig. 1 Phase image and absorption image of the mouse paw. (a) phase image, (b) absorption image. Bar: 1mm.

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x^{n e x t} = x^{c u r r e n t} - \frac{A_{i} x^{c u r r e n t} - p^{i}}{{‖ A_{i} ‖}^{2}} A_{i}^{T},

To evaluate the proposed method, a reference image was reconstructed by FBP using all 360 projections. Then the images were reconstructed by FBP, Projection onto Convex Sets-TV (POCS-TV), DRS-TV [22] and statistic iterative reconstruction (SIR) [15] using 25% rates of the full data set. POCS-TV aims to solve Eq. (12). In the algorithm, it first enforced the reconstructed image x consistency with the projection data and then set positive constraints on 𝓍 and did TV gradient descent. In Eq. (15), x means the reconstructed image, $A_{i}$ denotes the forward model for phase reconstruction, and $p^{i}$ means the i^th projection. DRS-TV solves Eq. (12) by applying Douglas–Rachford Splitting (DRS) algorithm.

SIR is only for phase image reconstruction and optimizes the function:

\min_{x} \frac{1}{2} {‖ A x - b ‖}_{2}^{2} + λ_{1} {‖ F x ‖}_{1} + λ_{2} R (x, x_{a b s o r p t i o n}, δ_{B} / μ_{B}),

where regularization term R contains a bone mask to restrict the quadratic diﬀerence between x and

x_{a b s o r p t i o n}

. Here, x is coupled to

x_{a b s o r p t i o n}

through a proportionality constant

δ_{B} / μ_{B}

, which is calculated based on the X-ray index of refraction of bone material. The parameters in the proposed method are optimized as: λ₁ = 0.4, λ₂ = 0.3. The data set used in our dual-dictionary training step is a GPCT of a mouse body.

Reconstructed images were quantitatively compared in terms of root-mean-square error (RMSE) and structural similarity index (SSIM). RMSE is frequently used to measure the differences between the reconstructed and the reference images, and SSIM to measure the similarity between them. A SSIM value approaching 1 indicates a higher similarity between the two images. SSIM and RMSE for phase image are illustrated in Fig. 2 and Fig. 3 respectively. For absorption image assessment, the quantitative metrics is shown in Fig. 4. In addition, enlarged regions of interest of phase images are also displayed in Fig. 5 for visual assessment.

Fig. 2 SSIM of reconstructed phase images.

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Fig. 3 RMSE of reconstructed phase images.

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Fig. 4 SSIM and RMSE of reconstructed absorption images.

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Fig. 5 Zoomed-in images of region of interest of the mouse paw (a) Reference, (b) Proposed, (c) DRS-TV, (d) FBP, (e) POCS-TV, (f) SIR.

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Figure 2 and Fig. 3 show the quality metrics of reconstructed phase images using different methods. It can be seen that DRS-TV has relatively higher SSIM and lower RMSE as compared to POCS-TV and FBP. However, with the combination of total variation and dictionary-learning, the proposed algorithm can make further improvement to the quality of reconstructed image, and achieve the lowest reconstruction error and highest structural similarity in both bone and soft tissues.

Besides, from Fig. 4 we can see that the absorption image was also improved simultaneously, which could not be realized in other algorithms. The noise in the absorption image was significantly reduced. Most importantly, the proposed method recovered small structures of the sample which could not be seen by other methods. The advantage of the proposed method is apparent when we use Fig. 5 for visual assessment. The reconstructed image using proposed method demonstrated the greatest similarity with the reference. It exhibited detailed features such as the structure of the bone (as shown by the arrow in Fig. 5) and had smoother soft tissue features compared with DRS-TV method. The image reconstructed by FBP was significantly degraded by noise and showed a large number of pervasive streak artifacts. To some extent the POCS-TV algorithm can suppress streak artifacts and recover smooth regions as compared with FBP. However, image reconstructed by POCS-TV had blurring artifact and low contrast structures were almost invisible. The SIR algorithm can recover small structures of the bone, but soft tissues had limited improvement.

4. Conclusion

The proposed research reveals the potential of TV and DDL in reconstruction of GPCT and confirms the feasibility of applying existing CT reconstruction methods in phase contrast imaging. In our work, a specimen with both bones and soft tissues was investigated in the experiment. Through dual-dictionary learning, the absorption image and the phase image can provide complementary information for each other. Thus the quality of both images can be improved simultaneously. Using this method, high similarity and low deviation between the reconstructed image and the reference image were achieved. Especially, small structures in the high-absorption bones can be eventually restored.

To reduce the radiation and scanning time, GPCT reconstruction combining dictionary learning and TV optimization can provide a promising direction in insufficient data reconstruction. The proposed method uses phase and absorption information jointly to reconstruct tomographic images of the refractive index. The correlations between the phase and absorption images are fully exploited in the reconstruction. Consequently, a small number of X-ray projections are sufficient to restore detailed information of the sample.

Future work will focus on the usage of the scattering data in the phase image reconstruction. Since the scattering data reveal the scattering features of the sample, which is usually strong in the boundary of structures, this data could be helpful for edge enhancement and metal artifacts reduction.

Funding

National Natural Science Foundation of China (grant No. 11575115; grant No. U1732119); National Key Research and Development Program (grant No. 2016YFC0104608); Shanghai Jiao Tong University Medical Engineering Cross Research Funds (Grant No. YG2017QN51; Grant No. YG2016QN65).

Acknowledgments

Thanks to the BL13W beamline of Shanghai Synchrotron Radiation Facility.

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	Mouse paw
Rotation range (°)	0-180
Number of angles	360
Pixel size(µm)	9 × 9
Phase steps	16
X-ray energy (keV)	20
Projection exposure time (ms)	15
Total exposure time (s)	86

New reconstruction method for few-view grating-based phase-contrast imaging via dictionary learning

Abstract

1. Introduction

2. Method

2.1 First dual-quality dictionary learning: from low-resolution to high-resolution

2.2 Second dual-modality dictionary learning: from absorption to phase image

2.3 Estimated image Guided reconstruction

3. Results and discussion

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (5)

Tables (1)

Equations (16)

Optics Express