Monochromatic-beam-based dynamic X-ray microtomography based on OSEM-TV algorithm

Abstract

Monochromatic-beam-based dynamic X-ray computed microtomography (CT) was developed to observe evolution of microstructure inside samples. However, the low flux density results in low efficiency in data collection. To increase efficiency, reducing the number of projections should be a practical solution. However, it has disadvantages of low image reconstruction quality using the traditional filtered back projection (FBP) algorithm. In this study, an iterative reconstruction method using an ordered subset expectation maximization-total variation (OSEM-TV) algorithm was employed to address and solve this problem. The simulated results demonstrated that normalized mean square error of the image slices reconstructed by the OSEM-TV algorithm was about 1/4 of that by FBP. Experimental results also demonstrated that the density resolution of OSEM-TV was high enough to resolve different materials with the number of projections less than 100. As a result, with the introduction of OSEM-TV, the monochromatic-beam-based dynamic X-ray microtomography is potentially practicable for the quantitative and non-destructive analysis to the evolution of microstructure with acceptable efficiency in data collection and reconstructed image quality.

Keywords

Image reconstruction techniques dynamic microtomography quantitative analysis limited projections X-ray imaging

1 Introduction

Synchrotron radiation X-ray microtomography (SR-X μCT) is a non-destructive technique widely used for visualizing the internal information of samples, and for evaluating quantitative information on their three-dimensional (3D) structures [1]. Along with the rapid development of detector techniques and the high flux output of the third generation synchrotron radiation facility, the spatial resolution of SR-X μCT has reached sub-micron and the exposure time of single image has been milliseconds. All these progresses make the four-dimensional (4D) tomography possible [2] and it has been applied in many fields with a few seconds temporal resolution, such as observing the bubble growth in hydrated basaltic melts [3], investigating the influence of intermetallics on solidification defects in alloys [4] and so on. In SR-X μCT, there are two illumination modes, pink/white beam or monochromatic beam. The high flux of pink/white beam makes the lower exposure time possible. Using this illumination mode, a drastic change in respiratory tract of a living worm in three dimensions was observed dynamically with a temporal resolution of 0.5 s [5]. And a moving screw-and-nut-type hip joint in the insect Sitophilus granarius (grain weevil) was captured at a 50 milliseconds temporal resolution [6]. However, the disadvantages of pink/white beam are also apparent, including the poor phase contrast, hardly quantitative reconstructed results and high radiation dose delivered to samples. For the investigation of dynamic processes in live samples, the high dose is fatal. Thus, these in vivo 4D tomography experiments above continued only a few seconds before the samples were dead. The monochromatic beam is the alternative method, which has good phase contrast, quantitative reconstructed results and low radiation dose. The disadvantage of the monochromatic beam is the low flux and that results in a higher exposure time and then a lower temporal resolution, especially for imaging with relative larger field of view at a beamline with wiggler source. In order to increase the data-collection efficiency of the dynamic microtomography based on monochromatic X-ray beam, decreasing the projection number is a good choice. Thus a challenge of reconstructing high quality image from these limited projections appears.

The filtered back projection (FBP) algorithm is a typical analytical CT reconstruction algorithm which uses Fourier slice theory to arrive at a closed form solution, it is computationally fast, easily implemented, accurate, and widely used in commercial instrumentations [7]. However, there are inherent limitations to FBP that is it requires high completeness of projection data which means the number of projections should satisfy the Shannon sampling theorem [8]. It posits that the minimal number of tomographic projections, required for reconstruction of a noiseless image is π×K/2. Here K is the width in pixels of the field of view. If limited projections are used to reconstruct a slice using the FBP algorithm, streaking artifacts will appear in the reconstructed slice. These streaking artifacts will deteriorate the images and will influence the consequent image analysis such as image segmentation. To deal with the limited projections data, equally sloped tomography (EST) is one option. EST is an exact tomographic method utilizing an oversampling iterative Fourier-based reconstruction [9 –11]. For the FBP reconstructions, all projections were calculated at equal-angle increments, uniformly distributed across 180°, while for EST reconstructions, the projections were calculated along equally sloped lines of the pseudopolar grid distributed across 180° at constant slope increments. However, dynamic X-ray microtomography usually needs to acquire a series of CT data continuously, which means the sample should rotate continuously at a high speed. Therefore, the synchronizing signal between the data acquisition and the angle of the sample stage is a problem. Then, iterative reconstruction methods should be another good choices [12]. In addition, the piece-wise constancy assumption of the desired slice is reasonable and then the optimization of the total variation (TV) will help suppress the streaking artifacts [13 –17]. Then a large number of pioneering studies came up [18 –22]. In this paper, we employed the ordered subset expectation maximization method with the minimization of the total variation (OSEM-TV) to reconstruct the image. The OSEM method is a fast statistical iterative algorithm that has been proven to be accurate in positron emission tomography (PET) reconstruction and to be able to reconstruct images with limited projections data [23, 24]. Stsepankou et al. evaluated robustness of OSEM-TV in cone-beam CT reconstruction through adding three different and (for clinical application) typical classes of errors [25]. Ahn et al. found that OSEM-TV was useful for reducing the occurrence of artifacts due to gaps between detector modules in small-diameter PETscanners [26].

In this paper, we demonstrated the practicability of using OSEM-TV to reconstruct images efficiently from few projections acquired from monochromatic-beam-based dynamic X-ray microtomography. We first presented numerical results under different noise levels. We then evaluated image reconstruction quality using the experimental results.

2 Methods

2.1 Total variation

The image reconstruction process of dynamic microtomography can be described as solving the linear system of equations AX = Y. Here X represents the desired image, A is the system matrix which can be determined by the imagery architecture [27]. Y is the projection data acquired by measurements. For the limited projections problem, the rows of A are much less than the columns, which makes the system of equations ill-posed. In order to solve the ill-posed problem, the TV is introduced and is defined as $\begin{matrix} TV (X) = \sum_{ij} \nabla_{ij} X \\ = \sum_{ij} \sqrt{(X (i + 1, j) - X (i, j))^{2} + (X (i, j + 1) - X (i, j))^{2}} \end{matrix}$ (1) where X (i,j) means the image value at position (i,j) and ∇_ij is the discrete gradient operator. Then the image reconstruction could be achieved by solving the following non-linear constrained minimization problem: $min_{X} TV (X), s . t . Y = AX$ (2)

2.2 OSEM-TV algorithm

The convex optimization problem equation (2) could be solved with an iterative algorithm. In the first part of the iterative progress, the image X is updated by the OSEM algorithm to get a new image $X_{OSEM}^{(k)}$ based on AX = Y. In the second part, the standard steepest descent method is used to make the TV minimum. The pseudo-code is shown as below:

Initialization of the image for OSEM

k = 1 and X⁽⁰⁾ = 0

OSEM process:

\begin{matrix} X_{i}^{l^{'} + 1} = \frac{X_{i}^{l^{'}}}{\sum_{j} A_{ji}} \sum_{j \in S_{l}} A_{ji} \frac{Y_{j}}{\sum_{k} A_{jk} X_{k}^{l^{'}}} \\ S_{l} = [A_{1}, A_{2}, \dots, A_{n}], Y = AX \end{matrix}

(3) where A _i,j is the element of the system matrix that is defined by the data acquisition system. The system matrix was divided into n subsets and each subset has data from M/n projection angles such as [θ, θ + 180n/M, …, θ + 180 (1 - n/M)]. In addition, M is the number of projections. One OSEM process is completed when l = n.

Initialization of the steepest descent image

m = 1 and ${\tilde{X}}^{(0)} = X^{(n)}$

The steepest descent process

\begin{matrix} {\tilde{X}}^{(m)} = {\tilde{X}}^{(m - 1)} - α \cdot | {\tilde{X}}^{(0)} - X^{(0)} | \cdot \frac{β}{| β |} \\ β = \partial {| X |}_{TV} / \partial X \end{matrix}

(4) where m = 1, 2, ..., N, α and N are adjustable parameters. β/|β| is the normalized gradient descent direction. The distance

| {\tilde{X}}^{(0)} - X^{(0)} |

provides a measure for degree of convergence of the OSEM process and it could be the self-adapting searching step-size of the steepest descent process.

Initialization of the next iterative process

X^{(0)} = {\tilde{X}}^{(N)}

The step (b) ∼ (e) are a complete iterative process.

3 Materials and methods

3.1 Simulations

The performance of the FBP and OSEM-TV algorithms is first investigated by simulation. The phantom is a discrete 1024×1024 Shepp-Logan phantom. For the projection data, 60 projection angles within 180 degrees is generated. The incident rays are set to quasi planes waves with series of statistical noise levels and the noise type is Poisson noise, which are equivalent to different exposure times. The noise strength is defined as σ/μ, where σ means the standard variance of the incident ray and μ means the mean value of the incident ray. In this simulation, the noise strengths were set to 0, 0.01, 0.04, 0.07, 0.10 and 0.15 respectively.

3.2 Experiments

The experimental data were collected at BL13W at the Shanghai Synchrotron Radiation Facility, which is a wiggler beamline and dedicates to X-ray imaging [28 –30]. The polymer samples containing polypropylene, rods of polystryrene (Φ= 1.6 mm), nylon (Φ= 1.6 mm) and PMMA globules (Φ= 0.6 mm) were investigated. The data sets were collected using a series of statistical noise levels by tuning the filter and the exposure time. Nine data sets were collected, and the mean pixel values of the background were calculated as indicators of flux. The parameters are summarized in Table 1. The energy of the incident rays is an important parameter. We chose the energy that made sure the transmission was 50%. The sample-to-detector distance (SDD) is another important parameter. A longer SDD gives a stronger image edge enhancement, however, the spatial resolution decreases. In this experiment, we chose a proper SDD carefully, which gave an obvious edge enhancement without reducing imaging quality seriously. In the end, the photon energy of 16 keV, sample-to-detector distance of 10 cm and a sCMOS detector with an effective pixel size of 3.25 μm were used to acquire all data sets for the statistical noise study. Within a 180-degree CT scan range, 96 projections were collected. Table 2 shows the nominal δ values (refractive index decrements) of these four materials, and the δ values are the reconstructed physical quantities.

Table 1
Pixel values in the background for the data sets collected under different exposure times

Data index 1 2 3 4 5 6 7 8 9

Pixel value 396 532 740 943 2040 2703 4559 6456 9202

Exposure time[ms] 17.5 29 40 52 105 155 270 385 550

Data index	1	2	3	4	5	6	7	8	9
Pixel value	396	532	740	943	2040	2703	4559	6456	9202
Exposure time[ms]	17.5	29	40	52	105	155	270	385	550

Table 2

δ values (refractive index decrements) of different materials

Materials	Polypropylene	Polystryrene	Nylon	PMMA
δ value [*10^–7]	8.3	9.2	10.1	10.4

The real sample was an alive bell cricket and was settled in a ventilated plastic container at a normoxic state and at room temperature. The plastic container was used to ensure that movements of the cricket were negligible during the imaging process. Before the experiments, the cricket stayed still in the container for half an hour to adapt to the new surroundings. To obtain optimal results, the experimental parameters were carefully chosen. The photon energy was set to 14 keV, the sample-to-detector distance was set to 50 cm, and the region of interest was 2048×700 pixels (horizontal×vertical). Under these parameters, the acquisition time of one data set was 500 ms. The experiment lasted 75 s, and 150 data sets were obtained.

3.3 Data processing

First, the nine experimental data sets of the phantom were processed with Paganin phase retrieval algorithm [31] via PITRE software package [32]. Next, the pre-processed projections were reconstructed by the FBP algorithm and the OSEM-TV algorithm respectively. The iterative stopping criterion of the OSEM-TV was that the difference of the normalized mean square errors (NMSEs) between adjacent slices was less than 1*10^–2 or the iterations reached 40. In addition, the simulated data with no need of pre-processed had the same data processingprocedure.

3.4 Data analysis

To evaluate the quality of the reconstructed slices quantitatively, the normalized mean squared errors (NMSE) and the contrast-to-noise ratio (CNR) are introduced, which are defined as: $\begin{matrix} NMSE = \sqrt{\frac{\sum_{i, j = 1}^{N} {(X_{i, j} - Y_{i, j})}^{2}}{\sum_{i, j = 1}^{N} Y_{i, j}^{2}}} \\ CNR = \frac{| S_{a} - S_{b} |}{\sqrt{σ_{a}^{2} + σ_{b}^{2}}} \end{matrix}$ (5) where X and Y denote the reconstructed and original N*N images. Definitely, the smaller the NMSE the better the image quality. For the CNR, S_a and S_b are the mean values of two homogeneous region of interest (ROI) in the slices, where a and b represent two different materials; while σ_a and σ_b denote the standard deviations of the values in these two ROIs. Moreover, the histograms of the reconstructed slices under different projection number and exposure time are also given, in which different materials could be discriminated by a serial of peaks. In this way, so-called density resolution could be revealed. That is to say, the width of the peak directly represents the density resolution: the narrower the peak, the higher the density resolution [31].

4 Results and discussion

4.1 Simulations

The reconstructed slices of phantoms by FBP and OSEM-TV algorithms are shown in Fig. 1 respectively. Visual inspection of these slices suggests that, under conditions of limited projections and different statistical noise levels, the OSEM-TV algorithm can effectively suppress streak artifacts, better than the FBP algorithm. Effect of noises on the reconstructed image quality is investigated. CNR values are shown in Table 3 and the ROIs are the white rectangles marked in Fig. 1(b1). As one might expect, the CNR values decrease with the increase of the noise level. The OSEM-TV algorithm obtains the best image quality as a general trend. As shown in Fig. 2(a), NMSE value of slices reconstructed by OSEM-TV is about 1/4 of that by FBP, which means that much better image quality can be achieved by OSEM-TV at all levels of noises. During the simulation, the stopping iteration number of 30 is selected for OSEM-TV algorithm according to the convergence curves as shown inFig. 2(b).

Fig.1

Reconstructed slices from the FBP algorithm (a1-a3) and the OSEM-TV algorithm (b1-b3). From the left to the right, the noise strengths are 0, 0.04, and 0.15 respectively.

Fig.2

Analysis of NMSEs of reconstructions using the OSEM-TV and FBP. (a) The NMSE of reconstructed slices under different statistical noise levels using the FBP and OSEM-TV algorithms. (b) The convergence curves of the OSEM-TV algorithm under different statistical noise levels.

Table 3

CNR values for reconstructed slices of phantom under different statistical noise levels

	FBP			OSEM-TV
Noise Strength	CNR_ab	CNR_ac	CNR_bc	CNR_ab	CNR_bc	CNR_ac
0	1.49	4.60	3.12	66.98	245.22	186.64
0.01	1.48	4.57	3.10	59.63	220.92	160.74
0.04	1.41	4.34	2.94	30.83	116.05	79.88
0.07	1.23	3.77	2.63	17.99	61.95	49.04
0.10	1.17	3.46	2.26	12.98	44.14	37.38
0.15	0.88	2.74	1.74	9.47	34.89	24.02

4.2 Experimental results

In order to improve the efficiency of data collection, the exposure time for a single projection should be as short as possible. However, the signal-to-noise ratio decreases with the photons reducing. In practical applications, a compromise should be made between the density resolution of the CT and data-collection efficiency. A phantom made up of polymer fibers enclosed in a tube with similar density (see Table 2) is employed to explore this effect. The slices reconstructed by the two algorithms are shown in Fig. 3, in which the data sets indexed with 1, 5 and 9 (see Table 1) are selected and a small projection number of 96 is used. As shown in Fig. 3, the OSEM-TV algorithm still works much better than the FBP algorithm, which confirms the results of simulation. It is also obvious that the fake image of streaks can be suppressed effectively by OSEM-TV at all noise levels. At lower exposure time, the boundary of the polymer samples can still be distinguished clearly, though noises increase apparently inside the homogeneous materials.

Fig.3

Reconstructed slices of polymer sample, where (a1)-(a3) are slices reconstructed from data set indexed 1, 5 and 9 respectively (see Table 1) using the FBP algorithm; (b1)-(b3) are slices reconstructed from the same data set using the OSEM-TV algorithm. The projections used for reconstruction are all 96.

The CNR values for polystyrene, nylon and PMMA are presented in Table 4. For calculating the CNR value, each two of the three material are selected and the ROIs are the rectangles marked in Fig. 3(a2). As expected, the CNR values decrease as the exposure time reduces. With the increasing of exposure time, the image quality based on OSEM-TV algorithm grows much faster than the FBP algorithm based.

Table 4

CNR values of polymer sample

	FBP			OSEM-TV
Index	CNR_ab	CNR_ac	CNR_bc	CNR_ab	CNR_bc	CNR_ac
1	1.41	4.84	3.27	2.28	8.89	5.93
2	1.84	5.82	3.98	3.52	11.17	7.02
3	2.12	6.79	4.58	3.72	13.61	9.25
4	2.52	7.21	4.71	4.75	14.09	8.72
5	3.04	8.90	5.78	5.47	19.33	11.58
6	3.18	8.69	6.12	5.05	16.46	13.47
7	3.56	9.90	7.19	6.81	19.49	18.46
8	3.67	10.93	7.31	5.95	22.40	16.78
9	4.37	11.42	8.04	7.99	24.43	23.48

^aPMMA, ^bnylon, ^cpolystyrene.

To evaluate the density resolution of the OSEM-TV algorithm further, histograms of the reconstructed slices from data sets indexed 1–9 respectively are given in Fig. 4, in which the projection number used for the reconstructions is 96. For the longest exposure time corresponding to the data set indexed 9, four peaks in the histogram that are corresponding to PP (polypropylene), PS (polystyrene), Nylon and PMMA respectively, can be revealed obviously. This means that the four kinds of polymer samples could be resolved easily in the reconstructed slices. With the decrease of exposure time, the height of the peak declines while the correspondent width of the peak expands. Up to data set indexed 5, the four peaks are just distinguishable. For shorter exposure time, the peaks attenuate gradually. At the shortest exposure time of 17.5 ms, all the peaks disappear except for that of PP that has the largest sectional area. As to PMMA, which has smaller size, the correspondent peak decays quickly with the decrease of exposure time, which means that for higher spatial resolution more photons are needed. For experimental applications, according to the above analysis, the least pixel value in the background of each projection should be about 2040 as indicated in the data set indexed 5.

Fig.4

Histograms of reconstructed slices from data sets indexed 1–9, corresponding to exposure time of 17.5, 29, 40, 52, 105, 155, 270, 385, 550 ms respectively, using the OSEM-TV algorithm, where bg denotes background; PP: polypropylene; PS: polystyrene.

An experiment with a bell cricket was also carried to evaluate the algorithms. Fig. 5(a) shows the photo of the bell cricket. Fig. 5 (c) and (d) give the reconstructed slices of the bell cricket, corresponding to the abdomen of the bell cricket, using the FBP and OSEM-TV algorithms respectively. The normal projection number for FBP reconstruction of the sample structure should be 1200. However, in this experiment, projection number for one CT data set is limited to 68, which means that the data collection efficiency could be improved by 18 times. Then, the selected CT data set is reconstructed by FBP and OSEM-TV respectively. Shown in Fig. 5 (c) is the slice reconstructed by FBP algorithm, in which the image quality deteriorates due to the severe streaking artefacts. Obviously, it is hard to proceed with further quantitative analysis based on such a data set. Furthermore, the reconstruction by OSEM-TV algorithm is shown in Fig. 5 (d).

Fig.5

Picture of the bell cricket and reconstructed results, (a) The photo of the bell cricket used in experiments, (b) The 3D rendering image of the live bell cricket, reconstructed slices using FBP algorithm (c) and OSEM-TV algorithm (d) respectively.

It is obvious that the streaking artefacts are suppressed effectively by the OSEM-TV algorithm, while the microstructure of the sample, including the tracheae inside legs, the abdomen and the wing, remains. The correspondent three-dimensional rendering image is also shown in Fig. 5(b), from which the whole structure of the bell cricket is reconstructed without deterioration of image quality due to the streaking artefacts. Using the data captured in dynamic CT, the three-dimensional structure evolution of the live bell cricket has been got, in which the compression and expansion of the abdomen can be observed, as shown in the Supplementary Movie 1.

5 Conclusions

In this paper, we employed OSEM-TV, an efficient iterative reconstruction method, for the monochromatic-beam-based dynamic X-ray microtomography, which was evaluated by simulations and experiments respectively. According to the simulation on a standard Shepp-Logan phantom, NMSE value of slices reconstructed by OSEM-TV is about 1/4 of that by FBP. Concerning image quality and reconstruction efficiency, OSEM-TV is practicable for the processing of massive data in the monochromatic-beam-based dynamic X-ray microtomography. Experimental results for a phantom made up of polymer fibers confirmed the simulation results and density resolution was high enough to resolve different polymer materials at the least pixel value of 2040 in the projections. In experiments for a live bell cricket, three-dimensional structure evolution of the live bell cricket was revealed successfully, in which the compression and expansion of the abdomen could be observed explicitly. As a result, we conclude that with the use of OSEM-TV, the monochromatic-beam-based dynamic X-ray microtomography is practicable for the quantitative and non-destructive analysis to the evolution of microstructure with acceptable efficiency.

Footnotes

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (No. 11375257 and 81430087), the Joint Funds of the National Natural Science Foundation of China (No. U1232205), the state key research and development program (No. 2017YFA0206004 and 2017YFA0403800), Youth Innovation Promotion Association of Chinese Academy of Sciences (No. 2017304) and the CAS-CSIRO cooperation research project (No. GJHZ1303).

The Supplementary Movie is available in the electronic version of this article: .

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Monochromatic-beam-based dynamic X-ray microtomography based on OSEM-TV algorithm

Abstract

Keywords

1 Introduction

2 Methods

2.1 Total variation

3.1 Simulations

3.2 Experiments

Table 1 Pixel values in the background for the data sets collected under different exposure times Data index 1 2 3 4 5 6 7 8 9 Pixel value 396 532 740 943 2040 2703 4559 6456 9202 Exposure time[ms] 17.5 29 40 52 105 155 270 385 550

3.4 Data analysis

4.1 Simulations

Footnotes

Acknowledgments

References

Table 1
Pixel values in the background for the data sets collected under different exposure times

Data index 1 2 3 4 5 6 7 8 9

Pixel value 396 532 740 943 2040 2703 4559 6456 9202

Exposure time[ms] 17.5 29 40 52 105 155 270 385 550