Swinging multi-source industrial CT systems for aperiodic dynamic imaging

Weiwen Wu; Hengyong Yu; Changcheng Gong; Fenglin Liu

doi:10.1364/OE.25.024215

1. Introduction

The demand for high quality control of key components brings a great pressure to scientists and engineers to gain a deep understanding of a series of industrial phenomena, such as the process of additive manufacturing (AM) [1–4], defect formation of casting cooling processes [5–7], deformation and fracture of material [8,9], dynamic liquid infiltration in tiny loose solid substance [10]. Furthermore, a better understanding of formation mechanism of these phenomena is needed for exploring performance properties of materials or improving and optimizing procedure conditions. These phenomena involve complex interactions among a multitude of transient physical phenomena, such as mass transport, heat transfer, solidification, microstructural evolution, fluid flow, and thermal distortion [11]. Because the changes usually disappear within the object, it brings another great challenge for observation. Visualizing and analyzing the characteristics of these processes are extremely thorny, and it usually is limited if we employ destructive measurement methods. The dilemma with destructive methods lies in the interference between the internal changeable region and realistic measurements. Fortunately, the nondestructive measurement methods can overcome this limitation.

In order to investigate the dynamic behaviors inhibited by the opacity of complex structures, many nondestructive testing methods were proposed. Arthur and Cheverton designed systems, apparatus and methods to visualize a three dimensional manufacturing process by combining different data from several high-speed cameras [12]. A smoothed-particle hydrodynamics method [13] and large-eddy simulation method [14] were designed to predict and visualize the dynamic development of short-term fluid flow and oxide content at casting. To qualify the microporosity during cooling solidification of aluminum-copper alloys, three techniques were utilized i.e., x-ray temperature gradient stage (XTGS), x-ray microtomography and mesoscales simulations [15]. Sun et al. investigated the post-yield deformation process of cortical bone by using a high magnification optical microscopy and atomic force microscopy (AFM) [16]. A microscopic particle image velocimetry technique was proposed to study interactions of dynamic immiscible liquid–liquid in a porous micromodel [17]. All of the aforementioned techniques have limitations. For example, the AFM can only be used for observing surface structures, and the smoothed-particle hydrodynamics method is limited to low spatial resolution. By analyzing the aforementioned processes, common features can be obtained in the following aspects: i) the changeable process usually lasts a short period of time (several minutes or hours); ii) only small sub-regions within the object are changeable for a given inter-arrival time.

Aiming at obtaining an indirect measurement of the spatial distribution of a physical quantity (i.e. density) within the scanned object by the way of x-ray radiation, the x-ray computed tomography (XCT) technique was developed. It can provide accurate images clearly showing the interior tiny structures of scanned object. Compared with the direct measurement techniques, XCT cannot reconstruct an accurate image before a group of complete projection data is collected. Therefore, the temporal resolution becomes a main disadvantage of XCT for some applications. As reported in [15,18], XCT systems equipped with a single x-ray source/detector pair have been applied to visualize and quantify changes. Because the scanned object is assumed motionless during the course of acquiring a full scan data set, these dynamic structures might not be accurately reconstructed from projections acquired at a time frame.

To gain a higher temporal resolution, it is natural to accelerate gantry rotation speed and boost x-ray source energy [19] in medical CT. Nevertheless, both x-ray source power and gantry rotation speed have engineering limits [20]. Alternatively, we can equip multiple x-ray source/detector pairs to reduce the necessary scan range to improve the temporal resolution. For example, dual-source CT scheme was used to improve temporal resolution for cardiac imaging [21]. In the past years, many multi-source CT structures and the corresponding reconstruction algorithms were reported for medical applications [22–26]. However, if the multi-source CT architecture proposed for small animal imaging in [26] is applied to visualize the above industrial aperiodic phenomena, it would be limited by the following three aspects:

▲This scheme employed a slip ring and several source/detector pairs. Because the X-ray tube voltage usually is no more than 150kv for clinical/preclinical applications, the system can improve the temporal resolution [27]. However, due to the diversity of industrial requirements, the voltage of X-ray source ranges from hundreds kV to several MV [28].Thus, if such a system is used for industrial imaging, it will be greatly limited by the huge transmission power and complicated control and data signals [29].
▲Because different industrial applications correspond to different system configurations and testing parameters, a series of special slip rings are needed to satisfy the practical needs. Therefore, it is unsuitable to adopt the slip ring technology for industrial aperiodic dynamic imaging.
▲The multi-source CT architecture is usually integrated into other device (3D print facility, tensile testing machine and cooling vessel). An oversized slip ring may be also a limitation in practice.

Motivated by the aforementioned facts, we propose a swinging multi-source CT scheme (SMCT) to study dynamic changes of the AM process, defect formation in casting cooling, deformation and fracture of substances, liquid infiltration on the porous ceramic materials, etc. The proposed “swinging CT” is different from the multi-source system in [26]. First, the system emphasizes the concept of “swinging”. That is, this system consists of multiple source/detector pairs, each of which only swings a small angle forth and back so that it is easy to be implemented in engineering. Because each of the source/detector pair only needs to swing a small angle, our system does not need a slip ring. Second, the swinging CT structure is flexible to adapt different system configuration and industrial testing requirements since it is equipped with a bearing. The bearing has advantages in terms of concise structure, simple technique and low price, and it is suitable for large-scale serial production. Therefore, depending on specific objects, the bearing could be easily replaced. Third, adopting the “swinging” technology is good for reducing the system size and further making it easy for integration.

The image reconstruction algorithms play an important role in XCT, and they have significantly advanced in recent years. Particularly, the compressive sensing (CS) theory has led to a powerful tool to sample compressible signals at a rate that much less than the Nyquist. This allows us to accurately reconstruct images from less measurements [30]. In past years, the prior image constrained compressed sensing (PICCS) algorithm has been proved to have a good performance in 4D CBCT [19,31–33]. To further improve temporal resolution, the PICCS algorithm was used in dual-source CT and generated the TRI-PICCS algorithm [34]. Therefore, to improve temporal resolution for our SMCT, we will modify the PICCS algorithm for the SMCT structure, leading to an SM-PICCS algorithm.

The rest of this paper is organized as follows. In section II, we present the structure of SMCT and briefly analyze data acquisition and prior image. In section III, we present three different algorithms for the SMCT system with an emphasis on the SM-PICCS algorithm. In section IV, we show the results of numerical studies from the SMCT systems with 9 and 7 source/detector pairs. In section IV, we discuss some related issues and conclude the paper.

2. SMCT system

2.1. Imaging system

As shown in Fig. 1, the SMCT system consists of $Q$ ( $Q$ being an odd number, e.g. $Q = 9$ ) pairs of x-ray source/detector which are distributed uniformly along a circumference on a hollow swing table. The detector of source/detector pair can be either a linear array detector or a flat panel detector (FPD). The scanning object within the field of view (FOV) is placed on a stage concentrically with the swing stage. A servo motor with a speed reducer is used to drive the system for swing CT scanning. As shown in Fig. 2, the radius of the FOV is denoted as $R$ . $Q$ is odd to avoid the interference of the x-ray beams, the x-ray sources and the detectors. During a 360° CT scan, the swing system only needs to be rotated a fixed small angle for projection data acquisition (e.g. each x-ray source only needs to swing 40 degrees if $Q = 9$ ).

Fig. 1 Illustration of an SMCT configuration.

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Fig. 2 Illustration of an SMCT fan-beam geometry.

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As shown in Fig. 3, one swing process can be divided into three sections of movement as acceleration, uniform rotation and deceleration. In an industrial CT system, because its spatial resolution is always an important parameter for inspection, it is a general configuration to use smaller x-ray source focal spot and detector cells. As a result, the sampling time to acquire projection data usually is longer than a medical CT scanner, and it is always equal or greater than100ms corresponding to a scanning rotation speed below 10.0 rpm [35]. Although the commercialized radiotherapy system called “tomotherapy” is used in clinics with MV beams and the slip-ring technique for both imaging and therapy, it cannot improve the system temporal resolution and spatial resolution simultaneously so that it is unsuitable for industrial applications. As shown in Fig. 1, suppose the motor speed reduction ratio is 100, it is not difficult to limit the time for acceleration and deceleration within 100 ms with a rotation speed less than 10.0 rpm. That is, the data sampling time can match the swing acceleration and deceleration time. In fact, the data acquisition is synchronous with the rotation angle. That is, the projection data is collected in terms of uniform rotation locations rather than uniform time distribution, and the projection sequence, corresponding to different rotation angles, can be calibrated by the recorded uneven time sequence. Therefore, such a SMCT system is feasible in practice.

Fig. 3 A full swing process includes forward swing and backward swing processes. T represents a complete swing cycle, and t_a and t_d are acceleration and deceleration times.

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Because the system only depends on a simple swinging motion, it does not need a slip ring. Indeed, such a system can improve temporal resolution for aperiodic dynamic imaging compared with the conventional single or dual x-ray source industrial CT systems. If the imaging object remains motionless during the scanning process, our system can acquire a group of complete data set for image reconstruction when the gantry rotates an angle of $2 π / Q$ . in one direction. This equals to the classical image reconstruction from a full circular scan. Assuming the distance from x-ray source to object is h and the distance between object and detector is d. Let S_i ( $i = 1, 2, 3, \dots, Q$ ) be the location of i^th x-ray source. If the detector length is given as L, we can find the maximum fan-angle of detector $ϕ$ at the origin

ϕ_{\max} = 2 π / Q,

and the minimum value of d as

d_{\min} = \frac{L}{2} \tan (\frac{ϕ_{\max}}{2}) .

Then, the minimum value of h is

h_{\min} = \sqrt{L^{2} / 4 + d_{\min}^{2}} .

As a result, we can obtain the minimum radius of object

R_{\min} = \frac{L h_{\min} / 2}{\sqrt{L^{2} / 4 + {(h_{\min} + d_{\min})}^{2}}},

and a ratio

r_{a t i o} = \frac{h_{\min}}{d_{\min}} = \frac{1}{\cos (ϕ_{\max} / 2)} .

Because

R = \frac{L h / 2}{\sqrt{L^{2} / 4 + {(h + d)}^{2}}},

we can select an appropriate

d > d_{m i n}

and let

h > d r_{a t i o}

in practice. For a fixed relationship

h = d r_{a t i o}

, it is easy to verify that

\partial R / \partial d > 0

. Similarly, for a fixed d, we can increase h and

\partial R / \partial h > 0

. This implies that R_min is always be available by adjusting the values of d and h. Especially, if h approaches to infinity, R would be L/2.

2.2. Data acquisition

To further understand the advantage of the SMCT structure in improving temporal resolution, we can assume a continuously moving region within the object, as shown in Fig. 2. All sources are initially positioned in the red points in Fig. 4 to simultaneously record momentary status of imaging object at time frame 0. These sources are rotated to their adjacent locations (yellow points in Fig. 4) to acquire projection data at time frame 1. In this way, while they reach the locations of black points in Fig. 4, the trajectories of all sources form a circle for all time frames, and this can be called the first half cycle (FHC). At the end of FHC, all sources will reach their spatial limit. In the next time frames, all sources will return to the red points and form the second half cycle (SHC).

Fig. 4 Data acquisition process of an SMCT system.

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2.3. Prior image

The biggest challenge to recover high quality images in the context of dynamic aperiodic imaging is that there are only a few projections acquired at the same time point. To reconstruct a high quality image from such an undersampling data set, it usually needs extra information to regularize the relationship among different time frames. Thus, incorporating a prior image to constrain the image space is a simple strategy. The prior image quality has a great impact on the reconstructed images. The higher quality of prior image is, the higher the quality of reconstructed image is. Thus, to accurately obtain a high quality prior image, a natural yet fast way is to employ the conventional filtered back-projection (FBP) algorithm [36, 37] to reconstruct the object image from a full scan data set and then treat it as the prior image. However, because the object is changeable with the time scale, such a complete data set cannot be achieved even for the SMCT. Note that only some small sub-regions within the object are dynamic for an appropriate time window. If the time of FHC or SHC is shorter than the selected time window, the projection data set of different time frames from each FHC (SHC) can be considered as a “quasi-complete” data set. Here, because the “quasi-complete” projections are obtained from all time frames during the FHC (SHC), such a “quasi-complete” data set means the projection view is sufficient but it is corrupted by the data inconsistency. Therefore, we can reconstruct the prior image from such a “quasi-complete” data set using the FBP method. In fact, the “quasi-complete” data set are stored in computer in a chronological order and the distribution of data is illustrated in Fig. 5. On the basis of the above sub-section, we need to preprocess the projection data when they are applied to the FBP algorithm. That is, we need to rearrange the “quasi-complete” data set in x-ray source order.

Fig. 5 The data distribution of an SMCT system.

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3. Reconstruction algorithm

3.1. Algebraic reconstruction technique

To explain the idea of iterative reconstruction, we start with a relatively simple example that is algebraic reconstruction technique (ART). If we ignore the noise on the measurement data, the objective of image reconstruction is to solve the following matrix equation,

A f = P,

where

A

and

f

represent the system matrix and measured projection vector, and

f

represents the image vector to be reconstructed. In fact, the reconstructed image

f

is a two-dimensional matrix

f_{i, j}

whose width and height are

I

and

J

, and the index

1 \leq i \leq I

and

1 \leq j \leq J

are integrates. We can always rearrange

f_{i, j}

into a vector by defining

f_{n} = f_{i, j}, n = (i - 1) \times I + j .

Now, let be the ray-sum measured along the m^th ray path. It can be expressed as

P_{m} = \sum_{n = 1}^{N} a_{m n} f_{n}, m = 1, 2, 3, ..., M,

where

a_{n m}

is the contribution of the n^th pixel to the m^th x-ray and M is the total number of measurements. In fact,

A

and

P

can be expressed as

A = {(a_{m n})}_{M \times N}

and

P = {[P_{1}, P_{2}, ..., P_{M}]}^{T}

, respectively. Equation (7) can be iteratively solved by the classic ART as

f_{n}^{(k + 1)} = f_{n}^{(k)} + λ \frac{P_{m} - \sum_{n = 1}^{N} a_{m n} f_{n}^{(k)}}{\sum_{n = 1}^{N} a_{m n}^{2}} a_{m n}, k = 0, 1, 2, \dots,

where k is the iteration number,

m = \mod (k, M)

, and

λ \in (0, 2)

is a relaxation factor, which play an important role in the algorithmic convergence and image quality. In our experiments,

λ

is optimized for all the reconstruction methods in the subsection 4.1.

3.2. Total variation minimization with steepest descent search

The CS-based algorithms have a huge impact on the reconstructed images from sparse and locally truncated projections. Because the discrete gradient transform (DGT) of an image is often sparse, its ℓ₁-norm, known as total variation (TV), is introduced to optimize the system Eq. (7)

m i n TV (f), s . t . A f = P

where

TV (f) = \sum_{i = 2}^{I} \sum_{j = 2}^{J} \sqrt{{(f_{i, j} - f_{i - 1, j})}^{2} + {(f_{i, j} - f_{i, j - 1})}^{2}} .

The solution of Eq. (11) can be obtained by an alternative minimization method. It includes two major steps. In the first step, an intermediate image is obtained by incorporating the data fidelity term using the ART, simultaneous algebraic reconstruction technique (SART), order-subset SART (OS-SART) or some other reconstruction algorithms. In this study, we employ the ART method mentioned in 3.1. In the second step, a steepest descent search method is employed to minimize the TV of the intermediately reconstructed image. In this study, similar to [26], we also named this total variation minimization method with the steepest descent search technique as TVM-SD.

3.3. Swinging multi-source PICCS (SM-PICCS) algorithm

In order to obtain high temporal resolution images from the SMCT system, we modify the prior image constrained compressed sensing (PICCS) and apply it to our case. The PICCS algorithm pays much more attention on reconstructing a sparse image from a given highly incomplete data set with the constraint of a high quality prior image. The prior is from the same object, and it can be reconstructed from a complete data by FBP. Assuming we have obtained a high quality prior image $f^{p}$ , it can be utilized to constrain the object image $f$ . As a result, the PICCS algorithm is designed to iteratively solve the following minimization problem:

m i n κ TV (f) + (1 - κ) TV (f - f^{p}), s . t . A f = P

where

κ

(

0 \leq κ \leq 1

) is an empirical parameter which can be determined by the noise level in the prior image

f^{p}

. When the prior image contains less noise,

κ

should be approach to 1. Conversely, if the prior image is highly contaminated,

κ

would trend to 0. In fact,

κ = 0.5

can be used for most of the cases if the prior image is obtained by the conventional FBP algorithm [34].

There are several approaches to solve Eq. (13). In this study, it is solved by adopting a similar method to the TVM-SD. Considering the SMCT system, we modify the PICCS and generate a swinging multi-source PICCS algorithm (named as SM-PICCS). Previously literatures demonstrated that the TVM-SD algorithm has a great potential for limited angle image reconstruction of a multisource interior imaging system. The SM-PICCS algorithm is different from the TVM-SD. The SM-PICCS algorithm can compute a mixed gradient for both the current image $f^{(k)}$ and difference image $f^{(k)} - f^{p}$ .

4. Experimental results

To evaluate the performances of the proposed SMCT scheme and the SM-PICCS algorithm for aperiodic dynamic image reconstruction, the ART, TVM-SD and SM-PICCS algorithms are implemented in Matlab. Extensive numerical simulations and realistic specimen studies are performed on a PC (8.0 GB memory, 3.6 GHz CPU). The initial guess images of TVM-SD and ART are set as zero. The initial guess image of SM-PICCS is set as the prior image.

4.1. Aperiodic dynamic phantom simulation study

To evaluate the performance of the proposed SMCT scheme and the SM-PICCS algorithm, we first use a numerical disk to simulate the generation and disappear of air bubbles in casting cooling. The cylindrical casting phantom consists of 15 circular bubbles, as shown in Fig. 6. To further simulate aperiodic dynamic structures, we assume that there are four linearly varying bubbles during 50 time frames. The radii of four marked bubbles are changed continuously according to the corresponding curves in Fig. 7. The rest small bubbles with different sizes are used to evaluate spatial resolution. We further assume the number of x-ray sources is 7/9 (e.g. $Q = 7$ or $Q = 9$ ) in this architecture. In fact, we mainly consider the following two aspects to choose the number of X-ray sources. On one hand, the number of X-ray sources depends on different imaging requirements. Because the swinging multi-source CT architecture only plays an auxiliary role in the monolithic construction, more specific structures should be determined by specific applications. On the other hand, due to the limitation of physics and FOV, the number of X-ray sources cannot be too large. Meanwhile, to improve the temporal resolution of SMCT, the number of X-ray source cannot be too small. Therefore, we only analyze the SMCT scheme with 7/9 sources. The detailed simulation parameters are specified in Table 1. We can infer from Table 1 that700 views are uniformly sampled in the system of 7 sources, and 720 views are sampled in the system of 9 sources. The sampling time is set as 0.1s in this study. Again, each x-ray source can sample 100 views during FHC or SHC when $Q = 7$ (80 views when) $Q = 9$ .

Fig. 6 Three representative images of the dynamic phantom at different time frames 6, 25, and 45. The four bubbles marked 1-4 are changing and the display window is [0 1000].

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Fig. 7 The original and reconstructed radii of four dynamic bubbles.

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Table 1. Numerical Simulation Parameters

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In order to evaluate the SM-PICCS algorithm for SMCT with 9 sources, three undersampling factors $w = 10, 20, 40$ are chosen to generate data using an analytic method [37] for each x-ray source. This corresponds to 72, 36 and 18 views in each time frame within a full scan, respectively. For the case of 7 sources, we choose $w = 10, 25, 50$ , which results in 70, 28 and 14 views in each time frame within a full scan, respectively. The undersampled views are nonuniformly distributed in each time frame, and the data acquisition is interleaved in the same time frame (Fig. 4). The combination of the interleaved projections yields a complete data set which enables reconstruction of prior image, as shown in Fig. 8. Obviously, the dynamic bubbles have severe artifacts caused by data inconsistency [34].

Fig. 8 The prior image reconstructed from projections with an undersampling factor 50 and 7 x-ray sources.

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To estimate the reconstructed radii of the dynamic bubbles with the SM-PICCS algorithm, the SMCT results with 7 sources/detector pairs and an undersampling factor 50 are shown in Fig. 7. Overall, the estimated dynamic radii are consistent with the original curves. Without loss of generality, representative images are shown in Figs. 9 and 10 at time frame 6 using different algorithms after 300 iterations. All parameters in our TVM-SD algorithm are the same as those in [26]. The SM-PICCS reconstruction parameter $κ$ is set as 0.51. The number of TV and the maximal step for the steepest descent are set to 5 and 0.015, respectively. From Figs. 9 and 10, one can observe severe limited angle artifacts in the images reconstructed by the ART and TVM-SD algorithms. However, the SM-PICCS algorithm can obtain high quality images even from 18 views of 9 sources.

Fig. 9 Representative images reconstructed from noise-free projections acquired with 7 x-ray sources. The undersampling factors are 50, 25 and 10 for the upper, middle and bottom rows, respectively. From the left to right columns, the images are reconstructed by the ART, TVM-SD and SM-PICCS algorithms, respectively. Each reconstructed image consists of 256 × 256 pixels, and the display window is [0 1000].

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Fig. 10 Respective images reconstructed from noise-free projections with 9 x-ray sources. The undersampling factors are 40, 20 and 10 for the upper, middle and bottom rows, respectively. From the left to right columns, the images are reconstructed by the ART, TVM-SD and SM-PICCS algorithms, respectively. Each reconstructed image consists of 256 × 256 pixels, and the display window is [0 1000].

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To characterize the antinoising performance of the SM-PICCS, uniformly distributed Gaussian noise is super imposed to the projections, and $10 %$ of the maximum value of noise-free projections data is selected as the standard variance. This equals to employing a bowtie filter to make the same expected photon number at all the detector elements. The results are displayed in Figs. 11 and 12. From Figs. 11 and 12, one can see that the SM-PICCS algorithm has a strong capability to suppress noise. Because the prior image in SM-PICCS suffers from noise induced by FBP, the anti-noise ability of the SM-PICCS is a little worse than TVM-SD method.

Fig. 11 Same as Fig. 9 but reconstructed from noisy projections.

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Fig. 12 Same as Fig. 10 but reconstructed from noisy projections.

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In order to quantify the image reconstruction accuracy, the root mean square error (RMSE) is employed. In our numerical simulations, the ideal phantom image is selected as reference. However, because the ideal image is unknown in practice, an image reconstructed from a complete data set by FBP or ART can serve as the reference. Noting that the reference image from real data may contain noise and artifacts, it is insufficient for realistic image reconstruction to compare the RMSEs as that in numerical simulations. The quantitative results from numerical simulations are listed in Table 2. We can see that the mechanical architecture with 9 x-ray source-detector pairs is superior to that with 7. This is because the 9 sources/detectors mode equals to simultaneously acquire more views compared with 7 sources/detectors mode. Noting that the greater the source number is, the smaller the field of view is for a fixed distance between detector and origin. For a given number of x-ray sources, if the spin angle of each source is greater during a time frame, the reconstructed image is closer to the ground truth However, a greater rotation angle can reduce the temporal resolution. Thus, we need to compromise the number of source/detector pairs, field of view and temporal resolution.

Table 2. RMSEs of Reconstructed Phantom Images.

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In order to investigate the convergence of the SM-PICCS and other comparative methods, 1000 iterations are used. Without loss of generality, we only consider the convergence curves of all methods for the SMCT with 7 source/pairs and an undersampling factor 10. The RMSEs and data fidelity term vs. iteration number are shown in Fig. 13. Compared with the ART and TVM-SD techniques, the SM-PICCS can converge to a solution quickly with a smaller RMSE. From the enlarged version of the SM-PICCS convergence curve (Fig. 13 (b)), one can see that the RMSE decrease rapidly at first and then it is subsequently increase. There is a turning point. This is because the object function is not the RMSE during the iteration. At the very beginning, although the RMSE decreases to a local minimum then increases, it does not mean the image quality become worse. In fact, the image is blurry at the beginning, and it becomes clearer and clearer with the increase of iteration number.

Fig. 13 The convergence curves of different algorithms within 1000 iterations.

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To study the effect of relaxation factor $λ$ on the reconstructed image quality and algorithmic convergence, we first consider the SMCT scheme with 7 source/detector pairs and an undersampling factor 10. We optimize the relaxation factor by manually adjusting the value of $λ$ and comparing the image quality. The RMSEs of the reconstructed images and algorithmic convergence curves for three methods (i.e. ART, TV and SM-PICCS) with respect to $λ$ are shown in Fig. 14. From Fig. 14, we can infer that a proper relaxation factor not only results in the best image quality but also accelerates the algorithm convergence. To obtain high-quality reconstructed images, $λ$ should be selected as 0.6 for all the algorithms for noise-free projections, and it should be 0.4, 0.6 and 0.4 for noisy reconstruction for the ART, TVM-SD and SM-PICCS algorithms, respectively. Then, for the rest of other configurations, the corresponding relaxation factors are optimized using the same strategy.

Fig. 14 The convergent curves of different algorithms with different relaxation factors. The top row is for noise-free projections by using ART (a), TVM-SD (b) and SM-PICCS (c). The bottom row is the counterpart of top row for noisy projections.

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All the reconstruction methods are implemented under the same conditions. For the SMCT scheme with 7 source/detector pairs and an undersampling factor $w = 50$ , the ART, TVM-SD and SM-PICCS take 0.07, 1.01, 1.96 seconds per iteration, respectively. The TVM-SD optimization needs remarkably longer time than the ART because it is implemented a TV step after each ART iteration. For the SM-PICCS method, the more computational cost is required for the mixed TV optimization, and this time-consuming task can be accelerated by adopting hardware techniques such as GPU.

4.2. Realistic specimen study

To demonstrate the usefulness and feasibility of the proposed SMCT and the associated SM-PICCS algorithm in practical applications, a small specimen is scanned by a micro cone-beam CT system (μCT) including one x-ray source and one flat panel detector in Chongqing University. In this study, the distance from the nano-tube x-ray source to the flat panel detector is 1128.0 mm, the distance between x-ray source and rotation axis is 126.9 mm, the sampling time is 0.5 s, and 1000 projections are uniformly acquired over a full scan circular trajectory with a sampling angle of 0.36°. Because the magnification ratio is so large in this configuration that it can obtain ultra-high spatial resolution. The plat panel detector consists of 1024 × 1024 elements, each of which covers an area of 0.2 × 0.2 mm². The detector offsets are 0.1mm and 3.1 mm along the rotation axis and lateral directions, respectively. Figure 15 demonstrates the 3D specimen structure reconstructed by a classical FDK algorithm. Because of the structural similarity of specimen along the rotation axis direction, most parts of the detailed structures are not changeable except for some small regions. Therefore, we consider the specimen along z-axis as a dynamic 2D object with time. We extract 50 reconstructed image slices close to the middle plane to model a continuously changing test specimen, as shown in Fig. 16. The specimen data set is resampled according to the parameters in Table 3. Similar to the numerical simulations, for a group of given x-ray sources number and undersampling factor, we can further extract and form new sinograms. Because the new sonograms are corrupted by noise during the course of data acquisition on the real microCT, the reconstructed images are noisy, and the dynamic parts are blurred (Fig. 17). In this experiment, we only analyze the results from time frame 4. Figure 16 shows the reconstructed images from original full scan sinograms by the FBP method, which are considered as reference images. It is observed that the reference images are corrupted by noise. This can affect the quantitative analysis results. Figure 17 shows representative prior images from realistically synthesized sinograms with noise and data inconsistency. Similarly, images of time frame 4 are reconstructed by the ART, TVM-SD and SM-PICCS algorithms, and the results are shown in Figs. 18 and 19. The parameters are same as in the aforementioned numerical simulations for both TVM-SD and SM-PICCS algorithms except for the relaxation factor.

Fig. 15 3D structure of a specimen reconstructed by an FDK algorithm. (a), (b) and (c) are the central slices of xy, xz and yz planes, respectively. Each image slice consists of 512 × 512 pixels, and each pixel covers an area of 45.1 × 45.1 μm². The display window is [0 0.1].

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Fig. 16 Representative image slices to mimic different time frames of a dynamic object in a display window is [0 0.1]. Each slice consists of 512 × 512 pixels.

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Table 3. Parameters for Realistic Specimen Simulation

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Fig. 17 Prior images reconstructed from realistically synthesized projections for time frame 4. (a) is a synthesized sinogram for a prior image assuming a system with 7 x-ray sources; (b) is same as (a) but a system with 9 x-ray sources; (c) is reconstructed from (a) with an undersampling factor 50; (d) is the difference image between (c) and the original image reconstructed from (a); (e) is reconstructed from (b) with an undersampling factor 40; and (f) is the difference image between (e) and the original image reconstructed from (b) using FDK. The display window of (c) and (e) is [0 0.1], and the display window of (d) and (f) is [-0.03, 0.03]. The size is 256x256 for all the images.

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Fig. 18 Images reconstructed from 7 x-ray source/detector pairs for time frame 4 with different undersampling factors in a display window [0 0.1]. From the top to bottom rows, the undersampling factors are 50, 25, 10 and 5, respectively. From the left to right columns, the images are reconstructed by the ART, TVM-SD and SM-PICCS algorithms after 300 iterations, respectively. Each image consists of 256 × 256 pixels.

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Fig. 19 Same as Fig. 18 but from 9 x-ray source/detector pairs for time frame 4.

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From the above results, one can see that the SM-PICCS has strong capability to reconstruct high quality images even from severe undersamped data set compared with the ART and TVM-SD. For example, the reconstructed images of SM-PICCS from 28 views (7 source/detectors pairs) or 36 views (9 source/detectors pairs) are significantly better than the counterparts of ART and TVM-SD. To verify the effectiveness of different algorithms, the profiles along the yellow line marked in Fig. 18 are given in Fig. 20. We can infer that the smaller the undersampling factor is, the closer the grayscale value to original value is. To quantify the reconstruction errors, RMSEs are listed in Table 4, which clearly shows the advantages of SM-PICCS. Because the reference images contain noise, the RMSEs in Table 4 are not the optimal measures to compare three methods in reconstructing images from the undersampling data sets.

Fig. 20 Representative profiles along the yellow line marked in Fig. 18. (a), (b), (c) and (d) are profiles for the same undersampling factor and different algorithm.

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Table 4. RMSEs of the Reconstructed Specimen Images (unit: 10⁻³)

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To verify the outstanding performance of the SM-PICCS for aperiodic dynamic imaging, we demonstrate the image reconstruction of the SM-PICCS algorithm for the time frame 0. Compared with the third column of Fig. 18, we can see in Fig. 21 that the result from the SM-PICCS method is the best when the variation across time is minimal. However, if there are significant changes during the sample across time, the quality of prior image would be worse. Because the quality of prior image plays an important role in the PICCS algorithm [33,34], a poor prior image would comprise the performance of the SM-PICCS algorithm.

Fig. 21 Same as the third column of Fig. 18 but for time frame 0.

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5. Conclusions and discussions

To visualize the dynamic behaviors inhibited by the opacity of complex industrial processes, we propose a multiple-sources swinging CT imaging architecture and develop the associated image reconstruction algorithm. Both numerical simulations and realistic specimen experiments are performed to demonstrate the feasibility of the proposed method. Because the testing objects are continuously changeable with different time frames, highly undersampled data sets from the SMCT can lead to severe motion artifacts by the classic algorithms, such as ART and TVM-SD. Based on the PICCS algorithm, we modified PICCS for the SMCT structure. In our method, the CT images from inconsistent projections serve as initial guesses, and an alternating minimization strategy is used to reduce data inconsistency artifacts. Both qualitative and quantitative evaluation results from realistic specimen experiments confirm the SM-PICCS algorithm outperforms the ART and TVM–SD methods.

The PICCS introduces high-quality prior image into the CS framework and further improves the temporal resolution. To investigate the effect of different undersampling factors, we draw plots for the RMSEs v.s. undersampling factors in Fig. 22. From Fig. 22, we can infer that for a given RMSE, we can obtain a maximum undersampling factor. When the critical undersampling factor decreases to 20, the RMSE would be decline slowly in our numerical studies. The critical value is about 10 in the specimen study. In our numerical simulation study, the sampling time of the SMCT architecture is 0.1s. The SMCT scheme with 7 source/detector pairs collect 700 projections for a scan. This implies a temporal resolution of 10.0s. With an undersampling factor 20, the temporal resolution can reach 0.5s. If the undersampling factor is 50, the temporal resolution would further increase to 0.2s. In other words, the higher the temporal is, the lower the quality of reconstructed image is. Thus, we have to compromise the temporal resolution and image quality for the SMCT structure. For the realistic data experiment, the sampling interval is 0.50s for the micro industrial system with a single x-ray source and detector. Because each scanning collects 1000 projections, the temporal resolution of this system can be as worse as 500s. However, if we adopt the SMCT scheme with 9 source/detector pairs and an undersampling factor 10, the temporal resolution can be as high as 5.56s.

Fig. 22 Illustration of the relationship between RMSEs and undersampling factors. (a) is for numerical simulations and (b) is for realistic experiment.

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The SMCT consists of multiple x-ray source/detector pairs, and this structure suffers from complex x-ray scatterings. The x-ray scattering signals can be enhanced both within and across the individual imaging chains. Projections from each detector would be contaminated by the “forward scattering” photons originally from the associated x-ray source and the “cross-scattering” photons generated by the x-ray sources in other imaging chains [38]. Therefore, it is necessary to develop scatter correction techniques to address the scatter artifacts for the proposed SMCT in engineering implementation in the next step. As it is pointed out in [26], for a fixed distance between detector and origin, the greater the number of the source/detector pairs is, the smaller the radius of the FOV to be imaged is. This means there may be only a small ROI will be illuminated by the SMCT. Therefore, we will consider to conduct ROI image reconstruction in the near future. In this way, we can move the ROI to system origin and perform interior reconstruction aided by a scout scan. Indeed, the PICCS algorithm has been extended to interior ROI image reconstruction [32]. It is a challenge for us on how to obtain high quality prior ROI image.

It is well known that the TVM-SD can recover high-quality images from few-view projections [39–42]. However, the problem in this paper is more challenging. Except the issue of few-view projections, we also face the data inconsistency caused by the dynamic process and Gaussian noise. Besides, the related parameters play a key role in reconstructing high-quality images in iterative algorithms, and there are many parameters which affect the final reconstructed images. Although we have done our best to elaborate the parameters in the TVM-SD technique, the reconstructed image using the TVM-SD may be not as good as reported in [26] due to the aforementioned reasons. In the future, we will continue to refine the parameters using the reported strategies in [39–42]. Nevertheless, there are sufficient convincing evidences to prove the SM-PICCS method outperforms the TVM-SD algorithm for dynamic imaging [32,33]. In this study, the prior image is reconstructed by FBP from projections collected during a FHC or SHC (Fig. 5). A data set corrupted by data inconsistency is employed to reconstruct prior initial image. To further improve image quality, we can obtain the prior image by collecting a data set similar to short-scan. Assuming an undersampling factor of 10 from FHC or SHC for 7 source/detector pairs, if we adopt the mode in Fig. 23 to acquire prior image, the temporal resolution of prior image would be improved and it can lead to high-quality images.

Fig. 23 The data distribution of the SMCT system from a scan similar to short-scan.

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The SM-PICCS method mainly focuses on the first order total variation minimization. In most of the industrial applications, the imaging objects are homogenous, and they can be well approximated by the piecewise constant model. As a result, the first order total variation regularizer is the optimal choice to preserve edges. However, if the object consists of complicated regions, such a regularizer will be ineffective. Due to the advanced performance of high order total variation [43,44], we will consider the higher order total variation image reconstruction method in the near future. In this paper, as a preliminary study, we only focus on the feasibility of SMCT and consider the fan-beam geometry. The realistic specimen data set is obtained from a micro cone-beam system with a single x-ray source and detector pair. Because there are changeable local 3D regions within the specimen along the rotation axis, it is reasonable for us to mimic a time dimension using one of the spatial dimensions. Note that a single measurement of SMCT has contributions from multiple “time” steps. The projection data set from SMCT is gained from the multiple “time” steps rather than different slices along rotation axis. Based on the changeable regions within the object, we replace the time dimension with a spatial dimension. While such an approach only work for parallel beam or fan-beam geometry in theory, the circular cone-beam geometry should be a good approximation of 3D parallel-beam geometry when the object is very small compared to the distance from the source to origin. Particularly, this is true for most of the nano-CT and micro-CT scanners.

In conclusion, we propose a multi-source swinging CT architecture and develop the associated SM-PICCS method to study dynamic process of AM and so on. The SMCT scheme can satisfy the requirements for visualizing the dynamic process and the SM-PICCS can reconstruct the aperiodic moving object. Our results show that the SM-PICCS algorithm can effectively remove data-inconsistency artifacts and reconstruct high quality image. This will be extremely useful for aperiodic dynamic imaging in industrial applications.

Funding

National Natural Science Foundation of China (NSFC) (No. 61471070), National Instrumentation Program of China (No. 2013YQ030629); NSF CBET CAREER award (No. 1540898).

Acknowledgments

The authors would like to thank Mr. Lei Ran for helping prepare structure figure of the proposed SMCT system.

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Parameters	$Q = 7$	$Q = 9$
Distance from source to rotation axis h (mm)	640.0	822.0
Distance from detector to object d (mm)	459.2	569.5
Radius of FOV R (mm)	60.0	60.0
Efficient detector array length L (mm)	206.4	203.2
Detector pixel size (mm)	0.8	0.8
Sampling interval of the same source	$π / 350$	$π / 360$

The number of x-ray sources		$Q = 7$			$Q = 7$
Undersampling factors		1	25	50	10	20	40
	ART	227.95	238.00	243.85	211.50	216.97	222.25
Noise-free projections	TVM-SD	213.21	224.25	236.53	193.5	203.08	211.81
	SM-PICCS	55.64	55.79	67.98	54.71	56.84	62.90
	ART	234.08	244.08	248.48	215.59	222.52	228.06
Noisy projections	TVM-SD	214.62	226.35	238.22	197.88	206.70	214.32
	SM-PICCS	80.06	80.09	88.99	77.12	78.90	84.33

Parameters	$Q = 7$	$Q = 9$
Distance from source to rotation axis h (mm)	533.3	520.2
Distance from detector to object d (mm)	328.2	439.1
Radius of FOV R (mm)	20.0	20.0
Efficient detector array length L (mm)	65.2	73.8
Detector pixel size (mm)	0.2	0.2
Sampling interval of the same source	$π / 350$	$π / 360$

Number of x-ray sources		$Q = 7$				$Q = 9$
Undersampling factor		5	10	25	50	5	10	20	40
	ART	3.5	5.4	6.8	7.9	3.0	4.3	5.9	6.8
Time frame 4	TVM-SD	2.4	3.9	6.5	7.2	2.3	3.7	5.0	6.1
	SM-PICCS	2.0	2.0	3.2	4.3	2.0	2.0	2.8	4.4

Parameters	$Q = 7$	$Q = 9$
Distance from source to rotation axis h (mm)	640.0	822.0
Distance from detector to object d (mm)	459.2	569.5
Radius of FOV R (mm)	60.0	60.0
Efficient detector array length L (mm)	206.4	203.2
Detector pixel size (mm)	0.8	0.8
Sampling interval of the same source	$π / 350$	$π / 360$

Swinging multi-source industrial CT systems for aperiodic dynamic imaging

Abstract

1. Introduction

2. SMCT system

2.1. Imaging system

2.2. Data acquisition

2.3. Prior image

3. Reconstruction algorithm

3.1. Algebraic reconstruction technique

3.2. Total variation minimization with steepest descent search

3.3. Swinging multi-source PICCS (SM-PICCS) algorithm

4. Experimental results

4.1. Aperiodic dynamic phantom simulation study

4.2. Realistic specimen study

5. Conclusions and discussions

Funding

Acknowledgments

References and links

Cited By

Figures (23)

Tables (4)

Equations (13)

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