Image gradient L 0 -norm based PICCS for swinging multi-source CT reconstruction

: Dynamic computed tomography (CT) is usually employed to image motion objects, such as beating heart, coronary artery and cerebral perfusion, etc. Recently, to further improve the temporal resolution for aperiodic industrial process imaging, the swinging multi-source CT (SMCT) systems and the corresponding swinging multi-source prior image constrained compressed sensing (SM-PICCS) method were developed. Since the SM-PICCS uses the L 1 -norm of image gradient, the edge structures in the reconstructed images are blurred and motion artifacts are still present. Inspired by the advantages in terms of image edge preservation and fine structure recovering, the L 0 -norm of image gradient is incorporated into the prior image constrained compressed sensing, leading to an L 0 -PICCS algorithm. The experimental results confirm that the L 0 -PICCS outperforms the SM-PICCS in both visual inspection and quantitative analysis.


Introduction
Dynamic computed tomography (CT) has been used to image a changeable object with time in industrial applications [1][2][3][4]. To improve the temporal resolution with high image quality for the dynamic CT system, one key problem is how to obtain sufficient projections in short time. However, limited by the constraints of the mechanical gantries, the scanning speed of the conventional CT can reach no more than one rotation per 200ms [5]. To further improve the scanning, the multi-source CT system is developed, in which multiple source-detector pairs installed on a gantry simultaneously rotate around the imaging object to acquire projections. For example, a novel technique called real time tomography (RTT) system [6] was developed by adopting a circular array of X-ray source rather than the complicated mechanical scanning. The C-arm CT was developed for dynamic imaging to improve the temporal resolution by reducing the scan angle range. Those systems have achieved good performance in medical applications. However, the complicated structure and expensive cost make them difficult to be applied in industrial applications.

SMCT sy
To image the proposed to im and the corres pairs of X-ray concentric be The SMCT s system, it can not changeab source/detecto always chang in poor quali SM-PICCS re The CT im can be approx ystem industrial ape mprove the tem sponding fan-b y source/detec aring structure ystem is drive n obtain Q un ble with time or pair rotates eable with tim ity of reconstr econstruction m maging model ximated as: eriodic process mporal resolutio beam geometry ctor and they a e. The imaging en by a servo niform circular e, the system an angle of 2π e, it is difficult ructed images. method was pro Assuming the ons when eac ever, since the rojections, and cted image qu 1. noise-free proje stem was CT system d integer) ble with a ting table.
a SMCT e object is ch X-ray e object is d it results uality, the ections, it (1) where I and J are the height and width of reconstructed image and their product equals to N .
, i j f represents the pixel value located at the image grid (i, j). Here, the image gradient values of image boundary are set to 0.

L 0 -PICCS model
The image gradient L 0 -norm was proposed to calculate the number of non-zeros in image gradient image. It can be defined as || || (| | | |), where ( ) ϕ ⋅ is a function and it can be read as Compared with the TV, the image gradient L 0 -norm counts the number of non-zeros for gradient image. Consequently, the image edge information and sharp structure can be effectively retained. Thus, the L 0 -norm of image gradient is employed to replace the L 1 -norm in the SM-PICCS algorithm. The reconstruction model of L 0 -PICCS can be formulated as following 0 0 arg min || || (1 ) || ( ) || . .
. Af P u u where 1 t and 2 t represent two error feedback variables respectively. The problem Eq. (10) can be divided into three sub-problems by employing the split-Bregman strategy [29]: Sub-problem 1: Sub-problem 2: Sub-problem 3: Because the mathematical model of Eq. (11a) is a strictly convex problem, the sub-problem 1 can be easily solved. To determine the minimization point of Eq. (11a), the derivative of Eq. (11a) should be equivalent to zero. We have, which can be simplified as Therefore, the ( ) The sub-problem 2 includes an L 0 -norm minimization and it results in non-convex and NP-hard problem. Fortunately, this problem can be solved by utilizing an approximate method [28].

End for
Output: reconstruction result

Experiments
The more X-ray source/detector pairs mounted on SMCT system, the better reconstructed image quality and the higher the temporal resolution that can be obtained [14]. However, a large number of X-ray source/detector pairs can reduce radius of the field of view (FOV). In addition, the large number of source/detector pairs can increase the manufacturing costs of the system. Meanwhile, the scattering effect of X-ray source will be aggravated from different directions, which will degrade the reconstructed image quality. Therefore, the number of source/detector pairs are set as 7 and 5 and the corresponding sampling views are 700 and 500 in the simulation. In the realistic data set experiment, the sampling views are 450 and 448 respectively. Noise-free

M-PICCS algo
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Realistic data set
To demonstrate the usefulness of the L 0 -PICCS algorithm in practical applications, a specimen is scanned by a micro cone-beam CT system with one X-ray source and one flat panel detector (FPD) in Chongqing University. In this study, the tube voltage and current are set as at 60 kV and 200 mAs. The FPD consists of 1024 × 1024 units and each of them covers an area of 0.127 × 0.127 mm 2 . The distance starting from the X-ray source to FPD and object are 502.0 mm and 132.3 mm respectively. 450 projections are uniformly acquired over a full scan range. Figure  6(a) shows the reconstructed 3D structure of the specimen using FDK algorithm and Fig. 6(b) shows the 50 slices around the middle plane. From Fig. 6, it can be seen that the structure of the specimen is similar along rotation axis direction, i.e., most parts of structure have not changed except for some small regions. Since the distance from the source to object far outweigh that between source and object, the circular cone-beam geometry can be treated as a good approximation of 3D parallel-beam geometry for such micro-CT system. In fact, this is true for most of the nano-CT and micro-CT scanners. Hence, the extracted 50 slices of the FPD along the z-axis can be treated as the 2D dynamic object. Some representative slices reconstructed from different FPD rows are shown in Fig. 7.
To imaging the aperiodic dynamic process in industrial applications, we extract 450 and 448 projections for SMCT systems equipped with 5 and 7 source/detector pairs respectively. The undersampling factors are set as 10, 18, 30 for SMCT system with 5 source/detector pairs and result in the number of projections being 45, 25, 15. Similarly, the undersampling factors are set as 8, 16, 32 for SMCT system with 7 source/detector pairs and lead to the number of projections being 56, 28, 14. To clarify the projection extraction, here the undersampling factor is assumed as 10 for SMCT with 5 source/detector pairs. In the case, the interlaced strategy is adopted to extract 45 projections for one row of FPD.  previously ing based problem, prove the rated into c data set ality with e gradient L 1 -norm minimization can lead to a global smoothing by punishing image gradients coefficients. In contrast, the image gradient L 0 -norm minimization penalizes the non-zero number of the small image gradient rather than the amplitude, which can be beneficial for image edge information protection and finer structure recovering. From the results with different undersampling factors in numerical simulations, it can be seen that the reconstructed image quality is decreased with the undersampling factor increased. Especially, the undersampling factor is degraded to 25, the reconstructed images from L 0 -PICCS and SM-PICCS are degraded than those obtained from undersampling factor 10. However, the images reconstructed by the L 0 -PICCS provide a higher image quality than those obtained by the SM-PICCS. As the aforementioned, increasing the undersampling factor will improve the temporal resolution. Thus, L 0 -PICCS is able to acquire better images quality than SM-PICCS when the temporal resolution increased. Again, compared with the SM-PICCS, the proposed L 0 -PICCS method can reconstructed a satisfied image with fewer projections. It indicates the developed method has a potential to reduce the number of the X-ray sources with the same temporal resolution. When the undersampling factor is fixed, the results of L 0 -PICCS from 5 X-ray sources are more accurate than that obtained by SM-PICCS from 7 X-ray sources. The realistic data set experiments demonstrate that L 0 -PICCS can acquire a better image quality than SM-PICCS. The realistic data set experiments demonstrate that if there are changing region rapidly within the object, both L 0 -PICCS and SM-PICCS may fail to reconstruct high quality images. For example, the undersampling factor 32 for SMCT system equipped with 7 X-ray source-detector pairs, the projections are contaminated by data inconsistency and result in image quality are further comprised. Fortunately, the proposed L 0 -PICCS can still achieved better reconstructed image with clear image edge and finer structures.
Although the L 0 -PICCS algorithm has obtained excellent performance, there are still some issues. First, the L 0 -PICCS method contains four parameters, which may become the biggest problem in practical applications. In this study, those parameters are optimized based on RMSE and visual evaluation. Figure 10 further demonstrates the results in terms of the RMSE v.s. four parameters in the case of undersampling factor 10 for SMCT with 5 source/detector pairs. The theoretical analyses and furtherly optimizations are still open problems that need to be fully addressed in our future work [31]. Second, the realistic data set is collected from the micro-CT system equipped with one X-ray source/detector pair. The system ignores the forward and backward X-ray scattering effects. Thus, how to obtain a high quality reconstructed image is a challenge considering scattering effect in practical. Third, the L 0 -PICCS is first proposed for SMCT system in this study. In fact, it can be also applied to cardiac CT (4DCT) [1,32,33], spectral CT [34,35]. Particularly, when the proposed L 0 -PICCS is applied to spectral CT reconstruction, the averaged image can be considered as the prior image.
In summary, based on the advantages of the image gradient L 0 -norm minimization in image edge protection and finer features recovering, it was employed to improve SMCT system temporal resolution by combining the prior image. Considering the image gradient L 0 -norm optimization is a non-convex and NP-hard problem, the split-Bregman strategy was used to solve the L 0 -PICCS model. The experiment results demonstrate the proposed L 0 -PICCS can obtain better reconstructed image quality than other comparisons. This will be extremely meaningful for dynamic CT reconstruction.   Reeder, and algorithm,"