A Curvelet based Sinogram Correction Method for Metal Artifact Reduction

In the X-ray computed tomography (CT) imaging, metal artifact occurs often due to the non linear attenuation effect of the metallic subject with respect to the photon energy. Especially high attenuated material, such as plumbum (Pb.) can easily distort the sinogram due to the few remaining of the information on the metal traces. The remedy of such a circumstance, an image inpainting algorithm is useful to ﬁll the missing information in the sinogram. Most metal artifact reduction (MAR) algorithm based on inpainting consist of the ﬁlling the missing values using only informations around the matal traces. Without the consideration of the continuity of the sinogram, the inpainting method can generate addition artifact as a side effect. In this paper, we propose a curvelet based inpainting method to promote the continuity of the sinogram, induced by the rotation of the X-ray scan system. Numerical simulation and phantom experiment are provided to support the our assertion.


Introduction
CT images with metallic objects suffer from various artifact, for example, beam hardening, scattering, and photon starvation.There has been a lot of methods and studies for metal artifact reduction over the last three decades, which can be roughly classified into two approaches: metal-induced beam hardening correction (BHC) method and inpainting based method.BHC method aims to correct the non-linear relation between the attenuation of the metallic subject and the photon energy.For example, [3] have proposed the correction formulation for the beam hardening artifact, using rigorous mathematical observation.However, high attenuated subject, such as plumbum (Pb.) causes a severe photon starvation artifact, and such a circumstance, the suggested correction formulation in [3] is hard to applied to image restoration.Inpainting methods treats the metal trace in sinogram as missing data and estimate it from the neighboring pixels in sinogram.Linear and polynomial interpolations [4,5], total variation inpainting [6], and curvature driven inpainting algorithm [7] are proposed for the sinogram inpainting.Those methods are, unfortunately, used only local information near the distorted metal traces.Therefore, it does not guarantee the continuation of the sinogram coming from the unique property of the rotation measurement in X-ray CT systems.There was an attempt to use the information that is distributed in entire sinogram [8].The authors observed that a single pixel information in the image domain is propagated along the sinusoid curve in the projection domain.To manipulate the pixel information along the sinusoid curves, multi-scale approach were used but it could not provide the continuity from the rotation of X-ray CT scan system.Moreover, the suggested algorithm has to solve the ill-posed linear system, therefore, the reconstructed image quality strongly depends on the regularization constraint.In [9], the sinusoid-liked curve based inpainting algorithm were suggested to suppress the streaking artifact coming from sparse sampling and detector gaps.The authors provided an algorithm that the approximation of the sinusoid curve and the estimation of a eigenvector-guided interpolation to preserve the sinogram texture continuity.The algorithm, however, had some complicated due to the consistance of several steps.In this paper, we propose a curvelet based inpainting method to handle the continuity of the sinogram along the sinusoid curves by rotating of the X-ray CT scan system.Under the observation of the sinogram as the bundle of sinusoid curves, the proposed algorithm is a minimization problem with the sparsity of the coefficient when curvelet transforms are applied in the sinogram.We provide the numerical simulation and phantom experiment to support the our assertion.

Methods
The wavelet is commonly used method to represent the image with small number of the coefficients generated by the multi-scale decomposition, however, it deteriorates the edge information when we use in the image restoration problem [10,11].To overcome such a disadvantage, the curvelet is motivated by the mathematical representation of lines or edges in the mage processing [12].Consider the generation of the sinogram.It is easily observed that single pixel in the image domain traces as the shape of sinusoid curve in the projection domain, that is, the sinogram can be the composition of sinusoid curves.Therefore, we may have the sparsity property of the coefficient efficiently when a sinogram is represented by a curve-shaped basis.Under the observation, we propose the minimization problem to promote the l 1 -sparsity of the coefficient of the curvelet transformed sinogram.min 1 More info about this article: http://www.ndt.net/?id=23715 9th Conference on Industrial Computed Tomography, Padova, Italy (iCT 2019) where C is the curvelet transform, u is the recovered data, u 0 is the metal contaminated data, Ω is the projection domain, and M is the metal trace.Since the equation ( 1) is not easy to solve, we adopt the augmented Lagrangian method [13] by taking the augmented variable v replacing of C u.
where α is a positive real number and , is the inner product.Since the analytic solution is not obtained easily, we solve the equation 2 with an iterative manner; See Algorithm 1.

Algorithm 1 Curvelet based Sinogram Inpainting
Initialization: Let u 0 be the contaminated image.Set the initial values in the metal region of u 0 and p 0 = 0. 1: while until convergence with respect to the given tolerance do Apply u k = u 0 in the Ω \ M.

4:
Update the Lagrange multiplier p k such that 5: end while 3 Experiments In this section, we provide the numerical simulation and phantom experiment.First, we simulate the X-ray projection of the circular object containing a metallic circular inclusion with high attenuation.Figure 1  Second, we perform the experiment for the phantom containing three metallic inserts.We use the X-ray CT scan system with 450 KV in DUKIN Co. Ltd., Korea.In this study, we only consider the mid-plane in our cone-beam CT scanner.The detector dimension is 3072 × 3072 with pixel size 0.099 × 0.099 mm 2 .Total number of projection views are 720 for 360 degree rotation.
The experimental phantom is circular shaped cylinder with several sized holes.In this phantom, we put three plumbum inserts to generate severe photon starvation artifacts.In Figure 2, (a) shows the sinogram contaminated by metallic objects.(b) shows the result applying the inpainting method using linear interpolation.(c) is the result applying the proposed method.Comparing the linear interpolation based algorithm, the proposed method completes the missing informations in the sinogram, continuously.Cross section profiles of sinograms help to explain our assertion.(d), (e) and (f) in Figure 2 are cross section profiles marking with green, red, and blue lines on Figure 2-(a).We observe that the proposed algorithm can fill the inpainting region with smooth curves.2, respectively.Comparing (b) and (c), we observe that the reconstruction image using curvelet inpainting algorithm reduce the streaking artifact rather than linear inpainting one.We provide cross section profiles along the lines indicated in Figure 3-(a).The contrast of the reconstructed image are improved when we use the curvelet based inpainting algorithm in (d).In Figure 3 (e) and (f), we see that there exist the curving artifact in the metal region, generated by the linear interpolation inpainting as a side effect.On the contrary, the proposed algorithm can reduce the curving artifact effectively.

Conclusion
We propose the curvelet based inpainting algorithm for metal artifact reduction to promote the continuity of the sinogram.Without considering the continuity of the sinogram, it may produce additional artifact as a side effect.To handle the continuity of the sinogram as considering the bundle of the sinusoid curves, we adopt the minimization problem to promote the l 1 -sparsity of the coefficient of the curvelet transform.In the phantom experiment, the reconstructed image obtained from the linear inpainting algorithm includes additional streaking and curve artifacts that do not appear in the original image.The proposed algorithm, as expected, can improve the reconstructed image quality without additional artifact by considering the continuity of the sinogram.However, the proposed iterative method is computationally expansive.Therefore, future works will be focused on the acceleration algorithm to solve the proposed algorithm in the practical sense.

Figure 1 :
Figure 1: (a) sinogram with metal.(b) inpainting result of (a) using the linear interpolation.(c) inpainting result of (a) using the proposed method.

Figure 2 :
Figure 2: (a) sinogram containing metallic objects.(b) inpainting result of (a) using the linear interpolation.(c) inpainting result of (a) using the proposed method.(d) is the cross section profile along the green line.(e) is the cross section profile along the red line.(f) is the cross section profile along the blue line.

Figure 3 :
Figure 3: Reconstructed image using sinograms in Figure 2. (a) is the cross section profile along the green line.(b) is the cross section profile along the red line.(c) is the cross section profile along the blue line.