Weighted Total Variation Constrained Reconstruction for Reduction of Metal Artifact in CT Yanbo Zhang, Xuanqin Mou and Hao Yan Abstract–TV constrained reconstruction could obtain perfect results from incomplete data, and has been applied to reduce metal artifact by assuming that the projection contaminated by metal is missing. In TV constrained reconstruction, the selection of a proper step parameter for TV minimization procedure is a key point. However, this parameter is usually selected empirically, and it is a constant for all pixels in the whole image domain, regardless of the difference of missing projection quantity at different pixels. By analyzing the relationship between the missing projections and pixels position, a Weighted Total Variation (WTV) constrained reconstruction method is proposed to reduce metal artifact in this paper. For WTV constrained method, the parameters are no longer the same, but vary over image domain as the introduced information miss rate. The simulation results show that the proposed method is more effective than current TV constraint to reduce metal artifact. Moreover, WTV constrained method is extended to other incomplete projection problems.
I. INTRODUCTION artifact is a major problem in medical computed Mtomography (CT). It is mainly caused by beam hardening, ETAL
noise, the non-linear partial volume effect, and scatter [1]. Various algorithms have been proposed to reduce metal artifact during last three decades. A class of algorithms assume that the projection contaminated by metal is missing and the image could be reconstructed from incomplete data iteratively [2]. Iterative reconstruction constrained by total variation (TV) could obtain perfect results from incomplete data [3], and has been applied to reduce metal artifact [4, 5]. In TV constrained reconstruction, the selection of a proper step parameter for TV minimization procedure is a key step. However, this parameter is usually selected empirically according to different imaged object, and it is hard to pick up an optimum value directly. Moreover, the TV parameter is a constant for all pixels in the whole image domain, regardless of the difference of missing projection quantity at different pixels. Distribution of artifacts varies at different pixels. By analyzing the relationship between the missing projections and pixel position, a Weighted Total Variation (WTV) constrained reconstruction method is proposed to reduce metal artifact. In this method, the parameters are no longer the same, but vary
over different pixels in image domain, and could be calculated accurately by backprojection procedure. II. WEIGHTED TOTAL VARIATION CONSTRAINED RECONSTRUCTION As shown in Fig. 1, the points next to the metal region have fewer available projection data and the points far from the metal region have relative more available projection data. Thus, when reconstructing an image by iterative type algorithms, the pixels near the metal region are less accurate and should have large TV parameters to reduce artifact, and the pixels far from the metal region are more accurate and should have small TV parameters to preserve the detail information of an image. Therefore, it is rational to adopt nonuniform parameters for pixels. A. Analysis of the Information Miss Rate (IMR) A fan-beam scanning geometry is shown in Fig. 1. It can be seen that the X-ray SP will penetrate through the metal region when the X-ray source S is on the dashed arc AB or CD , therefore, the projection information to reconstruct the point P ( x, y ) is missing. An Information Miss Rate (IMR) at P could be defined as follows: IMR( P ) =
AB + CD 2π
(1) It can be noted that it is hard to calculate the IMR at each pixel points according to the definition. In order to accomplish the calculation efficiently, a binary projection sinogram pˆ is introduced. pˆ is computed from projection p , the projections contaminated by metal (projections in the sector region SGH) are picked up and set to one, and other projections are set to zero. Actually, pˆ indicates that whether a projection penetrates metal region, namely: ⎪⎧1, pˆ ( P,φ ) = ⎨ ⎪⎩0,
for
S ∈ AB ∪ CD
for
S ∈ BC ∪ DA
(2)
where φ is projection view angle. Therefore, the formula (1) can be further rewritten as follows: D 1 ⎡ B pˆ (( x, y ),φ )dφ + ∫ pˆ (( x, y ),φ )dφ ⎤ ⎥⎦ C 2π ⎢⎣ ∫A (3) 1 2π pˆ (( x, y ),φ )dφ = ∫ 2π 0
IMR( x, y ) = Manuscript received May 10, 2010. This work was supported in part by National Natural Science Foundation of China (NSFC) under Grant No. 60551003 and No.60472004, and the Doctoral Fund of Ministry of Education of China under Grant No.20060698040. Yanbo Zhang, Xuanqin Mou (corresponding author) and Hao Yan are with the Institute of Image Processing & Pattern Recognition, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China. (e-mail:
[email protected],
[email protected],
[email protected]).
978-1-4244-9105-6/10/$26.00 ©2010 IEEE
Formula (3) indicates that the IMR for any image pixels can be calculated by backprojection procedure from the binary projection pˆ . Therefore, IMR could be obtained easily.
2630
in this paper. The iteration number is 50 for reconstruction, and in every iteration, there are 3 ART operations and 50 TV minimizing iterations. 1 0.9
Metals
0.8 0.7 0.6 0.5 0.4 0.3 0.2
Fig. 1. Illustration of a fan-beam scanning geometry consisting of an X-
B. Weight Function As analyzed above, the TV parameters should be set according to the IMR. Therefore a weight function with respect to IMR is introduced. According to the property of TV parameters, weight function is a monotonically nondecreasing function with respect to IMR, and function values must be no less than zero. Here, we adopt a linear weight function W , which mapping the maximum and minimum of IMR to two constants a and b respectively, as shown in the following: W ( x, y ) =
(a − b) IMR ( x, y ) + (b max{IMR} − a min{IMR}) , a>b>0 max{IMR} − min{IMR}
(4) where a is selected as 10 times of b in this paper. C. Implementation The implementation for WTV constrained reconstruction consists of two phases: weight values calculation and iterative reconstruction. For the former phase, the key point is to calculate pˆ , which can be obtained by two steps: a) to confirm the compact support of metal from projection, b) reprojecting and binarization. Whereafter, weight function values can be computed according to formula (3) and (4). For the later phase, WTV constrained iterative reconstruction is with the same framework in literature [3] and the constant TV parameter is replaced by WTV parameters.
0.1
Fig. 2. The phantom with two metals (left) and the corresponding IMR (right).
In the simulation, three WTV constrained reconstructions with TV parameters range [ b a ] (selected as [0.01 0.1], [0.1 1] and [1 10] respectively) are tested in our work. And Sidky’s TV constrained reconstruction method [3] is implemented as contrast experiment. To compare fairly, TV constrained reconstructions are taken for 19 times to search the optimum result by TV parameters traversal ranging from 0.005 to 7. Here, Root Mean Square Error (RMSE) is selected as assessment index. Fig. 3 gives the reconstruction RMSE comparison between WTV and TV with various parameters. Since parameters of WTV are no longer a constant but vary in a range, each WTV result is shown as a solid line in Fig. 3. It can be seen that WTV has similar RMSE in both TV parameter ranges [0.01 0.1] and [1 10], compared to the smallest ones of TV constraint in these ranges respectively; moreover, WTV has much smaller RMSE in [0.1 1] than TV with optimum parameter 0.5. Thus, WTV constraint has better RMSE performance than TV constraint. -1
10
-2
10
-3
10 RMSE
ray source S and the source trajectory ABCDA . The elliptical shadow region represents the object support with a point P( x, y ) , and the small dark region Rm is the support of metal region. Both lines AC and BD are tangent to Rm and intersect at P . The sector SEF is covered by X-rays, and the sector SGH denotes that X-rays penetrate through the metal region.
-4
10
-5
10
-6
10
III. SIMULATION RESULTS
-3
10
We adopt an equiangular fan-beam CT which is similar with in literature [1, 3], the number of projection views is 360, the number of detector cells is 850, the maximum fan-angle is 45 , and the distance from X-ray source to rotation centre is 500mm. The reconstructed image contains 256 × 256 pixels and the size of each pixel is 1mm × 1mm . A modified low contrast Shepp-Logan phantom with additional low contrast dots array is adopted in the simulation. The phantom has two metals as shown in the left of Fig. 2. And its corresponding IMR is shown in the right of Fig. 2. TV parameters present the same distribution with IMR due to the linear function adopted
-2
10
-1
10 TV Parameter
0
10
1
10
Fig. 3. Image performance reconstructed from noise-free projection datasets. Red circles denote TV constrained results and blue solid lines denote WTV constrained results, where the horizontal and vertical axes represent the TV parameter and RMSE.
Convergence curves of the TV and WTV constraint algorithms with various parameters are shown in Fig. 4. It can be seen that all of curves basically converge after 50 iterations. And WTV with parameter range [0.001 0.1] and [0.1 1] have trend to get smaller RMSE with more iterations.
2631
0
10
-1
10
-2
RMSE
10
-3
10
-4
10
TV parameter 0.1 TV parameter 0.5 TV parameter 1 TV parameter 5 WTV parameter [0.01 0.1] WTV parameter [0.1 1] WTV parameter [1 10]
-5
10
-6
10
0
5
10
15
20 25 30 Iteration number
35
40
45
50
Fig. 4. Convergence curves of the TV and WTV constraint with parameter 0.1, 0.5, 1, 5 for TV and parameter values range [0.01 0.1], [0.1 1], [1 10] for WTV, respectively, where the horizontal and vertical axes represent the iteration number index and corresponding RMSE, respectively.
The WTV and TV with various parameters constrained reconstructed results are shown in Fig. 5. TV constraint with a too small parameter cannot reduce metal artifact well around the metal regions (see the 1st row of Fig. 5); while a very large parameter will bring great error to the whole image (see the 2nd row of Fig. 5). Therefore, to reconstruct an image with less metal artifact, selecting a proper parameter is of paramount importance for TV constraint. We have found the optimum parameter 0.5 for TV constraint from Fig. 3, the corresponding reconstructed image in the 3rd row of Fig. 5. It can be seen that streak artifacts are reduced greatly, but still visible in the difference image. The parameters range of WTV is [0.1 1], and the mean value of WTV parameters is 0.1429. It can be seen from the difference image that a lot of strong streaks exist for TV with 0.1429 in the 4th row of Fig. 5; for WTV, the metal artifact is further reduced and hardly to be observed in the difference image in the first row of Fig. 5. It can be seen that the WTV constrained reconstructed result is more accurate than TV constrained reconstruction with the optimum parameter. In addition, WTV has better application prospect, since it is much easier to confirm a proper parameter values range than to find the optimum parameter.
Fig. 5. Simulation results for the metal artifact reduction from noise-free projection datasets. The left column is reconstructed images with display window [0.9 1.1] and the right column is the difference between the left column and the phantom with display window [-0.002 0.002]. The 1st and 2nd rows are TV with 0.05 and 5, respectively; the 3rd row is reconstructed results constrained by TV with 0.5, which is the optimum value in this simulation; the 5th row is reconstructed results constrained by WTV with parameter range [0.1 1], while the 4th row is reconstructed results constrained by TV with 0.1429, which is the mean value of WTV parameters in the 5th row.
To evaluate the noise characteristics and demonstrate the stability of TV and WTV methods, the signal-dependent Gaussian noise is add to the projection [6] since logarithm transformed projection data of low dose CT follow approximately a Gaussian distribution with a nonlinear dependence between variance and projection mean [7]. Fig. 6 gives the TV and WTV constrained reconstructed images from noisy projection datasets. It can be observed that there is less low frequency error for WTV and TV with the parameter 0.1429 than that for TV with the optimum parameter 0.5 in the difference images. While the RMSE of WTV, TV with parameter 0.1429 and 0.5 are 0.008134, 0.008297 and 0.008278, respectively. TV with 0.1429 has greatest RMSE while WTV has smallest one in this group of reconstructions, this is understandable because TV with small parameter cannot suppress noise well while WTV has a good trade-off between noise and metal caused errors. Thus it can be concluded from the results that WTV method is robust against noise for metal artifact reduction in CT.
2632
can be calculated from the binary projection by backprojection as shown in the right one of Fig. 7. There are various information missing types for incomplete CT projection, and to calculate IMR from binary missing projection is only one of them. IMR may be obtained according to the angle of available projection for a point to limited-angle problem. An illustration of 90 degree angle scanning is shown in the left one of Fig. 8. The X-ray source S only moves along the trajectory AB . Because the X-ray beam always cover the entire object (see the sector SEF in Fig. 8), the available projection number is same for arbitrary point in the object. However, the field angle of a point P to the two ends of X-ray source trajectory, namely ∠APB in Fig. 8, varies at different positions. In extreme cases for example, if ∠APB is equal to or greater than 180° , there is no information lost to this point, and the pixel value of P can be reconstructed accurately and analytically [8]; while if ∠APB is approximate to zero, the useful information for reconstructing the point P is similar to that of a single projection. So to evaluate lost information rate using missing projection number is not proper any more, and it is reasonable to use the angle ∠APB to calculate IMR as shown in the following:
Fig. 6. Counterpart of top three rows in Fig. 5 in the case of noisy projection datasets with display window [0.9 1.1] for left column and [-0.1 0.1] for right column.
⎧180° − ∠APB , if ∠APB < 180° ⎪ IMR( P ) = ⎨ 180° ⎪0, others ⎩
(5)
According to the formula (5), the corresponding IMR is finally obtained as shown in the left one of Fig. 8.
IV. DISCUSSIONS WTV constrained method can be extended and applied to various incomplete projection cases, in which the key point is to get the IMR according to CT scanning scheme. In this section, we apply WTV constrained algorithm to bad detector bins and limited-angle problems, and the corresponding IMRs are given. The WTV parameter can be obtained easily using the weight function proposed in section II.B, whereas the reconstruction in these incomplete projection cases is beyond the scope of this work and omitted here. 1
0.8
50
0.6 0.55 0.5 0.45 0.4 0.35 0.3
Fig. 8. Illustration of limited-angle problem (left) and the corresponding IMR (right). The X-ray source S can only move on the trajectory AB with a 90 degree central angle, and sector SEF is covered by X-rays. The available X-ray SP for a point P is limited in the angle ∠APB .
View
100 0.6
150 200
0.4
250 300
0.2
350 200
400 Bin
600
In the near future, we will further investigate WTV constrained reconstruction for other incomplete projection problems. Moreover, the impact of weight function selection on reconstruction performance will be studied.
800 0
Fig. 7. Projection sinogram with 10 bad detector bins (left) and the corresponding IMR (right).
Fig. 7 gives a WTV solution for bad detector bins problem. The No. 271 to No.280 detector bins are bad so the corresponding projection datasets are missing as shown in the projection sinogram in the left one of Fig. 7. Similar to metal caused projection missing issue, the missing projection is set to one and rest projection is set to zero. Whereafter, the IMR
V. CONCLUSIONS A Weighted Total Variation (WTV) constrained reconstruction method is proposed to reduce metal artifact in this paper. For WTV constrained method, the TV parameters vary over image domain as the introduced Information Miss Rate (IMR). The simulation results show that the proposed method is more effective than current TV constraint to reduce metal artifact. Furthermore, WTV constraint could be applied to other incomplete projection problems.
2633
REFERENCES [1] [2] [3] [4] [5] [6]
[7]
[8]
B. De Man, J. Nuyts, P. Dupont et al., “Metal streak artifacts in X-ray computed tomography: A simulation study,” IEEE Transactions on Nuclear Science, vol. 46, no. 3, pp. 691-696, 1999. G. Wang, D. L. Snyder, J. A. O’Sullivan et al., “Iterative deblurring for CT metal artifact reduction,” IEEE Transactions on Medical Imaging, vol. 15, no. 5, pp. 657-664, 1996. E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology, vol. 14, no. 2, pp. 119-139, 2006. X. Zhang, J. Wang, and L. Xing, "Metal artifact reduction in computed tomography by constrained optimization," Proc. of SPIE, pp. 76221T, 2010. J. Choi, M. Kim, W. Seong et al., “Compressed sensing metal artifact removal in dental CT,” Proc.of ISBI, pp. 334-337, 2009. Shaojie Tang, Xuanqin Mou, Qiong Xu et al., “Noise reduction by projection direction dependent diffusion for low-dose x-ray cone-beam CT,” 10th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp.381-384, Beijing, PRC, Sep. 2009. J. Wang, T. Li, H. Lu et al., “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose Xray computed tomography,” IEEE Transactions on Medical Imaging, vol. 25, no. 10, pp. 1272-1283, 2006. F. Noo, M. Defrise, R. Clackdoyle et al., “Image reconstruction from fan-beam projections on less than a short scan,” Physics in Medicine and Biology, vol. 47, pp. 2525, 2002.
2634
