Mathematical Methods in Biomedical Imaging 2014View this Special Issue
A CT Reconstruction Algorithm Based on L1/2 Regularization
Computed tomography (CT) reconstruction with low radiation dose is a significant research point in current medical CT field. Compressed sensing has shown great potential reconstruct high-quality CT images from few-view or sparse-view data. In this paper, we use the sparser L1/2 regularization operator to replace the traditional L1 regularization and combine the Split Bregman method to reconstruct CT images, which has good unbiasedness and can accelerate iterative convergence. In the reconstruction experiments with simulation and real projection data, we analyze the quality of reconstructed images using different reconstruction methods in different projection angles and iteration numbers. Compared with algebraic reconstruction technique (ART) and total variance (TV) based approaches, the proposed reconstruction algorithm can not only get better images with higher quality from few-view data but also need less iteration numbers.
Since computed tomography (CT) technique was born in 1973, CT has been widely applied in medical diagnose, industrial nondestructive detection, and so forth . In medical CT field, CT reconstruction with low radiation dose is a significant research problem, which needs the reconstruction of high-quality CT images from few-view or sparse-view data. Recently, compressed sensing (CS)  theory has been applied in CT image reconstruction; it is possible to reconstruct high-quality images from few-view data under the frame of CS. In CS theory, sparse signal is often regularized with regularization; L0 regularization is the sparest and most ideal regularization norm ( regularization denotes the number of nonzero signal elements). However, L0 regularization is susceptible to noise interference and it is difficult to solve equations, and regularization is usually used as the regularization operator. Theoretically, if regularization is closer to regularization could get higher-quality CT images in CT reconstruction.
Recently, Xu et al. proposed regularization  and a soft thresholding algorithm , and Xu and Wang proposed a hybrid soft thresholding algorithm . regularization has some theoretical properties, such as unbiasedness, sparse regularization, and Oracle. It is easier to produce sparser solution compared with the currentregular operator used in CT reconstruction. Theoretically, CT reconstruction method based on regularization will reconstruct higher-quality CT images with few-view data.
On solving regularization problem, Goldstein and Osher proposed Split Bregman method  which is derived from Bregman iterative algorithm . Split Bregman method uses an intermediate variable to split regularization and regularization into two equations; regularization equation can be solved by gradient descent method, and regularization equation can be solved by thresholding algorithm. Split Bregman method can accelerate iterative convergence and produce better results. Based on Split Bregman method, Vandeghinste et al. proposed Split Bregman-based sparse-view CT reconstruction  and iterative CT reconstruction using shearlet-based regularization . Chu et al. proposed multienergy CT reconstruction based on low rank and sparsity with the Split Bregman method (MLRSS) . Chang et al. proposed a few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction .
In this paper, we propose a CT reconstruction algorithm based on regularization and Split Bregman method. In the following section, regularization, Split Bregman method, and the proposed algorithm will be introduced. In the third section, we will use the proposed algorithm to analyze numerical phantom and real projection data. In the last section, we will discuss relevant issues and conclude the paper.
2. Materials and Methods
2.1. L1/2 Regularization
Generally, CT reconstruction algorithm can be divided into analytic reconstruction algorithm and iterative reconstruction algorithm; the current typical analytic reconstruction algorithm is filter back-projection (FBP), and iterative reconstruction algorithm contains algebraic reconstruction technique (ART) , simultaneous algebraic reconstruction technique (SART) , expectation-maximization (EM) , and so forth. Mathematical model  of CT image reconstruction can be expressed as where is the projection matrix, is the projection data, and is the reconstruction image.
For sparse-view data, it is difficult to reconstruct high-quality images using the conventional CT image reconstruction algorithms, especially for analytic reconstruction algorithms which require high completeness of data. Meanwhile, there are also some artifacts in the reconstruction images using the conventional iterative reconstruction algorithms. In 2006, Donoho put forward the compressed sensing (CS) theory ; its main idea is that most of signals are sparse in the proper orthogonal transform domain, which means most of signal transformation coefficients are close to zero or equal to zero in orthogonal transformation, such as gradient transformation  and shearlet transformation . In CS theory, an image can be reconstructed from a rather limited amount of data as long as an underlying image can be sparsely represented in an appropriate domain and determined from these data. CT sampling signal is a typical sparse signal; the x-ray attenuation coefficients of some regions of tested objects (i.e., human body) are similar or equal. Thus, CT reconstruction approaches based on compressed sensing can reconstruct high-quality CT images from sparse-view data.
CT reconstruction problem can be converted to a constrained optimization problem where is the regularization function, usually denoted by norm of wavelet, gradient, and so forth. In order to simplify (2), we can use the penalty function method in the optimization method to convert it into an unconstraint optimization problem where is the weight coefficient, can be denoted by total variation (TV) which is a research hotspot in current CT research field. TV method has been widely used in sparse-view and limited angle CT reconstruction .
In compressed sensing theory, norm is the most ideal regularization norm, but it is difficult to solve equations with norm, and regularization is easily interfered by noise in CT reconstruction, so norm is commonly replaced by norm. Theoretically, using a regularization norm which is closer to norm will reconstruct higher-quality CT images. The definitions of , , and regularization norm are where and denotes the nonzero elements number in matrix .
As shown in Figure 1, regularization norm is closer to L0 regularization norm than L1 regularization norm.
Xu et al. proposed a regularization fast algorithm ; it can be expressed as where is the regularization coefficient.
The definition of operator is
Then the optimal solution can be represented as where the operator is
Please see  for more proof details.
2.2. Split Bregman Method
In order to solve (3), Goldstein and Osher proposed Split Bregman method , using an intermediate variable to split L1 regularization and regularization into two equations; L2 regularization equation can be solved by gradient descent method and regularization equation can be solved by thresholding algorithm. Then the unconstrained optimal problem of (3) can be converted into where is the sparse transform and the common used sparse transform contains gradient, wavelet, shearlet, and so forth.
Equation (13) can split into two equations:
There are several advantages of Split Bregman method. Firstly, Split Bregman method can accelerate iterative convergence and calculate better results. Secondly, Split Bregman method can be widely used in CT reconstruction; it can not only solve regularization problem but also solve other regularization problems.
2.3. CT Reconstruction Algorithm Based on L1/2 Regularization
According to aforementioned methods, we propose a CT reconstruction algorithm based on regularization, where norm is used as the regularization norm and gradient as the sparse conversion; then (11) can be expressed as
Combine with Split Bregman method to solve (17) as follows.
Step 1. One has
Step 2. One has
Equation (21) is derivable and derivation of regularization as follows:
To derivate (21), we have
Then where represents iteration numbers and parameter can be acquired by the following equation:
3. Experimental Study
3.1. Numerical Simulation
In this section, we study the ART algorithm, TV based ART algorithm (ART-TV), and regularization based Split Bregman method (SpBr-) and analyze the reconstructed images. In this paper, we test Shepp-Logan phantom as shown in Figure 2, and the size of phantom image is 256 × 256. We assume that the CT system was viewed as a typical parallel-beam geometry, and the scanning range was from 0° to 180° with a angular increment; projection angles can be indicated as where is the number of projection angles.
We will compare the reconstruction results from noise-free and noise data and projection numbers . In the simulation, we add 0.01% Gaussian noise to noise-free projection data, and iteration number for every reconstruction algorithm is 50; the reconstruction results are shown in Figure 3.
From Figures 3 and 4, we can see that the reconstructed images using ART and ART-TV methods contain a lot of noise and artifacts from noise-free and noise data, while the reconstructed images using SpBr- method include less noise and artifacts and have clearer edges.
In order to evaluate the quality of reconstructed images, we use root mean square errors (RMSE) as the evaluation index. The definition of RMSE is where and are the reconstructed image and original image, respectively, and the image size is .
As shown in Table 1, the RMSE of reconstructed image using SpBr- method is much smaller than reconstructed images using ART and ART-TV methods, which means SpBr- can reconstruct higher-quality images.
As shown in Figure 5, the SpBr- algorithm can reconstruct high-quality images at less iteration numbers, which indicates that SpBr- algorithm can accelerate iterative convergence. From Figure 6, we can see that SpBr- algorithm can reconstruct high-quality images at less projection numbers.
3.2. Real Data Study
In this section, we reconstruct oral images using three different algorithms with real projection data, where projection numbers are 90 and iteration numbers are 50 while the original projection numbers are 360. And as shown in Figure 7(a), we evaluate three images with reconstructed image using ART with original projection data as a standard image. The reconstructed images and RMSE of three reconstructed images are shown in Figure 7 and Table 2.
As shown in Figure 7, the reconstructed images using ART and ART-TV method have more noise and artifacts, while the reconstructed image using SpBr- method has less noise and artifacts. And from Table 2, the RMSE of reconstructed image using SpBr- method is smaller than that of reconstructed images using ART and ART-TV methods, which means SpBr- can reconstruct higher-quality images with clearer details. From the comparison of reconstructed images using different algorithms, we can see that SpBr- can reconstruct high-quality images at less projection angles and less iteration numbers.
4. Discussions and Conclusion
There are several issues worth further discussion in the reconstruction study. Firstly, the thresholding algorithm was not applied to solve the regularization problem. There are two reasons. First, if the thresholding algorithm is applied to solve the regularization problem, the edge and details of reconstructed images will not be clear, while the SpBr- without thresholding algorithm reconstructs images with clear edges. Then, if the threshold value is settled to solve regularization problem, the reconstruction speed will be reduced a lot. Secondly, there are some artifacts in the reconstructed image using SpBr-method from noise data. The reason is that regularization is closer to regularization, and the ability of denoising of regularization is bad; therefore the ability of denoising of regularization is not good enough. Thirdly, in the reconstruction from real projection data, the reconstructed image using ART-TV method lost some details, while details of reconstructed image using SpBr- method are much clearer.
In the further research, we will try to use SpBr- algorithm in interior CT and study the region of interest (ROI) reconstruction, which will reduce radiation dose as much as possible.
In conclusion, we proposed a CT reconstruction algorithm based on regularization; the reconstructed results demonstrate that SpBr- can reconstruct high-quality images from few-view data. In the urgent demand of radiation reduction, SpBr- algorithm will have great potential in clinical application.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the National Natural Science Foundation of China Grants (61201346 and 61171157) and the Fundamental Research Funds for the Central Universities (106112013CDJZR120020). The authors would like to thank real projection data provider Dr. Zhichao Li from the Third Military Medical University, Chongqing, China.
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