_{1/2}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 L_{1/2} regularization operator to replace the traditional L_{1} 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 [_{0} regularization is the sparest and most ideal regularization norm (_{0} regularization is susceptible to noise interference and it is difficult to solve equations, and

Recently, Xu et al. proposed

On solving

In this paper, we propose a CT reconstruction algorithm based on

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) [

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 [

CT reconstruction problem can be converted to a constrained optimization problem

In compressed sensing theory,

As shown in Figure _{0} regularization norm than L_{1} regularization norm.

_{1}, and

Xu et al. proposed a

If

The definition of operator is

Then the optimal solution

Please see [

In order to solve (_{1} regularization and _{2} regularization equation can be solved by gradient descent method and

Using an intermediate variable

Equation (

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

According to aforementioned methods, we propose a CT reconstruction algorithm based on

Combine with Split Bregman method to solve (

One has

One has

One has

Equation (

To derivate (

Then

In this section, we study the ART algorithm, TV based ART algorithm (ART-TV), and

Shepp-Logan phantom.

We will compare the reconstruction results from noise-free and noise data and projection numbers

The reconstructed images using three different reconstruction algorithms from the noise-free and noise data. (a)–(c) Reconstructed images from noise-free data: (a) reconstructed image using ART method, (b) reconstructed image using ART-TV method, and (c) reconstructed image using SpBr-

From Figures

The comparison between reconstructed images using three different reconstruction algorithms and original Shepp-Logan image. (a) The profiles of line 128 in reconstructed images using ART and ART-TV methods from noise-free data and original Shepp-Logan image, (b) the profiles of line 128 in reconstructed images using SpBr-

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

As shown in Table

The RMSE of reconstructed images using three different algorithms from noise-free and noise data.

Methods | ART | ART-TV |
SpBr-L_{1/2} |
---|---|---|---|

Noise-free | 0.0305 | 0.0104 | 0.0044 |

Noise | 0.0388 | 0.0274 | 0.0102 |

As shown in Figure

The RMSE line of reconstructed images with different reconstruction algorithms at 60 projection angles and different iteration numbers, and the iteration numbers range from 1 to 100. (a) The RMSE of reconstructed images from noise-free projection data and (b) the RMSE of reconstructed images from noise projection data.

The RMSE line of reconstructed images with different reconstruction algorithms at 50 iteration numbers and different projection angles; the projection angles range from 1 to 180. (a) The RMSE of reconstructed images from noise-free projection data and (b) the RMSE of reconstructed images from noise projection data.

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

The RMSE of reconstructed images using three different algorithms with real projection data.

Methods | ART | ART-TV | SpBr-L_{1/2} |
---|---|---|---|

RMSE | 0.0270 | 0.0256 | 0.0236 |

The reconstructed images using three algorithms from real projection data; iteration numbers are 50. (a) Reconstructed image using ART with original projection data, (b) the reconstructed image using ART method, (c) the reconstructed image using ART-TV method, and (d) the reconstructed image using SpBr-

As shown in Figure

There are several issues worth further discussion in the reconstruction study. Firstly, the thresholding algorithm was not applied to solve the

In the further research, we will try to use SpBr-

In conclusion, we proposed a CT reconstruction algorithm based on

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.

_{1/2}regularization

_{1/2}regularization: a thresholding representation theory and a fast solver