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Reconstruction from few views is an important problem in medical imaging and applied mathematics.
In this paper, a combined energy minimization is proposed for image reconstruction.

Computed tomography (CT) is one of the most important advance in diagnostic radiology in recent decades. CT uses multiple X-ray images to build up cross-sectional and 3D pictures of structures inside the human body which enable doctors to view internal organs with unprecedented precision. However, the use of ionizing radiation in CT may induce cancer in the exposed individual after a latent period [

Some ways can be used to reduce the radiation dose from CT such as decreasing intensity of X-ray beam, handling scattered radiation, restricting exposure area. Reducing the X-ray exposing time is a simple one. Here we focus on the low-dose X-ray imaging strategy that only a limited number of projection images are taken for the reconstruction, which is called limited-view reconstruction [

CT reconstruction methods can roughly be categorized as analytic reconstruction methods and iterative reconstruction methods. The analytic reconstruction methods, such as filtered back-projection (FBP) methods [

Recently, the minimization of the image total variation (TV) has been introduced to divergent-beam CT and an new iterative image reconstruction algorithm was presented [

In this paper, a novel image reconstruction model is proposed. The image total variation and the

There are many approaches about tomographic reconstruction from limited views projection data [

However, it is known that the ray does harm to human body and abundant irradiation may lead to cancer [

Tomographic reconstruction from few views projection data is an efficient way to reduce the harm caused by ray irradiation, and there are some approaches about it [

As the gray image to be reconstructed can be denoted by below

The projection can be denoted as the following equations:

Unfortunately, this equations are indeterminate if the reconstruction was based on few views. In other words, the number of the equations are less than the number of variables (

The ART can be applied to solve this equation and it means to solve the following problem:

The total variation (TV) was first introduced by Rudin et al. [

The TV minimization can efficiently reduce errors and preserve features in the image reconstruction. In next section, we will concentrate on developing a new model to improve the convergence speed and reduce errors based on this TV model.

It is known that the convergence speed will be enhanced when the

The natural idea is to combine the

To improve the TV result of image denoising, Chambolle and Lions [

Based on the discrete form of the CL energy and the rearranged vector

With the Lagrange method applied, this constraint optimization problem can be rewritten as an unconstraint optimization problem of following combined Chambolle-Lions (CCL) energy:

The gradient of

Set a initial value

% Initialization

maxGrad =

% Iterations

while (

%Linear search

min

while (

if (

From the experiments results, we will find some advantages of the proposed model. These are chiefly due to the different optimization problems and it is related with the algorithm. More exactly, the first term in (

Reconstruction of Shepp-Logan phantom from 72 views.

The true image is taken to be the Shepp-Logan image shown in Figure

Reconstruction of Shepp-Logan phantom from 72 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

Reconstruction of Shepp-Logan phantom from 24 views.

The true image is still taken to be the same Shepp-Logan image as Example

Reconstruction of Shepp-Logan phantom from 24 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

Reconstruction of fruits image from 72 views.

A fruits image with size

Reconstruction of fruits from 72 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

Reconstruction of fruits image from 30 views.

The same fruits image as Example

Reconstruction of fruits from 30 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

Reconstruction of a synopsis phantom from 72 views.

A synopsis phantom with size

Reconstruction of a synopsis phantom from 72 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

Reconstruction of a synopsis phantom from 20 views.

The same synopsis phantom as Example

Reconstruction of a synopsis phantom from 20 views.

Original

Art result

TV result

CL result

Horizontal gray

Vertical gray

Result analysis of Example

Evolution of PSNR

Evolution of

In this paper, a novel model for image reconstruction from few views in parallel-beam data is proposed. First, the

The authors would like to thank the anonymous reviewers for their constructive feedback and valuable input. Due thanks are for the supports to their program from the TI, the XILINX, and the Software School of Xidian University. This program is partially supported by NSFC (Grant nos. 61072105, 61007011) and by the Open Projects Program of National Laboratory of Pattern Recognition. The Project is also partially supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program no. 2010JM8005) and Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 11JK0504).