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The nonmonotone alternating direction algorithm (NADA) was recently proposed for effectively solving a class of equality-constrained nonsmooth optimization problems and applied to the total variation minimization in image reconstruction, but the reconstructed images suffer from the artifacts. Though by the

Statistical and iterative reconstruction algorithms in computed tomography (CT) are widely applied since they yield more accurate results than analytic approaches for low-dose and limited-view reconstruction. These algorithms involve solving a linear system:

The theory of compressed sensing [

The

Researchers in this area have been seeking for other efficient and stable algorithms inspired by the compressed sensing theory, for example, solving an

The purposes of this paper are multifold:

(i) Presenting a combined

(ii) Adopting a newly developed alternating direction method NADA to efficiently solve minimization

(iii) Resolving the computation challenge problem caused from the

The numerical experiments demonstrate that the proposed algorithm improves the quality of the recovered images for the same cost of CPU time and reduces the computation time significantly while maintaining the same image quality compared with the

The rest of the paper is organized as follows. Section

The TV minimization of an image

For convenience, we denote

Minimization (

(i) Choose a direction

(ii) Select a step size

(iii) Set

(iv) Compute

Combined with the solution of (

The

A new approach for solving a more general minimization than the regularization minimization in [

In this section we present

The term

It is noted that the objective function

In this subsection we address how to calculate

Equation (

Let

Before we develop a procedure to calculate a positive root of

It is known that

If

The second derivative

Finally, we can determine the locations of zeros of

Let

Choose

Choose

We identify an interval containing a positive root of

Thus,

Based on the above discussion, the calculation process of the proposed

input

initialize

while

1. update

1.1. decent direction

1.2. step size

1.3. update

1.4. compute

2. update

for

2.1. set

2.2. if

else

select an initial guess

find a zero

end if

end for

3. update

4. if

5. update

end while

output

In this section, the proposed combined

Experiments are conducted to compare the reconstruction by the two algorithms after the same number of iterations. The original/reconstructed Shepp-Logan phantom images and the original/reconstructed cardiac images after same numbers of iterations are shown in Figure

Original and reconstructed images. First row: Shepp-Logan phantom after 150 iterations. Second row: cardiac image after 80 iterations.

The quality of images is evaluated using the relative error

Comparison of two algorithms after same number of iterations.

Shepp-Logan Phantom, after 70 iterations | ||||||
---|---|---|---|---|---|---|

Time(s) | Error | RMSE | NRMSD | NMAD | SSIM | |

| 9.5 | 0.121 | 0.030 | 0.031 | 0.124 | 0.826 |

| 6.3 | 0.067 | 0.016 | 0.017 | 0.068 | 0.912 |

Cardiac Image, after 50 iterations | ||||||
---|---|---|---|---|---|---|

Time(s) | Error | RMSE | NRMSD | NMAD | SSIM | |

| 5.1 | 0.131 | 0.047 | 0.033 | 0.152 | 0.696 |

| 4.9 | 0.116 | 0.042 | 0.029 | 0.131 | 0.735 |

Experiments are also conducted to compare iteration numbers and CPU time by the two algorithms when the same relative error is achieved and thus the image quality is same. The tolerance of the relative error to terminate the iteration is selected as 0.05. The average iteration numbers and CPU time from 100 tests of the

Comparison of two algorithms with the same tolerance of relative error 0.05.

Phantom | | | CPU Time Saved | ||
---|---|---|---|---|---|

No. Iter. | Time(s) | No. Iter. | Time(s) | ||

Shepp-Logan | 129 | 17.4 | 86 | 7.4 | 58% |

| |||||

Cardiac | 194 | 24.9 | 178 | 19.4 | 22% |

Relative error vs. iteration number in reconstruction by two algorithms to achieve the same tolerance of relative error 0.05. (a) Shepp-Logan phantom; (b) cardiac image.

Both types of the numerical experiments demonstrate that the proposed

The application of the

There are some aspects to be further investigated. It is a challenging problem to select good values of other parameters such as

The MATLAB numerical data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

_{0}-norm optimization

_{0}norm

_{0}norm

_{0}norm regularization for sparse-view X-ray CT reconstruction

_{0}-regularized gradient prior

_{0}gradient regularized model with box constraints for image restoration

_{2}regurizations for limited-angle CT reconstruction

_{1}minimization

_{1}-norm minimization

_{1}-problems in compressive sensing