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Optical computed tomography technique has been widely used in pathological diagnosis and clinical medicine. For most of optical computed tomography algorithms, the relaxation factor plays a very important role in the quality of the reconstruction image. In this paper, the optimal relaxation factors of the ART, MART, and SART algorithms for bimodal asymmetrical and three-peak asymmetrical tested images are analyzed and discussed. Furthermore, the reconstructions with Gaussian noise are also considered to evaluate the antinoise ability of the above three algorithms. The numerical simulation results show that the reconstruction errors and the optimal relaxation factors are greatly influenced by the Gaussian noise. This research will provide a good theoretical foundation and reference value for pathological diagnosis, especially for ophthalmic, dental, breast, cardiovascular, and gastrointestinal diseases.

Computed tomography (CT) was firstly proposed in 1970s. From then on, many research results on theoretical analysis and actual applications of CT were obtained [

In general, analytic method and iterative method are usually used to reconstruct the image in OCT. The foundation of analytic method is continuous signal model, which is sensitive to noise and requires complete projection data [

Compared with the analytic method, the iterative method can obtain good reconstruction quality in the case of low SNR and incomplete projection data, which is more suitable for clinical application. The common iterative algorithms include algebraic reconstruction technique (ART) [

As we know, for common reconstruction algorithms, the quality of the reconstruction image is related to the number of projection directions, the number of sampling points for each projection direction, the range of the field of view, and so forth. Besides these, some other factors are also important for iterative algorithms, such as relaxation factor, basic function, prior knowledge, and iterative times. Among these factors, the relaxation factor plays a very important role in the quality of the reconstruction image.

In this paper, the optimal relaxation factors of the ART, MART, and SART algorithms for bimodal asymmetrical and three-peak asymmetrical tested images are analyzed and discussed. The reconstructions with Gaussian noise are also considered to evaluate the antinoise ability of the above three algorithms; some related numerical simulation results are also presented.

According to the mathematical foundations of computed tomography [

The detailed steps of the ART are as follows.

Set the initial value of

According to the iterative formula, calculate

According to the iterative formula, calculate

Continue to iterate until the preconditions are satisfied and the iteration ends.

The iterative formula can be expressed as

It is not difficult to prove that the sequence generated by the iterative algorithm converges to a vector when the relaxation factor is appropriate.

From formula (

The detailed steps of MART are similar to that of ART, but the initial value of

From formula (

In the above two iterative algorithms, only one ray is considered for each iteration, and if the projection of this ray contains errors, the solution will also have errors. To avoid these errors, SART was proposed.

The iterative formula can be expressed as

Although the quality of the reconstruction image is related to many factors, the relaxation factor plays the most important role in it. In this numerical simulation, we fix the values of other parameters to evaluate the influence rules of relaxation factor on the above three iterative algorithms and figure out the optimal value of relaxation factor.

The number of projection directions is 4, the number of grids is 25 × 25, the range of the field of view is 180°, the basis function is sinc-function, the iterative time is 30, and the bimodal asymmetrical function and three-peak asymmetrical function are chosen as tested images.

Figure

Tested image 1, bimodal asymmetrical image, where the

Tested image 2, three-peak asymmetrical image, where the

To compare the reconstruction effect of the ART, MART, and SART, two error indexes are defined as shown below, where

The average relative error:

The root-mean-square relative error:

The numerical simulation results are shown as Figures

Comparison of the antinoise ability of the ART algorithm for two tested images.

Gaussian noise |
Bimodal asymmetrical image | Three-peak asymmetrical image | ||
---|---|---|---|---|

Average relative errors (%) | Root-mean-square relative errors (%) | Average relative errors (%) | Root-mean-square relative errors (%) | |

(0, 0.025) | 1.04 | 7.75 | 1.13 | 8.18 |

(0, 0.050) | 1.21 | 8.38 | 1.28 | 8.82 |

(0, 0.075) | 1.63 | 10.92 | 1.82 | 11.35 |

(0, 0.100) | 1.96 | 12.66 | 2.07 | 13.70 |

(0, 0.125) | 2.15 | 14.44 | 2.56 | 16.21 |

Comparison of the antinoise ability of the MART algorithm for two tested images.

Gaussian noise |
Bimodal asymmetrical image | Three-peak asymmetrical image | ||
---|---|---|---|---|

Average relative errors (%) | Root-mean-square relative errors (%) | Average relative errors (%) | Root-mean-square relative errors (%) | |

(0, 0.025) | 0.88 | 7.52 | 0.92 | 7.76 |

(0, 0.050) | 1.02 | 9.61 | 1.15 | 10.51 |

(0, 0.075) | 1.45 | 13.06 | 1.53 | 14.92 |

(0, 0.100) | 1.66 | 15.84 | 1.88 | 17.29 |

(0, 0.125) | 2.03 | 18.65 | 2.60 | 22.53 |

Comparison of the antinoise ability of the SART algorithm for two tested images.

Gaussian noise |
Bimodal asymmetrical image | Three-peak asymmetrical image | ||
---|---|---|---|---|

Average relative errors (%) | Root-mean-square relative errors (%) | Average relative errors (%) | Root-mean-square relative errors (%) | |

(0, 0.025) | 1.07 | 8.83 | 1.11 | 8.91 |

(0, 0.050) | 1.21 | 9.16 | 1.24 | 9.65 |

(0, 0.075) | 1.74 | 11.75 | 1.88 | 13.04 |

(0, 0.100) | 1.96 | 14.69 | 2.10 | 15.92 |

(0, 0.125) | 2.95 | 23.17 | 3.09 | 24.23 |

The reconstruction error of the ART algorithm for two tested images, where the blue squares represent the average relative error, and the red stars represent the root-mean-square relative error. (a) Bimodal asymmetrical image, (b) three-peak asymmetrical image.

The reconstruction error of the MART algorithm for two tested images, where the blue squares represent the average relative error, and the red stars represent the root-mean-square relative error. (a) Bimodal asymmetrical image, (b) three-peak asymmetrical image.

The reconstruction error of the SART algorithm for two tested images, where the blue squares represent the average relative error, and the red stars represent the root-mean-square relative error. (a) Bimodal asymmetrical image, (b) three-peak asymmetrical image.

The influence of Gaussian noise on the optimal relaxation factor of the ART algorithm for two tested images, where blue squares represent the optimal relaxation factor of bimodal asymmetrical image, and red stars represent the optimal relaxation factor of three-peak asymmetrical image.

The influence of Gaussian noise on the optimal relaxation factor of the MART algorithm for two tested images, where blue squares represent the optimal relaxation factor of bimodal asymmetrical image, and red stars represent the optimal relaxation factor of three-peak asymmetrical image.

The influence of Gaussian noise on the optimal relaxation factor of the SART algorithm for two tested images, where blue squares represent the optimal relaxation factor of bimodal asymmetrical image, and red stars represent the optimal relaxation factor of three-peak asymmetrical image.

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On the basis of Section

The reconstruction errors with noise or without noise of the ART, MART, and SART algorithms are related to the tested image. In this paper, we choose bimodal asymmetrical image and three-peak asymmetrical image as tested images, and the reconstruction error of three-peak asymmetrical image is always larger than bimodal asymmetrical image for its complexity. In clinical application, the optimal relaxation factor should be chosen according to the actual situation, and the research results of this paper can be used as reference.

For bimodal asymmetrical image reconstruction without noise, the optimal relaxation factor of the ART, MART, and SART algorithms is about 1.6, 0.2, and 12, respectively. For three-peak asymmetrical image without noise, the optimal relaxation factor of the ART, MART, and SART is about 1.8, 0.18, and 14, respectively.

The reconstruction error and the optimal relaxation factor of the ART, MART, and SART algorithms are greatly influenced by the Gaussian noise. In general, the optimal relaxation factor is getting smaller with the increase of Gaussian noise, and the reconstruction errors are getting larger with the increase of Gaussian noise. The MART algorithm has the worst antinoise ability, and the ART algorithm has the strongest antinoise ability for the above tested images.

For reconstruction with Gaussian noise or another type noise, we should reduce and control the noise as much as possible when higher reconstruction precision is required and meet the inequality

In summary, the optimal relaxation factors of the ART, MART, and SART algorithms with noise or without noise are discussed for bimodal asymmetrical and three-peak asymmetrical images. The numerical simulation results show that the reconstruction errors and the optimal relaxation factors are related to the tested images. To evaluate the antinoise ability of the above three algorithms, the reconstructions with Gaussian noise are also considered, and the reconstruction error and the optimal relaxation factors are greatly influenced by the Gaussian noise. The MART algorithm has the worst antinoise ability, and the ART algorithm has the strongest antinoise ability for the above two tested images. In clinical application, the best relaxation factor should be chosen according to the actual situation, and we should reduce and control the noise as much as possible.

To obtain better reconstruction results, our future work will focus on considering more influence factors and noise types. This research will provide a good theoretical foundation and reference value for pathological diagnosis and clinical medicine, especially for ophthalmic, dental, breast, cardiovascular, and gastrointestinal diseases.

The authors declare that they have no financial or personal relationships with other people or organizations that can inappropriately influence their work; there are no professional or other personal interests of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in or the review of our work submitted.

This work was supported by the ChunHui Project Scientific Research Foundation, Education Ministry of China, under Grant no. Z2015106, the Natural Science Foundation of China under Grant no. 61307063, and the Xihua University Young Scholars Training Program under Grant no. 01201402. W. B. Jiang also would like to acknowledge the Overseas Training Plan of Xihua University (09/2014-09/2015, University of Michigan, Ann Arbor, US).