Relaxation Factor Optimization for Common Iterative Algorithms in Optical Computed Tomography

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.

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 [5].However, the complete projection data are often not available in clinical application, which are restricted by these factors of the detection environment, detection time, and the characteristics of the detected object.For example, it is necessary to scan the breast quickly due to safe radiation dose, and only sparse projection data can be obtained [21][22][23].The complete projection data also cannot be obtained in dental diagnosis due to limited sampling angle [18][19][20].From the view of mathematics, the problem of reconstructing an image from incomplete data is usually considered as an underdetermined problem.
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) [28], multiplicative algebraic reconstruction technique (MART) [29], and simultaneous algebraic reconstruction technique (SART) [30].
As we know, for common reconstruction algorithms, the quality of the reconstruction image is related to the number 2 Mathematical Problems in Engineering 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.

Theory
According to the mathematical foundations of computed tomography [5,31], the optical computed tomography reconstruction can be expressed as where  represents the known image data vector which can be obtained from an imaging device, such as medical apparatus and instruments. represents the projection matrix which can be chosen through one or more optimal criterions, which is related to the basic function. represents the unknown data which we want to reconstruct. is the reconstruction error vector.The purpose of reconstruction is to minimize .In order to minimize , there are many algorithms, such as the Fourier transform algorithm [32,33], convolution opposite projection algorithm [34], and iterative algorithm [28][29][30].For incomplete projection data reconstruction, the iterative algorithm is the best from the aforementioned three methods.In this section, the reconstruction theories and iterative formulas of three common iterative algorithms are proposed, respectively.They have similar iterative steps but different iterative formulas.

Algebraic Reconstruction Technique (ART).
The detailed steps of the ART are as follows.
Step 1. Set the initial value of  (0) = arbitrary value.
Step 4. Continue to iterate until the preconditions are satisfied and the iteration ends.
The iterative formula can be expressed as where   = (mod  + 1);  is the iterative times;  is total number of projection rays, which is related to the number of projection directions and the number of sampling points for each projection direction;    is the th row of the projection matrix;  is the relaxation factor.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 (2), one can see that only the pixels passing through the   th ray are considered during the pixel values correction of  () .This correction method is called "line by line correction."

Multiplicative Algebraic Reconstruction Technique (MART).
The detailed steps of MART are similar to that of ART, but the initial value of  (0) cannot be set as 0. The iterative formula can be expressed as where  max is the maximum of the projection matrix; other parameters are the same as the ART algorithm.
From formula (3), one can see that only the pixels passing through the   th ray are considered during the pixel values correction of  () .The only difference is that formula (2) is corrected by adding a number, and formula (3) is corrected by multiplying by a number.This correction method is also called "line by line correction."

Simultaneous Algebraic Reconstruction Technique (SART).
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 From formula (4), one can see that all the rays are considered for each iteration step, some random errors are averaged, and the total errors are reduced.Besides this, the initial value of  (0) is quite different from the above two iterative algorithms, where  (0) =   .This correction method is also called "point by point correction."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.

Simulations
Figure 1 shows the bimodal asymmetrical image, which can be expressed by [35]  ( And Figure 2 shows the three-peak asymmetrical image, which can be expressed by [35]   (1) The average relative error: (2) The root-mean-square relative error: Furthermore, to evaluate the antinoise ability of the above three algorithms, the reconstructions with Gaussian noise (,  2 ) are also considered, where  represents the mean value and  2 represents the variance value of Gaussian noise.In this paper, the mean value () remains unchanged ( = 0) and the variance value ( 2 ) changes from 0.025 to 0.125.

Numerical Simulations.
The numerical simulation results are shown as Figures 3-8 and Tables 1-3, where Figures 3,  4, and 5 show the reconstruction errors of the ART, MART, and SART algorithm for two tested images, respectively.And Figures 6, 7, and 8 show the influence rule of Gaussian noise on the optimal relaxation factor.And Tables 1, 2, and 3 show the antinoise ability of the above three algorithms for two tested images, respectively.Figure 3 shows the reconstruction errors of the ART algorithm for two tested images, where the -axis is the relaxation factor, and the -axis is the average relative error and root-men-square relative error.The valid range of -axis is between 0 and 2, and the curve is graphed by polynomial fit based on several discrete data.From Figure 3(a), one can see that the optimal relaxation factor is about 1.6 for bimodal asymmetrical image, and the average relative error and the root-mean-square relative error are 0.85 and 6.23, respectively.From Figure 3(b), one can see that the optimal relaxation factor is about 1.8 for three-peak asymmetrical   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.image, and the average relative error and the root-meansquare relative error are 0.94 and 6.55, respectively.
Figure 4 shows the reconstruction errors of the MART algorithm for two tested images, where the -axis is the relaxation factor, and the -axis is the average relative error and root-men-square relative error.From Figure 4(a), one can see that the optimal relaxation factor is about 0.2 for bimodal asymmetrical image, and the average relative error and the root-mean-square relative error are 0.58 and 4.66, respectively.From Figure 4(b), one can see that the optimal 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.
relaxation factor is about 0.18 for three-peak asymmetrical image, and the average relative error and the root-meansquare relative error are 0.60 and 4.85, respectively.
Figure 5 shows the reconstruction errors of the SART algorithm for two tested images, where the -axis is the relaxation factor, and the -axis is the average relative error and root-men-square relative error.From Figure 5(a), one can see that the optimal relaxation factor is about 12 for bimodal asymmetrical image, and the average relative error and the root-mean-square relative error are 0.76 and 5.90,  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.respectively.From Figure 5(b), one can see that the optimal relaxation factor is about 14 for three-peak asymmetrical image, and the average relative error and the root-meansquare relative error are 0.80 and 5.23, respectively.
Figure 6 shows the influence rule of Gaussian noise on the optimal relaxation factor of the ART algorithm for two tested images, where the -axis is the Gaussian noise (,  2 ),  2 ranges from 0 to 0.15, and the -axis is the optimal relaxation factor.From Figure 6, one can see that the optimal relaxation factor is getting smaller with the increase of Gaussian noise when  2 < 0.125.For both bimodal asymmetrical image and three-peak asymmetrical image, the two curves have similar trends, with only a few differences.
Table 1 shows the antinoise ability of the ART algorithm for two tested images with Gaussian noise (,  2 ).From Table 1, one can see that both average relative error and root-mean-square relative error are getting larger with the increase of Gaussian noise.For bimodal asymmetrical image reconstruction, the average relative error is 1.42 times and 2.31 times as the reconstruction with no-noise and the rootmean-square relative error is 1.35 times and 2.03 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.For three-peak asymmetrical image reconstruction, the average relative error is 1.36 times and 2.20 times as the reconstruction with no-noise and the rootmean-square relative error is 1.35 times and 2.09 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.
Figure 7 shows the influence rule of Gaussian noise on the optimal relaxation factor of the MART algorithm for two tested images, where the -axis is the Gaussian noise (,  2 ),  2 ranges from 0 to 0.15, and the -axis is the optimal relaxation factor.From Figure 7, one can see that there are different trends of the optimal relaxation factor for bimodal asymmetrical image and three-peak asymmetrical image, where the optimal relaxation factor decreases linearly with the increase of Gaussian noise for bimodal asymmetrical image.
Table 2 shows the antinoise ability of the MART algorithm for two tested images with Gaussian noise (,  2 ).From Table 2, one can see that both average relative error and root-mean-square relative error are getting larger with the increase of Gaussian noise.For bimodal asymmetrical image reconstruction, the average relative error is 1.76 times and 2.86 times as the reconstruction with no-noise and the rootmean-square relative error is 2.06 times and 4.00 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.For three-peak asymmetrical image reconstruction, the average relative error is 1.92 times and 4.33 times as the reconstruction with no-noise and the rootmean-square relative error is 2.17 times and 4.64 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.
Figure 8 shows the influence rule of Gaussian noise on the optimal relaxation factor of the SART algorithm for two tested images, where the -axis is the Gaussian noise (,  2 ),  2 ranges from 0 to 0.15, and the -axis is the optimal relaxation factor.From Figure 8, one can see that the optimal relaxation factor is getting smaller with the increase of Gaussian noise when  2 < 0.125.For both bimodal asymmetrical image and three-peak asymmetrical image, the two curves have similar trends, with only a few differences.
Table 3 shows the antinoise ability of the SART algorithm for two tested images with Gaussian noise (,  2 ).From Table 3, one can see that both average relative error and root-mean-square relative error are getting larger with the increase of Gaussian noise.For bimodal asymmetrical image reconstruction, the average relative error is 1.59 times and 2.58 times as the reconstruction with no-noise and the rootmean-square relative error is 1.55 times and 2.49 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.For three-peak asymmetrical image reconstruction, the average relative error is 1.55 times and 2.63 times as the reconstruction with no-noise and the root-mean-square relative error is 1.54 times and 2.56 times as the reconstruction with no-noise when  2 = 0.05 and  2 = 0.1, respectively.

Results and Discussions
On the basis of Section 3.3, we can draw several conclusions as follows: (1) 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.
(2) 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.
(3) 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.
(4) 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  2 < 0.05 when lower reconstruction precision is required.

Conclusions
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.

3. 1 .Figure 1 :
Figure 1: Tested image 1, bimodal asymmetrical image, where the axis and -axis are reconstruction area, and the -axis is normalized intensity (no units).

Figure 2 :
Figure 2: Tested image 2, three-peak asymmetrical image, where the -axis and -axis are reconstruction area, and the -axis is normalized intensity (no units).

Figure 3 :Figure 4 :Figure 5 :
Figure 3: 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.

Figure 6 :
Figure6: 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.

Figure 7 :
Figure7: 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.

Figure 8 :
Figure 8:  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.

Table 1 :
Comparison of the antinoise ability of the ART algorithm for two tested images.

Table 2 :
Comparison of the antinoise ability of the MART algorithm for two tested images.

Table 3 :
Comparison of the antinoise ability of the SART algorithm for two tested images.