# Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions

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## Abstract

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## 1. Introduction

## 2. Datasets and Methods

#### 2.1. Datasets

#### 2.2. Theoretical Scheme of Fermi–Dirac Correction Function

#### 2.3. Experimental Framework

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The model estimations of proposed distributions. In the model estimations, the global mean value is ${\epsilon}_{F}=125$ and the value $\epsilon \in \left[1,255\right]$. (

**a**) Difference comparison of the three proposed distributions, and the minus global standard deviation is ${\epsilon}_{b}=-1$. (

**b**) Behavior of the Fermi–Dirac (FD) distribution with different ${\epsilon}_{b}$ values. (

**c**,

**d**) The same situations but with Bose–Einstein (BE) and Maxwell–Boltzmann (MB) distribution, respectively.

**Figure 2.**The framework of the D-Unet-based structure for the brain tumor image segmentation. The original image sizes are 240 pixels, and we rescaled the input sizes to 160 pixels to match the requirements of the adopted neural network models. The third-dimensional value of 160 × 160 × 4 used in the 2D Unet represents four-type images, include T1, T1-weighted contrast-enhanced (T1-CE), T2-weighted, and T2-fluid-attenuated inversion recovery (FLAIR), and so does that used in the 3D feature extraction procedure. The fourth dimension value of 160 × 160 × 4 × 1 in the 3D feature extraction procedure represents the number of trials. The dimension-transform-blocks were used to fuse the two- and three-dimensional features in the encoding procedure, and only fused- and two-dimensional features were used for the information decoding. Thus, these procedures can offer a trade-off between high-dimensional information and computational complexity.

**Figure 3.**The figure shows the visualized results from the employed preprocessing methods and the proposed FD-type correction functions. Two values of ${\epsilon}_{b}$ were individually employed to estimate the kernel, and the corresponding results are illustrated in the second and third columns. The proposed FD-type correction functions exhibit the capability not only for image intensity normalization but also for image component filtering. The following columns list the preprocessed results using the other employed methods.

**Figure 4.**The visualized comparison between the 3D global histogram equalization function and the proposed FD-type correction functions. (

**a**) The preprocessing results dealing with these functions, and (

**b**) the predicted results of the tumor image segmentation categorized into ET, TC, and WT, and their corresponding ground truths.

Parameter/Function | Value/Method |
---|---|

Number of epochs | 30 |

Batch size | 32 |

Learning rate (Initial value) | 0.00015 |

Loss function | 3D soft dice loss function |

Optimizer | Adam |

**Table 2.**Performance comparison between the conventional preprocessing methods and the proposed FD-type correction functions. This table only lists the dice scores of the whole tumor (WT). Numbers in bold show the best records.

Preprocessing Method | Dice Score (WT only) | Computational Time (min) |
---|---|---|

Null | 0.7183 | 91 |

z-score Normalization | 0.9296 | 142 |

3D Global Histogram Equalization | 0.9148 | 84 |

Gamma Correction ^{1} | 0.9319 | 93 |

${\mathrm{FD}}_{1}$ correction | 0.9431 | 88 |

${\mathrm{FD}}_{2}$ correction | 0.9347 | 90 |

^{1}The gamma parameter = 0.6.

**Table 3.**The confusion-matrix table. It shows the comparison of accuracy, sensitivity (recall), and precision between the proposed FD-type correction functions and the conventional correction methods. Numbers in bold show the best records.

Preprocessing Method | Validation Stage (WT Only) | ||||||
---|---|---|---|---|---|---|---|

TP | TN | FP | FN | Accuracy | Recall | Precision | |

Null | 15,306 | 789,342 | 2003 | 5886 | 0.9901 | 0.71 | 0.85 |

z-score Normalization | 19,512 | 789,570 | 1776 | 1680 | 0.9957 | 0.90 | 0.90 |

3D Global Histogram Equalization | 19,293 | 789,400 | 1947 | 1899 | 0.9952 | 0.89 | 0.89 |

Gamma Correction ^{1} | 18,603 | 790,100 | 1248 | 2562 | 0.9953 | 0.86 | 0.92 |

${\mathrm{FD}}_{1}$ correction | 19,180 | 790,010 | 1339 | 2012 | 0.9959 | 0.89 | 0.91 |

${\mathrm{FD}}_{2}$ correction | 19,471 | 789,500 | 1841 | 1721 | 0.9956 | 0.90 | 0.89 |

^{1}The gamma parameter = 0.6.

**Table 4.**Further performance comparison between the 3D global histogram equalization function and the proposed FD-type correction functions. The dice scores of the tumor components categorized as WT, tumor core (TC), and enhancing tumor (ET) are individually presented for the comparison. Numbers in bold show the best records.

Preprocessing Method | Dice Score | Computational Time (min) | ||
---|---|---|---|---|

Training Stage | Validation Stage | |||

3D Global Histogram Equalization | WT: | 0.9220 | 0.8002 | 138 |

TC: | 0.9419 | 0.7688 | ||

ET: | 0.9142 | 0.6365 | ||

${\mathrm{FD}}_{1}$ correction | WT: | 0.9491 | 0.8337 | 140 |

TC: | 0.9757 | 0.7976 | ||

ET: | 0.9559 | 0.6802 | ||

${\mathrm{FD}}_{2}$ correction | WT: | 0.9336 | 0.8433 | 141 |

TC: | 0.9773 | 0.8041 | ||

ET: | 0.9606 | 0.6848 |

Preprocessing Method | Validation Stage | |||||||
---|---|---|---|---|---|---|---|---|

TP | TN | FP | FN | Accuracy | Recall | Precision | ||

3D Global Histogram Equalization | WT: | 18,910 | 789,880 | 1465 | 2281 | 0.9953 | 0.89 | 0.89 |

TC: | 7818 | 801,309 | 1490 | 1918 | 0.9958 | 0.80 | 0.84 | |

ET: | 3141 | 807,914 | 700 | 780 | 0.9981 | 0.80 | 0.82 | |

${\mathrm{FD}}_{1}$ correction | WT: | 18,949 | 789,874 | 1470 | 2242 | 0.9954 | 0.89 | 0.93 |

TC: | 7627 | 801,691 | 1107 | 2110 | 0.9960 | 0.78 | 0.87 | |

ET: | 3007 | 808,156 | 459 | 914 | 0.9983 | 0.77 | 0.87 | |

${\mathrm{FD}}_{2}$ correction | WT: | 19,240 | 789,955 | 1340 | 1952 | 0.9959 | 0.91 | 0.93 |

TC: | 7966 | 801,841 | 958 | 1772 | 0.9966 | 0.82 | 0.89 | |

ET: | 3153 | 808,200 | 415 | 768 | 0.9998 | 0.80 | 0.88 |

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**MDPI and ACS Style**

Tai, Y.-L.; Huang, S.-J.; Chen, C.-C.; Lu, H.H.-S. Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions. *Entropy* **2021**, *23*, 223.
https://doi.org/10.3390/e23020223

**AMA Style**

Tai Y-L, Huang S-J, Chen C-C, Lu HH-S. Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions. *Entropy*. 2021; 23(2):223.
https://doi.org/10.3390/e23020223

**Chicago/Turabian Style**

Tai, Yen-Ling, Shin-Jhe Huang, Chien-Chang Chen, and Henry Horng-Shing Lu. 2021. "Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions" *Entropy* 23, no. 2: 223.
https://doi.org/10.3390/e23020223