# A Novel Adaptive Non-Local Means-Based Nonlinear Fitting for Visibility Improving

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## Abstract

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## 1. Introduction

## 2. The Feasibility Analysis of Adaptive Geometric Non-Local Means Denosing Theory

## 3. Design of Spatial Geometric Fractal Iterative Denoising Algorithm

#### 3.1. Adaptive Geometry Non-Local Means Denoising Model

#### 3.1.1. New Similarity Description Method

#### 3.1.2. New Adaptive Association Approximation Kernel Function

- (1)
- Calculating the fractal dimension D of non-local weighted image ${\mathrm{I}}^{\prime}$.
- (2)
- Creating a new image B, the image size is equal to the non-local weighted image ${\mathrm{I}}^{\prime}$. Set the initial value for each pixel to 0.
- (3)
- The first row and the first column of the new image B are equal to the original image ${\mathrm{I}}^{\prime}$, and the value of each pixel in the image from the second row and the second column $B\left(i,j\right)$ is:$$B\left(i,j\right)=a{I}^{\prime}\left(i,j\right)+\frac{1-a}{8}\left[{\displaystyle \sum _{\begin{array}{c}k=-1\\ k\in \left\{-1,0,1\right\}\end{array}}^{1}B\left(i-k,j-k\right)}\right]$$

#### 3.2. The Optimal Fitting Solution of Centre Pixel Approximation in Arbitrary Geometric Plane

_{i}is described in Equation (11).

_{i}and the average of the window gray mean variance should be as small as possible. It is also a reasonable assumption in the graph theory for the limited neighborhood of the clear image. Secondly, in order to improve the image edge preservation, the difference gradient of this neighborhood is as large as possible. Finally, it is assumed that the noise from the image is in a controllable range. In other words, the absolute value of the difference between each estimated value y

_{i}and real observation value ${y}_{0i}$ should be less than a preset threshold.

**Q**is a nine-dimensional vector in which all elements are 1. Introducing Equation (18) into Equation (17), we obtain

**A**is

**X**are different, therefore Equation (24) can be obtained

**A**is greater than zero, the first condition of Equation (22) defines the inner region of an ellipse which is expressed as $\left[\begin{array}{cc}\alpha & \beta \end{array}\right]A\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]+{R}^{T}\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]+{Y}_{0}{}^{T}{Y}_{0}-T=0$. Assuming that x

_{m}is maximum among

**X**and x

_{n}is minimum among

**X**, the second condition is made up of the area enclosed by the upside of ${x}_{m}\alpha +\beta =0$ and ${x}_{n}\alpha +\beta =0$. The restricted condition enclosed area is shown in Figure 8.

## 4. Experimental Analysis

#### 4.1. Computer Simulation Experiment

#### 4.2. Physical Image Experiment

_{xy}; f

_{max}is the maximum gray value of the image f.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Three possible continuous surfaces of pixel distribution: (

**a**) Conical surface; (

**b**) cylindrical surface and (

**c**) extruded surface.

**Figure 4.**Simulation experiments of eight different templates. (

**a**) Eight approximation templates, (

**b**–

**i**) eight approximation results.

y_{01} | y_{02} | y_{03} |

y_{04} | y_{05} | y_{06} |

y_{07} | y_{08} | y_{09} |

**Figure 9.**Lena noise simulation image and experimental result curve. (

**a**) $\varsigma =0$; (

**b**) $\varsigma =1$; (

**c**) $\varsigma =3$; (

**d**) $\varsigma =5$; (

**e**) $\varsigma =7$; (

**f**) $\varsigma =9$; (

**g**) the evaluation index curve; (

**h**) the PSNR comparison curve; and (

**i**) the edge preserving index comparison curve.

**Figure 10.**Cameraman noise simulation image and experimental result curve. (

**a**) $\varsigma =0$; (

**b**) $\varsigma =1$; (

**c**) $\varsigma =3$; (

**d**) $\varsigma =5$; (

**e**) $\varsigma =7$; (

**f**) $\varsigma =9$; (

**g**) the evaluation index curve; (

**h**) the PSNR comparison curve; and (

**i**) the edge preserving index comparison curve.

**Figure 11.**Saturn noise simulation image and experimental result curve. (

**a**) $\varsigma =0$; (

**b**) $\varsigma =1$; (

**c**) $\varsigma =3$; (

**d**) $\varsigma =5$; (

**e**) $\varsigma =7$; (

**f**) $\varsigma =9$; (

**g**) the evaluation index curve; (

**h**) the PSNR comparison curve; and (

**i**) the edge preserving index comparison curve.

**Figure 12.**Black and white edge map denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 13.**Chinese rural elderly lady image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 14.**Italy architectural image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 15.**Butterfly image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 16.**Christ Church image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 17.**Tiger image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 18.**Flower image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 19.**Bear image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 20.**Girl image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**) The proposed method.

**Figure 21.**Straw rope image denoising results by different method. (

**a**) Noisy image; (

**b**) BAI; (

**c**) WT; (

**d**) MTLS; (

**e**) ASGF; (

**f**) RAID; (

**g**) HVD and (

**h**)The proposed method.

**Table 1.**The quantitative analysis results for Figure 12 of the denoising effect of various algorithms.

Noisy Image | BAI | WT | MTLS | ASGF | RAID | HVD | The Proposed Method | |
---|---|---|---|---|---|---|---|---|

$\gamma ({f}_{En})$ | 0.86 | 0.64 | 0.59 | 0.51 | 0.39 | 0.33 | 0.35 | 0.24 |

Def | 6.33 | 7.98 | 6.89 | 9.11 | 9.98 | 9.19 | 10.09 | 11.21 |

${\sigma}_{f}$ | 62.78 | 71.57 | 73.4 | 81.05 | 89.30 | 88.86 | 79.25 | 95.76 |

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**MDPI and ACS Style**

Wu, H.; Jia, L.; Meng, Y.; Liu, X.; Lan, J.
A Novel Adaptive Non-Local Means-Based Nonlinear Fitting for Visibility Improving. *Symmetry* **2018**, *10*, 741.
https://doi.org/10.3390/sym10120741

**AMA Style**

Wu H, Jia L, Meng Y, Liu X, Lan J.
A Novel Adaptive Non-Local Means-Based Nonlinear Fitting for Visibility Improving. *Symmetry*. 2018; 10(12):741.
https://doi.org/10.3390/sym10120741

**Chicago/Turabian Style**

Wu, Hongtao, Lei Jia, Ying Meng, Xiao Liu, and Jinhui Lan.
2018. "A Novel Adaptive Non-Local Means-Based Nonlinear Fitting for Visibility Improving" *Symmetry* 10, no. 12: 741.
https://doi.org/10.3390/sym10120741