# Image Dehazing Based on Local and Non-Local Features

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**I**is the observed image in bad weather,

**J**is the ideal image,

**A**is the air-light, and

**t**is the transmission map. From Equation (1), the transmission map

**t**plays a core role in determining the clarity of a restored image. As the atmospheric light vector

**A**is unknown, the accurate estimation of the transmission map

**t**is a challenging problem.

**t**is divided into two stages: a rough transmission map, and a refined transmission map. The rough

**t**can be estimated via imaging model-based methods, including Dark Channel Prior (DCP) [2], Hazy-Line Prior (HLP) [3], or Color Attenuation Prior (CAP) [4]. However, the refined

**t**is difficult to estimate, and it plays a key role in image restoration, because

**t**can provide edge and texture information to image restoration in bad weather.

**t**, because the TV prior helps in preserving the sharp edges and textures of an image and prevents solutions from having oscillations. Therefore, numerous TV-based methods have been proposed for use in hazy weather. Bredies et al. [10] proposed the concept of total generalized variation (TGV) of a function without producing staircasing effects. Gilboa et al. [11] applied a complex diffusion process by using the Schrodinger equation. A linear solution was obtained for the complex diffusion process, which could estimate a high-quality restored image. Recently, Liu et al. [12] estimated the refined

**t**by using non-local total variation regularization. Jiao et al. [13] proposed a hybrid model, with first- and second-order variations, and they improved the hybrid model via adaptive parameters. Parisotto et al. [14] introduced higher-order anisotropic total variation regularization, and used a primal-dual hybrid gradient approach to approximate numerically the associated gradient flow. However, the TV-based methods suffer from two defects: a staircasing effect, and a non-convex and non-smooth term. Fortunately, fractional-order calculus [15] may be a powerful tool to alleviate these problems.

## 2. Materials and Methods

#### 2.1. Fractional Derivative

#### 2.2. Proposed Dehazing Model

**I**

_{CLAHE}is the hazy image that has been enhanced via contrast-restricted histogram equalization (CLAHE) [34], and $C\left(\xb7\right)$ and $T\left(\xb7\right)$ are the local and non-local information regularization terms, respectively. λ

_{1}, λ

_{2}, λ

_{3}, and λ

_{4}are the regularization parameters, which are positive numbers. ${\mathit{D}}^{\alpha}$ is the fractional-order operator, ${\mathit{D}}^{\alpha}={\left[{\left({\mathit{D}}_{x}^{\alpha}\right)}^{*}{\mathit{D}}_{x}^{\alpha},{\left({\mathit{D}}_{y}^{\alpha}\right)}^{*}{\mathit{D}}_{y}^{\alpha}\right]}^{T}.\widehat{\mathit{t}}$ and $\widehat{\mathit{J}}$ are the transmission map and the ideal image to be optimized. ${\left|\xb7\right|}_{1}$ and ${\left|\xb7\right|}_{2}$ are the 1-norm and 2-norm operators. Both the first term and the second term are original image data fidelity items. The difference is that the first term can restore the background of the image and prevent image over-enhancement, while the second term focuses on restoring the texture and detail. The third, fourth, and fifth terms are the regularization terms.

_{1}and η

_{2}are the penalty parameters; η

_{1}can be estimated via the proposed network architecture, and η

_{2}is estimated via repeated experiments. The non-constrained optimization problem can be split into four subproblems

_{1}is the number iterations. Equation (10) is the $\widehat{\mathit{t}}$-subproblem, which is obtained via fixed $\widehat{\mathit{J}}$,

**v**,

**w**to solve variable $\widehat{\mathit{t}}$.

**v**,

**w**to solve variable $\widehat{\mathit{J}}$. The $\widehat{\mathit{J}}$-subproblem, shown in Equation (11), could then be obtained.

**v**-subproblem, as shown in Equation (12), the variables $\widehat{\mathit{t}}$, $\widehat{\mathit{J}}$,

**w**were fixed, as shown in Equation (9), and the variable

**v**was solved.

**w**-subproblem, the variables $\widehat{\mathit{t}}$, $\widehat{\mathit{J}}$,

**v**were fixed as shown in in Equation (9), and the variable

**w**was solved to obtain the subproblem, as shown in Equation (13). From Equations (10)–(13), the HQS algorithm separates the data fidelity term, the fractional derivative regularization term, and the two data-driven regularization terms into four different subproblems.

#### 2.3. Subproblem **v**

**Q**,

**K**, and

**V**are the query, key, and value matrices, and ${d}_{k}$ is the size of the

**K**matrix.

**Θ**represents the parameters of the proposed network, θ represents the balance parameters, L is the number of the training dataset, E(·) is the proposed network, and η

_{1}, λ

_{3}, λ

_{4}are the parameters in Equation (12). As shown in Equation (15), the first term is the preserve-texture term, which extracts information about textures and edges via the fractional differential. The second term is the output of the proposed network. The proposed network estimates the

**v**

^{k+1}via minimizing the loss function, as shown in Equation (15).

#### 2.4. Other Subproblems

**E**is the unit matrix, and the division of Equations (16) and (17) is the element division. Then, the

**w**-subproblem, which is a non-convex optimization problem, is estimated. In our previous work [13], the non-convex and non-smooth variation was applied to restore underwater images. Analogously, the solution of w is shown as follows:

#### 2.5. The Estimation of Atmospheric Light

**V**is the atmospheric veil and

**t**is the transmission map. Given the fixed

**t**, the

**A**is proportional to

**V**. Hence, the high-precision

**t**can estimate high-precision atmospheric light

**A**. In this paper, the problem of estimated

**A**could be converted to the problem of estimated high-precision

**t**. To estimate

**A**, the model based on fractional-order calculus was proposed, which estimated

**A**via calculating

**V**. It is shown as:

_{5}is the regularization parameter. To solve Equation (20), an efficient alternating optimization algorithm based on variable splitting is proposed in this Section. In the proposed solution, the auxiliary variable

**X**was introduced to replace the fractional-gradient operators D

^{α}

**V**, and the Equation (20) could be rewritten as

_{3}is the parameter to control the similarity between auxiliary variable

**X**and fractional-gradient operators. The optimization problem shown in Equation (22) could be solved by iterative minimization with respect to

**V**,

**A**, and

**X**, respectively. The three subproblems are shown as follows

**V**-subproblem is obtained via fixed the variables

**A**and

**X.**Similarly, we fixed the variables

**V**and

**X**; to solve variable

**A**, the

**A**-subproblem is

**V**and

**A**to solve the variable

**X**; the

**X**-subproblem is:

**X**could be obtained by applying the fast shrinkage operator:

#### 2.6. The Steps of the Proposed Method

- Input: hazy image.
- Output: clean image.
- Step 1: Estimate the rough transmission map and the initial atmospheric light. Establish the nitial parameters, and set k
_{1}=1 and k_{2}=1; - Step 2: Solve Equations (16) and (17), the trained network, and Equation (18);
- Step 3: Solve Equation (19) and Equations (26)–(28);
- Step 4: Repeat Step 3, until the iteration exit condition of the atmospheric light estimation is satisfied;
- Step 5: Repeat Steps 2, 3 and 4, until the iteration exit condition of the transmission map is satisfied;
- Step 6: Output the transmission map and clean the image.

## 3. Results

#### 3.1. Evaluation of Initial Atmospheric Light

#### 3.2. Evaluation on Synthetic Images

#### 3.3. Evaluation on Real-World Images

#### 3.4. Running Time Analyze

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

No. | Symbol | Annotation | Type |
---|---|---|---|

1 | A | Atmospheric light | Vector |

2 | $C\left(\xb7\right)$ | Local regularization term | Function |

3 | ${D}_{x}^{\alpha}$ | Horizontal direction right fractional operator | Operator |

4 | ${\left({D}_{x}^{\alpha}\right)}^{*}$ | Horizontal direction left fractional operator | Operator |

5 | ${\left({D}_{x}^{\alpha}\right)}^{*}{D}_{x}^{\alpha}$ | Horizontal direction composite fractional operator | Operator |

6 | ${D}_{y}^{\alpha}$ | Vertical direction right fractional operator | Operator |

7 | ${\left({D}_{y}^{\alpha}\right)}^{*}$ | Vertical direction right fractional operator | Operator |

8 | ${\left({D}_{y}^{\alpha}\right)}^{*}{D}_{y}^{\alpha}$ | Vertical direction composite fractional operator | Operator |

9 | ${d}_{k}$ | Size of key in MSA | Number |

10 | $E\left(\xb7\right)$ | The proposed network | Function |

11 | E | Unit matrix | Matrix |

12 | I | Hazy image | Matrix |

13 | ${\mathit{I}}_{CLAHE}$ | Optimized hazy image via CLAHE | Matrix |

14 | I | Pixel coordinates | Number |

15 | J | Ideal image | Matrix |

16 | $\widehat{\mathit{J}}$ | Ideal image, variable | Matrix |

17 | J | Pixel coordinates | Number |

18 | K | Key of MSA | Matrix |

19 | K | Gird size of standard discretization technique | Number |

20 | ${k}_{1}$ | Iterations number of transmission map estimation | Number |

21 | ${k}_{2}$ | Iterations number of atmospheric light estimation | Number |

22 | L | Number of train data | Number |

23 | M | Size of image | Number |

24 | M | Number of grids of standard discretization technique | Number |

25 | N | Size of image | Number |

26 | Q | Query of MSA | Matrix |

27 | $sgn\left(\xb7\right)$ | Symbolic function | Function |

28 | T | Transmission map | Matrix |

29 | $\widehat{\mathbf{t}}$ | Transmission map, variable | Matrix |

30 | $T\left(\xb7\right)$ | Non-local regularization term | Function |

31 | V | Value of MSA only in Equation (14) | Matrix |

32 | V | Atmospheric veil | Matrix |

33 | V | Auxiliary variable | Matrix |

34 | W | Auxiliary variable | Matrix |

35 | X | Auxiliary variable | Matrix |

36 | $\alpha $ | order | Number |

37 | ${\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4},{\lambda}_{5}$ | Regularization parameter | Number |

38 | ${\eta}_{1},{\eta}_{2},{\eta}_{3}$ | Penalty parameter | Number |

39 | Θ | The proposed network parameters | Matrix |

40 | $\theta $ | Balance parameter in Equation (15) | Number |

41 | ${\left|\xb7\right|}_{1},{\left|\xb7\right|}_{2}$ | 1-norm, 2-norm | Function |

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**Figure 1.**The architecture of the proposed two-streams network. To solve Equation (12), the first term of Equation (12) was designed as the upper stream, the second term was designed as the lower stream, and the third term could be calculated directly.

**Figure 3.**The architecture of the two successive Swin Transformer blocks. W-MSA and SW-MSA are multi-head self-attention modules with regular and shifted windowing configurations, respectively.

**Figure 6.**The dehazing effect of an indoor image with nonuniform illumination. The first row, from left to right, shows the ground truth, the synthetic hazy image, and the dehazing of the DCP, the CAP, and the HLP. The second row, from left to right, shows the dehazing methods of the OTM, the FFA, RefineDNet, and the CLAHE, as well as ours (the authors’).

**Figure 7.**The dehazing effect of an indoor image with uniform illumination. The first row, from left to right, shows the ground truth, the synthetic hazy image, and the dehazing of the DCP, the CAP, and the HLP. The second row, from left to right, shows the dehazing methods of the OTM, the FFA, RefineDNet, and the CLAHE, as well as ours (the authors’).

**Figure 8.**The dehazing effect of an outdoor image with a sky region. The first row, from left to right, shows the ground truth, the synthetic hazy image, and the dehazing of the DCP, the CAP, and the HLP. The second row, from left to right, shows the dehazing of the OTM, the FFA, RefineDNet, and the CLAHE, as well as ours (the authors’).

**Figure 9.**The dehazing effect of an outdoor image without a sky region. The first row, from left to right, shows the ground truth, the synthetic hazy image, and the dehazing of the DCP, the CAP, and the HLP. The second row, from left to right, shows the dehazing of the OTM, the FFA, RefineDNet, and the CLAHE, as well as ours (the authors’).

**Figure 10.**The PSNR and the SSIM of synthetic hazy images. (

**a**) The PSNR of tested methods with different datasets. (

**b**) The SSIM of tested methods with different datasets.

**Figure 11.**The dehazing results of real-world hazy images. From top to bottom are “stack”, “flower”, “city”, “Tiananmen”, “train”, and “mansion”. (

**a**) The origin hazy images. (

**b**) The dehazing result of the DCP. (

**c**) The dehazing effect of the CAP. (

**d**) The dehazing effect of the HLP. (

**e**) The dehazing effect of the OTM. (

**f**) The dehazing effect of the FFA. (

**g**) The dehazing effect of RefineDNet. (

**h**) The dehazing effect of the CLAHE. (

**i**) The dehazing effect of the authors’ proposed method.

**Figure 12.**The dehazing results of the real-world hazy image. (

**a**) The average score of IE for the dehazing result. (

**b**) The average score of variances for the dehazing result. (

**c**) The average score of BRISQUE for the dehazing result. (

**d**) The average score of NBIQA for the dehazing result.

**Table 1.**Comparison of three rough transmission map and initial atmospheric light estimation methods.

Method | I-Hazy | O-Hazy | ||
---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | |

DCP | 16.42 | 0.67 | 15.94 | 0.55 |

CAP | 16.37 | 0.66 | 15.87 | 0.58 |

HLP | 16.48 | 0.64 | 15.89 | 0.56 |

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## Share and Cite

**MDPI and ACS Style**

Jiao, Q.; Liu, M.; Ning, B.; Zhao, F.; Dong, L.; Kong, L.; Hui, M.; Zhao, Y.
Image Dehazing Based on Local and Non-Local Features. *Fractal Fract.* **2022**, *6*, 262.
https://doi.org/10.3390/fractalfract6050262

**AMA Style**

Jiao Q, Liu M, Ning B, Zhao F, Dong L, Kong L, Hui M, Zhao Y.
Image Dehazing Based on Local and Non-Local Features. *Fractal and Fractional*. 2022; 6(5):262.
https://doi.org/10.3390/fractalfract6050262

**Chicago/Turabian Style**

Jiao, Qingliang, Ming Liu, Bu Ning, Fengfeng Zhao, Liquan Dong, Lingqin Kong, Mei Hui, and Yuejin Zhao.
2022. "Image Dehazing Based on Local and Non-Local Features" *Fractal and Fractional* 6, no. 5: 262.
https://doi.org/10.3390/fractalfract6050262