# An Iterative Weighted-Mean Filter for Removal of High-Density Salt-and-Pepper Noise

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Scheme of the Iterative Weighted-Mean Filter

#### 2.1. Stage 1: Noise Detection

- All pixels in this region have extreme intensity.
- About half of the noise pixels take the intensity 255, so the total number of pixels with an intensity of 255 is greater than the pixels with an intensity of 0. In other words, pixels with an intensity of 255 are the majority.

_{5}(g) be its neighborhood window of size 5 × 5. If I(g) = 255 or I(g) = 0, label g as a noise candidate pixel; otherwise, label g as a noise-free pixel. For a noise candidate pixel g

_{candidate}, if all pixels in W

_{5}(g

_{candidate}) with intensity 0 or 255, and N

_{255}> T, where N

_{255}is the number of pixels with an intensity of 255, and the optimal value of T is 20, then g

_{candidate}is in the white extreme intensity flat regions. If I(g

_{candidate}) = 255, it is considered a noise-free pixel, otherwise it is a noise pixel.

#### 2.2. Stage 2: Noise Removal

#### 2.2.1. Selection of Filtering Window

_{5}(g), and we define five different windows r

_{i}(r

_{1}, r

_{2}, ... r

_{5}) according to the value of r, as shown in Figure 2a. Let W be the filtering window, and we choose W according to the following three rules:

- If the number of noise-free pixels in W
_{5}(g) is greater than 3, then set W = r_{1}as the candidate filtering window. If the number of noise-free pixels in W is less than 3, then let W = r_{1}+ r_{2}. By analogy, increase W by r_{i}(r_{1}, r_{2}, ... r_{5}) until the number of noise-free pixels selected exceeds 2. - If the number of noise-free pixels in W
_{5}(g) is 1 or 2, then let W = W_{5}(g). - If all pixels in W
_{5}(g) are noise, then a suitable filtering window cannot be obtained. In this case, the pixel needs to be further detected by method 2 in Section 2.2.2.

#### 2.2.2. Calculation of Noise Pixels Restored Value

_{5}(g) is greater than 0, then the method in 1 is used for denoising; otherwise, the method in 2 is used.

- In the spatial filtering theory, corrupted pixels can be restored using the normalized weighted mean of all pixels in the neighborhood. The noise restored value can be calculated as (3). Replace the noise pixel value with the restored value, and set R(g) = 0.

- 2.
- If g is in the extreme intensity flat regions, then the recovery step is performed according to the formula (5) and set R(g) = 0; otherwise, the pixel is processed in Stage 3.

#### 2.3. Stage 3: Noise Removal by Iterative Approach

- For each pixel g with R(g) = 1, process g by the method proposed in stage 2.
- If R is not a zero matrix, repeat 1 until R becomes a zero matrix, but use the last reconstruct image as the input image. Otherwise, leave it unchanged. If all pixels in the image are noisy pixels, then the procedure should stop.

## 3. Simulation Results

_{f}and u

_{g}are the mean of f and g, respectively; σ

_{f}and σ

_{g}are the standard deviation of f and g, respectively; σ

_{fg}is the covariance between u

_{f}and u

_{g}; C

_{1}and C

_{2}are constants used to maintain stability, where L = 255 is the dynamic range of pixel values, and K

_{1}= 0.01, K

_{2}= 0.03.

#### 3.1. Evaluate by Visual Perception and Quantitative Measurements

#### 3.2. Evaluate by Computational Time

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Illustrated images: (

**a**) Distribution of windows r

_{i}(r

_{1}, r

_{2}, ... r

_{5}), (

**b**) Weighted matrix X.

**Figure 6.**Visual perception of IWMF versus different methods on Test018: (

**a**) Original image, (

**b**) Noisy image of density 0.9, (

**c**) AFMF, (

**d**) ASWMF, (

**e**) TSF, (

**f**) DBA, (

**g**) FSMMF, (

**h**) ERMI, (

**i**) AWMF, (

**j**) DAMF, (

**k**) MDBMF, and (

**l**) IWMF.

**Figure 7.**Visual perception of IWMF versus different methods on Test006: (

**a**) Original image, (

**b**) Noisy image of density 0.8, (

**c**) AFMF, (

**d**) ASWMF, (

**e**) TSF, (

**f**) DBA, (

**g**) FSMMF, (

**h**) ERMI, (

**i**) AWMF, (

**j**) DAMF, (

**k**) MDBMF, (

**l**) IWMF.

**Figure 8.**Visual perception of IWMF versus different methods on Lena: (

**a**) Original image, (

**b**) Noisy image of density 0.9, (

**c**) AFMF, (

**d**) ASWMF, (

**e**) TSF, (

**f**) DBA, (

**g**) FSMMF, (

**h**) ERMI, (

**i**) AWMF, (

**j**) DAMF, (

**k**) MDBMF, and (

**l**) IWMF.

**Figure 9.**Versus different methods on 100 natural images in terms of the average values of (

**a**) PSNR, and (

**b**) SSIM.

Noise Density, % | 10 | 30 | 50 | 70 | 90 | 10 | 30 | 50 | 70 | 90 |
---|---|---|---|---|---|---|---|---|---|---|

PSNR (dB) | SSIM (%) | |||||||||

AFMF | 37.9 | 34.9 | 31.8 | 29.0 | 23.1 | 97.8 | 94.9 | 91.6 | 85.8 | 70.2 |

DBA | 40.5 | 34.8 | 30.4 | 26.1 | 19.9 | 98.6 | 94.8 | 90.5 | 80.1 | 56.5 |

ASWMF | 42.2 | 36.1 | 32.3 | 29.0 | 23.8 | 98.9 | 96.3 | 92.3 | 85.7 | 67.2 |

TSF | 43.2 | 36.8 | 33.0 | 30.2 | 27.1 | 98.9 | 96.5 | 93.1 | 87.9 | 80.2 |

AWMF | 39.6 | 36.7 | 32.3 | 28.2 | 24.5 | 98.9 | 96.0 | 91.9 | 84.1 | 76.4 |

DAMF | 43.2 | 36.9 | 33.1 | 30.1 | 27.0 | 99.1 | 96.5 | 93.0 | 87.8 | 80.1 |

FSMMF | 40.9 | 34.3 | 30.5 | 27.9 | 23.9 | 98.7 | 95.2 | 89.6 | 83.7 | 73.2 |

ERMI | 42.2 | 36.9 | 31.7 | 29.5 | 25.7 | 99.0 | 96.6 | 91.7 | 86.7 | 75.3 |

MDBMF | 42.7 | 36.6 | 33.0 | 30.1 | 26.1 | 99.0 | 96.5 | 93.0 | 87.7 | 77.8 |

IWMF | 43.3 | 37.6 | 34.0 | 31.0 | 27.1 | 99.1 | 96.9 | 93.8 | 89.3 | 80.3 |

Noise Density, % | 10 | 30 | 50 | 70 | 90 | 10 | 30 | 50 | 70 | 90 |
---|---|---|---|---|---|---|---|---|---|---|

PSNR (dB) | SSIM (%) | |||||||||

AFMF | 37.5 | 36.5 | 32.8 | 30.5 | 24.9 | 96.2 | 97.2 | 94.7 | 89.3 | 78.6 |

DBA | 40.2 | 34.6 | 31.9 | 27.2 | 20.4 | 98.5 | 95.3 | 86.6 | 80.2 | 70.6 |

ASWMF | 43.0 | 37.0 | 33.3 | 30.1 | 25.4 | 99.3 | 97.5 | 94.5 | 89.2 | 71.9 |

TSF | 43.5 | 36.9 | 33.7 | 30.6 | 27.5 | 99.2 | 97.3 | 94.5 | 89.4 | 84.5 |

AWMF | 40.5 | 35.6 | 31.5 | 26.9 | 25.0 | 99.0 | 96.8 | 94.3 | 89.2 | 71.8 |

DAMF | 43.5 | 36.9 | 33.8 | 30.4 | 27.3 | 99.3 | 97.3 | 94.7 | 89.2 | 84.3 |

FSMMF | 41.3 | 34.9 | 31.8 | 29.5 | 26.8 | 99.1 | 96.2 | 92.2 | 87.2 | 78.5 |

ERMI | 42.1 | 36.7 | 32.5 | 30.1 | 27.8 | 99.2 | 97.4 | 92.9 | 88.6 | 80.7 |

MDBMF | 42.7 | 36.8 | 33.6 | 30.9 | 27.8 | 99.3 | 97.4 | 94.6 | 90.6 | 83.3 |

IWMF | 44.0 | 37.9 | 34.8 | 31.7 | 28.5 | 99.5 | 98.0 | 95.8 | 91.9 | 84.5 |

Noise Density, % | 10 | 30 | 50 | 70 | 90 | 10 | 30 | 50 | 70 | 90 |
---|---|---|---|---|---|---|---|---|---|---|

PSNR (dB) | SSIM (%) | |||||||||

AFMF | 34.3 | 32.1 | 30.7 | 27.1 | 23.5 | 95.3 | 92.9 | 88.0 | 79.8 | 68.2 |

DBA | 36.1 | 31.2 | 28.1 | 23.5 | 18.9 | 97.2 | 90.1 | 80.6 | 73.2 | 59.9 |

ASWMF | 38.9 | 33.4 | 30.1 | 27.3 | 23.1 | 98.6 | 95.0 | 89.5 | 81.2 | 60.8 |

TSF | 39.1 | 33.3 | 30.5 | 28.3 | 25.2 | 98.5 | 95.1 | 89.9 | 83.4 | 71.5 |

AWMF | 38.0 | 33.2 | 30.1 | 27.4 | 22.5 | 98.4 | 94.5 | 88.5 | 80.2 | 60.0 |

DAMF | 39.1 | 33.5 | 30.6 | 28.1 | 25.0 | 98.5 | 95.2 | 90.1 | 82.9 | 71.4 |

FSMMF | 38.1 | 32.0 | 28.8 | 26.6 | 24.4 | 98.3 | 93.4 | 86.3 | 78.1 | 65.5 |

ERMI | 39.0 | 33.9 | 29.4 | 27.4 | 25.2 | 98.6 | 95.3 | 86.7 | 79.1 | 66.5 |

MDBMF | 39.1 | 33.6 | 30.5 | 28.0 | 25.1 | 98.6 | 95.1 | 90.0 | 82.9 | 70.6 |

IWMF | 40.3 | 34.7 | 31.4 | 28.6 | 25.6 | 98.9 | 96.1 | 91.6 | 84.3 | 71.6 |

**Table 4.**The average processing times were obtained by the denoising methods for 100 natural images.

Noise Density, % | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
---|---|---|---|---|---|---|---|---|---|

Time (ms) | |||||||||

AFMF | 22.44 | 23.61 | 28.97 | 36.16 | 50.77 | 60.25 | 70.97 | 94.59 | 116.6 |

DBA | 12.07 | 12.66 | 13.25 | 13.25 | 13.31 | 13.25 | 13.31 | 13.42 | 12.84 |

ASWMF | 7.95 | 19.79 | 21.91 | 22.49 | 25.15 | 28.74 | 33.04 | 34.51 | 29.68 |

TSF | 2.53 | 6.07 | 7.71 | 7.77 | 8.18 | 8.95 | 9.37 | 7.25 | 7.18 |

AWMF | 28.38 | 26.62 | 26.21 | 25.79 | 24.79 | 24.85 | 24.44 | 24.03 | 23.67 |

DAMF | 2.54 | 6.05 | 7.68 | 7.72 | 8.07 | 8.99 | 9.18 | 7.19 | 7.21 |

FSMMF | 3.59 | 6.12 | 9.54 | 13.61 | 16.37 | 16.93 | 19.14 | 19.91 | 19.96 |

ERMI | 0.94 | 1.59 | 2.29 | 3.29 | 9.77 | 10.36 | 10.48 | 11.36 | 23.09 |

MDBMF | 2.29 | 5.59 | 7.42 | 7.01 | 7.95 | 8.54 | 8.66 | 6.18 | 6.01 |

IWMF | 0.76 | 1.06 | 1.35 | 1.82 | 2.29 | 3.01 | 3.59 | 4.24 | 4.59 |

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**MDPI and ACS Style**

Chen, F.; Huang, M.; Ma, Z.; Li, Y.; Huang, Q.
An Iterative Weighted-Mean Filter for Removal of High-Density Salt-and-Pepper Noise. *Symmetry* **2020**, *12*, 1990.
https://doi.org/10.3390/sym12121990

**AMA Style**

Chen F, Huang M, Ma Z, Li Y, Huang Q.
An Iterative Weighted-Mean Filter for Removal of High-Density Salt-and-Pepper Noise. *Symmetry*. 2020; 12(12):1990.
https://doi.org/10.3390/sym12121990

**Chicago/Turabian Style**

Chen, Fengyu, Minghui Huang, Zhuxi Ma, Yibo Li, and Qianbin Huang.
2020. "An Iterative Weighted-Mean Filter for Removal of High-Density Salt-and-Pepper Noise" *Symmetry* 12, no. 12: 1990.
https://doi.org/10.3390/sym12121990