# Image Denoising Based on Bivariate Distribution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Algorithm

**w**for the observation

**y**is

- (1)
- First implement the DTCWT to get ${\mathbf{y}}_{k}$.
- (2)
- Calculate ${\widehat{\sigma}}_{n}^{2}$ using Equation (9).
- (3)
- Estimate marginal variance ${\sigma}^{2}$ using Equation (8).
- (4)
- Calculate the parameter ${\widehat{\gamma}}^{2}$ using Equation (10).
- (5)
- (6)
- Reconstruct the estimated image by ${\widehat{w}}_{1}$.

## 3. Experimental Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Histogram of the high-band dual-tree complex wavelet coefficients of boat image. Solid line: fitted curve of the same coefficients scaled by the proposed marginal PDF, for $\theta =4$. Dashed line: fitted curve of Gaussian PDF.

**Figure 3.**New bivariate shrinkage function derived from the model proposed in (2).

**Table 1.**Peak signal-to-noise ratio values of denoised images different test images and noise levels (${\sigma}_{n})$ of noisy.

Noisy | Method in [2] | Method in [6] | Method in [3] | Method in [8] | Proposed Method | |
---|---|---|---|---|---|---|

Barbara | ||||||

${\sigma}_{n}=10$ | 28.13 | 31.96 | 32.73 | 34.03 | 33.29 | 33.4417 |

${\sigma}_{n}=15$ | 24.61 | 29.57 | 30.56 | 31.86 | 31.17 | 31.3435 |

${\sigma}_{n}=20$ | 22.11 | 27.91 | 28.80 | 30.32 | 29.66 | 29.8558 |

${\sigma}_{n}=25$ | 20.17 | 26.72 | 27.45 | 29.13 | 28.52 | 28.7222 |

${\sigma}_{n}=30$ | 18.59 | 25.77 | 26.36 | 28.10 | 27.61 | 27.8130 |

boat | ||||||

${\sigma}_{n}=10$ | 28.13 | 32.22 | 33.20 | 33.58 | 32.99 | 33.1018 |

${\sigma}_{n}=15$ | 24.61 | 30.37 | 31.61 | 31.70 | 31.23 | 31.3226 |

${\sigma}_{n}=20$ | 22.11 | 28.97 | 30.28 | 30.38 | 29.94 | 30.0276 |

${\sigma}_{n}=25$ | 20.17 | 27.88 | 29.17 | 29.37 | 28.93 | 29.0215 |

${\sigma}_{n}=30$ | 18.59 | 27.03 | 28.14 | 28.51 | 28.12 | 28.2095 |

Lena | ||||||

${\sigma}_{n}=10$ | 28.13 | 34.07 | 34.92 | 35.61 | 35.29 | 35.3183 |

${\sigma}_{n}=15$ | 24.61 | 32.20 | 33.24 | 33.90 | 33.57 | 33.5090 |

${\sigma}_{n}=20$ | 22.11 | 30.86 | 31.99 | 32.66 | 32.33 | 32.2410 |

${\sigma}_{n}=25$ | 20.17 | 29.86 | 31.00 | 31.69 | 31.35 | 31.2753 |

${\sigma}_{n}=30$ | 18.59 | 29.02 | 30.14 | 30.91 | 30.54 | 30.4862 |

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**MDPI and ACS Style**

Zhao, P.; Zhao, X.; Zhao, C.
Image Denoising Based on Bivariate Distribution. *Symmetry* **2020**, *12*, 1909.
https://doi.org/10.3390/sym12111909

**AMA Style**

Zhao P, Zhao X, Zhao C.
Image Denoising Based on Bivariate Distribution. *Symmetry*. 2020; 12(11):1909.
https://doi.org/10.3390/sym12111909

**Chicago/Turabian Style**

Zhao, Ping, Xingyu Zhao, and Chun Zhao.
2020. "Image Denoising Based on Bivariate Distribution" *Symmetry* 12, no. 11: 1909.
https://doi.org/10.3390/sym12111909