A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise
Abstract
:1. Introduction
2. Preliminaries
3. Some Related Works and Motivation
3.1. Review of Some Related Works
3.2. The Property of Telegraph Diffusion Based Methods
4. The FTDE Model and Its Theoretical Analysis
4.1. The Proposed FTDE Model
4.2. Existence and Uniqueness of the Solution
- (i)
- ,
- (ii)
- ,
5. Numerical Scheme and Experiments
5.1. Numerical Scheme
Algorithm 1 DFT-based algorithm of the proposed model |
Input: Noisy level L, original image , Initialize:, Output: |
5.2. Numerical Experiments and Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Image | Texture1 | Texture2 | Satellite1 | Satellite2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
L | 1 | 4 | 10 | 1 | 4 | 10 | 1 | 4 | 10 | 1 | 4 | 10 |
b | 6 | 4 | 4 | 6 | 3 | 4 | 3 | 3 | 3 | 5 | 4 | |
Parameters | |||
---|---|---|---|
Algorithm | |||
AA () | 200 | 400 | 600 |
SO () | |||
CV () | |||
TD () | |||
NTV () | |||
DD () | |||
FDE () |
Model | SO | CV | AA | TD | NTV | DD | FDE | BM3D-SAR | Ours |
---|---|---|---|---|---|---|---|---|---|
PSNR | 9.31 | 10.98 | 12.65 | 11.15 | 12.45 | 12.57 | 12.58 | 12.83 | 12.82 |
MAE | 68.50 | 49.91 | 46.68 | 52.57 | 46.39 | 47.55 | 46.18 | 41.63 | 45.56 |
SSIM | 0.15 | 0.31 | 0.25 | 0.21 | 0.31 | 0.25 | 0.40 | 0.36 | 0.40 |
IICC | 0.52 | 0.43 | 0.54 | 0.38 | 0.53 | 0.46 | 0.58 | 0.59 | 0.55 |
PSNR | 14.65 | 13.73 | 13.90 | 13.87 | 15.27 | 14.55 | 15.78 | 15.85 | 15.70 |
MAE | 34.81 | 39.11 | 41.75 | 38.42 | 32.15 | 36.66 | 31.62 | 29.31 | 32.22 |
SSIM | 0.57 | 0.45 | 0.37 | 0.40 | 0.63 | 0.56 | 0.69 | 0.67 | 0.68 |
IICC | 0.76 | 0.65 | 0.76 | 0.64 | 0.79 | 0.71 | 0.80 | 0.81 | 0.79 |
PSNR | 17.13 | 15.28 | 14.80 | 16.56 | 17.85 | 17.27 | 18.22 | 18.30 | 18.07 |
MAE | 25.98 | 33.09 | 37.54 | 27.30 | 23.41 | 26.28 | 23.48 | 21.81 | 24.14 |
SSIM | 0.73 | 0.59 | 0.48 | 0.63 | 0.80 | 0.77 | 0.82 | 0.82 | 0.81 |
IICC | 0.86 | 0.75 | 0.84 | 0.80 | 0.88 | 0.86 | 0.89 | 0.89 | 0.88 |
Model | SO | CV | AA | TD | NTV | DD | FDE | BM3D-SAR | Ours |
---|---|---|---|---|---|---|---|---|---|
PSNR | 10.96 | 12.56 | 14.35 | 13.92 | 14.92 | 15.30 | 15.01 | 14.19 | 15.52 |
MAE | 50.46 | 37.44 | 33.56 | 34.81 | 30.41 | 30.97 | 32.30 | 29.02 | 30.96 |
SSIM | 0.23 | 0.37 | 0.39 | 0.29 | 0.37 | 0.39 | 0.43 | 0.43 | 0.45 |
IICC | 0.72 | 0.62 | 0.75 | 0.64 | 0.74 | 0.74 | 0.76 | 0.71 | 0.76 |
PSNR | 17.66 | 16.55 | 15.38 | 16.61 | 17.99 | 17.56 | 18.06 | 17.47 | 18.17 |
MAE | 22.10 | 25.84 | 31.45 | 24.85 | 20.91 | 23.18 | 22.58 | 21.14 | 22.53 |
SSIM | 0.60 | 0.49 | 0.51 | 0.51 | 0.63 | 0.64 | 0.67 | 0.62 | 0.68 |
IICC | 0.87 | 0.82 | 0.87 | 0.82 | 0.88 | 0.85 | 0.88 | 0.86 | 0.88 |
PSNR | 19.80 | 18.28 | 15.74 | 19.07 | 20.45 | 20.04 | 20.62 | 20.22 | 20.48 |
MAE | 16.97 | 20.76 | 30.43 | 18.52 | 15.62 | 17.23 | 16.56 | 15.64 | 16.87 |
SSIM | 0.75 | 0.65 | 0.56 | 0.70 | 0.80 | 0.79 | 0.80 | 0.79 | 0.80 |
IICC | 0.92 | 0.88 | 0.91 | 0.90 | 0.93 | 0.92 | 0.93 | 0.93 | 0.93 |
Model | SO | CV | AA | TD | NTV | DD | FDE | BM3D-SAR | Ours |
---|---|---|---|---|---|---|---|---|---|
PSNR | 13.02 | 15.44 | 17.77 | 17.47 | 18.56 | 20.26 | 19.61 | 20.50 | 20.47 |
MAE | 42.07 | 25.80 | 23.09 | 22.33 | 20.57 | 17.75 | 18.84 | 16.55 | 17.46 |
SSIM | 0.28 | 0.36 | 0.64 | 0.46 | 0.46 | 0.73 | 0.73 | 0.74 | 0.76 |
IICC | 0.82 | 0.67 | 0.79 | 0.68 | 0.81 | 0.85 | 0.85 | 0.86 | 0.86 |
PSNR | 20.64 | 21.75 | 21.20 | 21.40 | 21.67 | 22.60 | 22.89 | 23.40 | 23.18 |
MAE | 16.15 | 14.59 | 15.39 | 14.31 | 14.15 | 13.13 | 12.94 | 11.59 | 12.58 |
SSIM | 0.78 | 0.60 | 0.86 | 0.80 | 0.66 | 0.86 | 0.88 | 0.88 | 0.88 |
IICC | 0.90 | 0.90 | 0.90 | 0.88 | 0.90 | 0.92 | 0.92 | 0.93 | 0.93 |
PSNR | 23.62 | 24.04 | 23.96 | 23.93 | 23.74 | 24.64 | 24.97 | 25.66 | 25.19 |
MAE | 11.33 | 11.03 | 11.05 | 10.74 | 11.17 | 10.30 | 10.17 | 8.93 | 9.91 |
SSIM | 0.89 | 0.90 | 0.92 | 0.92 | 0.93 | 0.93 | 0.93 | 0.94 | 0.93 |
IICC | 0.94 | 0.94 | 0.95 | 0.94 | 0.94 | 0.95 | 0.95 | 0.96 | 0.95 |
Model | SO | CV | AA | TD | NTV | DD | FDE | BM3D-SAR | Ours |
---|---|---|---|---|---|---|---|---|---|
PSNR | 11.03 | 13.74 | 16.32 | 15.38 | 16.52 | 17.74 | 16.33 | 17.90 | 17.77 |
MAE | 56.07 | 34.88 | 29.43 | 29.66 | 27.68 | 24.94 | 29.29 | 23.77 | 24.97 |
SSIM | 0.28 | 0.25 | 0.53 | 0.48 | 0.32 | 0.59 | 0.58 | 0.61 | 0.61 |
IICC | 0.78 | 0.65 | 0.79 | 0.68 | 0.78 | 0.81 | 0.78 | 0.82 | 0.81 |
PSNR | 18.68 | 18.90 | 17.43 | 18.76 | 19.18 | 19.53 | 19.67 | 20.07 | 19.90 |
MAE | 21.80 | 21.23 | 26.48 | 20.93 | 20.20 | 29.71 | 19.85 | 18.33 | 19.42 |
SSIM | 0.75 | 0.44 | 0.65 | 0.73 | 0.54 | 0.77 | 0.78 | 0.80 | 0.79 |
IICC | 0.88 | 0.86 | 0.88 | 0.85 | 0.88 | 0.88 | 0.89 | 0.89 | 0.89 |
PSNR | 21.07 | 20.65 | 17.86 | 20.88 | 21.10 | 21.34 | 21.60 | 21.92 | 21.72 |
MAE | 16.32 | 17.17 | 25.37 | 16.59 | 16.13 | 16.04 | 15.85 | 14.80 | 15.68 |
SSIM | 0.87 | 0.82 | 0.69 | 0.85 | 0.88 | 0.86 | 0.87 | 0.88 | 0.87 |
IICC | 0.92 | 0.91 | 0.91 | 0.91 | 0.92 | 0.92 | 0.93 | 0.93 | 0.93 |
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Bai, X.; Zhang, D.; Shi, S.; Yao, W.; Guo, Z.; Sun, J. A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise. Fractal Fract. 2023, 7, 64. https://doi.org/10.3390/fractalfract7010064
Bai X, Zhang D, Shi S, Yao W, Guo Z, Sun J. A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise. Fractal and Fractional. 2023; 7(1):64. https://doi.org/10.3390/fractalfract7010064
Chicago/Turabian StyleBai, Xiangyu, Dazhi Zhang, Shengzhu Shi, Wenjuan Yao, Zhichang Guo, and Jiebao Sun. 2023. "A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise" Fractal and Fractional 7, no. 1: 64. https://doi.org/10.3390/fractalfract7010064
APA StyleBai, X., Zhang, D., Shi, S., Yao, W., Guo, Z., & Sun, J. (2023). A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise. Fractal and Fractional, 7(1), 64. https://doi.org/10.3390/fractalfract7010064