Directional TGVBased Image Restoration under Poisson Noise
Abstract
:1. Introduction
2. The KLDTGV${}^{2}$ Model
3. Efficient Estimation of the Image Direction
Algorithm 1 Direction estimation. 

4. ADMM for Minimizing the KLDTGV${}^{\mathbf{2}}$ Model
4.1. Solving the Subproblem in $\mathbf{x}$
4.2. Solving the Subproblem in $\mathbf{z}$
4.2.1. Update of ${\mathbf{z}}_{1}$
4.2.2. Update of ${\mathbf{z}}_{2}$ and ${\mathbf{z}}_{3}$
4.2.3. Update of ${\mathbf{z}}_{4}$
4.3. Summary of the ADMM Method
Algorithm 2 ADMM for problem (7). 

5. Numerical Results
5.1. Direction Estimation
5.2. Image Deblurring
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Blur  SNR  Model  $\mathit{\lambda}$  RMSE  ISNR  MSSIM  Iters  Time 

phantom  
Outoffocus  43  DTGV  57.5  2.2558 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  9.5472  9.3007 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  86  10.95 
TGV  275  2.8043 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.6568  8.9887 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  89  11.33  
37  DTGV  3.25  3.7573 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.4431  8.5823 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  122  15.45  
TGV  22.5  4.1719 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.5339  8.4061 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  52  6.64  
Gaussian  43  DTGV  25  1.5530 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  9.1966  9.7829 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  56  7.17 
TGV  100  1.8100 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.8667  9.7200 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  45  5.76  
37  DTGV  3  2.5498 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  9.0841  9.2994 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  90  11.41  
TGV  17.5  3.0674 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.4788  9.0199 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  53  6.76  
grass  
Outoffocus  43  DTGV  60  3.6313 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.7364  8.7262 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  136  15.55 
TGV  550  3.6575 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.6738  8.7188 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  179  20.39  
37  DTGV  50  5.6164 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.7390  7.6165 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  160  18.56  
TGV  55  5.7604 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.5191  7.4566 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  72  8.31  
Gaussian  43  DTGV  65  2.9883 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.3343  9.2764 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  106  12.08 
TGV  650  3.0814 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.0676  9.2523 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  136  15.48  
37  DTGV  5.5  4.2274 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.7973  8.5615 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  98  11.13  
TGV  35  4.3936 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.4624  8.4795 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  54  6.18  
leaves  
Outoffocus  43  DTGV  125  6.2767 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  7.4978  8.2099 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  251  31.18 
TGV  1100  8.2397 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  5.1342  7.1557 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  435  53.74  
37  DTGV  12.5  9.5597 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.1497  6.3065 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  257  31.87  
TGV  90  1.1874 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  2.2665  4.3294 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  113  14.03  
Gaussian  43  DTGV  150  7.3332 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.8675  7.7456 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  236  29.13 
TGV  1750  8.0857 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.0190  7.3001 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  380  46.77  
37  DTGV  12.5  9.0999 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  3.3907  6.6469 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  148  18.36  
TGV  100  1.0308 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  2.3081  5.6534 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  103  12.85  
carbon  
Outoffocus  43  DTGV  150  1.8360 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  1.2830 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  9.4734 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  331  13.78 
TGV  850  2.3825 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  1.0567 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  9.3671 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  233  9.73  
37  DTGV  20  3.1682 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  8.2416  8.6294 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  171  7.07  
TGV  150  3.8840 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.4723  8.2237 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  155  6.55  
Gaussian  43  DTGV  250  2.0453 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  8.6178  9.5974 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  305  12.53 
TGV  950  2.4839 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.9302  9.5698 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  171  7.12  
37  DTGV  15  2.7995 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  6.2017  9.3007 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  128  5.36  
TGV  150  3.3061 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$  4.7572  8.9690 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$  118  4.73 
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di Serafino, D.; Landi, G.; Viola, M. Directional TGVBased Image Restoration under Poisson Noise. J. Imaging 2021, 7, 99. https://doi.org/10.3390/jimaging7060099
di Serafino D, Landi G, Viola M. Directional TGVBased Image Restoration under Poisson Noise. Journal of Imaging. 2021; 7(6):99. https://doi.org/10.3390/jimaging7060099
Chicago/Turabian Styledi Serafino, Daniela, Germana Landi, and Marco Viola. 2021. "Directional TGVBased Image Restoration under Poisson Noise" Journal of Imaging 7, no. 6: 99. https://doi.org/10.3390/jimaging7060099