# Salt and Pepper Noise Removal with Multi-Class Dictionary Learning and L0 Norm Regularizations
## 5 Conclusions  
A new salt and pepper impulse noise removal method is proposed by simultaneously exploring: (1) The adaptively sparse representation of images; (2) better regularizing the sparsity of representation and the data fidelity under impulse noise. The former is accomplished via fast orthogonal dictionary learning within multiclass geometric image patches while the latter is enforced by regularizing L\({}_{0}\) norms on both the sparsity and data fidelity constraints. Experimental results demonstrate that this \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Noise Level**} \\ \cline{3-10}  & & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** \\ \hline \multirow{2}{*}{_Boat_} & L\({}_{1}\) & 43.07 & 40.58 & 38.60 & 36.99 & 35.20 & 33.54 & 31.58 & 27.35 \\  & L\({}_{0}\) & 46.07 & 43.68 & 41.78 & 40.26 & 38.67 & 37.04 & 35.21 & 33.59 \\ \hline \multirow{2}{*}{_[PERSON]_} & L\({}_{1}\) & 43.75 & 41.58 & 39.80 & 38.17 & 36.71 & 35.18 & 33.12 & 28.04 \\  & L\({}_{0}\) & 46.09 & 43.97 & 42.32 & 41.01 & 39.63 & 38.49 & 37.07 & 35.27 \\ \hline \multirow{2}{*}{_[PERSON]_} & L\({}_{1}\) & 42.06 & 39.67 & 37.65 & 35.74 & 34.08 & 32.38 & 30.45 & 26.21 \\  & L\({}_{0}\) & 45.62 & 43.27 & 41.35 & 39.44 & 37.48 & 35.63 & 33.52 & 32.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PSNR performance under the salt and pepper noise using FDCLP with L\({}_{1}\) or L\({}_{0}\) norms. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Noise Level**} \\ \cline{3-10}  & & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** \\ \hline \multirow{2}{*}{_Boat_} & L\({}_{1}\) & 0.9856 & 0.9743 & 0.9601 & 0.9407 & 0.9055 & 0.8571 & 0.7752 & 0.5537 \\  & L\({}_{0}\) & 0.9900 & 0.9830 & 0.9745 & 0.9631 & 0.9461 & 0.9139 & 0.8501 & 0.7178 \\ \hline \multirow{2}{*}{_Lena_} & L\({}_{1}\) & 0.9822 & 0.9711 & 0.9569 & 0.9391 & 0.9159 & 0.8819 & 0.8232 & 0.6458 \\  & L\({}_{0}\) & 0.9849 & 0.9759 & 0.9650 & 0.9531 & 0.9376 & 0.9188 & 0.8856 & 0.8208 \\ \hline \multirow{2}{*}{_[PERSON]_} & L\({}_{1}\) & 0.9867 & 0.9769 & 0.9624 & 0.9385 & 0.9001 & 0.8220 & 0.7058 & 0.4427 \\  & L\({}_{0}\) & 0.9909 & 0.9849 & 0.9766 & 0.9644 & 0.9412 & 0.8987 & 0.7709 & 0.6736 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MSSIM performance under the salt and pepper noise using FDCLP with L\({}_{1}\) or L\({}_{0}\) norms. Figure 8: Comparisons on quantitative measure using FDLCP-L\({}_{1}\) and FDLCP-L\({}_{0}\). (**a**)–(**c**) are PSNRs of _Boat_, _Lena_, and _Barbara_ images, respectively; (**d**)–(**f**) are MSSIMs of _Boat_, _Lena_, and _Barbara_ images, respectively. proposed approach outperforms the compared ones in terms of better preserving image structures and higher objective denoising performance evaluation criteria. The theoretical prof of the advantage of L\({}_{0}\) norm over L\({}_{1}\) norm in impulse noise removal would be an interesting future work. Conceptualization, D.G. ; Data curation, Z.T. and J.W. ; Formal analysis, Z.T. and J.W. ; Funding acquisition, D.G. and X.Q. ; Investigation, D.G. and X.D. ; Methodology, D.G. ; Project administration, D.G. ; Resources, M.X. ; Software, J.W. ; Supervision, X.Q. ; Writing--original draft, D.G. ; Writing--review & editing, D.G., Z.T., X.D. and X.Q. This work was supported in part by National Natural Science Foundation of China (No. 61871341, 61672335, and 61601276), the Science and Technology Program of Xiamen (No. 3502Z20183053), the Natural Science Foundation of Fujian Province of China (No. 2016J05205 and 2018J06018), the Fundamental Research Funds for the Central Universities (No. 20720180056). [PERSON] is grateful to [PERSON] for hosting the visit at University of Washington when writing this work and China Scholarship Council (No. 201808350010) to support the visit.