## 6 Conclusions and Prospects  
In the field of image enhancement, smoothing methods are used to correct singularities. Among these methods, we are interested in the p(x)-Laplacian. It is generally used in the literature with a fixed control parameter. This choice does not allow for finer processing over the entire image. We proposed to apply this nonlinear regularisation operator, which we have made adaptive, in order to emphasise important visual information. We therefore  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Poisson-Distributed Noise**} \\ \hline
**Images** & **Filtering Method** & **DSSIM (sd)** & **PSNR (sd)** \\ \hline \multirow{4}{*}{Berkeley} & Proposal (2) & 0.019 (0.007) & 42.338 (1.364) \\ \cline{2-4}  & Tchumperle (1) & 0.016 (0.007) & 51.841 (0.957) \\ \cline{2-4}  & Gaussian filtering (4) & 0.069 (0.021) & 27.027 (2.571) \\ \cline{2-4}  & Peer group filtering (3) & 0.020 (0.007) & 37.488 (2.792) \\ \hline \multirow{4}{*}{KODAK} & Proposal (2) & 0.007 (0.003) & 43.908 (1.079) \\ \cline{2-4}  & Tchumperle (1) & 0.005 (0.003) & 51.311 (1.205) \\ \cline{2-4}  & Gaussian filtering (4) & 0.045 (0.020) & 29.017 (2.963) \\ \cline{2-4}  & Peer group filtering (3) & 0.007 (0.003) & 39.948 (2.914) \\ \hline \multirow{4}{*}{USC-SIPI} & Proposal (2) & 0.014 (0.007) & 44.434 (1.716) \\ \cline{2-4}  & Tchumperle (1) & 0.013 (0.007) & 51.266 (1.857) \\ \cline{2-4}  & Gaussian filtering (4) & 0.036 (0.019) & 30.971 (3.030) \\ \cline{2-4}  & Peer group filtering (3) & 0.015 (0.007) & 42.167 (3.261) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Results of the evaluation of the application of smoothing methods on the Berkeley, KODAK, USC-SIPI noisy image databases using the Poisson-distributed noise generated method.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Gaussian-Distributed Additive Noise**} \\ \hline
**Images** & **Filtering Method** & **DSSIM (sd)** & **PSNR (sd)** \\ \hline \multirow{4}{*}{Berkeley} & Proposal (2) & 0.019 (0.007) & 42.495 (1.385) \\ \cline{2-4}  & Tchumperle (1) & 0.016 (0.007) & 55.810 (2.214) \\ \cline{2-4}  & Gaussian filtering (4) & 0.069 (0.021) & 27.028 (2.571) \\ \cline{2-4}  & Peer group filtering (3) & 0.020 (0.007) & 37.556 (2.840) \\ \hline \multirow{4}{*}{KODAK} & Proposal (2) & 0.006 (0.003) & 44.052 (1.085) \\ \cline{2-4}  & Tchumperle (1) & 0.005 (0.003) & 54.110 (2.205) \\ \cline{2-4}  & Gaussian filtering (4) & 0.045 (0.020) & 29.020 (2.963) \\ \cline{2-4}  & peer Group filtering (3) & 0.007 (0.003) & 40.094 (2.988) \\ \hline \multirow{4}{*}{USC-SIPI} & Proposal (2) & 0.014 (0.007) & 44.559 (1.733) \\ \cline{2-4}  & Tchumperle (1) & 0.013 (0.007) & 53.738 (3.184) \\ \cline{2-4}  & Gaussian filtering (4) & 0.035 (0.019) & 30.972 (3.031) \\ \cline{2-4}  & Peer group filtering (3) & 0.015 (0.007) & 42.382 (3.374) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Results of the evaluation of the application of smoothing methods on the Berkeley, KODAK, USC-SIPI noisy image databases using the Gaussian-distributed additive noise method.  
proposed an adaptive control criterion that controls smoothing as a function of the image gradient, which we exploited using statistics. In order to evaluate this new model, we used the PSNR and DSSIM parameters. The former indicates a good reconstruction quality when its value is above 24 dB, and the latter assesses the quality of the filtering. From our test database, the results obtained put the average PSNR at 24.80 dB, with the worst result around a reasonable reconstruction. According to the DSSIM parameter, the quality of the image improved, provided that it had undergone many degradations, such as lossy compression.  
Conceptualization, J.-L.H., J.N. and J.V.; Writing--original draft, J.-L.H., J.N., J.V. and I.-P.M.; Visualization, J.-L.H., J.N. and J.V. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.