## 4 Prediction of OOP Existence and the Parameters in It
### Curve Fitting Details
An example of such a situation is given in Figure 14, where both the predicted and the true \(\Delta PSNR\) and \(\Delta PSNR-HVS-M\) are negative (the predicted values are equal to \(-0.71\) dB and \(-6.34\) dB whilst the true values are equal to \(-0.73\) dB and \(-5.09\) dB, respectively). As one can see, the compressed image is slightly smeared. \begin{table}
\begin{tabular}{c c c} \hline
**Dependence** & **Expression** & **Parameters** \\ \hline \multirow{6}{*}{\(\Delta PSNR\) on \(P_{2\sigma}\)} & \multirow{6}{*}{\(f(x)=(p_{1}x+p_{2})/(x^{3}+q_{1}x^{2}+q_{2}x+q_{3})\)} & \(p_{1}=1.533\times 10^{4}\), \\  & & \(p_{2}=-1.112\times 10^{4}\), \\  & & \(q_{1}=75.71\), \\  & & \(q_{2}=-6291\), \\  & & \(q_{3}=639\) \\ \hline \multirow{6}{*}{\(\Delta PSNR\) on \(P_{2\sigma}\)} & \multirow{6}{*}{\(f(x)=(p_{1}x+p_{2})/(x^{2}+q_{1}x+q_{2})\)} & \(p_{1}=-6.162\times 10^{5}\), \\  & & \(p_{2}=1.077\times 10^{5}\), \\  & & \(q_{1}=2.843\times 10^{5}\), \\  & & \(q_{2}=4501\) \\ \hline \multirow{6}{*}{\(\Delta PSNR-HVS-M\) on \(P_{2\sigma}\)} & \multirow{6}{*}{\(f(x)=(p_{1}x+p_{2})/(x^{2}+q_{1}x+q_{2})\)} & \(p_{1}=2.895\times 10^{5}\), \\  & & \(p_{2}=-2.549\times 10^{5}\), \\  & & \(q_{1}=-1.72\times 10^{4}\), \\  & & \(q_{2}=2.263\times 10^{4}\) \\ \hline \multirow{6}{*}{\(\Delta PSNR-HVS-M\) on \(P_{2\sigma}\)} & \multirow{6}{*}{\(f(x)=(p_{1}x+p_{2})/(x^{3}+q_{1}x^{2}+q_{2}x+q_{3})\)} & \(p_{1}=-10.97\), \\  & & \(p_{2}=0.558\), \\ \cline{1-1}  & & \(q_{1}=-1.99\), \\ \cline{1-1}  & & \(q_{2}=1.82\), \\ \cline{1-1}  & & \(q_{3}=0.048\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the fitted curves. Figure 12: The scatter-plot \(\Delta PSNR\) vs. \(P_{2\sigma}\) and the fitted curve of the model Fourier 3 (RMSE is about \(0.42\) and the R\({}^{2}\) and AdjR\({}^{2}\) are about \(0.976\)). One can argue that the prediction is not accurate for \(\Delta PSNR-HVS-M\) when \(P_{2\sigma}\) is about 0.2. Consequently, in this case, \(\Delta PSNR-HVS-M\) can be from \(-16\) dB to \(-7\) dB depending on the image at hand whilst the predicted values are about \(-10.5\) dB (see Figure 13). Such an accuracy of prediction seems low since the errors can be up to 5.5 dB; however, in practice, this is not a problem for several reasons. First, such a small \(P_{2\sigma}\) usually corresponds to images corrupted by low intensity noise which is invisible. Then, no positive effect of noise filtering can be seen anyway. Second, such \(\Delta PSNR-HVS-M\) values usually correspond to \(PSNR-HVS-M^{n}\) about 55...60 dB. Then, after compression, one has a \(PSNR-HVS-M^{tc}\) about 45...50 dB, i.e., the introduced distortions are invisible. Analyzing the scatter-plots for \(\Delta PSNR-HVS-M\), one can expect that the prediction accuracy for \(P_{2\sigma}>0.5\), which is of more interest, is considerably better than for \(P_{2\sigma}\leq 0.5\) (see, e.g., the scatter-plot in Figure 9). To check this hypothesis, we have calculated the RMSE of the fitted curves in the considered intervals. The RMSE for \(P_{2\sigma}\leq 0.5\) equals to 2.34 whilst it is equal to 1.21 for \(P_{2\sigma}>0.5\), i.e., our assumption is valid. Then, one can carry out quite an accurate prediction in the interval \(P_{2\sigma}>0.5\) where it is especially important to undertake decisions on the OOP's existence and \(Q\) setting.