## 4 Prediction of OOP Existence and the Parameters in It
### Curve Fitting Details  
Having, in general, shown the possibility of a good curve fitting, we come to the next questions--what the best (appropriate) fitting is and how to characterize its accuracy. There are well developed theories of LMSE and robust fitting [46; 47]. A visual analysis of the scatter-plots in Figures 7-10 shows that there are no obvious outliers in the data; thus, there is no obvious necessity to apply a robust fitting. LMSE regression is implemented in many software tools including MATLAB and others; therefore, we can use one of them (the MATLAB Curve Fitting Tool in our case). To characterize the fitting accuracy, it is common [46] to use the root mean square (RMSE) that should be as small as possible, as well as a goodness-of-the-fit R\({}^{2}\) and/or adjusted goodness-of-the-fit AdjR\({}^{2}\) that have to be as large (close to the unity) as possible. Standard curve fitting tools usually provide several options of fitting functions including polynomials, exponential or weighted sum of exponents, rational functions, Fourier series, etc. Finding the best fitting function is more an engineering and heuristic rather than scientific task. Moreover, some solutions can be very close to each other according to quantitative criteria that does not allow for giving a unique practical recommendation. The curves fitted for the scatter-plots in Figures 8-11 have been obtained using rational functions. For the data in Figure 8, one has an RMSE about 0.37 and R\({}^{2}\) and AdjR\({}^{2}\) about 0.98. For the scatter-plot in Figure 9, the RMSE is about 1.72 and the R\({}^{2}\) and AdjR\({}^{2}\) are about 0.85. For the data in Figure 10, one has an RMSE about 0.43 and R\({}^{2}\) and AdjR\({}^{2}\) about 0.975. For the scatter-plot in Figure 11, the RMSE is about 1.66 and the R\({}^{2}\) and AdjR\({}^{2}\) are about 0.86. Supposing that the fitting is good enough, it is possible to conclude that the fitting for \(\Delta PSNR\) is considerably more accurate than for \(\Delta PSNR-HVS-M\) (this is not because of bad fitting but because of larger sparsity of data for \(\Delta PSNR-HVS-M\)). It is better to use \(P_{2\sigma}\) in predicting the \(\Delta PSNR\) and \(P_{2\sigma}\) in predicting the \(\Delta PSNR-HVS-M\). The parameters of the fitted curves are presented in Table 1.  
One might be interested whether or not other approximations are able to provide better results. The Fourier models, e.g., the Fourier model 3, are able to provide quite a good fitting in the sense of quantitative criteria (see Figure 12), but the obtained curves can be not monotonous, and that is not in agreement with our assumptions. The same relates to polynomial approximations. For polynomials of the order 2 and 3, the fitting accuracy parameters are worse than reported above, whilst for polynomials for the order 4 and 5, the curves become not monotonous. Finally, good approximations can be obtained for the weighted sums of two exponentials (see Figure 13 and compare it to Figure 9); thus, the use of rational functions or the sums of two exponentials can be considered an appropriate choice. Having the fitted curves at hand, it is possible to predict \(\Delta PSNR\) and \(\Delta PSNR-HVS-M\) for any image to be compressed. For this purpose, one has to calculate the input parameter and to substitute it into the approximating curve defined in Table 1. For example, the estimated \(P_{2\sigma}\) is equal to \(0.6\). Then, according to the data in Figure 7 (or Figure 12), the predicted \(\Delta PSNR\) is about \(-0.8\) dB, and the predicted \(\Delta PSNR-HVS-M\) is about \(-6.5\) dB according to the data in Figure 8 and about \(-6.1\) dB according to the approximation in Figure 13. In any case, one can conclude that the OOP for this image is absent and that it is not worth compressing it with \(Q_{\text{OOP}}\).