## 4 Prediction of OOP Existence and the Parameters in It
### Factors Affecting the Accuracy of Prediction  
There are also other factors affecting the accuracy of prediction. A first factor is noise realization. It is clear that, for a given test image and AWGN variance, the \(\Delta PSNR\)  
Figure 14: The noise-free test image Fr03 (**a**), this image corrupted by AWGN with noise variance equal to 25 (**b**) and the image compressed using \(Q_{\text{OOP}}=29\) (**c**), CR = 5.84. Figure 13: The scatter-plot \(\Delta PSNR-HVS-M\) vs. \(P_{2\sigma}\) and the fitted curve (the weighted sum of two exponentials, RMSE is about 1.64 and the R\({}^{2}\) and AdjR\({}^{2}\) are about 0.87). \(\Delta PSNR-HVS-M\), and even \(Q_{OOP}\) can vary from one noise realization to another. To study this aspect, we have analyzed three images of different complexity corrupted by AWGN with three different values. The values of \(\Delta PSNR\) and \(\Delta PSNR-HVS-M\) have been measured for \(Q_{OOP}\) and the variances of these parameters have been calculated. It has been established that variances of these parameters are about 0.001; even in the worst case the maximal variance was equal to 0.0033 for \(\Delta PSNR\) and 0.0046 for \(\Delta PSNR-HVS-M\). This means that the influence of realization can be neglected (recall that the RMSE of fitting is about 0.4 for \(\Delta PSNR\) and about 1.2 for \(\Delta PSNR-HVS-M\) in the area under interest). If the \(Q_{OOP}\) was different for different realizations, the differences were equal to 1 (e.g., an OOP was observed for Q equal to either 33 or 34) and the difference in values of \(\Delta PSNR\) or \(\Delta PSNR-HVS-M\) in the neighbor points was negligible (e.g., analyze the dependences in Figure 3 near their peaks). Thus, in practice, the influence of noise realization on the parameters of the OOP can be ignored. One more parameter that can influence the prediction is a possible variation of the input parameter. Having an approximation \(\Delta Metr(Q)\) and variance of the input parameter \(\sigma_{p}^{2}\), a variance of \(\Delta Metr(Q)\) can be estimated as \(Var_{\Delta Metr}^{2}\approx\sigma_{p}^{2}\left(\frac{d\Delta Metr(Q)}{dQ} \right)^{2}\). Then, \(\Delta Metr(Q)\) can be large in places where the absolute values of \(\frac{d\Delta Metr(Q)}{dQ}\) are large and if \(\sigma_{p}^{2}\) is quite large. The absolute values of \(\frac{d\Delta Metr(Q)}{dQ}\) are large for a large \(P_{2\sigma}\) or, equivalently, a small \(P_{2,7\sigma}\); however, we are more interested in places where the approximation curves cross the zero level. The absolute values of the derivatives there are about 50; thus, we need to understand what the values are of \(\sigma_{p}^{2}\). It is possible to expect that \(\sigma_{p}^{2}\) might depend on several factors, namely, the number of blocks \(M\), the properties of images and noise variance, and how the blocks are positioned in a considered image. To avoid possible problems with the images with some regular structures, we propose to apply a random positioning of blocks where indices of their left upper corner are (rounded-off) random variables in the limits from \(i=1\) till \(i=I_{Im}-7\) and from \(j=1\) till \(j=J_{Im}-7\), with a uniform distribution. Within this approach, we have first analyzed three test images of different complexity with three different noise variance values. A set of realizations of the random block positions have been generated for \(M=500\). A variance \(\sigma_{P}^{2}\) for the input parameter \(P_{2\sigma}\) was in the limits from \(5\times 10^{-6}\) till \(8\times 10^{-5}\). There is no obvious dependence on the image complexity but there is a tendency of \(\sigma_{P}^{2}\) decreasing if the AWGN variance increases (\(\sigma_{P}^{2}\) is about 0.00001 for an AWGN variance about 200 that usually corresponds to \(P_{2\sigma}\geq 0.5\)). The latter is a positive factor since \(\left(\frac{d\Delta Metr(Q)}{dQ}\right)^{2}\) is larger just for a large \(P_{2\sigma}\).