## 2 Materials and Methods
### The Design of the No-Reference Combined Metrics for RS Images  
Improvement of no-reference metrics is a demanding topic of research. Robust and stable results still pose a challenge and are required for many tasks. One of the most effective approaches to increase their performance is the combination of elementary metrics. The design of the NR combined metrics that would reflect the distortions met in RS images is based on the \"Noise and Actual\" (NA) subset from the TID2013 database, concerning 13 selected types of distortions listed in Section 2.1. This subset contains 1625 images out of 3000 present in the whole database (it is still more than included in some other commonly used datasets, such as LIVE or CSIQ [52]). Considering the combination based on the weighted product, the generalized formula of the combined metric may be presented as:  
\[CM=\prod_{i=1}^{N}Q_{i}^{w_{i}}, \tag{1}\]  
assuming the elementary metrics \(Q_{i}\). The application of the second investigated approach utilizing the weighted sum expressed as:  
\[CM^{+}=\sum_{i=1}^{N}\bigl{(}a_{i}\;Q_{i}^{w_{i}}\bigr{)}, \tag{2}\]  
with additional weights \(a_{i}\), increasing the flexibility of the designed combined metric, makes it possible to increase its correlation with subjective scores. Such an approach to metrics' combination, applied previously for full-reference metrics, has led to encouraging results [30]. In both cases, the values of all parameters (weights) for the selected set of elementary metrics are obtained as the results of optimization using the direct search method based on the Nelder-Mead simplex. For this purpose, MATLAB _fminsearch_ function may be effectively used. The experiments have been started by the calculation of the SROCC values of elementary metrics for the NA subset. As shown in Table 1, the best accuracy of visual quality estimation for the NA subset is provided by IL-NIQE [25], which value reaches 0.72 by SROCC. Due to the relatively small number of possible combinations, each pair of metrics has been subject to both types of combinations, according to Equations (1) and (2) with optimization of parameters using the SROCC as the criterion. The further choice of elementary metrics for the combination using three or more of them has been based on the correlation of their subset: the five best combinations of two metrics have been selected as the basis and the third metric has been combined with them (all metrics have been checked and weights have been optimized). Then five best combinations of three metrics have been selected and each of the other elementary metrics has been checked as the potential fourth element, and so on. The reason for the choice of the five best combinations for each \(N\) is caused by the observation in previous experiments with full-reference metrics applied to multiply distorted images [30]. In some cases, the second or third \"best\" combination of \(N\) elementary metrics has led to better results in combination with the \((N+1)\)-th metric. An important element in the design of the combined metrics is also the calculation time. To verify the possible limitations some experiments have been conducted calculating some elementary metrics using the Intel i7 10 gen laptop processor. Although the execution time of many metrics is below 1 s, it is worth mentioning some of the \"slowest\" algorithms, namely BLIINDS2 (nearly 16 s), NJQA (over 8 s), IL-NIQE (over 6 s), ARISM/ ARISMc (about 5 s), OG-IQA (over 4 s), or CORNIA (nearly 3 s). Nevertheless, since the processing time may vary, the precise measurement results have not been presented, limiting this thread to the indication of the most computationally demanding metrics verified using an exemplary computer for comparison purposes. The graphical illustration of the differences is presented in Figure 1 in logarithmic scale with respect to the fastest metric.