## 5 Methods for Estimation of HFR Video Characteristics
### Detection Model and Evaluation  
In the analysis of multifractal properties, different features were examined, like the left endpoint of a spectrum or the endpoint of a right tail [41]. Feature values directly rely on the content being analyzed. Thus, it is expected that properly selected features would enable a differentiation between temporally recovered sequences from corresponding original sequences. So, it is convenient to have a reference or a pseudoreference which will enable such differentiation. However, this is not an easy task in practice, where different  
Figure 4: An example of multifractal spectrum and corresponding characteristics. motion or entropy is related to various signals/sequences. Thus, multifractal features are performed on cumulative sums of trace sequences. Feature extraction is performed on the corresponding spectra in order to make differentiation between no frame rate change and the temporal recovery case [78]. In this paper, a TRR detection model is proposed. The focus is on differentiation between 120 fps original video and corresponding TRR video recovered from 60 fps compressed with HEVC and characterized by six crf values. The bases of the proposed approach are new multifractal features, representing primaly the relative width and height of multifractal spectra, calculated as  
\[w_{rel}=\frac{w_{r}}{w}=\frac{\alpha_{\max}-\alpha_{0}}{\alpha_{\max}-\alpha_{ \min}}, \tag{8}\]  
\[h_{rel}=\frac{a}{f(\alpha_{\max})}=\frac{f(\alpha_{\min})-f(\alpha_{\max})}{f( \alpha_{\max})}. \tag{9}\]  
The different quality of signals represented through crf values may show similar trends in the case of relative trends. Besides the slope corresponding to right side of a spectrum [79], the left side was examined utilizing information dimension. That is, a distance between two characteristic points is applied to describe the curve shape:  
\[d_{\inf}=\Big{(}(\alpha_{1}-\alpha_{0})^{2}+(f(\alpha_{1})-f(\alpha_{0}))^{2} \Big{)}^{1/2}. \tag{10}\]  
It is thought that it is possible to identify differences between the two groups, meaning original and TRR, i.e., to detect whether temporal recovery is performed or not in practice. Four multifractal features are selected in experimental analysis in order to develop a detection model without a reference using machine learning. Extracted features are utilized for machine-learning-based temporal recovery detection. Several classifiers are tested, like Support Vector Machine (SVM), decision tree and k-nearest neighbors (kNN) [80; 81; 82]. SVM is often applied for classification, with the ability to generate a hyperplane to perform differentiation between original and nonmodified data. The decision tree (DT) also represents one of the common choices when it comes to classification, where results are obtained according to a learned tree based on selected features. Also, one of the most popular picked solutions is the kNN algorithm. During the training/learning process, the algorithm tries to find k samples that are the closest to the targets. According to the selected metric, the distances between the specific point and the target point in the known category/group are calculated in a nonparametric approach. The smallest distances from the specific point (\(x_{q},y_{q}\)) are found, where the label of a point is found using majority voting:  
\[\hat{y}_{q}=\argmax_{j}\sum_{x_{i}\in kNN(x_{q})}I(y_{i}=j), \tag{11}\]  
where \(kNN\) in (11) denotes the neighborhood, and \(I\) is the indicator function. The main concept of kNN classification is illustrated in Figure 5a. Here, a modified kNN or weighted kNN (WkNN)-based model is proposed for TRR detection, and it is presented in Figure 5b. After generating trace sequences, where each video from the database contributes to each group (TRR and non-TRR meaning original), feature extraction is applied using the multifractal features explained in the previous paragraph.