Feature‐based no‐reference video quality assessment using Extra Trees

IET Image ProcessingVolume 16, Issue 6 p. 1531-1543 ORIGINAL RESEARCH PAPEROpen Access Feature-based no-reference video quality assessment using Extra Trees Hatef Otroshi-Shahreza, Hatef Otroshi-Shahreza Electrical Engineering Department, Sharif University of Technology, Tehran, Iran School of Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, SwitzerlandSearch for more papers by this authorArash Amini, Corresponding Author Arash Amini [email protected] orcid.org/0000-0002-7082-9581 Electrical Engineering Department, Sharif University of Technology, Tehran, Iran Correspondence Arash Amini, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran. Email: [email protected]Search for more papers by this authorHamid Behroozi, Hamid Behroozi Electrical Engineering Department, Sharif University of Technology, Tehran, IranSearch for more papers by this author Hatef Otroshi-Shahreza, Hatef Otroshi-Shahreza Electrical Engineering Department, Sharif University of Technology, Tehran, Iran School of Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, SwitzerlandSearch for more papers by this authorArash Amini, Corresponding Author Arash Amini [email protected] orcid.org/0000-0002-7082-9581 Electrical Engineering Department, Sharif University of Technology, Tehran, Iran Correspondence Arash Amini, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran. Email: [email protected]Search for more papers by this authorHamid Behroozi, Hamid Behroozi Electrical Engineering Department, Sharif University of Technology, Tehran, IranSearch for more papers by this author First published: 15 February 2022 https://doi.org/10.1049/ipr2.12428AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract With the emergence of social networks and improvements in the internet speed, the video data has become an ever-increasing portion of the global internet traffic. Besides the content, the quality of a video sequence is an important issue at the user end which is often affected by various factors such as compression. Therefore, monitoring the quality is crucial for the video content and service providers. A simple monitoring approach is to compare the raw video content (uncompressed) with the received data at the receiver. In most practical scenarios, however, the reference video sequence is not available. Consequently, it is desirable to have a general reference-less method for assessing the perceived quality of any given video sequence. In this paper, a no-reference video quality assessment technique based on video features is proposed. In particular, a long list of video features (21 sets of features, each consisting of 1 to 216 features) is considered and all possible combinations ( 2 21 − 1 ${2^{21}-1}$ ) for training an Extra Trees regressor is examined. This choice of the regressor is wisely selected and is observed to perform better than other common regressors. The results reveal that the top 20 performing feature subsets all outperform the existing feature-based assessment methods in terms of the Pearson linear correlation coefficient (PLCC) or the Spearman rank order correlation coefficient (SROCC). Specially, the best performing regressor achieves PLCC = 0.786 ${\text{ PLCC}=0.786}$ on the test data over the KonVid-1k dataset. It is believed that the results of the comprehensive comparison could be potentially useful for other feature-based video-related problems. The source codes of the implementations are publicly available. 1 INTRODUCTION With the excessive growth of video content and video applications, the quality assessment of video sequences has become of great importance. According to ref. [1], video content will account for 82 % $82\%$ of global internet traffic in 2022 from which 17 % $17\%$ is dedicated to live streams. Depending on the involved processes, the quality of video sequences might be different. Most notably, optical distortions imposed by the camera at the time of recording, distortions caused by the data compression, and the data loss within the communication channel affect the overall quality. In addition, at the receiving end, the original/reference video file is rarely available. This eliminates the possibility of using a full-reference video quality assessment (FR-VQA) algorithms. In this case, the class of no-reference video quality assessment (NR-VQA) algorithms is the only option. These methods predict the quality of a video file according to certain quality measures (e.g. mean opinion score, or shortly MOS) solely based on the available data. The prediction procedure oftentimes relies on the statistical features of the video files. Some of the known no-reference assessment methods are V-BLINDS [2], VIIDEO [3], V-CORNIA [4], and STFC [5]. For measuring the performance of an assessment method, it is common to check the outcome of the method on a database which is already rated by the users. The LIVE video quality assessment database [6] has been a frequently-used database for evaluating the performance of NR-VQA methods in the past. This database contains ten video files each accompanied with 15 synthetically distorted versions and the ratings of 38 users. Methods such a [2, 3, 7] have reported very good performances over this dataset. Aside from the small size of this dataset, different distortion types are separately applied to the source files, which is somehow different from the practical scenarios where a video file usually suffers from multiple distortion types with various degrees. Recently, the KonVid-1k database with 1200 video files is made available [8] which contains real distorted video files. With respect to the size and distortion types of the files, this database is expected to provide a more realistic test-bed for NR-VQA methods. It is counterintuitive to mention that some of the successful methods on the LIVE Video Quality database perform poorly on the KonVid-1k database [8]. Most NR-VQA techniques rely on learning the quality measure from a training dataset. For the learning process, one can either use the full video content or its reduced version in form of a number of features. The latter approach, besides the advantage of less computational cost, can be applied to arbitrary video frame sizes (if the features are properly chosen). In this paper, we first cover a list of 21 feature sets and consider all 2 21 − 1 $2^{21}-1$ combinations of these feature sets to train an Extra Trees regressor; we should highlight that the support vector regressor (SVR) has been the dominant regressor in the existing techniques and the use of Extra Trees regressor in this work is its first appearance in the VQA-related context. Based on the achieved performance of the trained regressors, we discuss how the features can be efficiently selected to achieve good results while avoiding high computational costs. By dividing the features into two categories of frame-level (within one video frame) and spatiotemporal (considering the frames over time), we find out that the statistics of the luma channel is the most informative frame-level feature (indeed, the most informative among all features). The results also assign significant weight to frame-level HOSA [9] and CORNIA scores as features. Furthermore, the most relevant spatiotemporal feature set is generated by the V-BLINDS method. We also define certain angular-type spatiotemporal features in this paper that enhance the V-BLINDS features. Contributions In this paper, we present a machine-learning based study on the problem of NR-VQA. To this end, we provide a list of video features employed in the existing NR-VQA methods, as well as some new features defined in this work. Next, we evaluate the performance of NR-VQA models formed by combining different features. This study consists of evaluating the effect of both the regressor type and the feature combinations. While the choice of the regressor is often ignored in the NR-VQA literature, we observe that it has a significant impact on the performance. To identify important features in predicting the video quality, we perform several experiments and comparisons. Specifically, we find that a success model needs to have a balanced combination of frame-level and spatiotemporal features. We further propose new spatiotemporal features that could be efficiently combined with frame-level features. In brief, the contributions of this paper are We explore the effect of the regressor on the outcome of the NR-VQA on a small-size problem. The best performing regressor (Extra Trees regressor) is then used for the remaining large-size problems. We introduce new spatiotemporal features by considering the video content as a 3D cube; some of these features later appear in high-performance NR-VQA feature combinations. We study the performance of all possible subsets of the features to identify the significance of each feature in the overall performance of a NR-VQA method; in particular, we report the performance of the pairs of the feature sets to introduce the best matching features. We investigate the importance of each feature group through a decision tree based regressor. The feature groups achieving the highest scores in this experiment are interpreted as most relevant feature groups in predicting video quality. Interestingly, the results match that of the comprehensive feature subset experiment (feature groups appearing in the top performing subsets). We introduce several feature subsets with small size that perform very well; the restriction on the size of the feature subsets helps us to cut the unnecessary computational cost. The top 20 performing feature subsets have competitive performance with the state-of-the-art feature-based NR-VQA (even deep-learning approaches) Structure of the paper We first review the related works in Section 2. Next, we describe the proposed method in Section 3 and provide experimental results in Section 4. Finally, the paper is concluded in Section 5. 2 RELATED WORKS In most of the previous studies of NR-VQA, the quality measure of mean opinion score (MOS) is predicted by applying the SVR method on a set of statistical features extracted from the video data. For instance, ref. [2] uses natural video statistics (NVS), frame-level natural scene statistics (NSS) (i.e. features introduced in ref. [10]), and a number of defined motion-related features. In ref. [3], the statistics of frame differences directly determine the quality measure (without regression). Although remarkable performance is reported on the LIVE VQA database, the performance of this method on the KonVid-1k database is disappointing [11]. A video file can be considered as the concatenation of multiple images. Based on this fact, the image quality assessment technique in ref. [4] is applied in ref. [12] to get the video quality metric by aggregation of frame-by-frame quality metrics. A set of five features (i.e. average frame blurriness, average frame contrast, average frame colourfulness, average spatial information in frames, and average temporal information) is introduced in ref. [11] to predict the MOS via SVR. Since most of the feature extractions in ref. [11] are performed frame-by-frame, in order to consider temporal variations, a number of spatiotemporal features are added to the previous five features in ref. [5]. Similarly, two types of regressions (namely, a SVR and a neural network) are trained in ref. [13] by using a set of features from spatiotemporal slices. The NARVAL model in ref. [14], also combines frame-level and video-level features. A modified version of this model called BB-NARVAL takes advantage of natural scene statistics reported in refs. [2, 15]. In ref. [16], the frame-level features are divided into two computational categories of low-complexity and high-complexity; the low-complexity features are extracted from all the frames, while the high-complexity features are extracted from only a subset of the frames. Low-complexity and high-complexity features are concatenated into a single feature vector for training a SVR and a random forest regression (named TLVQM-SVR and TLVQM-RFR, respectively). In ref. [17], several feature-based NR-VQA models are employed to extract statistical video features which are then combined to train a quality prediction SVR. The deep-learning approaches for video quality assessment are quite recent due to their high computational requirements. In refs. [18, 19], a 2D convolutional neural network (CNN) with recurrent neural networks are proposed. A 3D-CNN with recurrent units is also presented in ref. [20]. Combining the feature-based techniques and deep-learning approaches, ref. [21] uses a pooling of the frame-level feature maps of Inception-V3 [22] and Inception-ResNet-V2 [23] to train a SVR for each model. Several slices of the spatiotemporal cube are used to train a CNN in ref. [24] which generates the overall score by averaging over the patch scores. In ref. [25], a CNN is employed to extract frame-level features which are then, fed to a gated recurrent unit (GRU). Similarly, in refs. [26, 27], hand-crafted frame-level features are fed to long short-term memory (LSTM) unites. In ref. [28], a deep neural network is proposed which consists of a quality degradation learning sub-network (with convolutional layers) and a motion effect modelling sub-network (with recurrent layers). In ref. [29], several features (12 feature values per frame and 108 pooled feature values per video sequence) are extracted from each video and then, the number of features are reduced after a feature selection using the Extra Trees. Finally, a random forest regression is applied to predict the video quality score. However, no analytical results is reported for the selected features and the importance of the selected features. As explained above, the main distinguishing part in different feature-based methods (not using deep learning) is the set of used features. Indeed, the SVR has been the dominant regressor. In this paper, we consider a collection of features in the existing works and apply different regressors in a single test. By choosing the best regressor in this test (Extra Trees), we find the optimal feature-set selection by an exhaustive search. 3 PROPOSED METHOD Overview As mentioned earlier, we consider machine learning approaches for the purpose of no-reference video quality assessment. After describing several video quality metrics in Section 3.2, we shall train several regressors. From machine-learning point of view, each studied method consists of a number of features and a regressor, which translates the feature values into a single quality score. To distinguish between the impacts of the features and the regressors in the final performance, we consider two experiments. In the first experiment, by including a full list of features, we train various regressors. The goal of this experiment is to evaluate the effect of the regressor and to identify the best choice. In the second experiment, we fix the regressor (the best performer in the first experiment) and repeat the training phase by considering all combinations of the features. The result of the latter experiment is likely to identify the features with the highest impact on the quality assessment task. Based on this experiment, we can also find the frequency of features in the top performing models and use that as an importance metric for each feature. In addition, as a special case, we check the performance of single and pairs of features to provide a better intuition about the overlapping or complementary nature of the information encoded by the features. As another metric for the importance of each feature, we train an Extra Trees regressor and find the mean decrease impurity (MDI) score [30] over all trees in the trained regressor. Video features A video file can be considered as a cube of spatiotemporal data indexed by x $x$ , y $y$ , and t $t$ axes; x $x$ and y $y$ represent the horizontal and vertical spatial axes, while t $t$ represents the time axis. The standard frames of the video are simply the x y $xy$ slices of this cube. Similarly, one can define x t $xt$ and y t $yt$ slices as spatiotemporal frames. As each video pixel consists of luma and chroma components, we can decompose the video cube into one luma cube and two chroma cubes. For better illustration, a sample video cube and the mentioned slices are shown in Figure 1. FIGURE 1Open in figure viewerPowerPoint A video file can be viewed in a 3D spatiotemporal space, where two of the axes indicates the spatial dimensions ( x $x$ and y $y$ ) and the third axis represents the time ( t $t$ ). (a) representation of RGB colour space in spatiotemporal space. (b) Representation of luminance channels in spatiotemporal space. (c) Spatial and spatiotemporal slices of the video In this work, we consider various features extracted from the frames ( x y $xy$ slices), frame differences (temporal features) or spatiotemporal slices ( x t $xt$ and y t $yt$ ). To make the method independent of the video length and frame size, we use the statistics of these features (e.g. mean and variance) instead of their raw values. The summary of considered features and their execution time1 are provided in Table 1. The explanation of these features is provided below. TABLE 1. Summary of video features Feature # features Exe. Time 1 Frame rate 1 0 2 Spatial gradient 24 1.9 3 Spatiotemporal gradient 48 9.9 4 Spatial Laplacian 12 1.8 5 Spatiotempral Laplacian 24 7.3 6 Spatial gradient amplitude 12 2.0 7 Spatiotempral gradient amplitude 24 10.1 8 Spatial angular information 8 2.0 9 Spatiotempral angular information 16 10.1 10 Luma information 12 2.3 11 Chroma information 24 3.9 12 Temporal information 12 2.6 13 Colourfulness 42 2.9 14 LMSCN statistics 216 39.7 15 BRISQUE scores 6 39.8 16 V.BLINDS 46 771.7 (16_NSS) (37) (99.1) (16_NVS) (7) (539.5) (16_Motion) (2) (133.1) 17 VIIDEO features 72 225.4 18 HOSA scores 6 62.8 19 CORNIA scores 6 4.5 3.2.1 Frame rate The number of frames within each second of video. This value is a single-element feature set which can identify low quality video files, when the frame rate is below the standards. 3.2.2 Spatial gradient Each video frame represents an image; the horizontal and vertical gradients (derivatives) of this image reveal the edges. For this feature, we apply two 3 × 3 $3\times 3$ filter kernels (horizontal and vertical directions) on the luminance channel of the frame. Next, we compute the mean and standard deviation for each frame (four values in total). The final features in this set are determined by the minimum, maximum, mean, standard deviation, skewness, and kurtosis of these values along the video (total of 24 features). 3.2.3 Spatiotemporal gradient The introduced spatial gradient features are directly extracted from the frames. As explained earlier, we can use spatiotemporal slices ( x t $xt$ and y t $yt$ ) instead of the frames and apply the same strategy as above to obtain 48 features; 24 for x t $xt$ slices and 24 for y t $yt$ slices. While the gradient of the frames reveal spatial edges, the gradient of the spatiotemporal slices can reveal scene changes and temporal movements of the objects. 3.2.4 Spatial Laplacian The Laplacian of a frame is a particular second order derivative of the data and is known to be rotation and scale-invariant. We implement the Laplacian operator via a 5 × 5 $5\times 5$ filter kernel applied to the luminance channel of the frame. We find the mean and standard deviation of the result over each frame and use the minimum, maximum, mean, standard deviation, skewness, and kurtosis of these values along the video as the features (total of 12 features). 3.2.5 Spatiotemporal Laplacian To provide uniformity along all the three axes of the video cube, we also include the features related to the Laplacian of the spatiotemporal slices. Then, similar to the procedure explained for spatial Laplacian features, we obtain 24 features representing the spatiotemporal Laplacian (12 features for each of the x t $xt$ and y t $yt$ slices). 3.2.6 Spatial gradient amplitude The gradient of an image is a 2D vector at each pixel (vertical and horizontal derivatives). We define the amplitude of this vector as the spatial information at a pixel: ∥ g ∥ = g x 2 + g y 2 , \begin{equation} \Vert \mathbf {g} \Vert =\sqrt { g_x^2+g_y^2}, \end{equation} (1)where g x $g_x$ and g y $g_y$ are the horizontal and vertical derivatives of the luminance channel of the frame at a given pixel, respectively. We should highlight that the information of the gradient components is already included in the gradient features. However, the relationship between the gradient components and their amplitude is nonlinear. Therefore, the quality estimates of a regressor based on the gradient amplitude features might be better than the ones based on the gradient components, and vice versa. Hence, we include both sets of features. Similar to the previous features, we calculate the mean and standard deviation of these amplitudes over a frame and report the minimum, maximum, mean, standard deviation, skewness, and kurtosis of them along the video as the features (total of 12 features). 3.2.7 Spatiotemporal gradient amplitude We repeat the above procedure for spatiotemporal x t $xt$ and y t $yt$ slices to obtain two sets of 12 features (24 features in total). 3.2.8 Spatial angular information As we included the features related to the components of the gradient and its amplitude separately, it is logical to include a number of features that describe the angle of the gradient. Indeed, the direction of the gradient vector at each pixel reveals the local orientation of the edges, and human visual system is sensitive to angles [31]. We initially define the gradient angle at each pixel as θ g = tan − 1 g y g x . \begin{align*} \theta _{\mathbf {g}} = \tan ^{-1}{\left(\tfrac{g_y}{g_x}\right)}. \end{align*} Next, we discard the pixels at which the gradient amplitude value is below a threshold (this threshold is set as 20). This is due to the fact that the orientation of the gradient vector does not carry much information when the gradient amplitude is small. For the remaining pixels, we calculate the mean and standard deviation over the frame and report the mean, standard deviation, skewness, and kurtosis of them along the video as the features (total of eight features). It is worth mentioning that the angular information is differently represented in ref. [31]. 3.2.9 Spatiotemporal angular information Similar to the previous parts, we repeat the procedure for spatial angular features on x t $xt$ and y t $yt$ slices to obtain 16 spatiotemporal angular features (two sets of eight features). 3.2.10 Luma information The RGB representation of image pixels is one of the popular representations in which the colour and brightness information are simultaneously encoded. The decomposition into brightness and colour information is also very common for encoding video frames. The luma, the luminance, or the brightness component includes the morphological and geometrical content of the video. For instance, the sharpness (or blurriness) of the frames could be directly inferred from the luma channel. For this purpose, we compute the mean and standard deviation of the luminance channel of each video frame. Then, we report the minimum, maximum, mean, standard deviation, skewness, and kurtosis of these values along the video as the features (total of 12 features). It is conceptually possible to define the luma channel for x t $xt$ and y t $yt$ slices; however, we ignore such spatiotemporal luma features in this paper. 3.2.11 Chroma information Complementary to the luma channel, each video frame ( x y $xy$ ) consists of two chroma components that encode the colour content of the frame. The quality metrics related to the colour content are oftentimes derived from these components. To provide uniformity with the luma channel, we derive the mean and standard deviation for each of the chroma components separately, and use the minimum, maximum, mean, standard deviation, skewness, and kurtosis of these values along the video as the chroma-related features (total of 24 features). 3.2.12 Temporal information The difference between neighbouring pixels within a single frame is considered as spatial information and is properly captured using the derivative-type operators (such as gradient). In contrast, the temporal information of the video is understood as the pixel variations along the t $t$ axis. To extract the temporal features, we first subtract each luma frame from its previous luma frame in a pixel-by-pixel fashion (pixels at the same spatial locations). Next, we form the temporal features by evaluating the mean and standard deviation of each difference frame, and then, evaluating the minimum, maximum, mean, standard deviation, skewness, and kurtosis of these values along the video (total of 12 features). 3.2.13 Colourfulness The introduced chroma features mainly capture the intensity histogram of the present colours in the video. The colourfulness metric M ( 3 ) $M^{(3)}$ in ref. [32] measures how varied the colours are in a video file. To define this metric, let r g = R − G $rg=R-G$ and y b = 0.5 ( R + G ) − B $yb=0.5(R+G)-B$ , where R $R$ , G $G$ , and B $B$ stand for the red, green, and blue components of each pixel. Now, we have: M ( 3 ) = σ r g y b + 0.3 μ r g y b , \begin{align} M^{(3)}=\sigma _{rgyb}+0.3\mu _{rgyb}, \end{align} (2)where σ r g y b = Var ( r g ) + Var ( y b ) \begin{align} \sigma _{rgyb} &=\sqrt {{\rm Var}(rg)+{\rm Var}(yb)} \end{align} (3) μ r g y b = mean ( r g ) 2 + mean ( y b ) 2 . \begin{align} \mu _{rgyb} &=\sqrt {{\rm mean}(rg)^2+{\rm mean}(yb)^2}. \end{align} (4)( Var $\rm Var$ and mean $\rm mean$ are acting over the values of each x y $xy$ frame) To provide a more comprehensive set of colourfulness features, we keep Var ( r g ) ${\rm Var}(rg)$ , mean ( r g ) ${\rm mean}(rg)$ , Var ( y b ) ${\rm Var}(yb)$ , mean ( y b ) ${\rm mean}(yb)$ , σ r g y b $\sigma _{rgyb}$ , and μ r g y b $\mu _{rgyb}$ in addition to the overall colourfulness measure M ( 3 ) $M^{(3)}$ . For each of these seven frame-level values, we evaluate the minimum, maximum, mean, standard deviation, skewness, and kurtosis along the video (total of 42 features). 3.2.14 LMSCN statistics The Luma information explains the statistics of the brightness globally over the frames. For a local (spatially) measure of brightness we use the luminance channel in the form of mean subtracted contrast normalized (MSCN). For a mathematical description, let F ( i , j ) $F(i,j)$ represent a M × N $M\times N$ frame (luminance channel) with i ∈ { 1 , 2 , … , M } $i\in \lbrace 1,2,\ldots,M\rbrace$ , j ∈ { 1 , 2 , … , N } $j\in \lbrace 1,2,\ldots,N\rbrace$ . The MSCN frame F ̂ ( i , j ) $\widehat{F}(i,j)$ is defined as F ̂ ( i , j ) = F ( i , j ) − μ ( i , j ) σ ( i , j ) + C , \begin{equation} \widehat{F}(i,j)=\frac{F(i,j)-\mu (i,j)}{\sigma (i,j)+C}, \end{equation} (5)where μ ( i , j ) $\mu (i,j)$ and σ ( i , j ) $\sigma (i,j)$ are the weighted local mean and standard deviation of the frame μ ( i , j ) = ∑ k = − 3 3 ∑ l = − 3 3 w k , l F ( i + k , j + l ) , \begin{align} \mu (i,j) &= \sum _{k=-3}^{3} \sum _{l=-3}^{3} w_{k,l} F(i+k,j+l), \end{align} (6) σ ( i , j ) = ∑ k = − 3 3 ∑ l = − 3 3 w k , l [ F ( i + k , j + l ) − μ ( i , j ) ] 2 , \begin{align} \sigma (i,j) &= \sqrt {\sum _{k=-3}^{3} \sum _{l=-3}^{3} w_{k,l} [F(i+k,j+l)-\mu (i,j)]^2}, \end{align} (7)and C = 1 $C=1$ is a regularizing constant to avoid instability. The 7 × 7 $7\times 7$ weight matrix w = [ w k , l ] | k | , | l | ≤ 3 $\mathbf {w}=[ w_{k,l} ]_{|k|, |l|\le 3}$ mimics a symmetric 2D Gaussian filter with w k , l = e − 18 49 ( k 2 + l 2 ) ∑ | k | , | l | ≤ 3 e − 18 49 ( k 2 + l 2 ) . \begin{align*} w_{k,l} = \frac{ {\rm e}^{-\frac{18}{49} (k^2+l^2 )} }{ \sum _{|k|,|l|\le 3} {\rm e}^{-\frac{18}{49} (k^2+l^2 )} }. \end{align*} The 18 49 $\frac{18}{49}$ constant is chosen such that the main lobe of the Gaussian filter ( ± 3 σ $\pm 3\sigma$ ) fits within the 7 × 7 $7\times 7$ window. The histogram of F ̂ ( i , j ) $\widehat{F}(i,j)$ is known to be fairly described by the generalized Gaussian distribution (GGD) of the form [10, 15]: f ( x ; α , β ) = α 2 β Γ ( 1 α ) exp − | x | α β α , \begin{align*} f(x;\alpha ,\beta )=\frac{\alpha }{2\beta \Gamma (\frac{1}{\alpha })} { \exp {\left(- \tfrac{|x|^{\alpha }}{\beta ^{\alpha }} \right)}