## 3 Wiener-Granger Causality Analysis Theoretical Fundamentals
### Stochastic Autoregressive Model  
We consider carrying out analysis with two signals, \(C1\) and \(C2\), being easily extensible to \(n\) signals-textures. Each stationary process represents a texture of the scene and can be represented by an auto-regressive model as follows:  
\[C1(t)=\sum_{k=1}^{\infty}K_{C1}^{1}(k)\cdot C1(t-k)+\eta_{C1}^{1}, \text{with}\quad\sum_{C1}^{1}=\text{var}(\eta_{C1}^{1}), \tag{1}\] \[C2(t)=\sum_{k=1}^{\infty}K_{C2}^{1}(k)\cdot C2(t-k)+\eta_{C2}^{1}, \text{with}\quad\sum_{C2}^{1}=\text{var}(\eta_{C2}^{1}), \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Area** & **Application/Reference** \\ \hline Medical & \(\bigstar\) Computer-aided diagnosis of mammography masses based on a supervised content-based image retrieval approach [4]. \\  & \(\bigstar\) Relevance feedback for enhancing content based image retrieval and automatic prediction of semantic image features: Application to bone tumor radiography [3]. \\ \hline Vegetable & Vegetable Image Retrieval with Fine-tuning VGG Model and Image Hash [32]. \\ \hline COVID-19 & Deep metric learning-based image retrieval system for chest radiography and its clinical applications in COVID-19 [33]. \\ \hline Academic/Educational & Design and implementation of CBIR system for academic/educational purposes [5]. \\ \hline Video/Image processing & Comparative analysis on different degrees of jpeg compression used in CBIR systems [34]. \\ \hline Design & Diagram Image Retrieval and Analysis: Challenges and Opportunities [35]. \\ \hline Biology & Indexing and Image Search by the Content according to the Biological Base of the Cognitive Processing of Information using a Neural Sensor [36]. \\ \hline Cloud Repositories & \(\bigstar\) Practical Privacy-Preserving Content-Based Retrieval in Cloud Image Repositories [37]. \\  & \(\bigstar\) Towards Privacy-Preserving Content-Based Image Retrieval in Cloud Computing [38]. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Examples of some CBIR application fields. where \(\eta^{1}_{\text{C1}}\) and \(\eta^{1}_{\text{C2}}\) are the Gaussian random noise with zero mean value and unitary standard deviation; \(K^{1}_{\text{C1}}(k)\) and \(K^{1}_{\text{C2}}(k)\) are the coefficients of the regression model to the textures \(C1\) and \(C2\), respectively. The common auto-regressive model for the two textures is defined by the Equations (3) and (4)  
\[C1(t)=\sum_{k=1}^{\infty}K^{1,1}_{\text{C1}}(k)\cdot C1(t-k)+\sum _{k=1}^{\infty}K^{1,2}_{\text{C2}}(k)\cdot C2(t-k)+\eta^{2}_{\text{C1}},\\ \text{with}\sum_{\text{C1}}^{2}=\text{var}(\eta^{2}_{\text{C1}}) \tag{3}\]  
\[C2(t)=\sum_{k=1}^{\infty}K^{2,1}_{\text{Y1}}(k)\cdot C1(t-k)+\sum _{k=1}^{\infty}K^{2,2}_{\text{C2}}(k)\cdot C2(t-k)+\eta^{2}_{\text{C2}},\\ \text{with}\sum_{\text{C2}}^{2}=\text{var}(\eta^{2}_{\text{C2}}) \tag{4}\]  
where \(\sum_{\text{C1}}^{2}\) and \(\sum_{\text{C2}}^{2}\) are the variances of the residual terms \(\eta^{2}_{\text{C1}}\) and \(\eta^{2}_{\text{C2}}\), respectively. Now we analyze the variances/covariances residual terms \(\eta^{2}_{\text{C1}}\) using the equation in matrix form \(\Sigma\) (5):  
\[\Sigma=\left( \begin{array}{cc}\Sigma^{2}_{\text{C1}}&\text{Y}_{1,2}\\ \text{Y}_{2,1}&\Sigma^{2}_{\text{C2}}\end{array} \right) \tag{5}\]  
where \(\text{Y}_{1,2}\) is the covariance between \(\eta^{2}_{\text{C1}}\) and \(\eta^{2}_{\text{C2}}\) (defined as \(\text{Y}_{1,2}=cov(\eta^{2}_{\text{C1}},\eta^{2}_{\text{C2}})\)). Starting from the previous conditions, and using the concept of statistical independence between two random processes at the same time (in pairs), causality can be defined over time. An example of the causality between \(C1\) and \(C2\) is the expression given by Equation (6). \[F_{\text{C2,C1}}=\ln\left[\frac{\Sigma^{1}_{\text{C1}}\times\Sigma^{1}_{\text{ C2}}}{\Sigma^{2}_{\text{C1}}\times\Sigma^{2}_{\text{C2}}}\right] \tag{6}\]  
Equation (6) is commonly known as time domain causality.