## 3 Proposed DVC Framework for Online Correlation Noise Model
### Mathematical Representation of Phase Interpolation for Side Information Generation
#### 3.2.2 Mathematical Representation of Proposed Online Correlation Noise Model Estimation  
In conventional video coding, the distribution of the motion-compensated residual coefficients is modeled using the Laplacian distribution [34]. Different distribution models such as Gaussian distribution are found in the literature; however, the Laplacian distribution is often chosen to obtain a good trade-off between model complexity and accuracy [35]. Due to this trade-off feature, the Laplacian distribution is widely adopted in DVC for modeling the CNM. Ideally, the calculation of the noise correlation parameter can either be done with the SI at the encoder or the actual WZF at the decoder. However, this is practically not possible, so for the online correlation noise parameter estimation, the different codecs exploited the temporal correlation at the decoder [18] at the cost of extra computation. In this work, we propose a novel frame-level online CNM parameter calculation method using transformed coefficient bands. For offline CNM, the Laplacian distribution of the residual of WZF and SI transformed coefficient at \((x,y)\), with Laplacian distribution parameter, \(\alpha\), is found using (20). \[p[WZ(x,y)-SI(x,y)]=\frac{\alpha}{2}\exp[-\alpha\ |WZ(x,y)-SI(x,y)|] \tag{20}\]  
where \(\alpha=\sqrt{\frac{2}{\sigma^{2}}}\) is used to determines the error distribution between \(WZ(x,y)\) and \(SI(x,y)\), where \(\sigma^{2}\) defines the variance [19]. To compute the online CNM using the mean of transformed coefficients of a band at the decoder, it is necessary to estimate the Laplacian distribution parameter which is necessary to determine the error distribution between the original frame and SI. Therefore, for the proposed frame-level Laplacian distribution model, the estimation of the Laplacian distribution parameter is a crucial step. Since, for our proposed method, the Laplacian distribution model is computed at the frame level, the Laplacian distribution parameter is denoted as \(\alpha_{f}\). The steps involved in estimating \(\alpha_{f}\) are elaborated next. Let \(\mu^{R}\) be the set of means of residuals of the coefficient bands, \(\mu^{R}_{\ b}\). Therefore, for a total of 16 coefficient bands in a frame,  
\[\left\{\mu^{R}_{\ b}\middle|\mu^{R}_{\ b}\in\mu^{R},\ b=1,2,..,16\ \right\} \tag{21}\]  
and  
\[\mu^{R}_{\ b}=\mu^{WZ}_{b}-\mu^{SI}_{b},\ b=1,2,\ldots,16, \tag{22}\]  
where \(\mu^{WZ}_{b}\) and \(\mu^{SI}_{b}\) are the means of the WZF and SI transformed coefficient bands, respectively. The \(\mu^{WZ}_{b}\), is calculated at the encoder using  
\[\mu^{WZ}_{b}=\frac{1}{M}\sum_{j=1}^{M}X_{DCT,j}(j). \tag{23}\]In (23), \(X_{DCT,b}(j)\) is the transformed coefficient of the WZF for the \(j\)-th coefficient in band \(b\). On the other hand, the value, \(\mu_{b}^{SI}\), is the mean of transformed coefficients of SI band and is calculated at the decoder by using  
\[\mu_{b}^{SI}=\frac{1}{M}\sum_{j=1}^{M}Y_{DCT,b}(j). \tag{24}\]  
In (24), \(Y_{DCT,b}(j)\) is the transformed coefficient of the SI for the \(j\)-th coefficient in band \(b\). In both (23) and (24), \(j=1,2,\ldots,M\) where \(M\) defines the total number of coefficients in a band. Next, the variance, \(\sigma_{f}^{2}\), for the proposed residual means, \(\mu^{R}\), is calculated using the equation  
\[\sigma_{f}^{2}=E\bigg{[}\left(\mu^{R}\right)^{2}\bigg{]}-\left(E\left[\mu^{R} \right]\right)^{2}, \tag{25}\]  
where \(E\left[\mu^{R}\right]\) and \(E\left[\left(\mu^{R}\right)^{2}\right]\) are computed using Equations (26) and (27), respectively. \[E\left[\mu^{R}\right]=\sum_{b=1}^{16}\mu_{b}^{R}\ P_{b} \tag{26}\]  
\[E\bigg{[}\left(\mu^{R}\right)^{2}\bigg{]}=\sum_{b=1}^{16}\left(\mu_{b}^{R} \right)^{2}P_{b} \tag{27}\]  
In (26) and (27), \(P_{b}\) denotes the probability of occurrence of \(\mu_{b}^{R}\) in \(\mu^{R}\). Finally, the Laplacian distribution parameter \(\alpha_{f}\) is calculated using  
\[\alpha_{f}=\left\{