## 3 Proposed DVC Framework for Online Correlation Noise Model
### Mathematical Representation of Phase Interpolation for Side Information Generation
#### 3.2.1.3 Phase Shifting and Correction  
To avoid the phase ambiguity which happens due to large displacement corresponding to a phase difference of more than \(\pi\), and since the phase is periodic, the phase difference is normalized to between \([-\pi,\,\,\pi]\). Interpolation works accurately as long as the shift computed on a particular level mainly captures the frequency content corresponding to the correct motion. Therefore, shift correction based on confidence estimation is deployed to correctly estimate the motion even for high-frequency content. This approach robustly interpolates the accurate motion for high frequency by taking all available shift information into account. This approach assumes that phase difference between two resolution levels does not differ arbitrarily, i.e., phase differences between levels can be used as a confidence measure that quantifies whether the estimated phase shift is admissible. The phase shift correction is performed on the level \(l\) if the computed shift at level \(l\) differs from the coarser level \(l+1\) by more than a threshold. The phase shift correction is performed by first adding multiples of \(\pm 2\pi\) to \(\mathcal{O}_{diff}\). This leads to absolute differences between the phase values of consecutive levels that can never be greater than a given tolerance. The \(\pi\) is used as tolerance distance which modifies the phase values in such a way that the phase difference of a pixel between two levels can never be larger than \(\pi\). This step allows a meaningful range extension because the original phase difference are truncated to the range \([-\pi,\,\pi]\). The actual shift correction depends on the difference between two levels, which is taken under consideration for confidence estimation and is calculated using Equation (14),  
\[\varphi=\text{atan2}\Big{(}\sin(\mathcal{O}_{diff}^{l}-\mathcal{O}_{diff}^{l+1 }),\cos(\mathcal{O}_{diff}^{l}-\mathcal{O}_{diff}^{l+1})\Big{)}, \tag{14}\]  
where the coarser level phase value is scaled according to the arbitrary pyramid scale factor \(\lambda>1\). When \(\left|\varphi\right|>\frac{\pi}{2}\), the shift correction is performed to obtain the corrected phase difference using (15) and considerably leads to a better interpolation result. \[\widetilde{\mathcal{O}}_{diff}^{l}=\lambda\mathcal{O}_{diff}^{l+1} \tag{15}\]  
Although the phase shift correction can model large motion, it still has the limitation of presenting blur artifacts in some motions. Therefore, an additional enhancement step isperformed to limit even admissible phase shifts into well representable motions. Therefore, the phase difference is limited by a constant, \(\mathcal{O}_{limit}\), defined by Equation (16)  
\[\mathcal{O}_{limit}=\tau\pi\lambda^{L-l} \tag{16}\]  
where \(\tau\) defines the percentage of limitation and \(\tau\in(0,1)\), \(L\) is the total number of levels, \(\lambda\) is the scale factor and \(l\) is the current level. At the coarser level, if the magnitude of phase difference exceeds \(\mathcal{O}_{limit}\) then the corrected phase difference is set to zero. The next step is a smooth-phase interpolation between phases of input image \(1\), \(\mathcal{O}_{1}\), and input image \(2\), \(\mathcal{O}_{2}\). For convenience, from here onwards we will drop the superscript \(l\) to represent the pyramid level. As a result of shift correction, there is no surety that \(\mathcal{O}_{1}+\tilde{\mathcal{O}}_{diff}\) matches \(\mathcal{O}_{2}\) or any of the equivalent multiples of \(\mathcal{O}_{2}+\gamma 2\pi\) where \(\gamma\in\mathbb{N}_{0}\). For smooth interpolation, the original phases \(\mathcal{O}_{1}\) and \(\mathcal{O}_{2}\) are preserved along with shift-corrected phase difference \(\tilde{\mathcal{O}}_{diff}\).