## 3 Proposed DVC Framework for Online Correlation Noise Model
### Mathematical Representation of Phase Interpolation for Side Information Generation  
The standard interpolation approaches, e.g., optical flow, require accurate pixel correspondences between images to interpolate the in-between frames. In DVC, motion-compensated temporal interpolation (MCTI) is usually used for SI generation. However, in this paper, we implemented the phase-based interpolation for SI at the decoder. This technique represents motion as phase shifts in individual pixels and the computational cost is a fraction of that of any of the traditional SI interpolation techniques. In particular, the phase-based technique decomposes the frames into local phase and amplitude parameters using the complex-valued steerable pyramid. This complex steerable pyramid decomposes the one-dimensional (1D) or two-dimensional (2D) signal into spatial scale, orientation, and position [28]. In videos, the phase-based interpolation method manipulates motion by analyzing the signal of the local phase over time in different spatial scales and orientations. It mitigates the need for highly complex global computation. The computation efficiency is offered by phase-based representation, as it only performs per-pixel modifications to represent the pixel motion by shifting its phase [29]. Further efficiency is achieved due to the phase-shift correction method that combines the phase information across different levels of a multi-scale pyramid. The successfulness of image transition is achieved due to the deployment of a correction algorithm which adapts both amplitude and phase shift of the image. Before starting the part of actual image interpolation, we first explain its working concept for a basic one-dimensional (1D) signal.  
The Fourier shift theorem motivates the assumption that motion can be encoded using the phase differences. Let us consider the 1D case, in which any function \(f(x)\) can be represented in the Fourier domain as a sum of complex sinusoids over all frequencies \(\omega\) by Equation (1),  
\[f(x)=\sum_{\omega=-\infty}^{\omega=+\infty}A_{\omega}e^{i\omega x}=\sum_{ \omega=-\infty}^{\omega=+\infty}A_{\omega}e^{i\Omega_{\omega}} \tag{1}\]  
where the \(\Omega_{\omega}\) and \(A_{\omega}\) represent the phase and amplitude of the complex sinusoid, respectively. Then the shifted version of function \(f(x)\) shifted by spatial displacement \(\delta(t)\) of is given by Equation (2).  
\[f(x+\delta(t))=\sum_{\omega=-\infty}^{\omega=+\infty}A_{\omega}e^{i\omega(x+ \delta(t))} \tag{2}\]  
The phase difference, \(\Omega_{diff}^{\omega}\), between the original and shifted functions can be represented by Equation (3).  
\[\Omega_{diff}^{\omega}=\omega x-\omega(x-\delta(t))=\omega\delta(t) \tag{3}\]The phase shift \(\mathcal{O}_{shift}\), which corresponds to the actual spatial displacement between translated functions is defined in terms of the phase difference \(\mathcal{O}_{diff}^{\omega}\) between two phase curves scaled by angular frequency \(\omega\) and is given in Equation (4).  
\[\mathcal{O}_{shift}=\frac{\mathcal{O}_{diff}^{\omega}}{\omega} \tag{4}\]  
In the Fourier transform domain, the shifted function is defined as the sum of complex sinusoids over all frequencies \(\omega\), and is given by Equation (5),  
\[f(x+\delta(t)\ )=\sum_{\omega=-\infty}^{\omega=+\infty}R_{\omega}(x,t) \tag{5}\]  
where each sinusoid represents one band, i.e., \(R_{\omega}(x,t)=A_{\omega}e^{i\omega(x+\delta(t))}\). Now for the intermediate sinusoids representing the translational functions, the phase difference is modified according to the intermediate positions between functions that are defined by weight \(\beta\in(0,1)\). Therefore, the modified bands \(\tilde{\mathrm{R}}_{\omega}(x,t)\) with corresponding modified phase \(\mathcal{O}_{\omega}=\omega(x+\beta\delta(t))\) w.r.t \(\beta\) is given by Equation (6).  
\[\tilde{\mathrm{R}}_{\omega}(x,t)=A_{\omega}e^{i\omega(x+\beta\delta(t))} \tag{6}\]  
The in-between functions are then taken by integrating the modified bands following Equation (5).