## 3 Proposed Lagrange Multiplier Selection Algorithm
### Analyzing the Distortion Relationship between Temporal Subbands and Reconstructed Frames
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Frame Mode** & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) \\ \hline bi-direction & \(-1/2\) & \(1\) & \(-1/2\) & \(1/4\) & \(1\) & \(1/4\) \\ uni-left & \(-1\) & \(1\) & \(0\) & \(1/2\) & \(1\) & \(0\) \\ uni-right & \(0\) & \(1\) & \(-1\) & \(0\) & \(1\) & \(1/2\) \\ intra & \(\mathrm{N}/\mathrm{A}\) & \(\mathrm{N}/\mathrm{A}\) & \(\mathrm{N}/\mathrm{A}\) & \(\mathrm{N}/\mathrm{A}\) & \(1\) & \(\mathrm{N}/\mathrm{A}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficients \(\alpha_{\mathrm{\tau}}\) and \(\beta_{\mathrm{\tau}}\) depending on the mode of the current frame. \[\varepsilon_{\mathrm{f}_{2i+1}}=-(1/8)\cdot\varepsilon_{\mathrm{H}_{i-1}}+(1/2) \cdot\varepsilon_{\mathrm{L}_{i}}+(3/4)\cdot\varepsilon_{\mathrm{H}_{i}}+(1/2) \cdot\varepsilon_{\mathrm{L}_{i+1}}-(1/8)\cdot\varepsilon_{\mathrm{H}_{i+1}}, \tag{13}\]  
where \(\varepsilon_{\mathrm{f}}\) denotes the reconstruction error of frame f. The assumption supposes that the distortion in each T-band can be modeled as independent and identically-distributed zero-mean additive white noise. Let \(\sigma_{\mathrm{L}}^{2}\) and \(\sigma_{\mathrm{H}}^{2}\) be the error variance of pixels in low-frequency subbands and that of pixels in high-frequency subbands, respectively. Also, let \(\sigma_{\mathrm{f}_{2i}}^{2}\) and \(\sigma_{\mathrm{f}_{2i+1}}^{2}\) denote the error variance in the reconstructed video frames. Accordingly, the distortions in the reconstructed frames are expressed as:  
\[\sigma_{\mathrm{f}_{2i}}^{2}=1\cdot\sigma_{\mathrm{L}}^{2}+(1/8)\cdot\sigma_{ \mathrm{H}}^{2},\ \ \sigma_{\mathrm{f}_{2i+1}}^{2}=(1/2)\cdot\sigma_{\mathrm{L}}^{2}+(19/32)\cdot \sigma_{\mathrm{H}}^{2}. \tag{14}\]  
The distortion associated with different T-bands will contribute differently to distortion in the reconstructed frames. Typically, this difference is quantified by analyzing the weighting factor for inverse MCTF. The weighting factor indicates how much a unit distortion in a specified subband contributes to the overall distortion in the reconstructed video. The derivation of its weighting factors is given as:  
\[\omega_{\mathrm{L}}=\left(1/2\right)^{2}+\left(1\right)^{2}+\left(1/2\right)^ {2}=1.5, \tag{15}\]  
\[\omega_{\mathrm{H}}=\left(-1/8\right)^{2}+\left(-1/4\right)^{2}+\left(3/4 \right)^{2}+\left(-1/4\right)^{2}+\left(-1/8\right)^{2}\approx 0.7188. \tag{16}\]  
This results in distortion fluctuation across the reconstructed frames after one-level inverse MCTF. When multi-level MCTF is involved, the weighting factor of each temporal subband is modeled as:  
\[\omega_{\mathrm{LL}}=\omega_{\mathrm{L}}\cdot\omega_{\mathrm{L}},\ \omega_{\mathrm{LH}}=\omega_{\mathrm{L}}\cdot\omega_{\mathrm{H}},\ \omega_{\mathrm{LLLL}}=\omega_{\mathrm{LL}}\cdot\omega_{\mathrm{L}},\ \omega_{\mathrm{LLH}}=\omega_{\mathrm{LL}}\cdot\omega_{\mathrm{H}},\ldots \tag{17}\]  
Therefore, the distortion relationship between the T-bands and the reconstructed frame can be derived in the same way but with an iterative calculation. That is, the distortion for the original video sequences (denoted as D) after T levels MCTF decomposition can be derived by:  
\[ \begin{array}{l}\mathrm{D}=\sum\limits_{i=1}^{\mathrm{K}}\mathrm{D}_{i}\\ =\sum\limits_{i=1}^{\mathrm{K}}\left(\sum\limits_{t=1}^{\mathrm{T}}\prod \limits_{j=1}^{t-1}\omega_{i\_{\mathrm{L}}\mathrm{H}}^{(j)}\mathrm{d}_{i\_{ \mathrm{L}}\mathrm{H}}^{(t)}+\prod\limits_{t=1}^{\mathrm{T}}\omega_{i\_{ \mathrm{L}}\mathrm{d}_{i\_{\mathrm{L}}\mathrm{L}}^{(\mathrm{T})}}\right)\,\end{array}  \tag{18}\]  
where \(\mathrm{d}_{i\_{\mathrm{L}}\mathrm{H}}^{(t)}\), and \(\mathrm{d}_{i\_{\mathrm{L}}\mathrm{L}}^{(\mathrm{T})}\) are the reconstruction distortion (namely error variance) of the high-frequency subbands in the \(t\)th (\(1\leq t\leq\mathrm{T}\)) temporal level, and low-frequency subbands in the highest temporal level, respectively.