## 3 Proposed Lagrange Multiplier Selection Algorithm
### Adaptive Lagrange Multiplier Selection
So we can obtain the \(\mathrm{M}_{i,i+1}^{\mathrm{Y}}\) value as shown below:  
\[\mathrm{M}_{i,i+1}^{\mathrm{Y}}=-\sum\limits_{\mathrm{x}=0}^{\mathrm{L}-1} \sum\limits_{\mathrm{y}=0}^{\mathrm{L}-1}\mathrm{P}_{i,i+1}^{\mathrm{Y}}( \mathrm{x},\mathrm{y})\log\Bigl{[}\mathrm{P}_{i,i+1}^{\mathrm{Y}}(\mathrm{x},\mathrm{y})/\mathrm{P}_{i}^{\mathrm{Y}}(\mathrm{x})\mathrm{P}_{i+1}^{ \mathrm{Y}}(\mathrm{y})\Bigr{]}, \tag{23}\]  
where \(\mathrm{P}_{i}^{\mathrm{Y}}(\mathrm{x})\) and \(\mathrm{P}_{i+1}^{\mathrm{Y}}(\mathrm{y})\) are the possibilities that a pixel with gray level x in frame \(\mathrm{f}_{i}\) and a pixel with gray level y in frame \(\mathrm{f}_{i+1}\), respectively. The gradient based subband content complexity measure has been considered in our Lagrange multiplier selection scheme. Here, the gradient per pixel (GPP) of the temporal subband is defined by:  
\[\mathrm{GPP}_{i}=\frac{1}{\mathrm{M}\times\mathrm{N}}\sum\limits_{\mathrm{n} =0}^{\mathrm{N}-1}\sum\limits_{\mathrm{m}=0}^{\mathrm{M}-1}(|\mathrm{I}_{i}( \mathrm{m},\mathrm{n})-\mathrm{I}_{i}(\mathrm{m}+1,\mathrm{n})|)+(|\mathrm{I}_ {i}(\mathrm{m},\mathrm{n})-\mathrm{I}_{i}(\mathrm{m},\mathrm{n}+1)|), \tag{24}\]  
where M and N are the width and height of the temporal subband, and \(\mathrm{I}_{i}(\mathrm{m},\mathrm{n})\) is the pixel value at position \((\mathrm{m},\mathrm{n})\). With the statistical analysis, we observe that, if the subband belongs to the homogeneous regions, the number of bits will be a smaller value, and it will relevantly increase with the complex of the texture. Hence, the texture homogeneity of each subband is measured by calculating the ratio between variance and mean of the subband, which is given by:\[\text{Ratio}_{k}=\frac{\text{Variance}}{\text{Mean}}=\frac{\text{Variance}}{ \frac{1}{256}\sum\limits_{n=0}^{\text{N}-1}\sum\limits_{m=0}^{\text{M}-1}I_{i}(m,n )}, \tag{25}\]  
where Variance and Mean are the variance and mean value of each temporal subband, respectively. Let W denote the synthesis gain matrix with regard to target video sequence. Therefore, the associated temporal subband weighting factor \(\Omega=(\omega_{1},\omega_{2},\ldots\omega_{\text{K}})\) for prediction can be calculated as:  
\[\omega_{i}=\omega_{\text{filter},i}\cdot(\alpha\cdot\text{GPP}_{i}+\beta\cdot \text{Ratio}_{i})^{-\gamma\cdot\text{MI}_{i}}\cdot\text{W}_{i\to i+1}, \tag{26}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are model parameters updated by multiple linear regression analysis. For each temporal subband frame \(i\), \(\omega_{\text{filter},i}\) is the filter-based weighting factor using Equations (15)-(17); \(\text{MI}_{i}\), \(\text{GPP}_{i}\), and \(\text{Ratio}_{i}\) can be computed by Equations (23)-(25), respectively. As pointed out in [31], \(\text{W}_{i\to i+1}\) is the cross-subband error propagation, which reflects the importance of coefficients in each subband. Similar to JPEG2000 [47], we assume that the distortions from various temporal subbands are approximately additive. Equation (18) reveals that the total distortion is simply a linear combination of the distortions of all the temporal subbands. It is noticeable that the wavelet coefficients within each temporal subband obey the Gaussian distribution [48; 49], hence, the distortion of the transform coefficients can be expressed as follows:  
\[\text{d}_{i}=\text{k}^{\prime}\frac{\tau\text{e}}{6}\left(\prod\limits_{q=1}^ {\text{Q}}\sigma_{q}^{2}\right)^{1/\text{Q}}2^{-\eta\tau_{i}}\omega_{i}, \tag{27}\]  
where \(\text{d}_{i}\) and \(\tau_{i}\) stand for the distortion and bitrate for the \(i\)th temporal subband frame, respectively. \(\sigma_{q}^{2}\) is the variance of the \(q\)th coefficient, Q the number of coefficients concerned. Besides, \(\text{k}^{\prime}\) and \(\eta\) are the parameters related to the temporal decomposition scheme used and the corresponding slope for SVC, respectively. However, in practice, it is very difficult to determine the variance for a given coefficient, as required by Equation (27).