## 3 Proposed Lagrange Multiplier Selection Algorithm
### Adaptive Lagrange Multiplier Selection  
As described in Subsection 2.2, the Lagrange multiplier \(\lambda\) is usually used to guide the bit allocation of each subband so that the overall distortion can be minimized and also with balanced the rate and distortion among the reconstructed frames. To our best knowledge, a larger \(\lambda\) will result in a higher coding distortion with less coding bits. Contrarily, a smaller \(\lambda\) will lead to a lower coding distortion with more coding bits. Therefore, we can allocate more bits to the T-bands with more detailed information by decreasing the subband-level Lagrange multiplier. We define a weighting factor vector as \(\Omega=(\omega_{1},\omega_{2},\ldots\omega_{\mathrm{K}})\), which indicates the contribution of a unit quantization error in each temporal subband to the overall distortion in the reconstructed video sequence. To obtain the weighting factor of the MCTF process, we need to employ a novel derivation, which involves the complicated motion compensation process and temporal subband content characteristics statistics. For computing these weighting factors, we utilize the mutual information, gradient per pixel, and texture homogeneity to measure the temporal subband content features. Mutual information (MI) is a measurement of similarity between frames which can detect the differences among successive frames [44]. In this paper, we utilize MI to measure the similarity between the temporal subband frames. A large difference between frames (corresponding to the high motion activity) leads to a low MI value, while a small change between frames responds to a high MI value [45; 46]. Let X be a discrete random variable with a set of possible outcomes \(\mathrm{A}_{X}=\{\mathrm{a}_{1},\mathrm{a}_{2},\ldots,\mathrm{a}_{L}\}\) with possibilities \(\{\mathrm{p}_{1},\mathrm{p}_{2},\ldots,\mathrm{p}_{L}\}\), \(\mathrm{p}_{X}(\mathrm{x}=\mathrm{a}_{I})=\mathrm{p}_{II}\geq 0\), and \(\sum\limits_{\mathrm{x}\in\mathrm{A}_{X}}\mathrm{p}_{X}(\mathrm{x})=1\). According to the information theory, the entropy of X is:  
\[\mathrm{H}(\mathrm{X})=-\sum\limits_{\mathrm{x}\in\mathrm{A}_{X}}\mathrm{p}_{ X}(\mathrm{x})\log\mathrm{p}_{X}(\mathrm{x}). \tag{19}\]  
The joint entropy of discrete random variables X and Y is:  
\[\mathrm{H}(\mathrm{X},\mathrm{Y})=-\sum\limits_{\mathrm{x},\mathrm{y}\in \mathrm{A}_{X},\mathrm{A}_{Y}}\mathrm{p}_{XY}(\mathrm{x},\mathrm{y})\log[ \mathrm{p}_{XY}(\mathrm{x},\mathrm{y})]. \tag{20}\]  
The MI between random variable X and Y is given by:  
\[\mathrm{I}(\mathrm{X},\mathrm{Y})=-\sum\limits_{\mathrm{x},\mathrm{y}\in \mathrm{A}_{X},\mathrm{A}_{Y}}\mathrm{p}_{XY}(\mathrm{x},\mathrm{y})\log\frac{ \mathrm{p}_{XY}(\mathrm{x},\mathrm{y})}{\mathrm{p}_{X}(\mathrm{x})\mathrm{p}_{ Y}(\mathrm{y})}. \tag{21}\]  
The relation between MI and joint entropy is given by:  
\[\mathrm{I}(\mathrm{X},\mathrm{Y})=\mathrm{H}(\mathrm{X})+\mathrm{H}(\mathrm{Y })-\mathrm{H}(\mathrm{X},\mathrm{Y}). \tag{22}\]  
In a YUV formatted video sequence, let us consider a gray level video sequence with intensity value ranging from 0 to \(\mathrm{L}-1\) (e.g., L = 256 for 8-bit depth). For the luminance component Y, \(\mathrm{P}_{i,i+1}^{\mathrm{Y}}(\mathrm{x},\mathrm{y})\) (\(0\leq\mathrm{x},\mathrm{y}\leq\mathrm{L}-1\)) is the probability that a pixel with gray level x in frame \(\mathrm{f}_{i}\) has a gray level y in frame \(\mathrm{f}_{i+1}\). Let \(\mathrm{M}_{i,i+1}^{\mathrm{Y}}\) be the MI of the luminance component.