## 3 Proposed Lagrange Multiplier Selection Algorithm
### Analyzing the Distortion Relationship between Temporal Subbands and Reconstructed Frames  
The distortion fluctuation exhibited by the 3-D wavelet-based SVC codecs can be better understood by analyzing the distortion propagation during MCTF. In the ENH-MC-EZBC codec, the adaptive MCTF framework has been implemented using either the Haar or the 5/3 filter to improve the coding performance. As shown in Figure 2, the coefficients \(\alpha_{\mathrm{\tau}}\) and \(\beta_{\mathrm{\tau}}\) (\(\mathrm{\tau}=1,2,3\)) along the branches are filter-based weighting factors, which are given in Table 1.  
In the Haar MCTF, the original frames are filtered temporally with a two-tap Haar filter along the motion trajectory. For the connected pixels, the low-frequency subbands and high-frequency subbands are implemented by the following lifting structure:  
\[\mathrm{H}_{i}(\mathrm{m},\mathrm{n})=\mathrm{I}_{2i+1}(\mathrm{m},\mathrm{n}) -\widetilde{\mathrm{I}}_{2i}(\mathrm{m}-\mathrm{d}_{\mathrm{m}},\mathrm{n}- \mathrm{d}_{\mathrm{n}}), \tag{8}\]  
\[\mathrm{L}_{i}\Big{(}\mathrm{m}-\overline{\mathrm{d}}_{\mathrm{m}},\mathrm{n}- \overline{\mathrm{d}}_{\mathrm{n}}\Big{)}=\mathrm{I}_{2i}\Big{(}\mathrm{m}- \overline{\mathrm{d}}_{\mathrm{m}},\mathrm{n}-\overline{\mathrm{d}}_{\mathrm{ n}}\Big{)}+(1/2)\cdot\widetilde{\mathrm{H}}_{i}\Big{(}\mathrm{m}-\overline{ \mathrm{d}}_{\mathrm{m}}+\mathrm{d}_{\mathrm{m}},\mathrm{n}-\overline{\mathrm{d }}_{\mathrm{n}}+\mathrm{d}_{\mathrm{n}}\Big{)}. \tag{9}\]  
For the 5/3-based MCTF, the temporal analysis can be written as:  
\[\mathrm{H}_{i}(\mathrm{m},\mathrm{n})=\mathrm{I}_{2i+1}(\mathrm{m},\mathrm{n}) -(1/2)\cdot\widetilde{\mathrm{I}}_{2i}(\mathrm{m}-\mathrm{d}_{\mathrm{m}}, \mathrm{n}-\mathrm{d}_{\mathrm{n}})-(1/2)\cdot\widetilde{\mathrm{I}}_{2i+2}( \mathrm{m}+\mathrm{d}_{\mathrm{m}},\mathrm{n}+\mathrm{d}_{\mathrm{n}}), \tag{10}\]  
\[\mathrm{L}_{i}\Big{(}\mathrm{m}-\overline{\mathrm{d}}_{\mathrm{m}},\mathrm{n}- \overline{\mathrm{d}}_{\mathrm{n}}\Big{)}=\mathrm{I}_{2i}(\mathrm{m},\mathrm{n })+(1/4)\cdot\widetilde{\mathrm{H}}_{i}\Big{(}\mathrm{m}-\overline{\mathrm{d }}_{\mathrm{m}}-\mathrm{d}_{\mathrm{m}},\mathrm{n}-\overline{\mathrm{d}}_{ \mathrm{n}}-\mathrm{d}_{\mathrm{n}}\Big{)}+(1/4)\cdot\widetilde{\mathrm{H}}_{i -1}\Big{(}\mathrm{m}-\overline{\mathrm{d}}_{\mathrm{m}}+\mathrm{d}_{\mathrm{m }},\mathrm{n}-\overline{\mathrm{d}}_{\mathrm{n}}+\mathrm{d}_{\mathrm{n}}\Big{)}, \tag{11}\]  
where \(\mathrm{H}_{i}\) and \(\mathrm{L}_{i}\) denote the temporal high- and low-frequency subbands of the \(i\)th (\(i=1,2,\cdots\mathrm{K}\)) frame of a video sequence, respectively; \(\mathrm{I}_{2i}(\mathrm{m},\mathrm{n})\) represents the pixel value at position \((\mathrm{m},\mathrm{n})\) in frame \(\mathrm{f}_{2i};\widetilde{\mathrm{I}}_{2i}(\mathrm{m}-\mathrm{d}_{\mathrm{m}},\mathrm{n}-\mathrm{d}_{\mathrm{n}})\) and \(\widetilde{\mathrm{I}}_{2i+2}(\mathrm{m}+\mathrm{d}_{\mathrm{m}},\mathrm{n}+ \mathrm{d}_{\mathrm{n}})\) denote the interpolated values of pixel \((\mathrm{m},\mathrm{n})\) in the frame \(2i\) and \(2i+2\), respectively. \(\widetilde{\mathrm{H}}\) is the interpolated value of pixel in the high-frequency frame; \((\mathrm{d}_{\mathrm{m}},\mathrm{d}_{\mathrm{n}})\) is the motion vector in frame \(\mathrm{f}_{2i};\overline{\mathrm{d}}_{\mathrm{m}}\) is the closest integer value to \(\mathrm{d}_{\mathrm{m}}\), and \(\overline{\mathrm{d}}_{\mathrm{n}}\) is defined in the same way. For mathematical convenience, we only consider the case in the lifting structure of one-level 5/3 inverse MCTF in the following. Thus, according to the Equations (10) and (11), the errors in the reconstructed \(\mathrm{f}_{2i}(\mathrm{m},\mathrm{n})\) and \(\mathrm{f}_{2i+1}(\mathrm{m},\mathrm{n})\) can be formulated as:  
\[\epsilon_{\mathrm{f}_{2i}}=-(1/4)\cdot\epsilon_{\mathrm{H}_{i-1}}+1\cdot \epsilon_{\mathrm{L}_{i}}-(1/4)\epsilon_{\mathrm{H}_{i}}, \tag{12}\]