## 3 Subdivision Wavelets Using Lifting
### Wavelet synthesis and analysis  
Suppose \(M^{i}\) is arbitrary circular layer of the surface, we have:  
\[M^{i}=\sum_{m}v^{\prime}_{i,m}\varphi^{\prime}_{i,m}+\sum_{n}e^{\prime}_{i,n} \psi^{\prime}_{i,n}.\;\forall m,\forall n \tag{9}\]  
Figure 6: (a) Constructing wavelets as linear combination of lazy wavelets and scaling functions in circular layers; (b) Discrete subdivision masks of basis functions in circular layers (lazy wavelets and scaling functions). By inserting Equation (7) to Equation (9), we get:  
\[M^{i}=\sum_{i}v_{i,m}\varphi_{i,m}+\sum_{j}e_{i,n}\psi_{i,n}+\sum_{n}e_{i,n}(\sum _{k=-1}^{2}w_{k}\varphi_{i,n})\;\forall m,\forall n. \tag{10}\]  
?From Equation (10), we can construct a new wavelet synthesis from the original lazy wavelets by additional lifting operation prior to the subdivision masks:  
\[ \begin{split}& v^{\prime}_{i,j+k}=v^{\prime}_{i,j+k}+w_{k}e^{ \prime}_{i,j}\quad\forall k=-2,-1,...,3,\forall i,\\ & e^{\prime\prime}_{i,j}=e^{\prime}_{i,j}+\frac{1}{2}(v^{\prime }_{i,j}+v^{\prime}_{i,j+1}),\quad v^{\prime\prime}_{i,j}=\frac{1}{4}(e^{ \prime\prime}_{i,j-1}+2v^{\prime}_{i,j}+e^{\prime\prime}_{i,j}).\end{split}  \tag{11}\]  
By inverting the rules of wavelet synthesis process, we can get the new rules for wavelet analysis in circular layers:  
\[ \begin{split}& v^{\prime}_{i,j}=\frac{1}{2}(4v^{\prime\prime}_{i,j} -e^{\prime\prime}_{i,j-1}-e^{\prime\prime}_{i,j}),\quad e^{\prime}_{i,j}=e^{ \prime\prime}_{i,j}-\frac{1}{2}(v^{\prime}_{i,j}+v^{\prime}_{i,j+1}),\\ & v^{\prime}_{i,j+k}=v^{\prime\prime}_{i,j+k}-w_{k}e^{\prime}_{i, j}\quad\forall k=-2,-1,...,3,\forall i.\end{split}  \tag{12}\]  
The rules of wavelet transforms in radial layer can be obtained by the similar methods. More discussion about the wavelets in radial layers is given in our previous work [20]. Here we only show the wavelet analysis rules in radial layers:  
\[ \begin{split}& v=\frac{1}{U}(v^{\prime}-\frac{\alpha}{n\beta} \sum_{k=0}^{n-1}e^{\prime}_{k}),\quad v_{j}=\frac{8}{5}\times(v^{\prime}_{j}- \frac{1}{8}v-\frac{1}{4}e^{\prime}_{2,j}),\\ & v_{i,j}=2\times(v^{\prime}_{i,j}-\frac{1}{4}(e^{\prime}_{i-1,j} +e^{\prime}_{i,j}))\quad\forall j,\forall i>1,\\ & e_{j}=e^{\prime}_{j}-(1-\beta)v-\sum_{k=0}^{n-1}v_{k}\gamma_{k -j},\quad e_{i,j}=e^{\prime}_{i,j}-\frac{1}{2}(v_{i-1,j}+v_{i,j})\quad\forall j,\;\forall\;i>1,\\ & v_{k}=v_{k}-t_{k}e_{j}\qquad\forall j,\;\forall k=0,...,n-1,\\ & v_{k}=v_{k}-s_{k}e_{i,j}\qquad\forall j,\;\forall i>1,\forall k =0,...,7.\end{split}  \tag{13}\]  
where \(t\) and \(s\) is the precomputed weights in radial layers, \(\alpha=\beta-\frac{1}{4}\), \(\beta=\frac{1}{2}\), \(\gamma_{k}=\frac{1}{n}(\beta-\frac{1}{2}+\frac{5}{8}c^{k}_{n}+(c^{k}_{n})^{2}+ \frac{1}{2}(c^{k}_{n})^{3})\), \(c^{k}_{n}=\cos(\frac{2k\pi}{n})\), \(U=(1-\alpha)-\frac{\alpha}{\beta}(1-\beta)\). By applying the rules of polar subdivision wavelet transforms, we can give the algorithms of synthesis and analysis. Since the synthesis algorithm is similar to analysis algorithm except the order of sub-steps. Here, we only show the analysis algorithm: For a given polar structure, one step of polar subdivision wavelets analysis includes. 1. Performing the wavelet analysis in circular layers. For each circular layer i (\(i>0\)), we perform: 1. lazy wavelet analysis: for each vertex \(v_{i,m}\) in circular layer i, denote \(v_{i,m}\) (m = 0, 2, 4, 6,...) as the \(v^{\prime\prime}_{i,j}\) (j = m/2), and \(v_{i,m}\) (m = 1, 3, 5,...) as \(e^{\prime\prime}_{i,j}\) (j=(m-1)/2). Calculate each \(v^{\prime}_{i,j}\) and each \(e^{\prime}_{i,j}\) from \(v^{\prime\prime}_{i,j}\) and \(e^{\prime\prime}_{i,j}\) through Equation (6); 2. lifting operations on \(v^{\prime}_{i,m}\); for each \(e^{\prime}_{i,j}\), calculate \(v^{\prime}_{i,j+k}\) around \(e^{\prime}_{i,j}\) through \(v^{\prime}_{i,j+k}=v^{\prime}_{i,j+k}-w_{k}e^{\prime}_{i,j}\quad\forall k=-2, -1,...,3,\forall i\). 2. Performing the wavelet analysis on radial layers based on the results of (i).