## 3 Subdivision Wavelets Using Lifting
### Lifting wavelets  
The lazy wavelets constructed by reversing subdivision scheme directly usually has a poor fitting quality (see Figure 3). In order to reduce the correlation of wavelet coefficients, we can use lifting to increase the orthogonality of lazy wavelets. Though global orthogonalization may lead to a better result, it is often unaffordable in computation. For efficiency, we use additional local lifting operations [9] prior to original lazy wavelets. Since the subdivision in radial layers and circular layers are performed independently, we performed the lifting independently too. In radial layers, as shown in Figure 4, we can construct new wavelets \(\psi^{\prime}\) and \(\psi^{\prime}_{j}\) by combining the lazy wavelets \(\psi\) and \(\psi_{j}\) with scaling functions whose corresponding vertices are neighbor to \(e\) and \(e_{j}\). The related subdivision masks of lazy wavelets in radial layers are shown in Figure 5. More details about lifting subdivision wavelets in radial layers can be found in our previous work [20]. Here we mainly focus on how to construct the lifting subdivision wavelets in circular layers.  
As shown in Figure 6(a), we construct new wavelets \(\psi^{\prime\prime}_{i,j}\) by combining the lazy wavelets \(\psi^{\prime}_{i,j}\) with scaling functions whose corresponding vertices are neighbors to \(e^{\prime}_{i,j}\) in circular layers. Since these lifted  
Figure 4: Constructing wavelets as linear combination of lazy wavelets and scaling functions in radial layers.  
Figure 5: Discrete subdivision masks of basis functions in radial layers (lazy wavelets and scaling functions).  
Figure 3: (a) original surfaces perturbed with white noise; (b) the surfaces performed lazy wavelet analysis once and twice; (c) the surfaces performed our scheme once and twice.  
wavelets should be orthogonalized with the scaling functions, so we need to obtain the weights \(w_{i}\), such that:  
\[\psi^{\prime\prime}_{i,j}=\psi^{\prime}_{i,j}+\sum_{k=-2}^{3}w_{k}\phi^{\prime}_ {i,j+k}\;and\;\langle\psi^{\prime\prime}_{i,j},\phi^{\prime}_{i,j+k}\rangle=0 \tag{7}\]  
To solve the Equation (7), we need to calculate the inner products of basis functions. A proper inner product could considerably improve the result quality and reduce the complexity of computation. Here we adopt the definition of inner product introduced by [PERSON] [11]. He suppose the scaling functions of finer resolution form an orthogonal basis without considering all correlation of finer-level coefficients. The mutual inner products of wavelets and scaling functions are defined as the sum of corresponding multiplications of their discrete subdivision masks. This kind of inner product can be calculated directly from the discrete subdivision masks, shown in Figure 6(b), and avoid complex computation needed by the inner product defined with continuous shape of basis functions. More details of local inner production are referred in [PERSON]'s work. Therefore, without considering the boundary of control nets, we have the inner products of lazy wavelets:  
\[\langle\psi_{i,j},\phi_{i,j-1}\rangle=\langle\psi_{i,j},\phi_{i,j+2}\rangle=1/ 32 \langle\psi_{i,j},\phi_{i,j+1}\rangle=23/32\] \[\langle\phi_{i,j},\phi_{i,j-2}\rangle=\langle\phi_{i,j},\phi_{i,j +2}\rangle=1/64 \langle\phi_{i,j},\phi_{i,j-1}\rangle=\langle\phi_{i,j},\phi_{i,j+1} \rangle=7/16\] \[\langle\phi_{i,j},\phi_{i,j}\rangle=35/32\]  
Insert the inner products to Equation (7). and we get equation systems:  
\[Aw=b\quad where\;a_{mn}=\langle\phi^{\prime}_{i,m},\phi^{\prime}_{i,n}\rangle,b_{mn}=-\langle\psi^{\prime}_{i,m},\phi^{\prime}_{i,n}\rangle \tag{8}\]  
After solving these systems, we get weights \(w\) which can be precomputed for efficiency.