## 5 Experimental Results  
Table 1 shows the precomputed weighs \(t_{i}\) and \(s_{i}\) in radial layers when the valence of polar vertex is 8. In radial layers, since the \(e_{i,j}\) is far away from the polar vertex when \(i>2\), there are no relations  
Figure 8: (a) Constructing wavelets as linear combination of lazy wavelets and scaling functions in radial layers of overlap regions; (b) Constructing wavelets as linear combination of lazy wavelets and scaling functions in circular layers of overlap regions; (c) Discrete subdivision masks of basis functions. Figure 7: Performing the compound wavelet analysis: (1) separating the input mesh and perform wavelet analysis in the overlap region; (2a) decomposing the polar structure in circular layer; (2b) decomposing the polar structure in radial layer; (3) decomposing the remainder. (4) Joining the resulted meshes after removal of overlapping facets. between the points in different radial directions. In this situation, \(s_{1}\), \(s_{3}\), \(s_{4}\) and \(s_{6}\) are zero. Table 2 shows the precomputed weighs \(w_{i}\) in circular layers. It can be further improved by combining more scaling functions \(\phi^{\prime}_{i,j}\) with \(\psi^{\prime}_{i,j}\). When performing the multiple levels of wavelet analysis, the stability is an important issue of fitting operations. In order to examine the stability of our polar subdivision based wavelets, we perform a noise-filtering experiment similar to [11, 13, 14]. First, we subdivide the mesh 4 times by polar subdivision, and perturb the subdivided mesh with white noise to all vertices of the final mesh. Then, the perturbed surface will be analyzed step by step by using our wavelet analysis algorithm 4 times. At each resolution, we subdivide the mesh to level 4 (the highest resolution) again without considering the wavelet coefficients of any higher resolution. By this way, we get a sequence of low-pass filtered versions of the noisy mesh, shown in Figure 9. We can compare the difference between low-pass filtered versions of noisy mesh and the original unperturbed mesh by calculating the corresponding \(L_{2}\)-norm errors. The test results in Tables 3 and 4 show that the error does not significantly increase after multiple steps of wavelet analysis operations. In contrast to the lazy wavelets, our wavelet scheme has a good performance of stability. Figure 10 shows the results of compound wavelet analysis. After wavelet analysis, the polar structures and the rest quadrilateral meshes are smoothly joined. \begin{table}
\begin{tabular}{c c c c} \hline \(t_{i}\) & \(s_{i}(v_{0}=v)\) & \(s_{i}(v_{0}\in v_{j})\) & \(s_{i}\) (\(v_{0}\in v_{i,j}\)) \\ -0.212173 & 0.060810 & 0.228947 & 0.189068 \\ -0.239089 & 0.026201 & 0.000000 & 0.000000 \\ -0.052675 & -0.560051 & -0.544587 & -0.525336 \\ 0.083197 & 0.026201 & 0.000000 & 0.000000 \\ 0.080547 & -0.011349 & 0.000000 & 0.000000 \\ 0.076314 & -0.506860 & -0.516979 & -0.525336 \\ 0.080547 & -0.011349 & 0.000000 & 0.000000 \\ 0.083197 & 0.182173 & 0.186000 & 0.189068 \\ -0.052675 & - & - & - \\ 0.098667 & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Precomputed weights in radial layers (n = 8). \begin{table}
\begin{tabular}{c c c c c c} \hline \(w_{-2}\) & \(w_{-1}\) & \(w_{0}\) & \(w_{1}\) & \(w_{2}\) & \(w_{3}\) \\ -0.0829 & 0.2265 & -0.5356 & -0.5356 & 0.2265 & -0.0829 \\ \hline \end{tabular}
\end{table} 
Table 2: Precomputed weights in circular layers. From the rules of polar subdivision wavelet transforms, we know that the computations of lifting are executed over each vertex. And the time complexity of computation over one vertex is O(1). So the time complexity of lifting operation only depends the number of vertices of polar structures. Since the time complexity of polar subdivision is O(n) (n is the number of vertices of polar structure), the time complexity of the whole wavelet transforms is O(n). Since it is not necessary to use the filter banks (they