## 5 Experimental Results
\begin{table}
\begin{tabular}{c c c c} \hline Level & Number of vertices & \(L_{2}\) error of our scheme (with) & \(L_{2}\) error of lazy wavelets (without) \\
4 & 55297 & 0.00194545 & 0.00198193 \\
3 & 13825 & 0.000941068 & 0.00477773 \\
2 & 3457 & 0.000538884 & 0.0179629 \\
1 & 865 & 0.000344581 & 0.0756179 \\
0 & 217 & 0.000270338 & 0.333253 \\ \hline \end{tabular}
\end{table} 
Table 4: \(L_{2}\) error of mushroom in different levels (with and without lifting). Figure 9: Progressive wavelet analysis of surfaces (with white noises) from level 4 to level 0. \begin{table}
\begin{tabular}{c c c c} \hline Level & Number of vertices & \(L_{2}\) error of our scheme (with) & \(L_{2}\) error of lazy wavelets (without) \\
4 & 24577 & 0.00192136 & 0.00192128 \\
3 & 6145 & 0.00096071 & 0.00475723 \\
2 & 1537 & 0.00061625 & 0.017283 \\
1 & 385 & 0.000485615 & 0.0720949 \\
0 & 97 & 0.000562381 & 0.251045 \\ \hline \end{tabular}
\end{table} 
Table 3: \(L_{2}\) error of hat in different levels (with and without lifting). often mean large matrices) to perform the wavelet transforms and the computation is fully in-place, there is no additional memory cost either. The space complexity of wavelet transforms also depends on the number of vertices of polar structure. The well-designed data structures based on the principle of polar structure can effectively reduce the memory cost further. Though the subdivisions in radial layers and circular layers are independent, more testing is necessary to determine which should perform first. We performed the experiments on both selective orders. We perturbed the original meshes with white noise, and performed wavelet analysis on the perturbed meshes. Then we subdivided the new meshes to original resolution, and compared them with the original unperturbed ones by calculating \(L_{2}\) errors. From the experimental results shown in Table 5, we find that there is only a small difference between the orders, especially if many vertices in the complex models. \begin{table}
\begin{tabular}{c c c c} \hline Mesh & A and B & A and O & B and O \\ mushroom & 0.00001449 & 0.00200506 & 0.00200448 \\ hat & 0.00018902 & 0.00577932 & 0.00577456 \\ hand & 0.00017053 & 0.00491544 & 0.00491222 \\ \hline \end{tabular}
\end{table} 
Table 5: The test results of performing polar subdivision wavelet transforms in different orders. For simple, we name the resulted meshes by performing analysis in radial layers first as A; the resulted meshes by performing analysis in circular layer first as B; O means original unperturbed meshes. The table lists the \(L_{2}\) error between any two of them. Figure 10: Mesh sequences in different resolution: (a) the original meshes; (b) the meshes by wavelet analysis once; (c) the meshes by wavelet analysis twice; (d) the surface of polar structures by wavelet analysis twice. [PERSON] _et al._[18] provided the revised polar subdivision with the \(C^{2}\) continuity on polar vertices, which synthesized the subdivision rules of radial and circular layers in a more tightly way. As the wavelet framework facilitates both multiresolution analysis and synthesis efficiently, our proposed wavelet scheme inherits the advantages of subdivision and provides the stable shape preservation for analysis, by performing the local lifting operations.