## 4 Bicubic B-spline Subdivision Wavelets  
Formally a realistic model may be composed of the polar structures and the quadrilaterals surrounding the polar structures, shown in Figure 1. In fact the polar structures can also be considered as the combination of extraordinary mesh points surrounded by triangles, and quadrilaterals that have points of valence four. The extraordinary mesh points need only be separated by one layer of points of valence four. For the quadrilaterals, we can apply the classical subdivisions, such as Catmull-Clark subdivision. Both Catmull-Clark and polar subdivision generalize bicubic spline subdivision. They can form a powerful combination for smooth object design: while Catmull-Clark subdivision can be applied to the points of valence four, polar subdivision can be applied to the regions surrounding polar vertex. [PERSON] _et al._[17] have shown how to combine the meshes of these two generalizations of bicubic B-spline subdivision. Based on their work, we can develop a compound subdivision wavelets on the meshes with quadrilateral faces. Catmull-Clark subdivision wavelet can be applied to the ordinary points while polar subdivision wavelets can be applied on extraordinary points.  
The Catmull-Clark subdivision wavelets have been constructed with local lifting by [PERSON] [9] and [PERSON] _et al._[13]. These schemes can be applied directly to the regions of common quadrilaterals(not include the quadrilaterals in polar structures). In [PERSON]'s scheme, the overlap regions between common quadrilaterals and polar structures are both subdivided by Catmull-Clark subdivision and polar subdivision,and drop the boundary facets of the meshes subdivided in polar subdivision and Catmull-Clark subdivision and join them by identifying the resulting boundary vertices. He proved that the jointed subdivision surface is \(C^{1}\) continuity and has bounded curvatures. Because of the different local lifting schemes applied to Catmull-Clark subdivision and polar subdivision, we need make an adjustment to the local lifting schemes in order to keep the continuity of two parts.  
The process of wavelet analysis in overlap region is shown in Figure 7. We consider the overlap regions as the extend part of polar structures and adopt the way similar to the construction of polar subdivision wavelets. The stencils for local lifting are shown in Figure 8. By applying this scheme, we can get the decomposition rules in the overlap regions:  
\[\begin{split}& v_{i,j}=\frac{1}{2}(4v^{\prime}_{i,j-1}-e^{\prime} _{i,j}),\quad e_{i,j}=e^{\prime}_{i,j}-\frac{1}{2}(v_{i,j-1}+v_{i,j}),\\ & v_{i,j+k}=v_{i,j+k}-w_{k}e_{i,j}\quad\forall k=-1,0,1,2,\forall i.\end{split} \tag{14}\]and reconstruction rules in the overlap regions:  
\[\begin{split}& v_{i,j+k}=v_{i,j+k}+w_{k}e_{i,j}\quad\forall k=-1,0,1, 2,\forall i,\\ & e^{\prime}_{i,j}=e_{i,j}+\frac{1}{2}(v_{i,j}+v_{i,j+1}),\quad v ^{\prime}_{i,j}=\frac{1}{4}(e^{\prime}_{i,j-1}+2v_{i,j}+e^{\prime}_{i,j}). \end{split} \tag{15}\]  
When performing the wavelet analysis on the model, we should separate the polar structures and the remaining quadrilateral meshes, and perform the wavelet analysis in the overlap region at first. After that we can decompose the polar structures and the rest meshes separately. The wavelet synthesis are processed in the reverse order of analysis.