### subdivision wavelets  
The subdivisions can iteratively refine a control mesh \(M^{0}\) so that the sequence of increasingly faceted polyhedra \(M^{1},M^{2},...\) converge to some limit surface \(M^{\infty}\). The property of subdivision makes it possible to define a collection of refinable scaling functions. Based on nested linear spaces spanned by the refinable scaling functions and an inner product designated, we can construct subdivision wavelets. Subdivision wavelets not only inherit the performance advantage of wavelets in shape reservation, but also overcome the shortcomings of classical wavelets that are hardly applied in 3D domains. They can be applied on surfaces of arbitrary topology type. Comparing with subdivisions, the subdivision wavelets transform can perform both decimation and refinement operations.  
The theoretical basis for constructing wavelets on subdivision surfaces was established by [PERSON] _et al._[1]. They also proposed a class of subdivision based wavelets. By generalizing the uniform subdivision in topology to a new irregular subdivision scheme, [PERSON] and [PERSON] [2, 3] extended the work of [PERSON] _et al._ and proposed a wavelet-based multiresolution analysis that can be applied directly to irregular meshes whose connectivity is unchanged in the wavelet analysis. [PERSON] _et al._[4] showed how to use least-squares data fitting to reverse subdivision rules and constructed the wavelets by straightforward matrix observations. [PERSON] _et al._[5] constructed multiresolution surfaces of arbitrary topologies by locally reversing the Doo subdivision scheme. [PERSON] _et al._[6] developed the Loop subdivision wavelets with small support for progressive geometry compression. Since the lifting scheme proposed by [PERSON] [7] can generate new biorthogonal wavelets from the classic wavelets and lazy wavelets, it is an important tool to construct subdivision wavelets. Based on lifting scheme, [PERSON] and [PERSON] showed how to construct lifting wavelets on the sphere with customized properties [8]. Using local lifting operations performed on polygonal meshes, [PERSON] _et al._[9, 10] gave a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two manifold topology and introduced the generalized B-spline subdivision-surface wavelets.  
Using local lifting and the discrete inner production, [PERSON] [11] constructed a biorthogonal wavelet on the Loop subdivision. [PERSON] _et al._[12] proposed unlifted Loop subdivision wavelet by optimizing free parameters in the extended subdivisions. [PERSON] _et al._[13] developed an effective wavelet construction based on general Catmull-Clark subdivisions and the resulted wavelets have better fitting quality than the previous Catmull-Clark like subdivision wavelets. Recently [PERSON] _et al._[14] constructed the new biorthogonal wavelets based on \(\sqrt{3}\) subdivision over triangular meshes. The polar subdivision was first proposed by [PERSON] _et al._[15; 16]. [PERSON] _et al_[17; 18] extended the work of [PERSON] _et al_ and provided the polar subdivision schemes, which can be combined with Catmull-Clark subdivision. Our work is mainly based on the scheme of [PERSON] _et al._  
The rest of paper is organized as follows: The rules of bicubic polar subdivision and its reformulation are introduced in Section 2. We explain how to construct the polar subdivision based wavelets in radial/circular layers in Section 3, and show how to combine the polar subdivision wavelets with Catmull-Clark subdivision wavelet in Section 4. We present the experimental results for the wavelet analysis in Section 5. Finally, we conclude the work in Section 6.