√3-Subdivision-Based Biorthogonal Wavelets

Abstract

A new efficient biorthogonal wavelet analysis based on the principal square root of subdivision is proposed in the paper by using the lifting scheme. Since the principal square root of subdivision is of the slowest topological refinement among the traditional triangular subdivisions, the multiresolution analysis based on the principal square root of subdivision is more balanced than the existing wavelet analyses on triangular meshes, and accordingly offers more levels of detail for processing polygonal models. In order to optimize the multiresolution analysis process, the new wavelets, no matter whether they are interior or on boundaries, are orthogonalized with the local scaling functions based on a discrete inner product with subdivision masks. Because the wavelet analysis and synthesis algorithms are actually composed of a series of local lifting operations, they can be performed in linear time. The experiments demonstrate the efficiency and stability of the wavelet analysis for both closed and open triangular meshes with principal square root of subdivision connectivity. The principal square root of -subdivision-based biorthogonal wavelets can be used in many applications such as progressive transmission, shape approximation, multiresolution editing and rendering of 3D geometric models.

Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

“…For testing the stability of wavelet transform, we made a noise-filtering experiment, which is often used to examine the stability of approximate subdivision wavelets [10][11][12][13][14][15]. We first perturb all vertices of the mesh at highest resolution with white noise.…”

“…Wang et al [12] developed an effective wavelet construction based on general CatmullClark subdivisions and the resulted wavelets have better fitting quality than the previous Catmull-Clark like subdivision wavelets. They also constructed several new biorthogonal wavelets based on 3 subdivision over triangular meshes, and approximate and interpolatory 2 subdivision over quadrilateral meshes [13][14][15]. Zhang et al presented a biorthogonal wavelet approach based on dual Doo-Sabin subdivision with the aid of the barycenter of the V-faces corresponding to old vertices [16].…”

“…With this assumption, the mutual inner product of wavelets and scaling functions is defined as the sum of multiplications of corresponding coefficients (geometry coordinates of points) at finer resolution, and calculated directly from the subdivision template. It is maybe not accurate, but works well in many works [10][11][12][13][14][15]. Figure 5 shows the discrete mask of extraordinary…”

Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing

“…With the idea of lifting scheme, [1] introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully adopted to develop multiresolution algorithms for quad surface and √ 3-refinement triangle surface processing in [35] and [36] respectively. When considering the biorthogonality, these papers do not use the conventional L 2 (IR 2 ) inner product, and therefore, they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets.…”

“…In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and √ 2 refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1-2): [29][30][31][32][33][34][35][36][37][38][39]2004) and Wang et al :874-884, 2006; IEEE Trans Vis Comput Graph 13 (5): [914][915][916][917][918][919][920][921][922][923][924][925]2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al : [874][875][876][877][878][879][880][881][882][883][884]2006) for dyadic multiresolution quad surface processing.…”