Research Of Multiresolution Representation For Curves And Surfaces Based On Subdivision

Subdivision technique is an important method of geometric modeling and is the combination of parametric and polygonal modeling. With the advantages of algorithms like simple expressed, easy to realize and the ability to handle meshes with arbitrary topology, subdivision technique has been well used in many areas in recent years. Since there are more ways to obtain three-dimension digital geometry data than before, geometric modeling on different equipments is needed. As a result, multiresolution modeling is introduced. From the point of view of multiresolution modeling, subdivision refines low-resolution models to high-resolution ones. Combining multiresolution analysis with subdivision, reverse subdivision scheme can be constructed, which decreases resolution of models. So under the setting of subdivision, we can transform models from low-resolution to high-resolution or do the opposite process. Based on two subdivision schemes for curves generation and one scheme for surface, the corresponding reverse schemes are introduced in the paper. The concrete introduction are as follows:Based on an interpolating and an approximating ternary subdivision scheme, respectively, decomposition and reconstruction filters satisfying biorthogonal conditions are constructed, which are the key issues in realizing curves multiresolution representation. The decomposition process is related to reverse subdivision, and the reconstruction process is related to subdivision. Compared with methods based on binary subdivision, the low-resolution curves with similar levels of resolution obtained by ternary and binary methods, respectively, have the similar ability in approximating the original curves. However, ternary methods run smaller number of decomposition times than binary methods to get low-resolution results with similar levels of resolution for the same original curve.Meshes with subdivision connectivity, named semi-regular meshes, are necessary for the applications of subdivision wavelets and reverse subdivision. For any original high-resolution mesh, we first simplify it and obtain a base mesh. Then, the base mesh is subdivided by 3^1/2 subdivision scheme and the vertices are resampled according to the original mesh. Finally, we get the reconstructed mesh with 3^1/2 subdivision connectivity. By calculating the Hausdorff distance between reconstructed and original meshes, the reconstructed meshes with different levels of resolution can be verified as well approximations to the original mesh.Based on an approximating 3^1/2 subdivision scheme, its corresponding reverse scheme with one parameter is studied. The reverse subdivision scheme includes topology and geometry operations which are introduced to be applied on close and open mesh surfaces. With different values of parameter, the effect of low-resolution mesh surfaces in approximating the original surface can be adjusted. Under certain values of parameter, the reverse subdivision scheme can be used in mesh denosing. Compared with multiresolution analysis based on Loop subdivision scheme, since 3^1/2 subdivision scheme generates faces more slowly than Loop scheme, reverse 3^1/2 subdivision decreases faces with a slower speed and produces more levels of resolution for a mesh.