U-V System And Its Application In Curve And Surface Reconstruction

Curve and surface reconstruction has been one of the most important content of research in Computer Aided Geometric Design(CAGD). It has wide applications in industry manufacture, chemistry, meteorology, geology and study of medicine. In this thesis we will focus on the construction and properties of a special complete orthogonal function system called U-V systems, and it's application in planar curve reconstruction.In chapter 1, we begin with the history and current situation of CAGD and curve and surface reconstruction. We also discuss the advantages and disadvantages of the two main representation forms of curves and surfaces, the parametric and implicit representations. In chapter 2 we discuss the construction and properties of U-V system in detail and list some good properties such as the property of normal complete orthogonal, reproducibility and compactly supported as well. From the process of the construction of function generator it is known that U-V system has different level of continuous and discontinuous information, which is very important for the reconstruction of curves which are not smooth enough. In chapter 3, we give an algorithm for the reconstruction of planar curves with sharp features using V-system introduced in chapter 2. The algorithm puts the detection of sharp features and the reconstruction of sharp features together, and handle the C~0,C~1,C~2 discontinuous in one uniform framework as well. As the V-system is a complete orthogonal function system, we can get our control coefficients from numerical integration instead of least squares which is used in most process of parametric reconstruction, which obviously improves the efficiency of the algorithm. In chapter 4, we extend the construction of V-system to hierarchical T-meshes and obtain a complete orthogonal wavelet basis on hierarchical T-meshes, which provides a theoretical foundation for adaptive curve and surface reconstruction and adaptive signal and frequency analysis. We end this thesis with a short chapter of conclusions and future works.