Efficient Triangular Surface Approximations Using Wavelets and Quadtree Data Structures

We present a new method for adaptive surface meshing and triangulation which controls the local level-of-detail of the surface approximation by local spectral estimates. These estimates are determined by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semi-orthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonal approximation of the surface can be performed with larger triangles without loosing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierarchical meshing algorithm which transforms the initially regular data grid into a quadtree representation by rejection of unimportant mesh vertices. The optimum triangulation of the resulting quadtree cells is carried out by selection from a look-up table. The tree grows recursively as controlled by detail signals which are computed from a modified inverse WT. In order to control the local level-of-detail, we introduce a new class of wavelet space filters acting as “magnifying glasses” on the data. We show that our algorithm performs a low algorithmic complexity, so that surface meshing can be achieved at interactive rates, such as required by flight simulators. However, other applications are possible as well, such as mesh reduction in complex data, FEM or radiosity meshing. The method is applied on different types of data comprising both digital terrain models and laser range scans. In addition, quantitative investigations on error analysis are carried out.