In order to clarify terminology throughout this paper a 2D mesh is any mesh consisting of two dimensional elements, such as triangles, even if these elements are embedded in 3D. A 3D mesh represents a volume data set, and the mesh elements are three dimensional, e.g. tetrahedra, octahedra or hexahedra. We understand the problem of 3D mesh reduction as an approximation problem in appropriately normed function spaces. In the method we proposed a coarse partition of the domain is first constructed with a piecewise finite element space defined on it. Then, the best approximation to the input data on this space according to a chosen norm is computed.
Typical norms used in computer graphics are the norms, but also more general norms can be employed.
The next step is to determine the local contribution of each element to the global approximation error and to mark those elements for refinement, where the local error is above a given threshold . The mesh is adaptively refined by local element subdivision resulting in a better approximation space where a new solution is computed. An iterative application of these steps generates a sequence of nested approximating spaces. For each one of these spaces we have an optimal representation of our data set which can be used for analysis and visualization. The sequence of approximating spaces is built on top of a mesh hierarchy, which is generated by local mesh refinement operations.