Fast Polygonal Approximation ofTerrains and Height

Several algorithms for approximating terrains and other height elds using polygonal meshes are described, compared, and optimized. These algorithms take a height eld as input, typically a rectangular grid of elevation data H(x; y), and approximate it with a mesh of triangles, also known as a triangulated irregular network, or TIN. The algorithms attempt to minimize both the error and the number of triangles in the approximation. Applications include fast rendering of terrain data for ight simulation and tting of surfaces to range data in computer vision. The methods can also be used to simplify multi-channel height elds such as textured terrains or planar color images. The most successful method we examine is the greedy insertion algorithm. It begins with a simple triangulation of the domain and, on each pass, nds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. The mesh is updated either with Delaunay triangulation or with data-dependent triangulation. Most previously published variants of this algorithm had expected time cost of O(mn) or O(n log m + m 2), where n is the number of points in the input height eld and m is the number of vertices in the triangulation. Our optimized algorithm is faster, with an expected cost of O((m+n) log m). On current workstations, this allows one million point terrains to be simpliied quite accurately in less than a minute. We are releasing a C++ implementation of our algorithm. Contents