Fast Polygonal Approximation of Terrains and Height Fields

Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field 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 flight simulation and fitting of surfaces to range data in computer vision. The methods can also be used to simplify multi-channel height fields 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, finds 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 logm+m), where n is the number of points in the input height field and m is the number of vertices in the triangulation. Our optimized algorithm is faster, with an expected cost of O((m+n) logm). On current workstations, this allows one million point terrains to be simplified quite accurately in less than a minute. We are releasing a C++ implementation of our algorithm.