Computing Realistic Terrains from Imprecise Elevations

Abstract

It is ideal for triangulated terrains to have characteristics or properties that are realistic. In the imprecise terrain model, each vertex of a triangulated terrain has an imprecise eleva- tion value only known to lie within some interval. Under some objective function, the goal is to compute a precise terrain by assigning a single elevation value to each point, so that the objective function is optimized.

This thesis examines various objectives, such as minimizing the number of local extrema and minimizing the terrain’s surface area. We give algorithms in some cases, hardness results in other cases. Specifically, we consider four objectives: (1) minimizing the number of local extrema; (2) optimizing coplanar features; (3) minimizing the surface area; (4) minimizing the maximum steepness.

Problem (1) is known to be NP-hard, but we give an algorithm for a special case. For problem (2) we give an NP-hardness proof for the general case and a positive result for a special case. Meanwhile, problems (3) and (4) can be approximated using Second Order Cone Programming. We also consider versions of these problems for terrains one dimension down, where the output is a polyline. Here we give very efficient algorithms for all objective functions considered.

Finally, we go beyond terrains and briefly consider the distant representatives problem, where the goal is to choose precise points from segments to be as far from each other as possible. For this problem, we give a parameterized algorithm for vertical segments, prove NP-hardness for unit horizontal segments, and show hardness of approximation for vertical and horizontal segments.