Matching Polyhedral Terrains Using Overlays of Envelopes

For a collection F of d-variate piecewise linear functions of overall combinatorial complexity n, the lower envelope E(F) of F is the pointwise minimum of these functions. The minimization diagram M(F) is the subdivision of Rd obtained by vertically (i.e., in direction xd+1) projecting E(F). The overlay O(F, G) of two such subdivisions M(F) and M(G) is their superposition. We extend and improve the analysis of de Berg et al. [17] by showing that the combinatorial complexity of O(F, G) is Ω(nd α2 (n)) and O(nd+ε ) for any ε > 0 when d ≥ 2, and O(n2 α(n) log n) when d = 2. We also describe an algorithm that constructs O(F, G) in this time. We apply these results to obtain efficient general solutions to the problem of matching two polyhedral terrains in higher dimensions under translation. That is, given two piecewise-linear terrains of combinatorial complexity n in Rd+1 , we wish to find a translation of the first terrain that minimizes its distance to the second, according...

• Type: Conference
• Year: 2004
• Where: SWAT
• Authors: Vladlen Koltun, Carola Wenk
• Keywords: Algorithms, Combinatorial Complexity, Distance Measures, Overall Combinatorial Complexity, SWAT 2004