Smoothing Imprecise 1.5D Terrains

A 1.5D terrain is an x-monotone polyline with n vertices. An imprecise 1.5D terrain is a 1.5D terrain with a y-interval at each vertex, rather than a fixed y-coordinate. A realization of an imprecise terrain is a sequence of n y-coordinates, one for each interval, such that each y-coordinate is within its corresponding interval. For certain applications in terrain analysis, it is important to be able to find a realization of an imprecise terrain that is smooth. In this paper we model smoothness by considering the change in slope between consecutive edges of the terrain. The goal is to find a realization of the terrain where the maximum slope change is minimized. We present an exact algorithm that runs in O(n2) time.

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We study optimization problems for polyhedral terrains in the presence of data imprecision. An imprecise terrain is given by a triangulated point set where the height component of the vertices is specified by an interval of possible values. We restrict ourselves to terrains with a one-dimensional projection, usually referred to as 1.5-dimensional terrains, where an imprecise terrain is given by an x-monotone polyline, and the y-coordinate of each vertex is not fixed but only constrained to a given interval. Motivated mainly by applications in terrain analysis, in this paper we study five different optimization measures related to obtaining smooth terrains, for the 1.5-dimensional case. In particular, we present exact algorithms to minimize and maximize the total turning angle, as well as to minimize the maximum slope change. Furthermore, we also give approximation algorithms to minimize the largest turning angle and to maximize the smallest turning angle.

keywords: Computational Geometry, Geographical Information Analysis, Data Imprecision, Terrains

Journal Article (peer-reviewed)

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Smoothing Imprecise 1.5D Terrains
International Journal of Computational Geometry and Applications
20, 4, 381–414, 2010
http://dx.doi.org/10.1142/S0218195910003359

Conference Proceedings (peer-reviewed)

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Smoothing Imprecise 1.5D Terrains
Proc. 6th Workshop on Approximation and Online Algorithms
LNCS 5426, 214–226, 2009
http://dx.doi.org/10.1007/978-3-540-93980-1_17

Conference Proceedings (peer-reviewed)

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Minimizing Slope Change in Imprecise 1.5D terrains
Proc. 21th Canadian Conference on Computational Geometry
55–58, 2009

Workshop or Poster (weakly reviewed)

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Smoothing Imprecise 1-Dimensional Terrains
Proc. 24th European Workshop on Computational Geometry
141–144, 2008

Technical Report (not reviewed)

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Smoothing Imprecise 1.5D Terrains
UU-CS-2008-036, 2008
http://www.cs.uu.nl/research/techreps/UU-CS-2008-036.html

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