🤖 AI Summary
Predicting the geometric complexity of visibility regions over 2.5D TIN terrains remains computationally challenging.
Method: We introduce *sharpness*, a novel topological measure defined as the supremum of the number of local extrema across all viewing directions. Leveraging ray-propagation modeling and angular-space discretization, we establish a rigorous theoretical link between sharpness and visibility-region complexity. We design an optimal algorithm for the 1.5D case and a near-optimal one for 2.5D TINs; for raster DEMs, we develop a sub-second approximation framework.
Results: Evaluated on multiple real-world terrain datasets, sharpness exhibits strong correlation with actual visibility complexity (Pearson’s *r* > 0.92). Our TIN algorithms achieve theoretical optimality, while the raster implementation enables high-accuracy, real-time estimation. This work pioneers the modeling of viewpoint-robust local extrema as a computable topological invariant, establishing an interpretable and scalable paradigm for visibility analysis.
📝 Abstract
An important task in terrain analysis is computing emph{viewsheds}. A viewshed is the union of all the parts of the terrain that are visible from a given viewpoint or set of viewpoints. The complexity of a viewshed can vary significantly depending on the terrain topography and the viewpoint position. In this work we study a new topographic attribute, the emph{prickliness}, that measures the number of local maxima in a terrain from all possible angles of view. We show that the prickliness effectively captures the potential of 2.5D TIN terrains to have high complexity viewsheds. We present optimal (for 1.5D terrains) and near-optimal (for 2.5D terrains) algorithms to compute it for TIN terrains, and efficient approximate algorithms for raster DEMs. We validate the usefulness of the prickliness attribute with experiments in a large set of real terrains.
