Terrain prickliness: theoretical grounds for high complexity viewsheds

TL;DR

This work studies the complexity of terrain viewsheds and introduces prickliness, a global, affine-invariant topographic attribute defined as the maximum number of local maxima across all affine transformations of a terrain. It develops theoretical results showing no correlation between prickliness and viewshed complexity in 1.5D terrains, but a provable, near-linear bound between prickliness and viewshed complexity in 2.5D terrains, along with corresponding algorithms. The authors provide optimal or near-optimal algorithms to compute prickliness for 1.5D TINs ($O(n\log n)$) and 2.5D TINs ($O(n^2)$), plus an approximate method for DEMs, and prove 3SUM-hardness lower bounds. Extensive experiments on real terrains compare prickliness to existing topographic attributes (TRI, TSI, FD) and demonstrate prickliness as a strong predictor of viewshed complexity for TINs, with clearer results on DEMs for high-point viewpoints. The work includes code releases for prickliness and multi-viewpoint viewsheds, offering practical tools for terrain analysis and GIS applications.

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 and (under standard assumptions) near-optimal algorithms to compute it for 1.5D and 2.5D TIN terrains, respectively, and efficient approximate algorithms for raster DEMs. We validate the usefulness of the prickliness attribute with experiments in a large set of real terrains.

Figures (23)

• Figure 1: Part of a TIN with a high-complexity viewshed. The viewpoint (not shown) is placed at the center of projection. The relevant triangles of the TIN are the ones shown, which define $n$ peaks and ridges. The viewshed in this case is formed by $\Theta(n^2)$ visible regions.
• Figure 2: Left: a TIN (in $\mathbb{R}^2$) with three peaks and one viewpoint ($p$), with a viewshed composed of three parts (visible parts shown orange). Right: transformation of the terrain with no peaks (other than $p$) but the same viewshed complexity. Dotted segments show lines of sight from $p$.
• Figure 3: (left) A TIN $T$, with triangulation edges shown in black, and elevation indicated using colors. (right) A visualization of the prickliness of $T$ as a function of the angles $(\theta, \phi)$ that define each direction (circles indicate contour lines for $\theta$); color indicates prickliness. The maximum prickliness is $8$, attained at a direction of roughly $\theta=13^\circ$ and $\phi=60^\circ$ (north-east from the origin).
• Figure 4: Example showing $se(v)$ (shaded) for two vertices.
• Figure 5: Left: terrains with low prickliness can have high viewshed complexity. Right: a vase.

Theorems & Definitions (27)

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