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Algorithms for GIS:
Computing visibility on terrains
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Visibility on terrains
• Are two points (on a terrain) visible to each other? • What can one see from a given point (on a terrain)? • How much does the visible area increase if we stand on a 10ft ladder? • What is the point with largest visibility? • What is the point with lowest visibility? • How to place an ugly pipe in a scenic area? • How to place a scenic highway? • What is the cumulative visible area from these set of cell towers? • Find a set of tower locations to cover the terrain • …
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Visibility on terrains
Problem: • Terrain model T + viewpoint v • Compute the viewshed of v: the set of points in T visible from v
Sierra Nevada, 30m resolution
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Visibility on terrains
Input: terrain model • grid or TIN (triangulation)
Output: viewshed model • grid elevation model ==> grid viewshed • TIN elevation model ==> TIN viewshed
(…more later…)
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Visibility on grid terrains
Sierra Nevada, 30m resolution
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Visibility
(u,v) visible iff segment uv does not intersect T
uv is called line-of-sight (LOS)
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Basic viewshed algorithm
1. Find the 2D intersections between vp and the grid lines
2. Lift to 3D: find the height of p by linear interpolation
3. Check if slope(vp’) is below slope (LOS)
v
v
p
p
p'
p'
ay1
y2
y3
y1 = d1 tan ay2 = d2 tan a
d1d2 d3
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Viewshed on grids
• The straightforward O( n n) algorithm • uses linear interpolation • “exact” as much as data allows; uses all data available
• Can we do better (faster) without introducing any approximation?
• Van Kreveld[vK’96] • nearest neighbor interpolation <——- simpler • O (n lg n)
Grid of n points: n x n
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Grids
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Grids with linear interpolation
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Grids with nearest neighbor interpolation
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vertical slope(p,a) = (h_p - h_a) / d(a,p)
a
p
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new
old
common
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a1 a2
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future
dead
alive
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Van Kreveld’s radial sweep algorithm
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Van Kreveld’s radial sweep algorithm
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Van Kreveld’s radial sweep algorithm
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Van Kreveld’s radial sweep algorithm
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test grid: hemisphere viewshed with NN interpolation viewshed with linear interpolation
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Computing viewsheds
1. Straightforward algorithm • uses linear interpolation • can be adapted to other interpolations • O( n n)
2. Radial sweep approach • uses nearest neighbor interpolation • crucially exploits that cells are “flat” obstacles • uses radial sweep + augmented RB tree • has accuracy issues ==> undesirable • O(n lg n)
3. A different approach: concentric sweep using horizons
Grid of n points: n x n
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Viewshed on grids using a concentric sweep
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Viewshed on grids using a concentric sweep
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Viewshed on grids using a concentric sweep
L1
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Viewshed on grids using a concentric sweep
L2
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Viewshed on grids using a concentric sweep
L3
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Horizons
• Merriam Webster: • the line where the terrain and the sky seem to meet
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Horizons
• Merriam Webster: • the line where the terrain and the sky seem to meet
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Hv : [0, 2PI) ——> R Horizon (with respect to v) in direction a, Hv(a)
• cut the terrain with a vertical plane through a ray from v of azimuth a • Hv(a) is the maximum vertical angle (zenith) of all points intersected by this plane (all the points on T
whose projection on the xy-plane has azimuth a)
Horizon
v b
Hv(a) = b
z
v x
y
a
this point (its slope) appears on the skyline
• Note: a point beyond this point is visible if and only if it’s above the horizon
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Horizons
• Merriam Webster: • the line where the terrain and the sky seem to meet
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Given point p on the terrain Look at the cross-section of the terrain from v towards p, and consider only the points between v and p Point p defines the azimuth angle ==> We can define a horizon in the same way
• Hv(p) is the maximum vertical angle (zenith) of all points between v and p whose projection on the xy-plane has azimuth = azimuth(vp)
Horizon
v b
Hv(a) = b
z
v x
y
a
this point (its slope) appears on the skyline
• Note: a point beyond this point is visible if and only if it’s above the horizon
p
p
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Viewshed on grids using a concentric sweep
vertical slope (zenith)
c PI/4 PI/2 PI 2PI
L1H1
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Viewshed on grids using a concentric sweep
vertical slope (zenith)
c PI/4 PI/2 PI 2PI
L2
H2
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Viewshed on grids using a concentric sweep
vertical slope (zenith)
c PI/4 PI/2 PI 2PI
L2
H1
H2
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Viewshed on grids using a concentric sweep
vertical slope (zenith)
c PI/4 PI/2 PI 2PI
H1 H2
L2
L1 H12 =
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Viewshed on grids using a concentric sweep
Is point p in L3 visible ?
L3p
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Viewshed on grids using a concentric sweep
L3p
ap is visible if slope(vp) < H12(a)
Is point p in L3 visible ?
a
H12
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Computing viewsheds
• Straightforward algorithm: O( n sqrt n) with linear interpolation • O(n lg n) with nearest neighbor (NN) interpolation
• uses radial sweep + augmented RB tree • crucially exploits NN • loses some accuracy
• A different approach: concentric sweep using horizons • can be adapted for linear interpolation or nearest neighbor • can be used for triangulated terrains • fast in practice because horizons stay very small
n = nb. of cells in the grid
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Viewsheds on triangulated terrains
• viewshed(p) contains all points of the terrain that are visible from p
from: http://arxiv.org/pdf/1309.4323.pdf
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Viewsheds on triangulated terrains
• viewshed(p) contains all points of the terrain that are visible from p
• viewshed(p) may intersect a triangle • viewshed(p) may intersect a triangle multiple times • Question: what is the complexity of viewshed(p) in the worst case? • Theorem: The complexity (number of vertices) of a viewshed on a
triangulated terrain is O(n2).
from: http://arxiv.org/pdf/1309.4323.pdf
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from: HH, MdB, KT 2009
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• Based on horizon computation • similar to grids • idea: traverse triangles in order of increasing distance form viewpoint, and
update horizon. • bootstrap with divide and conquer
Computing viewsheds on triangulated terrains
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Summary
• Visibility is a fundamental problem • Straightforward solution is fairly intuitive/simple/fast
• works well even for very large terrains • Refinement of straightforward solution exposes elegant ideas
• radial sweep • augmented RB-trees • visibility via horizons
• starting point for all improved solutions • Accuracy
• Interpolation is important
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Visibility on terrains
• Viewsheds are starting point for many other problems • Given a set of viewpoints, compute their joint visibility
• aka multiple-source viewshed • Find locations of watch towers so that together they can guard the terrain • Find point of maximum/minimum visibility • Find viewshed count (VC) grid
• VC(i,j) = nb. visible points in viewshed(i,j) • …
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Viewshed count
• Input: elevation grid G • Output: VC grid
• VC(i,j) = size of viewshed(i,j)
• Sketch an algorithm to compute VC and its running time
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