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Higher-Order Delaunay Triangulations
Marc van Kreveld
Presentation based on joint work with:
Joachim Gudmundsson, Mikael Hammar, Herman Haverkort, Thierry de Kok,
Maarten Löffler, Rodrigo Silveira
Overview
• Motivation– Triangulation for terrains
• Higher order Delaunay triangulations– Basics
• First order Delaunay triangulation results• Minimizing local minima in terrains• Higher order triangulations of polygons
Polyhedral terrains, or TINs
• Points with (x, y) and elevation as input• TIN as terrain representation• Choice of triangulation is important
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1573
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1012
1573
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Realistic terrains
• Due to erosion, realistic terrains– have few local minima– have valley lines that continue
local minimum, interrupted valley line
after an edge flip
Terrain modeling in GIS
• Terrain modeling is extensively studied in geomorphology and GIS
• Need to avoid artifacts like local minima• Need correct “shape” for run-off models,
hydrological models, avalanche models, ...
6 12
1752
2124
local minimum in a TIN
Terrain modeling in GIS
• Terrain convexity/concavity in cross-sections also influencessurface flow interest inplan curvatureandprofile curvature
Delaunay triangulation
• Maximizes minimum angle• Empty circle property
Delaunay triangulation
• Does not take elevation into account• May give local minima• May give interrupted valleys• Does not consider curvature
Triangulate to minimize local minima?
Triangulate to minimize local minima?
Connect everything to global minimum and complete bad triangle shape & interpolation
Higher order Delaunay triangulations
• Compromise between good shape & interpolation, and flexibility (w.r.t. DT) to satisfy other constraints
• k -th order: allow k points in circle
1st order
0th order
4th order
Higher order Delaunay triangulations
• Introduced by Gudmundsson, Hammar, and van Kreveld (ESA 2000, CGTA 2004)
• Delaunay triangulation = 0-th order DT• A triangulation is k-th order
Delaunay if the circumcircleof each of its triangles has≤ k points inside
Higher order Delaunay triangulations
• All edges that may be in an order-4 Delaunay triangulation
Higher order Delaunay triangulations
• uv is an order-k Delaunay edge the order-(k +1) VD has cells for{u, p1,..., pk} and {v, p1,..., pk}
(= the bisector of uv exists in the order-(k +1) VD
u
vp1
p2
{u, p1, p2, p3}
{v, p1, p2, p3}
p3
Higher order Delaunay triangulations
• Useful order-k Delaunay edges: edges that can be used in an order-k DT
useful order 5 Delaunay edge
Higher order Delaunay triangulations
• Computing all useful order-k Delaunay edges takes O(nk log n + nk2) time:– Compute the order-(k +1) VD to get order-k edges– Test each edge in O(k + log n) time for usefulness
• Trace the edge in the DT• Determine the two circles• Query with them:
≤ k points inside?
find k th closestpoint from center
Higher order Delaunay triangulations
• A useful order-k Delaunay edge can be used in an order-k DT, just take the CDT with the edge
• But: two (or more) useful order-k Delaunay triangulations may give an order-(2k−2) DT– The CDT guarantees order 2k−2– Sometimes the CDT gives order 2k−2 but another
triangulation gives order k
Open: Given n edges, complete them to the lowest order DT
(solved if all edges have useful order at most 3)
Higher order Delaunay triangulations
• Gudmundsson, Hammar, vK (2000)– higher order Delaunay triangulations
• Gudmundsson, Haverkort, vK (2003)– constrained higher order Delaunay triangulations
• de Kok, vK, Löffler (2005)– local minima, NP-hardness, drainage, experiments
• vK, Löffler, Silveira (2006/2007)– first order DT, polynomial, NP-hardness, approximation
• Silveira, vK (2007)– polygons, dynamic programming, FPT, experiments
Minimizing local minima
• Minimizing local minima for order-k DT is NP-hard if k = (n
)
Open: - Given n points and k 2 (constant), is minimizing local minima over all order-k DT NP-hard?- Is there an approximation with a factor better than O(k 2)?
We study two heuristics (flip and hull) for reducing local minima on terrains, and one (valley) making contiguous drainage networks
Experiments on terrains
• Quinn Peak• Elevation grid of
382 x 468
• Random sample of 1800 vertices
• Delaunay triangulation
• 53 local minima
• Hull heuristic applied
• Order 4 Delaunay triangulation
• 25 local minima
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10
order
loca
l min
ima
Hull heuristic
Flip heuristic
Delaunay triangulation
Hull-8 + valley heuristic
Experimental results
• Hull and Flip reduce local minima by 60−70% for order 8; Hull is often better
• Hull and Flip are near-optimal for orders up to 8• Valley reduces the number of valley edge
components by 20−40% for order 8• Hull + Valley seems best
First order Delaunay triangulations
• First order Delaunay triangulations have a simple structure– all certain edges (Delaunay) give a subdivision in
triangles and quadrilaterals– all possible edges
are diagonals ofthe quadrilaterals
First order Delaunay triangulations
• Minimizing local minima is easy: choose the diagonal that connects to the lowest point ofthe quadrilateral
O(n log n) timefor any n-point set
8
7
12
9
4
5
2
First order Delaunay triangulations
• Also simple: measures that relate to individual edges or triangles (or is composed of it), like– min max triangle area– min max angle– min total edge length– min sum of inscribed
circle radii– ...
First order Delaunay triangulations
• Not trivial– min max vertex degree– min max area ratio
(across edges)– min max spatial angle
(across edges)– max no. of convex edges– min no. of mixed vertices
A vertex v is mixed if it does not lie onthe 3d convex hull of {v} neighbors(v)
= a plane through v exists with all neighbors to one side
terrain
plane
First order Delaunay triangulations
• Not trivial: NP-hard– min max vertex degree– min max area ratio
(across edges)– min max spatial angle
(across edges)– max no. of convex edges– min no. of mixed vertices
NP-hard to decide if degree ≤ 20 can be achieved; reduction from planar 3-SAT
NP-hard; reduction from planar 3-SAT
NP-hard; reduction from planarMAX-2-SAT
First order Delaunay triangulations
• Not trivial: approximation– min max vertex degree– min max area ratio
(across edges)– min max spatial angle
(across edges)– max no. of convex edges– max no. of non-mixed
vertices
No PTAS possible;3/2-approx exists
(1−)-approx in
2O(1/ ) ·n time
(1−)-approx in
2O(1/) ·n time
2
First order Delaunay triangulations
• Not trivial: polynomial– min max area ratio (across edges)– min max spatial angle (across edges)
• Still O(n log n) time:– Sort the O(n) candidate values– Do a binary search; each decision
involves building an O(n) size2-SAT formula where the diagonalrepresents true or false
xi = true
xi = false
xj = true
xj = false(xi xj)
Higher order DT for polygons
• Can we optimally triangulate a polygon P over all order-k DTs (min max area; min weight; ...) ?
Extension 1: there may be points outside P that influence the order of triangles in P
Extension 2: there may be points or components inside P
Higher order DT for polygons
• Optimal triangulation of a polygon by dynamic programming for decomposable measures, typically in O(n3) time (Klincsek, 1980)
u
v
wOPT(u,v) =
max / min
{ OPT(u,w) OPT(v,w) }
w between u,v
Higher order DT for polygons
• For order-k DT:– First determine all order-k Delaunay triangles in P– Only use these in the dynamic program
u
v
w
The O(nk 2) order-k Delaunay triangles can be determined in O(nk 2 log k + kn log n) time(also for extension 1)
Dynamic programming: O(nk) optimal subproblems, O(k) choices O(nk 2) time
Higher order DT for polygons
• When components are inside:– Connect topmost point of each component to get
one polygon, in all possible ways– Apply the DP algorithm for each polygon
For h components there are O(nh) connections, but for each component we can restrict ourselves to O(k) connections
The whole algorithm takes O(nk log n + k h+2 n) time, so FPT for constant k
Higher order DT for polygons
• Why only O(k) connections?– The topmost point t must have a Delaunay edge tu up– Any Delaunay edge intersects O(k) order-k Delaunay
edges (GHK 2000)– The lowest one, vw, used in OPT gives that vt and wt
are also in OPT; at least one of them is upward
So for one upward Delaunay edge tu from t, only try tu and all upper endpoints of the order-k Delaunay edges that intersect tu
u
wtv
Higher order DT for polygons
• For a point set, if the order is low, there are many fixed edges and few components
Order 4; blue edgesare in every order-4 DT
Conclusions and future work
• Theory:– NP-hardness of minimizing local minima for small k ?– Completion of edges to lowest order DT ?– The PTAS for max no. of convex edges for order-1 DT
extends to order k,but is doubly-exponential in k:
– The PTAS for max no. of non-mixed vertices does not seem to extend
• Practice:– Up to what order can terrain criteria be solved
optimally in reasonable time?– Can flow processes be modelled well enough?
2 (2 O(k)
/ 2)
Based on ...
• Gudmundsson, Hammar, vK (2000)Higher order DTs
• Gudmundsson, Haverkort, vK (2003)Constrained higher order DTs
• de Kok, vK, Löffler (2005)
Generating realistic terrains with higher-order DTs• vK, Löffler, Silveira (2006/2007)
Optimization for first order DTs• Silveira, vK (2007)
Optimal higher order DTs of polygons