Higher-Order Delaunay Triangulations Marc van Kreveld Presentation based on joint work with: Joachim Gudmundsson, Mikael Hammar, Herman Haverkort, Thierry.

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

1012

1573

78

2421

2529

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1012

1573

78

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2529

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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


• 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


• 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


• All edges that may be in an order-4 Delaunay triangulation


• 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


• Useful order-k Delaunay edges: edges that can be used in an order-k DT

useful order 5 Delaunay edge


• 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


• 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)


• 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


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


• 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


• 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– ...


• 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


• Not trivial: NP-hard– min max vertex degree– min max area ratio


(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


• Not trivial: approximation– min max vertex degree– min max area ratio


(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


• 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


• 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


• 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


• 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


• 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


• 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

Higher-Order Delaunay Triangulations Marc van Kreveld Presentation based on joint work with: Joachim Gudmundsson, Mikael Hammar, Herman Haverkort, Thierry.

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