Computational Geometry

Mark de BergTU Eindhoven

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(Two) Selected Topics

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Reconstructing Terrainsfrom Elevantion Data

TU e Reconstructing terrains from elevation data

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given: terrain elevation at sample points

goal: construct terrain surface

TU e reconstructing terrains from elevation data

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922elevation?

TU e reconstructing terrains from elevation data

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Idea: use elevation ofnearest sample point

Voronoi diagram

TU e reconstructing terrains from elevation data

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Idea: use elevation ofnearest sample point

Voronoi diagram

Does not give good result:surface not continuous

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Better: determine elevation using interpolation

reconstruction using triangulation

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triangulation

Better: determine elevation using interpolation

reconstruction using triangulation

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triangulation

Better: determine elevation using interpolation

reconstruction using triangulation

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triangulation

Better: determine elevation using interpolation

gives continuous surface

reconstruction using triangulation

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What is a good triangulation?

or or . . .

reconstruction using triangulation

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What is a good triangulation?

or or . . .

long and thin triangles are bad =⇒ try to avoid small angles

reconstruction using triangulation

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What is a good triangulation?

or or . . .

long and thin triangles are bad =⇒ try to avoid small angles

Algorithmic problem: how can we quickly computea triangulation that maximizes the angles?

reconstruction using triangulation

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Voronoi diagrams and Delaunay triangulations

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Voronoi diagram

connect points whoseVoronoi cells are neighbors

Delaunay triangulation:

triangulation that maximizesthe minimum angle!

Voronoi diagrams and Delaunay triangulations

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

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TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

Algorithm for computing Delaunay triangulations:

1. for every triple of points2. do if all other points lie outside Circle(p, q, r)3. then Add triangle ∆(p, q, r) to DT

TU e computing the Delaunay triangulation

∆(p, q, r) is triangle in Delaunay triangulation

Circle(p, q, r) does not contain any other point

⇐⇒

Algorithm for computing Delaunay triangulations:

1. for every triple of points2. do if all other points lie outside Circle(p, q, r)3. then Add triangle ∆(p, q, r) to DT

• running time O(n4): very slow!

• can we do better? yes: there are algorithms running in O(n log n) time

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Analyzing Trajectoriesof Moving Objects

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• GPS data, RFID tags, surveillance cameras, simulations

Moving objects (spatio-temporal data): now widely available

Movement data can be considered in two ways:

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• GPS data, RFID tags, surveillance cameras, simulations

Moving objects (spatio-temporal data): now widely available

Movement data can be considered in two ways:

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• Individual locations, which change over time, are relevant

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• GPS data, RFID tags, surveillance cameras, simulations

Moving objects (spatio-temporal data): now widely available

Movement data can be considered in two ways:

=⇒

• Individual locations, which change over time, are relevant

• Trajectories are relevant

TU e Trajectory analysis

• given two trajectories, how similar are they?

• given a collection of trajectories, report all subtrajectories similar toa query trajectory

• is there a certain motion pattern? (leader following, flocking, . . . )

Analyzing trajectories: examples of interesting questions

• behavorial studies on animals

• analysis of sports games

• analyzing traffic

Analyzing trajectories: applications

TU e Measuring the similarity of trajectories

Basic question: how to measure similarity of two trajectories

A

B

We need a distance function dist(·, ·) on the trajectories:the smaller dist(A,B), the more similar A and B are

TU e Measuring the similarity of trajectories

Basic question: how to measure similarity of two trajectories

A

B

We need a distance function dist(·, ·) on the trajectories:the smaller dist(A,B), the more similar A and B are

First attempt: Hausdorff distance

dist(A,B) = max ( , )maxa∈A

minb∈B

d(a, b) maxb∈B

mina∈a

d(b, a)

TU e Measuring the similarity of trajectories

Basic question: how to measure similarity of two trajectories

A

B

We need a distance function dist(·, ·) on the trajectories:the smaller dist(A,B), the more similar A and B are

First attempt: Hausdorff distance

dist(A,B) = max ( , )maxa∈A

minb∈B

d(a, b) maxb∈B

mina∈a

d(b, a)

TU e Measuring the similarity of trajectories

Basic question: how to measure similarity of two trajectories

A

B

We need a distance function dist(·, ·) on the trajectories:the smaller dist(A,B), the more similar A and B are

First attempt: Hausdorff distance

dist(A,B) = max ( , )maxa∈A

minb∈B

d(a, b) maxb∈B

mina∈a

d(b, a)

TU e Measuring the similarity of trajectories

Is the Hausdorff distance a good measure for trajectory similarity?

Hausdorff distance:

TU e Measuring the similarity of trajectories

Is the Hausdorff distance a good measure for trajectory similarity?

Hausdorff distance:Hausdorff distance:

TU e Measuring the similarity of trajectories

Is the Hausdorff distance a good measure for trajectory similarity?

Hausdorff distance does not takecontinuity of trajectories into account

Hausdorff distance:Hausdorff distance:

Hausdorff distance can be small, evenwhen trajectories are very different

TU e Measuring the similarity of trajectories

A better measure: Frechet distance

A

B

• take continuous mapping µ : A→ B

• take as distance maxa∈A d(a, µ(a))

a

µ(a)

TU e Measuring the similarity of trajectories

A better measure: Frechet distance

A

B

• take continuous mapping µ : A→ B

• take as distance maxa∈A d(a, µ(a))

a

µ(a)

TU e Measuring the similarity of trajectories

A better measure: Frechet distance

A

B

• take continuous mapping µ : A→ B

• take as distance maxa∈A d(a, µ(a))

a

µ(a)

Which mapping? Take the best one:

Frechet-dist(A,B) = minµ

maxa∈A

d(a, µ(a)),

where min is taken over all continuous 1-to-1 mappings

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Hausdorff distance:

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Hausdorff distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

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Frechet distance:

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

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B b1b2

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a1 a2a3 a4

free-space diagram (for given ε)

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a4

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

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B b1b2

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a1 a2a3 a4

free-space diagram (for given ε)

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what does continuous 1-to-1 mappingcorrespond to in diagram?

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

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B b1b2

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a1 a2a3 a4

free-space diagram (for given ε)

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what does continuous 1-to-1 mappingcorrespond to in diagram?

xy-monotone path from lower left to upper right

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

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a3

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what does continuous 1-to-1 mappingcorrespond to in diagram?

xy-monotone path from lower left to upper right

must decide if monotone path existssuch that distance is always ≤ ε

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

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forbidden region in diagram:(a, b) such that d(a, b) > ε

ε

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

a2

a3

a4

forbidden region in diagram:(a, b) such that d(a, b) > ε

ε

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

a2

a3

a4

forbidden region in diagram:(a, b) such that d(a, b) > ε

ε

ε

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

a2

a3

a4

forbidden region in diagram:(a, b) such that d(a, b) > ε

ε

Algorithm:Compute decomposition into free andforbidden regions, and check if mono-tone path exists in free region.

TU e Measuring the similarity of trajectories

Computing similarity: deciding if Frechet-dist(A,B) ≤ ε

A

B b1b2

b3

b4

a1 a2a3 a4

free-space diagram (for given ε)

b1 b2 b3 b4a1

a2

a3

a4

forbidden region in diagram:(a, b) such that d(a, b) > ε

ε

Algorithm:Compute decomposition into free andforbidden regions, and check if mono-tone path exists in free region.

Can be done in O(nm) time

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is aconnected of B similar to A?

A

B

subcurve

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is aconnected of B similar to A?

A

B B′

A

B

subcurve

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is aconnected of B similar to A?

A

B B′

A

B

subcurve

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is aconnected of B similar to A?

A

B B′

A

B

B′

subcurve

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is aconnected of B similar to A?

A

B B′

A

B

mapping from B′ ⊆ B to A

path from bottom to top in diagram

B′

subcurve

TU e Measuring the similarity of trajectories

ε

Partial matching: what if we want to decide if there is a con-nected subcurve of B that is similar to A?

A

B B′

mapping from B′ ⊆ B to A

path from bottom to top in diagram

B′

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what if exact locations are not important?

Recall definition of Frechet distance:

Frechet-dist(A,B) = minµ

maxa∈A

d(a, µ(a)),

where min is taken over all continuous 1-to-1 mappings

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what if exact locations are not important?

Recall definition of Frechet distance:

Frechet-dist(A,B) = minµ

maxa∈A

d(a, µ(a)),

where min is taken over all continuous 1-to-1 mappings

Euclidean distance

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what if exact locations are not important?

Recall definition of Frechet distance:

Frechet-dist(A,B) = minµ

maxa∈A

d(a, µ(a)),

where min is taken over all continuous 1-to-1 mappings

Euclidean distancedifference in direction of movement

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Fr’echet distance: summary

• good measuring similarity of curves (e.g. trajectories)

• can be computed fairly efficiently for polygonal curves

• many variations (partial matching, directional Frechet distance, . . . )