An Optimization-Based Approach for Continuous Map Generalization - - PowerPoint PPT Presentation

1

An Optimization-Based Approach for Continuous Map Generalization

Dongliang Peng

Chair of Computer Science I, University of W¨ urzburg, Germany

2-1 [Google Maps] Marienkapelle [source: Wikipedia]

2-2 [Google Maps] Marienkapelle [source: Wikipedia] scroll

2-3 [Google Maps] Marienkapelle [source: Wikipedia] scroll scroll

2-4 [Google Maps] Marienkapelle [source: Wikipedia] scroll scroll scroll

2-5 [Google Maps] Marienkapelle [source: Wikipedia] Buildings disappear suddenly! scroll scroll scroll

2-6 [Google Maps] Marienkapelle [source: Wikipedia] Buildings disappear suddenly! scroll scroll scroll Smooth changes provide better experience!

3-1

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map.

3-2

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]:

3-3

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination

3-4

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination Simplification

3-5

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination Simplification Aggregation

3-6

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination Simplification Aggregation

• Classifi. & Symboli.

3-7

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination Simplification Aggregation Exaggeration

• Classifi. & Symboli.

3-8

Map Generalization. . .

. . . is about deriving a smaller-scale map from an existing map. Typical generalization operators [ESRI 1996]: Elimination Simplification Aggregation Collapse Typification Exaggeration

• Classifi. & Symboli.

Displacement Refinement

4-1

Continuous Map Generalization...

t = 0 t = 1 . . . is to derive a series of maps with smooth changes. input input

4-2

Continuous Map Generalization...

t = 0 t = 0.2 t = 1 . . . is to derive a series of maps with smooth changes. input input

4-3

Continuous Map Generalization...

t = 0 t = 0.2 t = 0.4 t = 1 . . . is to derive a series of maps with smooth changes. input input

4-4

Continuous Map Generalization...

t = 0 t = 0.6 t = 0.2 t = 0.4 t = 1 . . . is to derive a series of maps with smooth changes. input input

4-5

Continuous Map Generalization...

t = 0 t = 0.6 t = 0.2 t = 0.8 t = 0.4 t = 1 . . . is to derive a series of maps with smooth changes. input input

5-1

Related Work

• Morph between polylines

[N¨

• llenburg et al. 2008]

5-2

Related Work

• Morph between polylines

[N¨

• llenburg et al. 2008]
• Generate a good sequence of maps

[Chimani et al. 2014]

5-3

Related Work

• Morph between polylines

[N¨

• llenburg et al. 2008]
• Generate a good sequence of maps

[Chimani et al. 2014]

• Data structure for continuous generalization

[van Oosterom et al. 2014]

space scale cube (SSC)

5-4

Related Work

• Morph between polylines

[N¨

• llenburg et al. 2008]
• Generate a good sequence of maps

[Chimani et al. 2014]

• Data structure for continuous generalization

[van Oosterom et al. 2014]

export space scale cube (SSC) zoom

• ut

6-1 Contents of Thesis Optim. Related Generalization

6-2 Contents of Thesis Optim.

Optimal sequence for aggregation

Related Generalization

6-3 Contents of Thesis Optim.

A

⋆

ILP Optimal sequence for aggregation

Related Generalization

6-4 Contents of Thesis Optim.

A

⋆

ILP Optimal sequence for aggregation

Related Generalization

Aggregation Classification

6-5 Contents of Thesis Optim.

A

⋆

ILP Optimal sequence for aggregation Administrative boundaires

Related Generalization

Aggregation Classification

6-6 Contents of Thesis Optim.

DP A

⋆

ILP Optimal sequence for aggregation Administrative boundaires

Related Generalization

Aggregation Classification

6-7 Contents of Thesis Optim.

DP A

⋆

ILP Optimal sequence for aggregation Administrative boundaires Elimination Simplification

Related Generalization

Aggregation Classification

6-8 Contents of Thesis Optim.

DP A

⋆

ILP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Elimination Simplification

Related Generalization

Aggregation Classification

6-9 Contents of Thesis Optim.

DP MST A

⋆

ILP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Elimination Simplification

Related Generalization

Aggregation Classification

6-10 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Elimination Simplification Exaggeration

Related Generalization

Aggregation Classification

6-11 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Elimination Simplification Exaggeration

Related Generalization

Aggregation Classification

6-12 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Elimination Simplification Exaggeration

Related Generalization

Aggregation Classification

6-13 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Elimination Simplification Exaggeration Simplification

Related Generalization

Aggregation Classification

6-14 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Choosing right data structures Elimination Simplification Exaggeration Simplification

Related Generalization

2ε p Aggregation Classification SortedDictionary, SortedSet, . . .

6-15 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Choosing right data structures Elimination Simplification Exaggeration Simplification

Related Generalization

2ε p Aggregation Classification SortedDictionary, SortedSet, . . .

6-16 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Choosing right data structures Elimination Simplification Exaggeration Simplification

Related Generalization

2ε p Aggregation Classification SortedDictionary, SortedSet, . . .

7-1

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-2

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-3

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-4

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-5

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-6

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-7

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-8

Research Problem

input input village, town, city ly sport facility ly swamp ly

7-9

Research Problem

Given aggregation costs:

What is an optimal sequence?

input input village, town, city ly sport facility ly swamp ly

8-1

Preliminaries

start map goal map

8-2

Preliminaries

p3 Polygon pi start map goal map p1 p2 p4 p5 : area on start map

8-3

Preliminaries

Patch ui u1 u5 u6 u7 u6 Polygon pi start map goal map u3 u1 u2 u4 u5 : connected set of areas : area on start map

8-4

Preliminaries

Patch ui Region Ri R1 R2 Polygon pi start map goal map R1 R2 R1 R2 : connected set of areas : area on start map : area on the goal map

8-5

Preliminaries

Patch ui Region Ri R1 R2 Aggregate the smallest patch with its neighbour Polygon pi start map goal map R1 R2 R1 R2 : connected set of areas : area on start map : area on the goal map

9-1

Interleave Aggregation Sequences

input input

9-2

Interleave Aggregation Sequences

input input input input

• utput

input input

• utput

Compute a sequence for each region

9-3

Interleave Aggregation Sequences

Interleave according to order of smallest areas (as merge sort)

input input

• utput
• utput
• utput

input input

• utput

input input

• utput

Compute a sequence for each region

9-4

Interleave Aggregation Sequences

Interleave according to order of smallest areas (as merge sort)

input input

• utput
• utput
• utput

input input

• utput

input input

• utput

Compute a sequence for each region

10-1

Subdivision

Subdivision Pt,i

P2,2 P2,1 P2,3 P3,2 P3,1 P3,3 P3,4 P3,5 Pgoal = P4,1 Pstart = P1,1 P2,4

: patches subdividing a region

10-2

Subdivision

Subdivision Pt,i

P2,2 P2,1 P2,3 P3,2 P3,1 P3,3 P3,4 P3,5 Pgoal = P4,1 Pstart = P1,1 P2,4

Size n : patches subdividing a region : #polygons on start map n = 4

10-3

Subdivision

Subdivision Pt,i

P2,2 P2,1 P2,3 P3,2 P3,1 P3,3 P3,4 P3,5 Pgoal = P4,1 Pstart = P1,1 P2,4

Size n #subdivions is exponential in n. : patches subdividing a region : #polygons on start map n = 4

11-1

Formalizing a Pathfinding Problem

start goal

11-2

Formalizing a Pathfinding Problem

• Each subdivision is represented as a node

start goal

11-3

Formalizing a Pathfinding Problem

• Each subdivision is represented as a node
• Find a shortest path w.r.t. cost functions

start goal

12-1

Cost Function

• Type change: ftype(Ps,i, Ps+1,j)

We wish to aggregate patches with similar types Ps,i Ps+1,j u v village, town, city ly sport facility ly swamp ly lake, pond

12-2

Cost Function

• Type change: ftype(Ps,i, Ps+1,j)

We wish to aggregate patches with similar types Ps,i Ps+1,j u v village, town, city ly sport facility ly swamp ly lake, pond

12-3

Cost Function

• Type change: ftype(Ps,i, Ps+1,j)

We wish to aggregate patches with similar types Ps,i Ps+1,j u v village, town, city ly sport facility ly swamp ly lake, pond ℓint(Ps,k) = 19.5

6.2 3.9 2.8 3.7 2.9

• Interior length: flength(Ps,k)

Less length, easier to perceive

12-4

Cost Function

• Type change: ftype(Ps,i, Ps+1,j)

We wish to aggregate patches with similar types Ps,i Ps+1,j u v village, town, city ly sport facility ly swamp ly lake, pond ℓint(Ps,k) = 19.5

6.2 3.9 2.8 3.7 2.9

• Interior length: flength(Ps,k)

Less length, easier to perceive

13-1

Cost Function

• Path Π = (P1,i1, P2,i2, . . . , Pt,it)

13-2

Cost Function

• Path Π = (P1,i1, P2,i2, . . . , Pt,it)

gtype(Π) =

t−1

• s=1

ftype(Ps,is, Ps+1,is+1) glength(Π) =

t−1

• s=2

flength(Ps,is)

13-3

Cost Function

• Path Π = (P1,i1, P2,i2, . . . , Pt,it)
• Combination of the two costs:

g(Π) = (1 − λ)gtype(Π) + λglength(Π) gtype(Π) =

t−1

• s=1

ftype(Ps,is, Ps+1,is+1) glength(Π) =

t−1

• s=2

flength(Ps,is)

13-4

Cost Function

• Path Π = (P1,i1, P2,i2, . . . , Pt,it)
• Combination of the two costs:

g(Π) = (1 − λ)gtype(Π) + λglength(Π)

λ = 0.5

gtype(Π) =

t−1

• s=1

ftype(Ps,is, Ps+1,is+1) glength(Π) =

t−1

• s=2

flength(Ps,is)

14-1

A

⋆ Algorithm

• A best-first search algorithm. Find a path from s to t

s t

14-2

A

⋆ Algorithm

• A best-first search algorithm. Find a path from s to t
• Cost function: F(u) = g(u) + h(u)

– g(u): exact cost of s-u path – h(u): estimated cost of shortest u-t path s u t g(u) h(u)

14-3

A

⋆ Algorithm

• A best-first search algorithm. Find a path from s to t
• Cost function: F(u) = g(u) + h(u)

– g(u): exact cost of s-u path – h(u): estimated cost of shortest u-t path

• Guarantees a shortest path if

h(u) is smaller than real cost s u t g(u) h(u)

14-4

A

⋆ Algorithm

• A best-first search algorithm. Find a path from s to t
• Cost function: F(u) = g(u) + h(u)

– g(u): exact cost of s-u path – h(u): estimated cost of shortest u-t path

• Guarantees a shortest path if

h(u) is smaller than real cost s u t g(u) h(u)

• Helps ignore some paths

15-1

Estimating Cost

• htype(Pt,i) = n−1

s=t ftype(Ps,is, Ps+1,is+1)

We assume: Each patch immediately gets the target type.

P2,i2 P3,i3 P4,i4 P5,i5

s u t g(u) h(u)

15-2

Estimating Cost

• hlength(Pt,i)
• htype(Pt,i) = n−1

s=t ftype(Ps,is, Ps+1,is+1)

We assume: Each patch immediately gets the target type.

P2,i2 P3,i3 P4,i4 P5,i5

s u t g(u) h(u)

16-1

Overestimation

• Try finding a path by exploring at most M = 200,000
• nodes. If fail, try again but increasing estimated costs.

start goal

16-2

Overestimation

• Try finding a path by exploring at most M = 200,000
• nodes. If fail, try again but increasing estimated costs.
• A path seems more expensive, thus may be ignored

start goal

16-3

Overestimation

• Try finding a path by exploring at most M = 200,000
• nodes. If fail, try again but increasing estimated costs.
• Not optimal anymore
• nce increasing estimated costs
• A path seems more expensive, thus may be ignored

start goal

17-1

Integer Linear Programming

Form of an integer linear program (ILP) minimize C Tx subject to Ex ≤ H, x ≥ 0 0, and x ∈ ZI,

17-2

Integer Linear Programming

Form of an integer linear program (ILP) minimize C Tx subject to Ex ≤ H, x ≥ 0 0, and x ∈ ZI, Given variables x, minimize a cost subject to some constraints.

17-3

Integer Linear Programming

Form of an integer linear program (ILP) minimize C Tx subject to Ex ≤ H, x ≥ 0 0, and x ∈ ZI, Given variables x, minimize a cost subject to some constraints.

18-1

Using Integer Linear Programming

start goal

• Model complete graph by setting variables and constraints

18-2

Using Integer Linear Programming

start goal

• Model complete graph by setting variables and constraints
• Solve ILP with minimizing total cost

18-3

Using Integer Linear Programming

start goal

• Model complete graph by setting variables and constraints
• Solve ILP with minimizing total cost
• Define path according to values of variables, known from

solution

19-1

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3

p r q p r q p r q

19-2

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x1,q,r = 0

p r q p r q

19-3

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x2,p,r = 1 x2,r,r = 1 x1,q,r = 0 x2,q,r = 0

p r q p r q

19-4

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x2,p,r = 1 x2,r,r = 1 x3,p,r = 1 x3,r,r = 1 x1,q,r = 0 x2,q,r = 0 x3,q,r = 1

p r q p r q

19-5

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x2,p,r = 1 x2,r,r = 1 x3,p,r = 1 x3,r,r = 1 x1,q,r = 0 x2,q,r = 0 x3,q,r = 1

p r q p r q

• Constraints:

p is assigned to only one polygon:

r∈P xt,p,r = 1

19-6

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x2,p,r = 1 x2,r,r = 1 x3,p,r = 1 x3,r,r = 1 x1,q,r = 0 x2,q,r = 0 x3,q,r = 1

p r q p r q

• Constraints:

p is assigned to only one polygon: Enforce aggregation:

• r∈P xt,p,r = 1
• r∈P xt,r,r = n − t + 1

19-7

Example Variable and Constraints

• Variable: xt,p,r ∈ {0, 1}

∀t ∈ T, ∀p, r ∈ P xt,p,r = 1 ⇔ p is assigned to r at time t.

t = 1 t = 2 t = 3 x1,p,r = 0 x1,r,r = 1

p r q

x2,p,r = 1 x2,r,r = 1 x3,p,r = 1 x3,r,r = 1 x1,q,r = 0 x2,q,r = 0 x3,q,r = 1

p r q p r q

• Constraints:

p is assigned to only one polygon: Enforce aggregation:

• r∈P xt,p,r = 1
• r∈P xt,r,r = n − t + 1
• In total,

5 sets of variables 17 sets of constraints

20-1

Case Study

• Environment: C#, CPLEX

20-2

Case Study

• Environment: C#, CPLEX

5,448 patches scale 1 : 50 k 734 patches (regions) scale 1 : 250 k

• Data

21

Comparison of A

⋆ and ILP 1–5 6–10 11–15 16–20 21–25 26–36 25 50 75 100 n: number of polygons percent (%) A

⋆

ILP160 s ILP600 s percentage of regions that were found optimal solutions

22-1

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-2

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-3

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-4

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-5

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-6

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-7

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-8

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-9

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-10

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-11

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-12

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-13

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-14

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-15

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-16

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-17

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

22-18

An Optimal Sequence by A

⋆ 300 m start (n = 17) goal

23 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Choosing right data structures Elimination Simplification Exaggeration Simplification

Related Generalization

2ε p Aggregation Classification SortedDictionary, SortedSet, . . .

24-1

Generalizing Buildings to Built-up Areas

400 m Input: buildings

24-2

Generalizing Buildings to Built-up Areas

400 m Input: buildings

25-1

Aggregate and Grow

• riginal buildings
• Aggregate buildings that are too close, when zooming out

25-2

Aggregate and Grow

• riginal buildings
• Aggregate buildings that are too close, when zooming out
• Bridges and buildings constitute

a minimum spanning tree (MST)

25-3

Aggregate and Grow

t = 0

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

25-4

Aggregate and Grow

grow t = 0 t = 0.4

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge

25-5

Aggregate and Grow

grow t = 0 t = 0.4

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge

25-6

Aggregate and Grow

grow grow t = 0 t = 0.4 t = 0.6

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge add bridge

25-7

Aggregate and Grow

grow grow t = 0 t = 0.4 t = 0.6

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge add bridge

25-8

Aggregate and Grow

grow grow grow t = 0 t = 0.4 t = 0.6 t = 1

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge add bridge add bridge

25-9

Aggregate and Grow

grow grow grow t = 0 t = 0.4 t = 0.6 t = 1

• riginal buildings
• Aggregate buildings that are too close, when zooming out

zoom out: t increases

• Bridges and buildings constitute

a minimum spanning tree (MST)

add bridge add bridge add bridge

26-1

Three Join Types of Buffering

building

26-2

Three Join Types of Buffering

building miter: keep right angles

26-3

Three Join Types of Buffering

building miter: keep right angles dG

26-4

Three Join Types of Buffering

building square: avoid long spikes miter: keep right angles dG

26-5

Three Join Types of Buffering

building square: avoid long spikes miter: keep right angles dG dG dG dG

26-6

Three Join Types of Buffering

building square: avoid long spikes miter: keep right angles dG round: detect if two buildings are too close dG dG dG

27-1

Simpifying Based on Dilation and Erosion

polygon d

27-2

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents d

27-3

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents erode with 2d: remove bumps d

27-4

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents erode with 2d: remove bumps dilate with d d

28-1

Case Study

• Runtime: O(n3),

n: total number of edges over all input buildings

28-2

Case Study

• Environment

C#, Clipper (for buffering, dilation, erosion, and merge)

• Runtime: O(n3),

n: total number of edges over all input buildings

28-3

Case Study

• Environment

C#, Clipper (for buffering, dilation, erosion, and merge)

• Runtime: O(n3),

n: total number of edges over all input buildings

• Data: 2.5 k buildings, n = 19 k edges, 1 : 15 k,

dG = 25 m (IGN)

28-4

Case Study

• Environment

C#, Clipper (for buffering, dilation, erosion, and merge)

• Runtime: O(n3),

n: total number of edges over all input buildings

• Data: 2.5 k buildings, n = 19 k edges, 1 : 15 k,

dG = 25 m (IGN)

• 12 min for computing a sequence of 10 maps

29-1

Animation

400 m zooming out

29-2

Animation

400 m zooming out

29-3

Animation

400 m zooming out

29-4

Animation

400 m zooming out

29-5

Animation

400 m zooming out

29-6

Animation

400 m zooming out

29-7

Animation

400 m zooming out

29-8

Animation

400 m zooming out

29-9

Animation

400 m zooming out

29-10

Animation

400 m zooming out

29-11

Animation

400 m zooming out

30 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP MST A

⋆

ILP LSA DP Optimal sequence for aggregation Administrative boundaires Buildings to built-up areas Morphing polylines Choosing right data structures Elimination Simplification Exaggeration Simplification

Related Generalization

2ε p Aggregation Classification SortedDictionary, SortedSet, . . .

31-1

Conclusion

• Studied four topics of continuous generalization

31-2

Conclusion

• Studied four topics of continuous generalization
• Used optimization methods to attain good results

31-3

Conclusion

• Studied four topics of continuous generalization
• Used optimization methods to attain good results
• Shared experience of using right data structures

31-4

Conclusion

• Studied four topics of continuous generalization
• Used optimization methods to attain good results
• Shared experience of using right data structures

Future work

• Improve our methods

31-5

Conclusion

• Studied four topics of continuous generalization
• Used optimization methods to attain good results
• Shared experience of using right data structures

Future work

• Improve our methods
• Usability testing

31-6

Conclusion

• Studied four topics of continuous generalization
• Used optimization methods to attain good results
• Shared experience of using right data structures

Future work

• Improve our methods
• Usability testing
• Work on complete maps

(with roads, buildings, ...) scroll

31-7

Conclusion

Thank you!

• Studied four topics of continuous generalization
• Used optimization methods to attain good results
• Shared experience of using right data structures

Future work

• Improve our methods
• Usability testing
• Work on complete maps

(with roads, buildings, ...) scroll