Map Generalization. . .

(1)

1

An Optimization-Based Approach for Continuous Map Generalization

Dongliang Peng

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

(2)

2-1 [Google Maps]

Marienkapelle [source: Wikipedia]

(3)

2-2 [Google Maps]

scroll

(4)

2-3 [Google Maps]

scroll

(5)

2-4 [Google Maps]

scroll

(6)

2-5 [Google Maps]

Buildings disappear suddenly!

scroll

(7)

2-6 [Google Maps]

Buildings disappear suddenly!

scroll

Smooth changes

provide better experience!

(8)

3-1

Map Generalization. . .

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

(9)

3-2

Map Generalization. . .

Typical generalization operators [ESRI 1996]:

(10)

3-3

Map Generalization. . .

Elimination

(11)

3-4

Map Generalization. . .

Elimination Simplification

(12)

3-5

Map Generalization. . .

Elimination Simplification Aggregation

(13)

3-6

Map Generalization. . .

Classifi. & Symboli.

(14)

3-7

Map Generalization. . .

Exaggeration Classifi. & Symboli.

(15)

3-8

Map Generalization. . .

Collapse

Typification Exaggeration

Classifi. & Symboli.

Displacement Refinement

(16)

4-1

Continuous Map Generalization...

t = 0

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

input

(17)

4-2

Continuous Map Generalization...

t = 0 t = 0.2

input

(18)

4-3

Continuous Map Generalization...

t = 0 t = 0.2 t = 0.4

input

(19)

4-4

Continuous Map Generalization...

t = 0

t = 0.6

t = 0.2 t = 0.4

input

(20)

4-5

Continuous Map Generalization...

t = 0

t = 0.6

t = 0.2

t = 0.8

t = 0.4

input

(21)

5-1

Related Work

• Morph between polylines [N¨ollenburg et al. 2008]

(22)

5-2

Related Work

• Generate a good sequence of maps [Chimani et al. 2014]

(23)

5-3

Related Work

• Data structure for continuous generalization

[van Oosterom et al. 2014]

space scale cube (SSC)

(24)

5-4

Related Work

• Data structure for continuous generalization

[van Oosterom et al. 2014]

export

space scale cube (SSC)

zoom out

(25)

6-1 Contents of Thesis Optim. Related Generalization

(26)

6-2 Contents of Thesis Optim.

Optimal sequence for aggregation

Related Generalization

(27)

A^? ILP Optimal sequence for aggregation

(28)

Aggregation Classification

(29)

Administrative boundaires

(30)

DP A^? ILP Optimal sequence for aggregation

(31)

(32)

Buildings to built-up areas

(33)

DP

MST A^? ILP Optimal sequence for aggregation

(34)

6-10 Contents of Thesis

Aggregation, Simplification, Elimination

Optim.

DP

Exaggeration

(35)

Optim.

DP

Morphing polylines

Exaggeration

(36)

Optim.

DP

MST A^? ILP

LSA DP Optimal sequence for aggregation

Morphing polylines

Exaggeration

(37)

Optim.

DP

MST A^? ILP

Morphing polylines

Exaggeration

Simplification

(38)

Optim.

DP

MST A^? ILP

Morphing polylines

Choosing right data structures

Exaggeration

Simplification

p 2ε

SortedDictionary, SortedSet, . . .

(39)

Optim.

DP

MST A^? ILP

Morphing polylines

Exaggeration

Simplification

p 2ε

(40)

Optim.

DP

MST A^? ILP

Morphing polylines

Exaggeration

Simplification

p 2ε

(41)

7-1

Research Problem

input

input village, town, city ly

sport facility ly swamp ly

(42)

7-2

Research Problem

input

(43)

7-3

Research Problem

input

(44)

7-4

Research Problem

input

(45)

7-5

Research Problem

input

(46)

7-6

Research Problem

input

(47)

7-7

Research Problem

input

(48)

7-8

Research Problem

input

(49)

7-9

Research Problem

Given aggregation costs:

What is an optimal sequence?

input

(50)

8-1

Preliminaries

start map goal map

(51)

8-2

Preliminaries

p₃ Polygon p_i

start map goal map

p₁ p₂

p₄ p₅

: area on start map

(52)

8-3

Preliminaries

Patch u_i

u₁

u₅

u₆

u₇

u₆ Polygon p_i

start map goal map

u₃ u₁ u₂

u₄ u₅

: connected set of areas : area on start map

(53)

8-4

Preliminaries

Patch u_i Region R_i

R₁ R₂ Polygon p_i

start map goal map

R₁ R₂ R₁ R₂

: area on the goal map

(54)

8-5

Preliminaries

Patch u_i Region R_i

R₁ R₂ Aggregate the smallest patch with its neighbour

Polygon p_i

start map goal map

R₁ R₂ R₁ R₂

: area on the goal map

(55)

9-1

Interleave Aggregation Sequences

input input

(56)

9-2

Interleave Aggregation Sequences

input input

input output input input output input

Compute a sequence for each region

(57)

9-3

Interleave Aggregation Sequences

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

input output output output input

(58)

9-4

Interleave Aggregation Sequences

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

input output output output input

(59)

10-1

Subdivision

Subdivision P_t_,i

P_2,2 P_2,1

P_2,3

P_3,2 P_3,1

P_3,3 P_3,4 P_3,5

P_goal = P_4,1 P_start = P_1,1

P_2,4

: patches subdividing a region

(60)

10-2

Subdivision

Subdivision P_t_,i

P_2,2 P_2,1

P_2,3

P_3,2 P_3,1

P_3,3 P_3,4 P_3,5

P_2,4

Size n

: patches subdividing a region : #polygons on start map

n = 4

(61)

10-3

Subdivision

Subdivision P_t_,i

P_2,2 P_2,1

P_2,3

P_3,2 P_3,1

P_3,3 P_3,4 P_3,5

P_2,4

Size n

#subdivions is exponential in n.

: patches subdividing a region : #polygons on start map

n = 4

(62)

11-1

Formalizing a Pathfinding Problem

start goal

(63)

11-2

Formalizing a Pathfinding Problem

• Each subdivision is represented as a node

start goal

(64)

11-3

Formalizing a Pathfinding Problem

• Each subdivision is represented as a node

• Find a shortest path w.r.t. cost functions

start goal

(65)

12-1

Cost Function

• Type change: f_type(P_s_,i , P_s_+1,j)

We wish to aggregate patches with similar types

P_s_,i P_s_+1,j u

v

village, town, city ly sport facility ly

swamp ly lake, pond

(66)

12-2

Cost Function

P_s_,i P_s_+1,j u

v

swamp ly lake, pond

(67)

12-3

Cost Function

P_s_,i P_s_+1,j u

v

swamp ly lake, pond

`_int(P_s_,k) = 19.5

6.2

3.9 2.8

3.7

• Interior length: f_length(P_s_,k) 2.9

Less length, easier to perceive

(68)

12-4

Cost Function

P_s_,i P_s_+1,j u

v

swamp ly lake, pond

`_int(P_s_,k) = 19.5

6.2

3.9 2.8

3.7

• Interior length: f_length(P_s_,k) 2.9

Less length, easier to perceive

(69)

13-1

Cost Function

• Path Π = (P_1,i₁, P_2,i₂, . . . , P_t_,i_t )

(70)

13-2

Cost Function

• Path Π = (P_1,i₁, P_2,i₂, . . . , P_t_,i_t )

g_type(Π) =

t−1

X

s=1

f_type(P_s_,i_s, P_s_+1,i_s+1)

g_length(Π) =

t−1

X

s=2

f_length(P_s_,i_s)

(71)

13-3

Cost Function

• Path Π = (P_1,i₁, P_2,i₂, . . . , P_t_,i_t )

• Combination of the two costs:

g(Π) = (1 − λ)g_type(Π) + λg_length(Π) g_type(Π) =

t−1

X

s=1

g_length(Π) =

t−1

X

s=2

f_length(P_s_,i_s)

(72)

13-4

Cost Function

• Path Π = (P_1,i₁, P_2,i₂, . . . , P_t_,i_t )

• Combination of the two costs:

g(Π) = (1 − λ)g_type(Π) + λg_length(Π)

λ = 0.5

g_type(Π) =

t−1

X

s=1

g_length(Π) =

t−1

X

s=2

f_length(P_s_,i_s)

(73)

14-1

A

^?

Algorithm

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

s

t

(74)

14-2

A

^?

Algorithm

• 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)

(75)

14-3

A

^?

Algorithm

• Guarantees a shortest path if h(u) is smaller than real cost

s

u

t g(u)

h(u)

(76)

14-4

A

^?

Algorithm

• Guarantees a shortest path if h(u) is smaller than real cost

s

u

t g(u)

h(u)

• Helps ignore some paths

(77)

15-1

Estimating Cost

• h_type(P_t_,i) = Pn−1

s=t f_type(P_s_,i_s, P_s_+1,i_s+1)

We assume: Each patch immediately gets the target type.

P_2,i₂ P_3,i₃ P_4,i₄ P_5,i₅

s

u

t g(u)

h(u)

(78)

15-2

Estimating Cost

• h_length(P_t_,i)

• h_type(P_t_,i) = Pn−1

s=t f_type(P_s_,i_s, P_s_+1,i_s+1)

We assume: Each patch immediately gets the target type.

P_2,i₂ P_3,i₃ P_4,i₄ P_5,i₅

s

u

t g(u)

h(u)

(79)

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

(80)

16-2

Overestimation

• A path seems more expensive, thus may be ignored

start goal

(81)

16-3

Overestimation

• Not optimal anymore

once increasing estimated costs

• A path seems more expensive, thus may be ignored

start goal

(82)

17-1

Integer Linear Programming

Form of an integer linear program (ILP) minimize C^Tx

subject to E x ≤ H, x ≥ 000, and x ∈ Z^I ,

(83)

17-2

Integer Linear Programming

subject to E x ≤ H, x ≥ 000, and x ∈ Z^I , Given variables x,

minimize a cost subject to some constraints.

(84)

17-3

Integer Linear Programming

subject to E x ≤ H, x ≥ 000, and x ∈ Z^I , Given variables x,

minimize a cost subject to some constraints.

(85)

18-1

Using Integer Linear Programming

start goal

• Model complete graph by setting variables and constraints

(86)

18-2

Using Integer Linear Programming

start goal

• Solve ILP with minimizing total cost

(87)

18-3

Using Integer Linear Programming

start goal

• Solve ILP with minimizing total cost

• Define path according to values of variables, known from solution

(88)

19-1

Example Variable and Constraints

• Variable: x_t_,p,r ∈ {0, 1} ∀t ∈ T , ∀p, r ∈ P x_t_,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

(89)

19-2

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_1,q,r = 0

p r

q

p r

q

(90)

19-3

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_2,p,r = 1 x_2,r_,r = 1 x_1,q,r = 0 x_2,q,r = 0

p r

q

p r

q

(91)

19-4

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_2,p,r = 1 x_2,r_,r = 1

x_3,p,r = 1 x_3,r_,r = 1

x_1,q,r = 0 x_2,q,r = 0 x_3,q,r = 1

p r

q

p r

q

(92)

19-5

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_2,p,r = 1 x_2,r_,r = 1

x_3,p,r = 1 x_3,r_,r = 1

x_1,q,r = 0 x_2,q,r = 0 x_3,q,r = 1

p r

q

p r

q

• Constraints:

p is assigned to only one polygon:P

r∈P x_t_,p,r = 1

(93)

19-6

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_2,p,r = 1 x_2,r_,r = 1

x_3,p,r = 1 x_3,r_,r = 1

x_1,q,r = 0 x_2,q,r = 0 x_3,q,r = 1

p r

q

p r

q

• Constraints:

p is assigned to only one polygon:

Enforce aggregation:

P

r∈P x_t_,p,r = 1 P

r∈P x_t_,r_,r = n − t + 1

(94)

19-7

Example Variable and Constraints

t = 1 t = 2 t = 3

x_1,p,r = 0 x_1,r_,r = 1

p r

q

x_2,p,r = 1 x_2,r_,r = 1

x_3,p,r = 1 x_3,r_,r = 1

x_1,q,r = 0 x_2,q,r = 0 x_3,q,r = 1

p r

q

p r

q

• Constraints:

p is assigned to only one polygon:

Enforce aggregation:

P

r∈P x_t_,p,r = 1 P

r∈P x_t_,r_,r = n − t + 1

• In total,

5 sets of variables

17 sets of constraints

(95)

20-1

Case Study

• Environment: C#, CPLEX

(96)

20-2

Case Study

• Environment: C#, CPLEX

5,448 patches scale 1 : 50 k

734 patches (regions) scale 1 : 250 k

• Data

(97)

21

Comparison of A

^?

and ILP

1–5 6–10 11–15 16–20 21–25 26–36 0

25 50 75 100

n: number of polygons percent (%)

A^?

ILP_{160 s}

ILP_{600 s}

percentage of regions that were found optimal solutions

(98)

22-1

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(99)

22-2

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(100)

22-3

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(101)

22-4

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(102)

22-5

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(103)

22-6

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(104)

22-7

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(105)

22-8

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(106)

22-9

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(107)

22-10

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(108)

22-11

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(109)

22-12

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(110)

22-13

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(111)

22-14

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(112)

22-15

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(113)

22-16

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(114)

22-17

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(115)

22-18

An Optimal Sequence by A

^?

300 m

start (n = 17)

goal

(116)

23 Contents of Thesis

Optim.

DP

MST A^? ILP

Morphing polylines

Exaggeration

Simplification

p 2ε

(117)

24-1

Generalizing Buildings to Built-up Areas

400 m

Input: buildings

(118)

24-2

Generalizing Buildings to Built-up Areas

400 m

Input: buildings

(119)

25-1

Aggregate and Grow

original buildings

• Aggregate buildings that are too close, when zooming out

(120)

25-2

Aggregate and Grow

original buildings

• Bridges and buildings constitute a minimum spanning tree (MST)

(121)

25-3

Aggregate and Grow

t = 0

original buildings

zoom out:

t increases

(122)

25-4

Aggregate and Grow

grow

t = 0 t = 0.4

original buildings

zoom out:

t increases

add bridge

(123)

25-5

Aggregate and Grow

grow

t = 0 t = 0.4

original buildings

zoom out:

t increases

add bridge

(124)

25-6

Aggregate and Grow

grow

t = 0 t = 0.4

t = 0.6

original buildings

zoom out:

t increases

add bridge

(125)

25-7

Aggregate and Grow

grow

t = 0 t = 0.4

t = 0.6

original buildings

zoom out:

t increases

add bridge

(126)

25-8

Aggregate and Grow

grow grow

grow

t = 0 t = 0.4

t = 0.6 t = 1

original buildings

zoom out:

t increases

add bridge

add bridge add bridge

(127)

25-9

Aggregate and Grow

grow grow

grow

t = 0 t = 0.4

t = 0.6 t = 1

original buildings

zoom out:

t increases

add bridge

add bridge add bridge

(128)

26-1

Three Join Types of Buffering

building

(129)

26-2

Three Join Types of Buffering

building

miter:

keep right angles

(130)

26-3

Three Join Types of Buffering

building

miter:

keep right angles

d_G

(131)

26-4

Three Join Types of Buffering

building

square:

avoid long spikes miter:

keep right angles

d_G

(132)

26-5

Three Join Types of Buffering

building

square:

keep right angles

d_G

d_G d_G

(133)

26-6

Three Join Types of Buffering

building

square:

keep right angles

d_G

round:

detect if two buildings are too close d_G

d_G d_G

(134)

27-1

Simpifying Based on Dilation and Erosion

polygon d

(135)

27-2

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents d

(136)

27-3

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents

erode with 2d: remove bumps d

(137)

27-4

Simpifying Based on Dilation and Erosion

polygon dilate with d: remove dents

erode with 2d: remove bumps

dilate with d d

(138)

28-1

Case Study

• Runtime: O(n³),

n: total number of edges over all input buildings

(139)

28-2

Case Study

• Environment

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

(140)

28-3

Case Study

• Environment

• Data: 2.5 k buildings, n = 19 k edges, 1 : 15 k, d_G = 25 m (IGN)

(141)

28-4

Case Study

• Environment

• Data: 2.5 k buildings, n = 19 k edges, 1 : 15 k, d_G = 25 m (IGN)

• 12 min for computing a sequence of 10 maps

(142)

29-1

Animation

400 m

zooming out

(143)

29-2

Animation

400 m

zooming out

(144)

29-3

Animation

400 m

zooming out

(145)

29-4

Animation

400 m

zooming out

(146)

29-5

Animation

400 m

zooming out

(147)

29-6

Animation

400 m

zooming out

(148)

29-7

Animation

400 m

zooming out

(149)

29-8

Animation

400 m

zooming out

(150)

29-9

Animation

400 m

zooming out

(151)

29-10

Animation

400 m

zooming out

(152)

29-11

Animation

400 m

zooming out

(153)

30 Contents of Thesis

Optim.

DP

MST A^? ILP

Morphing polylines

Exaggeration

Simplification

p 2ε

(154)

31-1

Conclusion

• Studied four topics of continuous generalization

(155)

31-2

Conclusion

• Used optimization methods to attain good results

(156)

31-3

Conclusion

• Shared experience of using right data structures

(157)

31-4

Conclusion

Future work

• Improve our methods

(158)

31-5

Conclusion

Future work

• Usability testing

(159)

31-6

Conclusion

Future work

• Work on complete maps (with roads, buildings, ...)

scroll

(160)

31-7

Conclusion Thank

you!

Future work

• Work on complete maps (with roads, buildings, ...)

scroll