Map Generalization of Road Networks

Jan Terje Bjørke

Norwegian Defence Research Establishement and Department of Mathematical Sciences and Technology, The Agricultural University of

Norway

decreasing amount of detail

less visual conflicts

start view Cartographic

zoom out

eliminates details

ordinaryzoom out

Zoom out

Cartographic zoomadds details

Zoom in

Generalization of road networks should consider

• The visual separation of the map symbols

• Method: solve the visual separation problem by an elimination procedure based on information theory.

• The semantics of the network like

–the connectivity of the network

–the route to travel

–the road classes

–etc.

• Method: Introduce the semantic constraints into the optimisation procedure.

Shannon entropy

)(2

log)()( ypypYHYy

The events for the entropy computation: points along the arcs

Information points

Shannon equivocation

YyXx

xypxypxpXYH )|(log)|()()|(2

where is the conditional probabilitythat map symbol x is interpreted as map symbol y.

)|( xyp

Similarity and conditional probability

,0),(

else

when ),(

xy

Ts(T-s)/Txy

Yy

xy

xyxyp

),(

),()|(

T

1.0

0 s

Shannon useful information

• R=H(Y) – H(Y|X) ; termed useful information

• R = amount of roads in the map minus the amount of visual conflicts between the roads

• The maximum value of R is termed the channel capacity.

Oslo, T=15 Oslo, T=35

Useful information

Bergen T=35Bergen T=15

Useful information

Elimination algorithm

• eliminate the most conflicting road,

• repeat the procedure until

– the R-value has reached its maximum value

– or if the topological constraints prevent any arc to being eliminated.

Generalized map, T=35. The hierarchy of the roads are considered. No topology constraints are introduced. 83% of the roads are eliminated.

 

Oslo road map

Main roads in red, secondary roads in black and other roads in blue.

Oslo road map

Main roads in red, secondary roads in black and other roads in blue.

   

Generalized map, T=35.The hierarchy and the topology of the roads are considered.49% of the roads are eliminated. .

Oslo

T=1524% eliminated

T=2543% eliminated

T=3549% eliminated20 sec.

120 sec.

350 sec.

T=1013% eliminated

T=3539% eliminated

T=2528% eliminated

Bergen

T=1027% eliminated

T=2535% eliminated

Trondheim

Running time for the algorithm

Time in seconds to generalize the Oslo road map

source, d=10source, d=20

Max R, d=20 Max R, d=10

Declutteringhouse symbols in a part of the Oslo region

d=8, from optimization

Optimize number of symbols and symbol size

Conclusions

The method presented has three application dependent parameters:

1. by tuning the similarity function we decide the visual separability of the road symbols;

2. the connectivity condition decides how important the topology of the network is;

3. the hierarchy of the roads is considered by a weight function.

Conclusions

• The method can serve as a basis for cartographic zoom since it is:

– automated;

– fast in unconditional mode;

– the implementation of the contraints in the conditional mode is critical to the time complexity of the algoritm. This step requires more experiments

Further research

• Apply the information theoretic approach to compute the optimal number of coloured depth intervals in sea floor maps.

• Speed up the algorithms to solve the optimization problem.

• Apply the method to maps composed of different information sources.