Tight Rectilinear Hulls of Simple Polygons Annika Bonerath, - PowerPoint PPT Presentation

1 Tight Rectilinear Hulls of Simple Polygons Annika Bonerath, Jan-Henrik Haunert and Benjamin Niedermann Institute of Geodesy und Geoinformation, University of Bonn

2 - 1 Motivation Main Goal : Simplification of a polygon P with a polygon Q . P Q Q P

2 - 2 Motivation Main Goal : Simplification of a polygon P with a polygon Q . Requirements for Q : P • simple • C -oriented Q • contains P • cannot be shrunk (formalization follows on the next slides) Q P

2 - 3 Motivation Main Goal : Simplification of a polygon P with a polygon Q . Requirements for Q : P • simple • C -oriented • contains P • cannot be shrunk (formalization follows on the next slides) P

2 - 4 Motivation Main Goal : Simplification of a polygon P with a polygon Q . Requirements for Q : P • simple • C -oriented Q • contains P • cannot be shrunk (formalization follows on the next slides) Q P Optimization Goal : few bends, small area, short length

2 - 5 Motivation Main Goal : Simplification of a polygon P with a polygon Q . Application : • schematization of plane graph drawings • travel-time maps that visualize reachable Q parts in a road network • schematic representation of point sets, instead of using bounding boxes as usually Q done in data management systems

3 This Paper Restriction to rectilinear simple input and output polygons.

4 - 1 Formalization of Tight Hulls Definition : The polygon Q ′ is a linear distortion of Q if each edge of Q ′ can be scaled and translated such that the polygon Q results. Q ′ Q

4 - 2 Formalization of Tight Hulls Definition : The polygon Q ′ is a linear distortion of Q if each edge of Q ′ can be scaled and translated such that the polygon Q results. Q ′ Q

4 - 3 Formalization of Tight Hulls Definition : The polygon Q ′ is a linear distortion of Q if each edge of Q ′ can be scaled and translated such that the polygon Q results. Q ′ = Q

4 - 4 Formalization of Tight Hulls Definition : A simple polygon Q is a tight hull of another polygon P if Q contains P and there is no linear distortion of Q that lies in Q and contains P . P P P Q = P Q Q = bounding box of P What is a good tight hull?

4 - 5 Formalization of Tight Hulls Definition : The tight hull Q of P is α -optimal if Q minimizes cost( Q ) = α 1 · length( Q ) + α 2 · area( Q ) + α 3 · bends( Q ) over all tight hulls Q ′ . P P P Q Q Q length( Q ) area( Q ) bends( Q )

5 - 1 Structural Properties Lemma 1 : Every vertex of Q on P is a vertex of the P Q maximally subdivided P . Idea of Proof : v is a vertex of P v is not a vertex of P P P e 3 Q Q w e 2 e 2 e 1 e 1 v v but then Q is not tight (scale e 1 and e 3 )

5 - 2 Structural Properties Lemma 1 : Every vertex of Q on P is a vertex of the P Q maximally subdivided P . ⇒ use vertices of P for the computation of Q

5 - 3 Structural Properties Definition : The polyline B is a bridge if, • it consists of one or two incident line segments forming an “L” • it starts and ends at vertices of P . Definition : The region enclosed by B and the polyline of P connecting the same vertices as B is the bag of B . B bag P B bag B bag P bag P

5 - 4 Structural Properties Definition : The polyline B is a bridge if, • it consists of one or two incident line segments forming an “L” • it starts and ends at vertices of P . ⇒ every tight hull can be represented by a set of bridges

5 - 5 Structural Properties Lemma 2 : The bounding box B of P is a tight hull and P Q any other tight hull of P is contained in B . B Idea : carve into the bounding box B to generate any tight hull

6 - 1 Decomposition bounding box B tight hull Q input P

6 - 2 Decomposition Decompose B into four independent subinstances defined by bridges B 1 , B 2 , B 3 and B 4 . B 1 B 2 B 4 B 3

6 - 3 Decomposition Decompose B into four independent subinstances defined by bridges B 1 , B 2 , B 3 and B 4 . B 1 B 2 Q 1 Q 2 Q 4 Q 3 B 4 B 3 Idea : Solve instances independently and compose solutions.

7 - 1 Decomposition B 1 Decomposition Q 1 k j b c a f i d e h g

7 - 2 Decomposition Decomposition tree of B 1 . B 1 k j b c a f i d e h g

7 - 3 Decomposition Decomposition tree of B 1 . B 1 k k j b c a f i d e h g

7 - 4 Decomposition Decomposition tree of B 1 . B 1 k k j b c a f i d e h g

7 - 5 Decomposition Decomposition tree of B 1 . B 1 k b j a k j b c a f i d e h g

7 - 6 Decomposition Decomposition tree of B 1 . B 1 k b j a k j b c a f i d e h g

7 - 7 Decomposition Decomposition tree of B 1 . B 1 k b j a c k j b c a f i d e h g

7 - 8 Decomposition Decomposition tree of B 1 . B 1 k b j a c k j b c a f i d e f h g

7 - 9 Decomposition Decomposition tree of B 1 . B 1 k b j a c k j b c a f i d e f h e d g

7 - 10 Decomposition Decomposition tree of B 1 . B 1 k b j a i c k j b c a f i d e f h e d g

7 - 11 Decomposition Decomposition tree of B 1 . B 1 k b j a i c k j b c a f i d e h f h g e d g

8 - 1 Decomposition Rules Decomposition of a bridge B into • one to three connected bridges C 1 , C 2 , and C 3 , • each C 1 , C 2 , and C 3 lies in the bag of B • the polyline defined by C 1 , C 2 , and C 3 connects the start and endpoint of B • C 1 , C 2 , and C 3 may not cross each other pairwise B B C 3 C 1 B C 2 C 1 C 1 C 2

8 - 2 Decomposition Rules Note : rules guarantee that Q is not self-intersecting

9 Computation Lemma : For each tight hull there exists a decomposition tree. Observation : Each decomposition of a bridge can be described by two additional points ⇒ all possible decompositions can be enumerated in polynomial time. Use dynamic programming approach to build a decomposition tree of an α -optimal tight hull in O ( n 4 ) time.

10 Non-Rectilinear Input Polygon Problem : vertices of Q are not necessarily vertices of the maximally subdivided P P Simple Approximative Approach : sample regularly distibuted vertices on the edges of P P

11 Conclusion and Outlook Conclusion : • non-self intersecting α -optimal tight rectilinear hull in O ( n 4 ) time and O ( n 2 ) space Future Work : • C -oriented tight hulls • optimal solutions for arbitrary (simple) input polygons P Q P Q

12 Questions? Feedback? bonerath@igg.uni-bonn.de niedermann@igg.uni-bonn.de haunert@igg.uni-bonn.de