(a,b), a

Topological Relations
between Convex Regions
Sanjiang Li, Welming Liu
Presented by
Theodoros Chondrogiannis
Advanced Artificial Intelligence
Academic Year: 2010-11

Some Definitions
• Qualitative Spatial Reasoning (QSR)
The challenge of the AI approach to Spatial Reasoning
• Spatial Relations & Calculus
A calculus introduces a number of finite restrictions in
order to represent spatial relations
• The Region Connection Calculus(RCC) is
the most well-know topological formalism
(Randell, Cui & Cohn, 1992)
2

Region Connection Calculus
• RCC support the definition of two spatial
Relation Algebras (RCC5 and RCC8, contain 5 and
8 basic relations respectively)
• RCC8 are the only atomic topological
relations between closed disks but not for
other objects.
• Are RCC8 enough to represent all
relations between spatial objects? NO
• Introduction of semi-algebraic sets
3

RCC8 Basic Relations
• For two regions a, b we have:
(a,b) ∈ DC if a ∩ b = ∅
(a,b) ∈ EC if a° ∩ b° = ∅ but a ∩ b ≠ ∅
(a,b) ∈ PO if a°∩ b° ≠ ∅ and a ⊈ b and b ⊈ a
(a,b) ∈ TPP if a ⊆ b but a ⊈ b°
(a,b) ∈ NTPP if a ⊆ b°
(a,b) ∈ EQ if a = b
• All relations along with the converses of
TPP and NTPP are jointly exhaustive
and pairwise disjoint.
4

RCC8 Examples
RCC8 - PO Relation
RCC8 - PO Relation
RCC8 - EC Relation
5
RCC8 - TPP Relation

Atomic Topological Relations
• Hom is the set of all homeomorphisms of R 2
• Let a, b be two plane regions. The relation:
αa,b = { f(a) , f(b) : f ∈ Hom }
is an atomic relation and is the smallest topological
relation which contains (a,b)
• An atomic topological relation is unique for any
two regions.
• There are uncountably many atomic topological
relations
6

Restriction to Convex Regions
• The RCC8 relations DC, NTPP, EQ and NTPP~ are all
atomic on convex regions.
• The EC relation contains exactly two atomic
topological relations
• For a, b convex regions with a ≠ b, each maximally
connected component (mcc):
• If mcc ∈ (a°∩∂b) OR (∂a ∩ b°) is homeomorphic
to (0,1)
• If mcc ∈ (∂a ∩ ∂b) is a single point or
homeomorhpic to [0,1]
7

Relations for Convex Regions
& String Representation
• For a, b, a ≠ b, every subset of ∂(a ∩ b) is:
- Type u - mcc of a° ∩ ∂b
- Type v - mcc of ∂a ∩ b°
- Type x - 0-D mcc of ∂a ∩ ∂b
- Type y - 1-D mcc of ∂a ∩ ∂b
• Every relation (a,b), a ≠ b, is represented by a
circular string s which consist of {u , v , x , y}
• If s represents (a,b) then if s’ is a circular rotation of
s, s’ also represents (a,b)
8

String Example - Triangles
uxvy
uxvxuxvx
9

Convex Polygons Algorithm
Require: Vertices of two convex polygons, clockwise, P1,...,Pm, Q1,...,Qn.
Output: string representation s.
O ← an interior point of both polygons;
αi ← ∠P1OPi;
βj ←∠P1OQj;
For each αk, find jk such that βjk#≤ αk < βjk+1;
For each βk, find ik such that αik#≤ βk < αik+1;
for k = 1,2,··· ,m do
Pk′ ← the intersection of ray OPk and Qjk Qjk+1;
δk ←T(|OPk′|−|OPk|);
for k = 1,2,··· ,n do
Q′k ← the intersection of ray OQk and Pik Pik+1;
θk ←T(|OQk||-|OQ′k||);
Merge sort {αi} and {βj}, meanwhile compose the corresponding δi and θj into a
circular string s;
Replace consecutive uʼs (vʼs) in s with one u (v);
Replace consecutive xʼs in s with one y;
Insert an x between each pair of neighboring u (or v) in s;
Output string representation s.
10

Convex Polygons Algorithm
• For a,b, a ≠ b, the string that represents
(a,b) has length at most 2(m+n)
• A common interior point can be found in
O(log(m+n)) (Chazelle and Dolkin, 1987)
• A first version of s is produced in O(m+n)
and the post processing needs also O(m+n)
• In conclusion, the time complexity of the
algorithm os O(m+n), (linear).
11

Paper Contribution &
Further Research
• A clear formulation of what is the topological
relation between two regions and a string
representation of that relation
• Linear algorithms for computing and
comparing relations between convex polygons
• Extending the model to measure the similarity
of arbitrary atomic topological relations
• Relations between 3D convex objects
12

Thank you!