Qualitative Spatial Reasoning overLine-Region Relations

Leena and Sibel

Knowledge RepresentationSeminar Presentation

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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Motivation

I Modeling spatial relations

I How do humans conceptualize spatial relations?

I Strong correlation between Perceptual space and LanguageSpace

I Understanding cognitive perceptual groupings

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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9-Intersection

GoalA computational model to describe conceptual neighborhoods andenable the definition of a similarity metric for line region relations.

Conceptual Similarity: Which pairs of relationships aresimilar?

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Formal Definitions

LineA sequence of 1...n connected 1-cells between two geometricallyindependent 0-cells such that they neither cross each other norform cycles.

I Interior, Boundary, Exterior

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Formal Definitions(contd.,)

Region

A region is defined as a connected, homogeneously 2-dimensional2-cell. Its boundary forms a Jordan curve separating the regionsexterior from its interior.

I Interior, Boundary, Exterior

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Adjacency

Topological adjacency

I Adjacent(Interior A0) = A

I Adjacent(Boundary A) = A0andA

I Adjacent(Exterior A) = A

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Adjacency

Topological adjacency

I Adjacent(Interior A0) = A

I Adjacent(Boundary A) = A0andA

I Adjacent(Exterior A) = A

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Adjacency

Topological adjacency

I Adjacent(Interior A0) = A

I Adjacent(Boundary A) = A0andA

I Adjacent(Exterior A) = A

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Adjacency

Topological adjacency

I Adjacent(Interior A0) = A

I Adjacent(Boundary A) = A0andA

I Adjacent(Exterior A) = A

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9-Intersection(contd..,)

Topological adjacency

I 9 intersections between the different topological parts of a lineand a region

The 9-intersection Matrix(M) L0 R0 L0 R L0 RL R0 L R L RL R0 L R L R

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9-Intersection(contd..,)

I Binary assignment to intersections(,)I 512 possible instances of M

I 19 of 512 instances can actually be realized.

Example

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9-Intersection(contd..,)

I Binary assignment to intersections(,)I 512 possible instances of M

I 19 of 512 instances can actually be realized.

Example

Compute the values of the matrix... 14/41

Geometric interpretations

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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Snapshot Model

I A model of conceptual neighborhood among topologicalrelations between a line and a region.

Characteristics

I No prior knowledge of the potential transformations thatcould lead from one configuration to the other.

I Comparison on the basis of a pre-defined distance metric

Differences of Intersections(= 0, = 1)

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Defining topological neighbors

Distance between any two relationships rA, rB is given by:

TrA,rB =i=0

j=0

|MA[i , j ]MB [i , j ]| (1)

I It is the count of differences of empty/non empty entries ofcorresponding elements in the 9 intersections.

I Shortest Nonzero distance between relations is 1

I Spatial relations with the shortest non zero distance areconsidered topological neighbors.

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Example 1

Are these relations neighbors according to the snapshotmodel?

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Example 2

Another example... Are these relations topological neighbors?

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Conceptual neighborhoods derived from thesnapshot model

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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Smooth-Transition Model

Smooth TransitionAn infinitesimally small deformation that changes the topologicalrelation between the line and the region

Examples and Counterexamples

conceptual neighborsconceptual neighbors

not conceptualneighbors

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Formalization

A smooth transition occurs by moving around the lines

1. boundary nodes

Q: Do they intersect with the same region part?Transition Rule 1 if YesTransition Rule 2 if No

2. interior

Transition Rule 3 to extend the intersection area andTransition Rule 4 to reduce it

What this means for the 9-intersection:An entry or its adjacent entries gets changed from to or v.v.

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One More Thing...

Definition (Extent of a line part i)

I Denoted by #M[i , ]

I Count of intersections betw. line part i and the region parts

I #M[i , ] in the interval [0 . . . 3]

Examples

I extent of the lines boundary is either 1 (if both nodes arelocated in the same region part) or 2 (if the nodes are locatedin different parts of the region)

I extent of a lines exterior is always 3

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Transition Rule 1

If the lines two boundaries intersect with the same region part,then extend the intersection to either of the adjacent region parts:

#M[, ] = 1 = i(M[, i ] = ) : MN [, adjacent(i)] :=

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Transition Rule 2

If the lines two boundaries intersect with two different regionparts then move either intersection to the adjacent region part:

#M[, ] = 2 = i(M[, i ] = ) :MN [, i ] := MN [, adjacent(i)] :=

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Transition Rule 3

Extend the lines interior-intersection to either of the adjacentregion parts:

i(M[, i ] = ) : MN [, adjacent(i)] :=

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Transition Rule 4

Reduce the lines interior intersection on either of the adjacentregion parts.

#M[, ] = 2 = i(M[, i ] = ) : MN [, i ] := #M[, ] = 3 = i(i 6= ) : MN [, i ] :=

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Additional Consistency Constraints

1. If the lines interior intersects with the regions interior andexterior, then the lines interior must also intersect with theregions boundary.

M[, ] = M[, ] = = M[, ] :=

2. If the lines boundary intersects with the regions interior(exterior) then the lines interior must intersect with theregions interior (exterior) as well.

M[, ] = = M[, ] := M[, ] = = M[, ] :=

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Resulting Neighborhood Graph

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Comparison

Snapshot Model Smooth-Transition Model

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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Setup

I 2 geometrically distinct placements of the line for each of the19 topologically distinct relations

I a total of 38 diagrams each showing a line and a region

I line road, region parkI parks in all diagrams same size and shape

I 28 participants

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Setup (cont.)

Q: Find the pair that is topologically identical from among allgeometrically distinct diagrams.

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Setup (cont.)

Q: Find the pair that is topologically identical from among allgeometrically distinct diagrams.

A: The right and middle examples in the lower row.

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Task

I arrange the sketches into several groups, such that you woulduse the same verbal description for the spatial relationshipbetween the road and the park for every sketch in each group

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Goal

Goal

I analyse how the subjects formed groups of similar relations

I check similarity with presented conceptual neighborhoodmodels

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Results

Each spatial relation could be grouped as many as 112 times (4pairs times 28 subjects) with each other relation.

Number of times conceptual neighbors are grouped:

. . . 1 min max mean %2

snapshot model only 2 10 16 13.0 11.6smooth-transition model only 12 0 66 17.3 15.4both models 26 0 78 33.6 29.5neither model 131 - - 6.0 5.3

1Number of relations that are conceptual neighbors2percentage = mean / 112

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Agenda

Motivation

9-Intersection

Snapshot Model

Smooth-Transition Model

Evaluation

Summary

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Summary

Two Conceptual Neighborhood Models:

1. Snapshot Model

2. Smooth-Transition Model

Finding: Almost identical Conceptual-Neighborhood Graphs

Findings from the Human-Subject Experiment:

I models correspond largely to the way humans conceptualizesimilarity about line-region relations

I smooth-transition model captures more aspects of thesimilarity of topological line-region relations than the snapshotmodel

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References I

I Egenhofer, Max J., and David M. Mark. Modellingconceptual neighbourhoods of topological line-regionrelations. International journal of geographical informationsystems 9.5 (1995): 555-565.

I Mark, David M., and Max J. Egenhofer. Modeling spatialrelations between lines and regions: combining formalmathematical models and human subjects testing.Cartography and geographic information systems 21.4 (1994):195-212.

I Egenhofer, Max J., and A. Rashid Shariff. Metric details fornatural-language spatial relations. ACM Transactions onInformation Systems (TOIS) 16.4 (1998): 295-321.

I Talmy, Leonard. How language structures space. Springer US,1983.

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References II

I Clark, Herbert H. Space, time, semantics, and the child.(1973).

I Moore, Timothy E. Cognitive Development and theAcquisition of Language. (1973).

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Motivation9-IntersectionSnapshot ModelSmooth-Transition ModelEvaluationSummary