Spatial Computing or how to design a right-brain hemisphere - - PowerPoint PPT Presentation

1

Spatial Computing

– or how to design a right-brain hemisphere Christian Freksa University of Bremen

2

Acknowledgments

Some Examples of Spatial Problems

 (How) can I get the piano into my living room?  How do I get from A to B?  Which is closer: from A to B or from A to C?  Which is (the area of) my land?  Is the tree (walkway, driveway) on my property or

• n your property?

3

Many / most spatial problems come without numbers

 Do we have to formulate spatial problems in terms of

numbers in order to solve them (‘left-brain computing’)?

 Or can we find ways to process spatial configurations

directly (‘right-brain computing’)?

4

Plan for my talk

 Qualitative temporal and spatial reasoning  Conceptual neighborhood  SparQ toolbox  From relations to configurations  Spatial computing (vs. propositional computing)

Interaction most welcome!

5

Starting Point: ‘Allen Relations’ (1983)

(Previously published by C. Hamblin, 1972)

6

7

13 Qualitative Interval Relations

Relation before – after equal meets – met by

• verlaps –
• verlapped by

during – contains starts – started by finishes – finished by Symbol < > = m mi

• oi

d di s si f fi Pictorial Example

Allen´s Composition Table for Temporal Relations

9

... applied to 1-D Perception Space, arranged by conceptual neighborhood

spatially inhomogeneous categories:

• intervals
• points

compare:

• human perception
• human memory
• human concepts
• human language

10

Interval relations characterized by beginnings and endings

Interval relations characterized by relations between beginnings and endings

11

Spatial and Conceptual Neighborhood

spatial conceptual neighborhood between locations neighborhood between relations static structure process structure

Features of Conceptual Neighborhood

 Coarse relations = CNs of fine relations  CNs define conceptual hierarchies for representing

incomplete knowledge

 Efficient non-disjunctive reasoning  Incremental refinement as knowledge is gained  Natural correspondence to everyday concepts  Spatio-temporal inferences form conceptual

neighborhoods

 Reduce computational complexity from exponential

to polynomial

 Can be defined at arbitrary granularity

12

13

Incomplete knowledge as coarse knowledge

Example: Disjunction of the relations before or meets or overlaps (<, m, o) can be considered incomplete knowledge as it cannot be reduced to a single interval relation. It can be considered coarse knowledge as the three relations form a conceptual neighborhood that defines the coarse relation

14

Coarse relations as semi-interval relations I

15

Coarse relations as semi-interval relations II

16

Neighborhood-based coarse reasoning

17

Composition Table for Coarse Reasoning

18

Inference based on coarse relations

19

Fine reasoning based on coarse relations

20

Closed composition table for fine and coarse relations

21

A Multitude of Specialized Calculi

 Topology

 4-intersection, 9-intersection (Egenhofer et al.)  RCC-5, RCC-8 (Randell, Cohn et al.)

 Orientation

 point-based (double cross, FlipFlop, QTC, dipole)  extended objects

 Position

 Ternary Point Configuration Calculus (TPCC)

 Measurement

 Delta-Calculus

22

Generic Toolbox SparQ for Spatial Qualitative Reasoning

D Wolter, F Dylla, L Frommberger, JO Wallgrün

 Calculus specification

 base relations / operations in list notation  or: algebraic specification (metric space)

 Functional list notation  Interfacing: command line or TCP/IP  Available under GNU GPL license

 www.sfbtr8.spatial-cognition.de/project/r3/sparq/  manual included

23

Modular SparQ Architecture

syntax: sparq <module> <calculus> <operation> <input>

24

Boat Race [Ligozat 2005] Example: qualify

sparq qualify point-calculus all

((A 0) (B 10.5) (C 7) (D 7) (E 17)) ((A < B) (A < C) (A < D) (A < E) (B > C) (B > D) (B < E) (C = D) (C < E) (D < E))

25

Boat Race Ex: compute-relation

sparq compute-relation point-calculus composition < < (<) sparq compute-relation point-calculus converse (< =) (> =)

26

Boat Race Ex: constraint-reasoning

sparq constraint-reasoning pc scenario- consistency first ((E > B) (A < B) (A < C) (D = C)) ((C (=) D) (A (<) D) (A (<) C) (B (>) D) (B (>) C) (B (>) A) (E (>) D) (E (>) C) (E (>) A) (E (>) B))

27

Boat Race Ex: constraint-reasoning

sparq constraint-reasoning pc scenario- consistency first ((E > B)(A < B)(A < C)(D = C) (X < C) (B < X))

 NOT CONSISTENT

28

Boat Race Ex: constraint-reasoning

sparq constraint-reasoning pc scenario- consistency all < < <five scenarios found>

29

sparq quantify flipflop ((A B l C) (B C r D)) ((A 0 0) (B 7.89 15.36) (C -4.98 1.14) (D -36.75 21.25))

Spatial Configurations Example: quantify

experimental

30

SparQ - Summary

 generic qualitative reasoning toolbox

 binary and ternary calculi

 algebraic calculus specification

 determines operations automatically  calculus verification

 qualitative reasoning more effective /

efficient than general theorem proving

 challenges are welcome!

 available under GNU GPL license

 www.sfbtr8.spatial-cognition.de/project/r3/sparq/

 manual included

Challenge

 Knowing which tool to select for a given problem  Meta-knowledge about spatial reasoning

31

Spatial Configurations

32

Computation by Abstraction

Example: Trigonometry

 Given: a=5; b=3; c=6  Compute: α, β, γ, A, ...

33

A

α β

γ

Computation by Diagrammatic Construction

34

Computation by Diagrammatic Construction: A Form of Analogical Reasoning

Universal properties of spatial structures:

 Trigonometric relations hold on all flat surfaces  Flat diagrammatic media provide suitable spatial

properties to directly ‘compute’ trigonometric relations

 Static spatial structures can replace computational

processes of geometric algorithms

 Computational operations are ‘built into’ spatial structures  Constraints in spatial structures act instantaneously;

i.e., no constraint solving procedures are required

35

Computing Space

36

Diagrammatic vs. Formal Reasoning

concrete vs. abstract

37

task stage solution stage

formal spatial

formal reasoning

no time (instantaneous) language / formal level image level

formal specification spatial configuration formal result formalization instantiation formalization instantiation

time

Elementary Entities of Cognitive Processing

configurations 

• bjects

 areas  lines  points

38

configurations 

• bjects

 areas  lines  points

geometry cognition

‘basic level’

Composition Aggregation Decomposition Refinement Composition Aggregation

Spatio-Visual Problems

39

Reasoning by Imagination

 How many degrees is the smallest turn that aligns

the cube with its original orientation (corners coincide with corners, edges coincide with edges) ?

Diagrammatic Approach

the cube viewed from above

Limitations of Spatial Computing?

42

Approach: Implementation of a Visuo-Spatial Sketch-Pad

43

Courtesy: Mary Hegarty

44

Thank you very much for your attention!

www.spatial-cognition.de

45

46

Application-Perspectives

21.12.2007 06:53 Uhr

Schiffsunglück bei Krefeld

Sojaschiff rammt Kerosin-Tanker

Auf dem Rhein in Krefeld sind Donnerstagnacht drei Schiffe

• kollidiert. Die Bergungsarbeiten dauern an, die Höhe des Schadens

ist noch unklar. Drei Motorschiffe sind am Donnerstagabend auf dem Rhein in Höhe des Krefelder Stadtteils Uerdingen kollidiert. Eines der beteiligten Schiffe drohte zu sinken, doch konnte dies von den Rettungskräften verhindert werden.

47

SailAway

 International

navigation rules regulate right of way for pairs of vessels

 What happens

when more than two vessels are involved?

48

SailAway: Vessels A and B

49

SailAway: Vessels B and C

50

SailAway: Vessels A and C

51

SailAway: Conflicting Rules

52

The Space of Qualitative Values

e.g. double cross calculus [Freksa 1992]

left front right front straight ahead right abeam left abeam right left left back right back straight back

spatially inhomogeneous categories:

• areas
• lines
• points

compare:

• human perception
• human memory
• human concepts
• human language