1 / 35

Qualitative spatial representations

Qualitative spatial representations. Tony Cohn. School of Computing The University of Leeds a.g.cohn@leeds.ac.uk http://www.comp.leeds.ac.uk/. Contents. Brief survey of qualitative spatial/spatio-temporal representations Motivation Some qualitative spatial representations

idalia
Download Presentation

Qualitative spatial representations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Qualitative spatial representations Tony Cohn School of Computing The University of Leeds a.g.cohn@leeds.ac.uk http://www.comp.leeds.ac.uk/

  2. Contents • Brief survey of qualitative spatial/spatio-temporal representations • Motivation • Some qualitative spatial representations • Spatial Change • Challenges

  3. Qualitative spatial/spatio-temporal representations • Naturally provides abstraction • Machine learning • Well developed calculi, languages, (often) semantics • Complementary to metric representations • Provide foundation for domain ontologies with spatially extended or indexed objects • Applications in geography, computer vision, robotics, natural language understanding, biology…

  4. What is QR? (1) • QR (about physical systems) • symbolic, not analogical • continuous scalar quantities mapped to finite discrete space (qualitative quantity space) • e.g... á-, 0, +ñ • model situation by relationships between these quantities • relative size; arithmetical relationships, ... • de Kleer, Kuipers, Forbus,…

  5. What is QR? (2) • relevant distinctions only • e.g. empty/full ... • - 0 + • Ambiguity • Not a replacement for Quantitative reasoning

  6. What is QSR? (1) • Develop QR representations specifically for space • Richness of QSR derives from multi-dimensionality • Consider trying to apply temporal interval calculus in 2D: < = m o s d f • Can work well for particular domains -- e.g. envelope/address recognition (Walischewski 97)

  7. What is QSR? (2) • Many aspects: • ontology, topology, orientation, distance, shape... • spatial change • Vagueness and uncertainty • reasoning mechanisms • pure space v. domain dependent • Formulations • FOPC, relation algebras, constraint languages, spatial (modal) logics

  8. Ontology of Space • extended entities (regions)? • points, lines, boundaries? • mixed dimension entities? • Open/closed/regular/non regular regions? • Multi-piece (disconnected)? Interior connected? • What is the embedding space? • connected? discrete? dense? dimension? Euclidean?... • What entities and relations do we take as primitive, and what are defined from these primitives?

  9. Mereology • Theory of parthood (Simons 87) • P(x,y) • Many theories • What principles should hold? • E.g. Weak supplementation principle: If x is a proper part of y, then there should be some other proper part z of y not identical with x. (not all mereologies obey this principle)

  10. Mereotopology • Combining mereology and topological notions • Usually built from a primitive binary conection relation, C(x,y) • Reflexive and symmetric • Several different interpretations in the literature • Can define many relations from C(x,y)

  11. RCC-8 • 8 provably jointly exhaustive pairwise disjoint relations (JEPD) DC EC PO TPP NTPP EQ TPPi NTPPi

  12. C(x,y) is very expressive • Can also define: • Holes, dimension, one pieceness • Topological functions • Boolean functions (sum, complement, intersection) • … • Touching v connection?

  13. An alternative basis: 9-intersection model (9IM) • 29 = 512 combinations • 8 relations assuming planar regular point sets • potentially more expressive • considers relationship between region and embedding space • Variant models discrete space (16 relations) • Egenhofer & Sharma, 93) • Dimension extended” method (DEM) • In the case where array entry is ‘¬’, replace with dimension of intersection: 0,1,2

  14. The 17 different L/A relations of the DEM

  15. Mereology and Topology • Which is primal? (Varzi 96) • Mereology is insufficient by itself • can’t define connection or 1-pieceness from parthood 1. generalise mereology by adding topological primitive 2. topology is primal and mereology is sub theory 3. topology is specialised domain specific sub theory Challenge: choosing primitives and inter-relating primitives in different theories

  16. Between Topology andMetric representations • What QSR calculi are there “in the middle”? • Orientation, convexity, shape abstractions… • Some early calculi integrated these • we will separate out components as far as possible • Some example calculi in next few slides • Mostly defined using algebraic techniques rather than logics, or only semi-formally. Challenge: finding expressive but efficient “semi-metric” calculi.

  17. Orientation • Naturally qualitative: clockwise/anticlockwise orientation • Need reference frame • deictic: x is to the left of y (viewed from observer) • intrinsic: x is in front of y • (depends on objects having fronts) • absolute: x is to the north of y • Most work 2D • Most work considers orientation between points or wrt directed lines Challenge: combining region based mereotopology with point based orientation calculi.

  18. Qualitative Positions wrt oriented lines • pos(p,li) = + iff p lies to left of li • pos(p,li) = 0 iff p lies on li • pos(p,li) = - iff p lies to right of li l1 l2 +-- --- ++- +-+ l3 --+ +++ -++ Note: 19 positions (7 named) -- 8 not possible

  19. Star Calculus (Renz and Ligozat) If more than 2 intersecting lines used for defining sectors, then easy to define a coordinate system and thus a geometry.

  20. Qualitative Shape Descriptions • boundary representations • axial representations • shape abstractions • synthetic: set of primitive shapes • Boolean algebra to generate complex shapes Challenge: finding useful qualitative shape calculi

  21. boundary representations | É • Hoffman & Richards (82): label boundary segments: • curving out É • curving in Ì • straight | • angle outward > • angle inward < • cusp outward  • cusp inward Á • Meathrel & Galton (2001) provide a hierarchical, representation calculus • Generalises all previous approaches > > • Ì • Ì < > | É • Ì > >

  22. Using Convex Hull to describe shape • conv(x) + C(x,y) • topological inside • Clot in circulatory system • geometrical inside • between bones • “scattered inside” • Discs between vertebrae • “containable inside” • Ball of femur in pelvis • ...

  23. Expressiveness of conv(x) • Constraint language of EC(x) + PP(x) + Conv(x) • can distinguish any two bounded regular regions not related by an affine transformation • Davis et al (97) • intractable (at least as hard as determining whether set of algebraic constraints over reals is consistent • Davis et al (97)

  24. Mereogeometries • Region Based Geometry (RBG) • 2nd order axiomatisation • P(x,y) + Sphere(x) • Categorical • (Region based version of Tarski’s geometry) • Borgo and Masolo (06) • Analysis of several other systems (eg de Laguna) • Four shown to be strongly semantically equivalent • Some work on on constraint systems • Less expressive but more tractable

  25. Qualitative Spatio-temporal representations • Capturing interactions between time and space • continuity • Many temporal calculi • Temporal modal logics, Allen’s calculus… • How to combine? • Ontology of space-time (3+1D v. 4D) • Computational issues Challenge: finding useful qualitative spatio-temporal calculi

  26. Continuity Networks/Conceptual Neighbourhoods • What are next qualitative relations if entities transform/translate continuously? • E.g. RCC-8 • If uncertain about the relation what are the next most likely possibilities? • Uncertainty of precise relation will result in connected subgraph (Freksa 91) • Can be used as basis of a qualitative simulation algorithm

  27. Qualitative simulation of phagocytosis • Uses Continuity network + composition table to check consistency

  28. Conceptual Neighbourhoods for other calculi • Virtually every calculus with a set of JEPD relations has presented a CN. • E.g.

  29. Decidable Spatiotemporal modal logics(Wolter & Zakharyashev) • Combine point based temporal logic with RCC8 • temporal operators: Since, Until • can define: Next (O), Always in the future ¤+, Sometime in the future ¦+ • ST0: allow temporal operators on spatial formulae • satisfiability is PSPACE complete • Eg ¤+P(brain,head) • The brain will always be part of the head • can express continuity of change (conceptual neighbourhood) • Can add Boolean operators to region terms • E.g. EQ(arms,leftarm+rightarm)

  30. Spatiotemporal modal logic (contd) • ST1: allow O to apply to region variables (iteratively) • Eg ð ¤+P(Ofemur,femur) • The femur will never contract • satisfiability decidable and NP complete • ST2: allow the other temporal operators to apply to region variables (iteratively) • finite change/state assumption • satisfiability decidable in EXPSPACE • P(blood, ¦+ leftarm) • all molecules in the blood will be part of the left arm (but not necessarily at the same time)

  31. First order qualitative representation of DNA

  32. Representation in QSR • Mereology: assemblies • Meretopology • Rings, strings, covalent linkage, … • … + convex hull • Beads, tunnels, threading… • Helix • RBG

  33. Recap • Surprisingly rich languages for qualitative spatial representation • symbolic representations • Topology, orientation, distance, ... • hundreds of distinctions easily made • Static reasoning: • composition, constraints, 0-order logic • Dynamic reasoning: continuity networks/conceptual neighbourhood diagrams

More Related