The foundations of spatial change. Mike Worboys Department of Spatial Information Science and Engineering University of Maine

Transcription

1 The foundations of spatial change Mike Worboys Department of Spatial Information Science and Engineering University of Maine

2 Things that involve change State (part of situation) Absence of change Process (1) Change as it is actually occurring, something going on Event A chunk of change picked out as an individual from the ongoing flux Process (2) A structured succession of events ThinkSpatial 2

3 The geospatial context of change ThinkSpatial 3

4 process event change happens to state change continuant ThinkSpatial 4

5 PROCESS growing in size, changing shape, changing color STATE CHANGE area shape color ThinkSpatial 5

6 STATE CHANGE area shape color EVENT growing in size, changing shape, changing color ON Christmas Day, /25/ /26/2010 ThinkSpatial 6

7 Kinds of change types of change location change size change shape change topological change dimension change identity change posture change semantic change viewpoint change ThinkSpatial 7

8 Perspectives on change egocentric vs. allocentric Eulerian vs. Lagrangian ThinkSpatial 8

9 Measurement of change quantitative vs. qualitative phase space vs. mode space ThinkSpatial 9

10 Motion Aristotle: motion is change of any sort, including qualitative change (change of space was given the more specific term, locomotion ). Newton s world view has motion as a central piece. STIS: change-based and movement-based models.

11 Absolute and relative motion

12 Individual or aggregate motion

13 Real or apparent motion

14 Fictive motion

15 Another type of movement

16 Another type of movement

17 The geometry of change Erlangen program (Felix Klein 1872) A geometry is the study of a those properties invariant under a particular class of changes. ThinkSpatial 17

18 Rigid body motion Euclidean geometry Those properties of shapes invariant under rotations, reflections, and translations. length, angle, parallelism, area, congruence ThinkSpatial 18

19 Scaling Affine geometry Those properties of shapes invariant under rotations, reflections, and translations. angle, parallelism, similarities ThinkSpatial 19

20 Change of viewpoint Projective geometry Those properties of shapes invariant under perspective transformations colinearity, conic sections, ThinkSpatial 20

21 Change of connectivity Topology (and it s subset graph theory) Those properties of shapes invariant under homeomorphisms connectedness, genus, dimension, compactness, ThinkSpatial 21

22 Time the container of change ThinkSpatial 22

23 Reasoning about time (e.g., Allen s interval calculus) ThinkSpatial 23

24 Time Temporal structure Allen s interval calculus Time-varying propositions Fluents Event type Predicates Occurs (event, time) HoldsAt (fluent, time) Initiates (event, fluent, time) Terminates (event, fluent, time) Theory examples A fluent is true once it has been initiated by an event. Reasoning about events (McCarthy, Hayes, Kowalski and Sergot, Allen) A fluent is false once it has been terminated and before it has been initiated. ThinkSpatial 24

25 Working with processes Robin Milner

26 Process Aggregation composition parallelism choice reaction/communication

27 Basic process and mobility concepts Process names and constructions Process equivalence Independence vs. reaction Synchronous vs. asynchronous Determinism vs. non-determinism Operations Composition a.p Disjunction P+Q Parallelism P Q Reaction ((in a)p+q) ((out a)r+s) P R Replication!P Ambient n[p]

28 Process model c q 0 a s c r b This process is deterministic, as there is at most one transition (q, a, q ), for each pair (q, a). Process notation: Q 0 == ar R == bs + cq 0 S == cq 0

29 Process model c a q 0 a s t c c r b This process is nondeterministic, as there is more than one transition a with start state q 0. Process notation: Q 0 == ar + at = a(r+t) R == bs + cq 0 S == cq 0 T == cq 0

30 Process model q 0 c a s t 0 a c r b d u These processes communicate via input action a and output action a. The combined process is Q 0 T 0 Process notation: Q 0 == ar R == bs + cq 0 S == cq 0 == au T 0 Q 0 T 0 U R U == dt 0

31 Topological change

32 Two continuous functions are called homotopic if one can be "continuously deformed" into the other. Such a deformation is called a homotopy between the two functions. Homotopy ThinkSpatial 32

33 Homotopy (formal definition) A homotopy between two continuous functions f and g from a topological space X to a topological space Y is a continuous function 0 f 0 H : X *0,1+ Y such that, if x X then H(x,0) = f(x) H(x,1) = g(x). g H ThinkSpatial 33

34 Homotopic equivalence ThinkSpatial 34

35 Continuous change ThinkSpatial 35

36 Egenhofer s relations on S 2 ThinkSpatial 36

37 ?

38

39

40

41 Seeking the atoms What are the points, lines and polygons of topological change

42

43 Tree morphisms to represent topological change n 0 n' 0 node to node n 1 n' 1 n 2 n' 2 n 3 n 4 n' 3

44

45 Atomic Changes n 0 n' 0 n 0 n' 0 n 1 n' 1 n' 1 T 1 T 2 n 1 n 2 n' 1 T 1 T 2 n' 0 n 2 T 1 T 2 n 0 atomic insert atomic merge I atomic merge II n 0 n' 0 n 0 n' 0 n1 n' 1 n' 0 n 1 n 1 n' 1 n' 2 T 1 T 2 T 1 T 2 n 0 n' 2 T 1 T 2 atomic delete atomic split I atomic split II

46 Specifying Complex Changes The Canonical Form Theorem Every complex change C can be written as a composition in a particular and unique way of inserts, splits, merges, and deletes. C = I 1 S 1 M 1 D 1

47 Further work: Spatio-semantic change

48 Different types of quality involved in topological change

49 or????

50 Detecting change using decentralized approaches ThinkSpatial 50

51 Sensors responding to a dynamic field

52 Dynamic fields There are many examples of fields that change through time pollution plumes ocean currents population movements ST temperature variations Can we mine the dynamic field for events?

53 Approach Triangulate the spatial field according to the disposition of the sensors. Provide a threshold to distinguish regions of high activity. Use distributed algorithms to determine significant events in the sensor network, e.g., region splitting.

54 Approximating the Scalar Field scalar field discretization approximation

55 Selecting a Threshold

56 Effect of WSN Density

57 The Communication Graph

58 Challenge Detection of salient events in scalar fields. Focus on changes to regions of high activity. Focus on topological changes

59

60

61

62

63 Components A basic transition leads to a partition of the spatial domain into different components: Positive components Negative components Transition region A basic transition A partition

64 Irrelevant components Components that are not adjacent to the transition region are irrelevant to the type of topological changes Hole Self-merge

65 Key Features Property of the transition region (Added / Removed) Properties of the C-components (Represented by a tree) C-component a vertex Adjacency an edge Background C-component root Signs of the C-components labels

66 Classification Basic transitions of the same type have the same properties Hole Self-merge Properties of the C-components Type of Transition region: removed

67 Classification Different types of basic transitions have different properties

68 Acknowledgements NSF Project IIS : Monitoring Dynamic Spatial Fields Using Responsive Geosensor Networks John Stell Jixiang Jiang Cheng Zhong Chris Farah Lisa Walton Danqing Zhao