change Mike Worboys Department of Spatial Information Science and - PowerPoint PPT Presentation

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

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

The geospatial context of change ThinkSpatial 3

process happens change continuant to event state change ThinkSpatial 4

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

EVENT growing in size, changing shape, changing color ON Christmas Day, 2010 STATE CHANGE area shape color 12/25/2010 12/26/2010 ThinkSpatial 6

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

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

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

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.

Absolute and relative motion

Individual or aggregate motion

Real or apparent motion

Fictive motion

Another type of “movement”

Another type of movement

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

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

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

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

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

Time – the container of change ThinkSpatial 22

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

Reasoning about events (McCarthy, Hayes, Kowalski and Sergot, Time Allen) 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. – A fluent is false once it has been terminated and before it has been initiated. ThinkSpatial 24

Working with processes Robin Milner 1934 – 2010

Process Aggregation • composition • parallelism • choice • reaction/communication

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]

c Process model q 0 a s c b r 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

c Process model a q 0 a s c t c b r 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

c Process model q 0 a s t 0 a c b r 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 | T 0 Q 0 == aR R == bS + cQ 0 U | R S == cQ 0 T 0 == aU U == dT 0

Topological change

Homotopy 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. ThinkSpatial 32

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

Homotopic equivalence ThinkSpatial 34

Continuous change ThinkSpatial 35

Egenhofer’s relations on S 2 ThinkSpatial 36

?

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

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

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

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  …

Further work: Spatio-semantic change

Different types of ‘quality’ involved in topological change

… or ?? ??

Detecting change using decentralized approaches ThinkSpatial 50

Sensors responding to a dynamic field

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?

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.

Approximating the Scalar Field scalar ar field discretiza retizatio tion approximatio oximation

Selecting a Threshold

Effect of WSN Density

The Communication Graph

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

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

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

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 + 1 2 - 3 +

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

Classification Different types of basic transitions have different properties

Acknowledgements • NSF Project IIS-0534429 : Monitoring Dynamic Spatial Fields Using Responsive Geosensor Networks • John Stell • Jixiang Jiang • Cheng Zhong • Chris Farah • Lisa Walton • Danqing Zhao