The identification of state transitions of fuzzy objects

4. The identification of state transitions of fuzzy objects

The procedure in the previous section identifies the regions that represent the spatial extents of ob- jects in one epoch. The regions at different epochs should be linked to form lifelines of the objects. This can be realised under two assumptions: 1. Each object has a convex fuzzy spatial extent, i.e., we are dealing with elementary fuzzy objects as defined in Section 2.4. 2. The natural phenomena are changing gradually, so that the changes of the objects from year to year are limited. This implies that the spatial extents of one and the same object at different epochs should have a larger overlap than the spatial extents of different objects. Under this assumption we can find the successor of a spatial extent at epoch t by calculating its spatial n overlaps with all the spatial extents at epoch t n q 1. The overlap of two regions S and S can be a b found through the intersection of their two cell sets, which is a simple raster-based operation. OVERLAP S ,S s Cells S l Cells S 5 Ž . Ž . Ž . Ž . a b a b Ž . Ž . Cells S and Cells S represent the sets of raster a b cells belonging to region S and S , respectively. a b The spatial extents of the objects are fuzzy, and the evaluation of their overlap should take care of Table 1 Identification and presentation of state transition that. The possibility of a grid cell to be part of the overlap of two fuzzy regions can be defined accord- Ž . ing to Dijkmeijer and De Hoop 1996 , Over S ,S N P s MIN Part P ,S ,Part P ,S 4 a b i j i j a i j b 6 Ž . w x w x Part P , S and Part P , S can be evaluated i j a i j b Ž . according to Eq. 4 . The size of P is considered to i j be 1 here, so that the size of a fuzzy region S is defined as Size S s Part P ,S where P g Cells S Ž . Ž . Ý i j i j P i j 7 Ž . The size of the overlap of two fuzzy regions is then SOVERLAP S ,S s Overl S ,S N P 8 Ž . Ž . Ý a b a b i j P i j Ž . Ž . where P g Cells S l Cells S . i j a b Let R be the set of regions at epoch T , and let i i S g R and S g R . The following indicators can a 1 b 2 be used to evaluate the relationships between regions at the two epochs. First of all, the relative fuzzy overlap between two regions can be defined as ROverl S N S s SOVERLAP S ,S rSize S Ž . Ž . Ž . b a a b a 9 Ž . ROverl S N S s SOVERLAP S ,S rSize S Ž . Ž . Ž . a b a b b 10 Ž . Ž . where ROverl S N S represents the relative overlap b a Ž . with respect to S , and ROverl S N S is the relative a a b overlap with respect to S . The similarity of two b fuzzy regions can be defined as SOVERLAP S ,S Ž . a b Similarity S ,S s 11 Ž . Ž . a b Size S Size S Ž . Ž . a b Using these indicators, object state transitions can be identified between two epochs. Seven cases are shown in Table 1. The combinations of indicator functions behave differently for these seven cases. State transitions can now be identified by the following process: Ž . For all S g R compute Size S b 2 b For all S g R do a 1 Ž . compute Size S a For all S g R b 2 Ž . compute SOVERLAP S ,S a b Ž . Ž . compute Roverl S N S , Roverl S N S , b a a b Ž . Similarity S ,S a b Ž . Ž . evaluate shift S ; S , expand S ; S , a b a b Ž . shrink S ;S a b Ž . Ž . evaluate split S ; . . . S , . . . , appear S a b b Ž . ev alu ate m erg e . . . , S , . . . ; S , a b Ž . disappear S a The split process implies that one region S g R a 1 splits into several regions S g R , and the merge b 2 process implies that many regions S g R merge a 1 into one region S g R . b 2

5. Dynamics of fuzzy objects