1. Kalman filter, and Logical revision: comparing them on GI
Geoffrey Edwards
(Séminaires du CRG - Mars 2001)

2. Outline Fusion and revision – definitions
Topographic fusion – the problem
Framing the problem formally
Kalman filtering
Comparison with Logical Revision
Conclusion

3. Fusion and Revision - definitions
Many terms
Fusion
Integration
Revision
Updating
What is the difference between fusion and integration?
Fusion is a subset of integration, a kind of « total » integration
What is the difference between revision and update
Revision is a subset of updating

4. Fusion and Revision - definitions
What is the difference between revision and fusion?
The size of the respective databases?
I.e. revision occurs if one database is substantially larger than the other – one revises the larger database with the smaller one
Many current « fusion » techniques are actually « revision » techniques

5. Fusion and Revision - definitions
An integration of two knowledge bases (or data sets) where the identity of the product (i.e. the set of characteristic properties) is different than the simple combination of the earlier identities (and their associated properties)
Revision
An integration of two knowledge bases where the identify of the product is the same as one of the earlier identities, although some (non-essential) properties might be different

6. Topographic fusion – the problem (CITS – roads & rivers)

7. Topographic fusion – the problem (CITS – roads & streams)

8. Topographic fusion – the problem (CITS – streams & contours)

9. Topographic fusion – the problem
Incompatabilities
Roads which pass across water bodies
Roads which cross streams with no bridges
Lakes on the flanks of mountains
Streams which flow uphill
Etc.

10. Framing the problem formally
Variables and constraints
Slope < x
No intersections between contours and lake boundaries
Contours have errors associated with them, which are different from errors for lake boundaries, roads, etc.
Upward watershed matched to flow rate (i.e. watershed area must remain roughly constant)
Water bodies must be connected
Streams follow the maximum slope
Roads cross important streams at bridges
Roads cannot intersect with the interior of lakes

11. Framing the problem formally
Define contours as closed forms, each embedded in the next
Partition space « azimuthally » to formalize stream flows
Express 2, 3, 4, 8 as intersections between closed contours and other elements
Slope < x
No intersections between contours and lake boundaries
Contours have errors associated with them, which are different from errors for lake boundaries, roads, etc.
Upward watershed matched to flow rate (i.e. watershed area must remain roughly constant)
Water bodies must be connected
Streams follow the maximum slope
Roads cross important streams at bridges
Roads cannot intersect with the interior of lakes

12. Kalman filtering Principle:
given several predicates x(k), x(k+1), … linked by some dependance model (dynamic model: F)
given a series z(k), z(k+1) infered from respective x(.) through an observation model: H,
let ’s try to reduce the uncertainty (that any one model may causes)
Classical mathematical representation:
Between step k and k+1, the evolution (dependance) is:
x(k+1) = F(k).x(k) + u(k) + v(k) (1)
Between a predicate x and its « observed » counterpart z:
z(k) = H(k).x(k) + w(k) (2)
(v is a « state noise » and w an « observation noise »)
(u is a « deterministic -certain- predicate, may be T, i.e. null)

13. Kalman filtering Where the « revision » takes place:
according to (1) and (2): z(k+1) can be infered twice:

14. Kalman filtering
Where the « dependance » takes place: possible locations (x) are ruled by topo, geology, … (model F)
Where the « revision » takes place: the observations (z) may be the amount of incorrect surface and may decrease or increase depending on trajectory F B A

15. Kalman filtering Analogy Revision - Kalmnan Filtering
One cause of uncertainty: the knowledge of some « target variable » x comes through the observation of z « observed variable », then applying some inference to derive x from z. We need « link » rules between x and z (set L) as well as rules, or « constraints » that govern the x according to our knowledge of this « target » (set C).
In the « revision scheme » we consider two sets of formulas: one large uncertain set A and a second B, smaller and trustworthy. Hence we use B to « revise » A, trying to restore a possibly damaged global (A and B) consistency.
The « dissymetry » of this scheme tells us to split C and L into strong (F and H) and weak parts (v and w), and put them into set A and B respectively.

16. Kalman filtering From the Kalman filter viewpoint:
The x are the « target variables » while the z are the « observed » ones
matrix F (evolution rules) is what we try to improve (revise), and H (observation) matrix plays the role of the rules L.
According to the evolution of z, we will revise F: by choosing a model with the shortest distance.

17. Comparison Modification of one fact at a time
Bayesian networks
Global update which minimizes errors
Kalman filtering
Recursive updating which minimizes errors
Modification of a collection of facts
Logical revision
Cumulative updating which rejects or minimizes incompatability
Kalman
The minimization is also global

18. Conclusion a Phd thesis subject to start
… next talk will be given by Gilles Cotteret