Kalman filter, and Logical revision: comparing them on GI

La géomatique pour des interventions et des décisions éclairées / Geomatics for Informed Decisions

Networks of Centres of Excellence / Réseaux de centres d’excellenceR

. Jea

nsou

lin,

G. E

dwar

ds &

G. C

otte

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Kalman filter, and Logical revision: comparing Kalman filter, and Logical revision: comparing them on GIthem on GI

Geoffrey Edwards (Séminaires du CRG - Mars 2001)



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



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Fusion and Revision - definitionsFusion 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



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Fusion and Revision - definitionsFusion 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



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Fusion and Revision - definitionsFusion and Revision - definitions Fusion

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



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



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



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



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Topographic fusion – the Topographic fusion – the problemproblem 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.



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Framing the problem formallyFraming the problem formally Variables and constraints

1. Slope < x

2. No intersections between contours and lake boundaries

3. Contours have errors associated with them, which are different from errors for lake boundaries, roads, etc.

4. Upward watershed matched to flow rate (i.e. watershed area must remain roughly constant)

5. Water bodies must be connected

6. Streams follow the maximum slope

7. Roads cross important streams at bridges

8. Roads cannot intersect with the interior of lakes



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Framing the problem formallyFraming 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 elements1. Slope < x2. No intersections between contours and lake

boundaries3. Contours have errors associated with them, which are

different from errors for lake boundaries, roads, etc.4. Upward watershed matched to flow rate (i.e. watershed

area must remain roughly constant)5. Water bodies must be connected6. Streams follow the maximum slope7. Roads cross important streams at bridges8. Roads cannot intersect with the interior of lakes



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Kalman filteringKalman filteringPrinciple:

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)



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Kalman filteringKalman filteringWhere the « revision » takes place:

according to (1) and (2): z(k+1) can be infered twice:

z(k+1)= H.x(k+1)= H.(F.x(k)+v(k)) + w(k+1)estimated observation

x(k+1)andx(k+1)

x(k)

HoFF

H

H

z(k)= H.x(k+1)

z(k+1)= H.x(k+1) actual observation

target domain

observed domain



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Kalman filteringKalman filteringWhere 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

A B



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Kalman filteringKalman filteringAnalogy 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.



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Kalman filteringKalman filteringFrom 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.



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ComparisonComparison 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



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ConclusionConclusion

a Phd thesis subject to start

… next talk will be given by Gilles Cotteret

Documents

Kalman filter, and Logical revision: comparing them on GI