In Markov networks this approach may lead to prob-
lems, since the probability at a node A that has been
derived with the above formula, is used in a later stage
to recompute the probability at one (or more) of its
neighbouring nodes. Pearl [1988, p. 149] gives a nice
example of this kind of circular reasoning:
“Imagine that a processor F, representing the event
Fire, communicates asynchronously with a second pro-
cessor S, representing the event Smoke. At time tl,
some evidence (e.g. the distant sound of a fire engine)
gives a slight confirmation to F, thus causing the prob-
ability of Fire to increase from P(F) to P1(F). At a later
time, t2, processor S may decide to interrogate F; upon
finding P1(F), it revises the probability of Smoke from
P(S) to P2(S) in natural anticipation of smoke. Still
later, at t3, processor F is activated, and upon find-
ing an increased belief P2(S) in Smoke, it increases
P1(F) to an even higher value, P3(F). This feedback
process may continue indefinitely, the two processors
drawing steady mutual reinforcement void of any em-
pirical basis, until eventually the two propositions, Fire
and Smoke, appear to be firmly believed."
This kind problem can be solved by keeping track of
the source of evidence. However, this involves a more
complex algorithm, such that the advantages of local
asynchronous probability updates are lost.
Certainty factors
Certainty factors CF1( A, B1) and CF2(A, B2) arising
from two observations B1 and B2 are used to derive a
combined certainty factor with [Buchanan and Short-
liffe, 1984]
CFI-ECP2—CEI- CE? ^ if CFT; CE2 50
CE — / CFI-CF24CF1.CF2 if CF1,CF2<0
CFi+CFa
1—min(|CF1},|CF2)) otherwise
Whereas single certainty factors can already be mis-
leading, combined certainty factors are even more dan-
gerous, since any correlation between observations is
neglected.
Dempster-Shafer theory
Given two sources of evidence, the mass functions m1
and m2 such that the combined probability of a subset
S, ml 4- m2(S), is the sum of the joint probabilities
of all combinations of two subsets (73, U;) which in-
tersection equals S. This sum is normalised by the
sum of the joint probabilities of all combinations of
two subsets which intersection is not an empty set.
This normalisation is required in order to take out the
so-called weight of conflicting evidence.
S ml(Ti)m2(U;)
{9|T;NU;=S}
m1(7,)m2(U;)
{1,317 NU; 0}
m1+m2( 5) =
This update formula shows many resemblances to com-
bining evidence from two independent sources with
Bayesian probability theory. The Dempster-Shafer up-
date formula is, however, controversial. Especially in
case of incomplete probabilistic models, i.e. Bel(S) +
Bel(+5) < 1, it may lead to curious results (see e.g.
[Pearl, 1988, p. 447]).
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
e Probabilistic logic
As more evidence becomes available, the theory of
probabilistic logic will use this information to further
constrain the space of all possible probability assign-
ments until the probabilistic model. In this way results
remain consistent with Bayesian probabilistic meth-
ods. Pearl [1988] therefore concludes that in case of
analysis problems with incomplete probability models
probabilistic logic should be preferred above Dempster-
Shafer theory.
e Possibilities
Possibilities of set membership are typically updated
with
poss(AA B) = min (poss(A), poss( B))
poss(AV B) = max(poss(A), poss( B))
These update rules are only equivalent to probabilis-
tic rules when A and B are completely dependent, i.e.
A — B or B — A. But if, e.g, À and B are mutu-
ally exclusive, it is clear that P(A A B) should be zero
[Cheeseman, 1984].
6 UNCERTAINTY IN EXTRACTING ROADS
Surprisingly, only a very few publications deal with automatic
updating of road maps. The usage of an old road database
as a valuable source of knowledge still is very uncommon.
Many more papers have been published on road extraction to
build up a database from scratch. Most of these publications,
however, pay very little or no attention to the uncertainty in
the extracted roads. It seems that, like in many areas of
image understanding, the results are too poor to seriously
consider to describe their quality.
In this section we will again make a distinction between the
verification and the detection step in the updating process.
For both steps several presented results will be shown and it
will be discussed how the uncertainty in these steps was dealt
with or could have been dealt with.
6.1 Verification
Four examples are discussed that compare the contents of
an aerial image with roads in a database. The first two are
aimed at verification. The goal of the last two papers was
the location of a road junction. However, the same strategy
might have been used for verification as well.
Gunst and Hartog [1994] and Gunst [1996] discuss the advan-
tages of a knowledge based interpretation strategy for updat-
ing road maps. The existence of an old road in the new
image is verified by submitting the cross correlation between
grey value profiles of road cross sections and an artificial road
profile to a statistical test. If the cross correlation is lower
than a threshold, a change is hypothesized. Problems arise
with (larger) cars and overhanging trees alongside the road.
Since the road model does not contain any knowledge about
possibly occluding objects, many false alarms result. Hence,
the uncertainty about the correctness of the verification re-
sults are mainly due to insufficient modeling of the road's
context.
Baumgartner et al. [1996] compare extracted linear features
to the road sides in a vector-based GIS. Checks are performed
on parallelism, straightness and symmetry. With some effort
in error analysis of the feature extraction process, conditional
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