Image Processing Reference
In-Depth Information
11.6. Single sensor prediction-combination
Associating combination and prediction functions is the basic mechanism for
achieving temporal data fusion. Its implementation depends essentially on the for-
malism used for representing the data (probability, possibility, evidence mass). The
best known is the Kalman filter, which is based on probability theory. See [ABI 92,
BAR 88, KAL 60] for a detailed description.
In more general terms, the method relies on the alternate use of prediction and
combination mechanisms. Let us assume that we have an evolutionary model M X
such as it was defined in the previous section by equation [11.1], as well as a model
for the sensor H X , such that for any acceptable value of X , we can infer the value Y
from the sensor's measurement. Finally, let us assume that we know the inverse model
H 1
X
of this sensor that can be used to estimate X from Y .
First, we initialize at the time t 0 the data X ( t 0 ) at a value as close as possible to
the actual value we wish to determine, which is based either on prior knowledge or on
a first measurement Y ( t 0 ). We also initialize the confidence Conf X ( t 0 ) of this first
value in terms of reliability and/or accuracy. A new measurement Y ( t 1 ) is acquired
at the time t 1 >t 0 , to which we assign a confidence Conf Y ( t 1 ). The data X ( t 0 ) is
predicted up until the time t 1 by using an evolutionary model M X :
X t 0
Conf X t 0
, Δ t = X t 1 /t 0
M X
Conf X t 1 /t 0
[11.2]
t 0 , X ( t 1 /t 0 ) is the prediction of X at the time t 1 knowing all of the
measurements up until t 0 and Conf X ( t 1 /t 0 ) is the prediction of Conf X at the time t 1
knowing all of the measurements up until t 0 .
where Δ t = t 1
We also calculate:
Y t 1
Conf Y t 1
H 1
X
At the time t 1 , the data X ( t 1 /t 1 ) and its confidence Conf X ( t 1 /t 1 ) are estimated
by a conjunctive combination Comb of the data's history, represented by X ( t 1 /t 0 ),
and the innovation resulting from the measurement Y ( t 1 ):
Comb X t 1 /t 0
Conf
,H 1 Y t 1
= X t 1 /t 1
Conf
[11.3]
Conf Y t 1
X t 1 /t 0
X t 1 /t 1
We notice in equation [11.3] that all of the variables are referenced at t 1 and can
therefore be combined. During the prediction phase, the confidence should decrease
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