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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