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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