
326
7 Restoration
where v(i,j) is zero-mean white noise of variance σ
ν
2
. If we arrange all the
observations and noise in a row into corresponding vectors, i.e., Ϋ(ι') =
[y(i,
1), y(/,
2),...,
y(i, N)]
1
and e(0 = [v(/, 1), v(/, 2),..., v(/, N)]
1
, then the
sequence of observed vectors can be expressed as
Y(0
= §(0 + e(/) (209)
for i = 1, 2, ..., M. e(f) is a sequence of zero-mean random vectors with
covariance
C
v
(/c)
= E{
e(i)e\i
+ k)} =
σ
ν
2
[/]
A(fc) (210)
where Δ is the Kronecker delta defined before.
Given the first-order dynamical model of (207) and (209), a Kaiman filter
for obtaining the optimum estimates S(k) in terms of Y(l), Y(2),...