Image Processing Reference
In-Depth Information
where
U
1
=
x
(
f
1
). The argmax of
G
(
f
,
r
1
)
now yields a first estimate of the frequency of the
second sinusoid,
f
2
. Next improve the estimates of both
f
1
and
f
2
by again solving the NLS
problem by minimizing the functional
R
(
f
)
over
f
= {
f
1
,
f
2
}.
Note that this overwrites the
previous estimate of the first frequency
f
1
. The amplitudes
a
1
and
a
2
are recalculated using
(8) with
U
2
given by
U
2
= [
Φ
x
(
f
1
),
Φ
x
(
f
2
) ]
(15)
for the latest values of
f
1
and
f
2
, Finally, in this iteration estimates of the first two sinusoids
are removed from
y
:
r
2
=
y
-
A
(
U
2
)
U
2
. (16)
If
K
, the total number of sinusoids present in the signal, is known, this process is repeated
K
times until
f
K
and
a
K
are obtained. In the absence of noise, the sum of these sinusoids solves
(5) exactly and
r
K
= 0.
Inputs:
CS Mixing Matrix
Φ
Measured data
y
Maximum number of sinusoids
K
or threshold
T
f
min
,
f
max
Oversampling ratio for dictionary
N
f
Initialize
U
= [ ]
r
0
= y
K
T
= K
W =
(ΦΦ
H
)
-1
f =
(
f
max
-f
min
)/(
N N
f
)
Do
i
= 1 to
K
f
i
= Argmax
G
(
f
,
r
i-1
)
over
{
f
min
,
f
min
+
f
,…
f
max
-
f
,
f
max
}
{
f
1
,
f
2
,…
f
i
} =
Argmin[
R
(
f
)
with initial value
f
= {
f
1
,
f
2
, …,
f
i
} ]
U
=
{Φ
x
(
f
1
), Φ
x
(
f
2
)… Φ
x
(
f
i
)}
r
i
=
y
-
A
(
U
)
U
If
r
i
H
Wr
i
< T:
K
T
= i
Break
End If
End Do
Output of Do:
K
T
{
f
1
,
f
2
, …
f
KT
}
{
a
1
,
a
2
,…
a
KT
}=
A
[{ Φ
x
(
f
1
), Φ
x
(
f
2
)… Φ
x
(
f
K
) }]
Table 1. OMP/NLS Algorithm.