Digital Signal Processing Reference
In-Depth Information
A wavelet filter is denoted as
g(
•
)
.
x
h
=
g(
x
)
is the result of filtering an observed
•
)
.
x
l
=
−
spectrum
x
with
g(
x
h
is the smooth component removed by the filter.
Because of the overlapping of
B
and
s
k
(
k
x
1
,...,K
), when the baseline
B
is
eliminated completely by the wavelet filter, the smooth and broad components of
metabolite spectra will also be lost at the same time. This can be denoted as:
x
h
≈
=
S
h
and
x
l
≈
B
+
S
l
, where
S
h
=
g(
S
)
and
S
l
=
S
−
S
h
. To represent
S
h
, all the basis
vectors
in the dictionary
D
analyzed above are also processed by
the same wavelet filter. A new dictionary
D
h
using the remaining components
g(
d
i
)
(
i
{
d
1
,
d
2
,...,
d
M
}
1
,...,M
) is then constructed. The signal
x
h
remaining after filtering can be
written as:
=
x
h
=
S
h
+
ξ
B
=
D
h
w
+
ξ
B
≈
D
h
w
,
(14.15)
where
ξ
B
is the remaining component of the baseline after wavelet filtering, which
should be as small as possible. Finally, the representation coefficient vector
w
of
mixed resonances
S
with respect to the dictionary matrix
D
in Eq. (
14.14
) can be
estimated by computing the sparsest solution of Eq. (
14.15
).
14.3.3 Resonance Estimation with FOCUSS Algorithm
For estimating the sparse representation in Eq. (
14.15
), an algorithm based on FO-
CUSS algorithm was developed with the consideration of the following particulari-
ties in this application. Firstly, basis functions in the same group have the same cen-
tral frequency, so strong correlations exist between them. Pursuit algorithms such as
greedy pursuit and basis pursuit, which have severe restriction on the correlations
between basis functions, perform poorly here. Secondly, the expected representation
coefficients are nonnegative, and the non-negativity constraint can be added.
With the non-negativity constraint, the optimization function of regularized FO-
CUSS algorithm can be modified as
arg mi
w
J(
w
),
where
J(
w
)
=
+
γ
E
(p)
(
w
)
and
2
0
.
(14.16)
By reference to the iterative form of regularized FOCUSS algorithm, the iterative
form of the optimization in (
14.16
) is then developed as follows:
w
=
D
h
w
−
x
h
∀
i
:
w
i
≥
λ
I
)
−
1
x
h
;
W
k
+
1
D
k
+
1
T
(
D
k
+
1
D
k
+
1
T
(a)
w
k
+
1
=
+
0 if
w
k
+
1
(i)<
0
,
w
k
+
1
(i)
if
w
k
+
1
(i)
=
(b)
w
k
+
1
(i)
≥
0
,
1
−
(p/
2
)
,...,
1
−
(p/
2
)
)
and
D
k
+
1
=
where
W
k
+
1
=
D
h
W
k
+
1
.At
each iteration step, the negative values of the updated solution
w
k
+
1
are set to zero
to ensure the non-negativity of
w
. Actually, the non-negativity constraint also in-
creases the sparsity of a solution to a certain degree. The regularization parameter
λ
is the function of the level of noise. In [24], three different criteria for choosing
λ
are investigated. They are (i) quality of fit; (ii) a sparsity criterion; (iii)
L
-curve.
diag
(
|
w
k
(
1
)
|
|
w
k
(M)
|
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