Biomedical Engineering Reference
In-Depth Information
line is likely to be located at one of its vertices, so a sparse solution is likely to be
obtained.
Figure
2.3
b shows the case of the
L
2
-norm minimization in Eq. (
2.52
). In this
figure, the broken line again represents the
x
1
and
x
2
that satisfy the measurement
equation
y
=
h
1
x
1
+
h
2
x
2
, and the circle represents the
L
2
-norm objective function
x
1
+
x
2
=
constant. In this case, the
x
1
and
x
2
on the broken line that minimizes
x
1
+
x
2
at which the circle
touches the broken line. An example of such solution is indicated by the small filled
circle. In this case, both
x
1
and
x
2
have nonzero values, and a non-sparse solution is
likely to be obtained using
L
2
-norm regularization.
Finally, Fig.
2.3
c shows a case of the general
L
p
normminimization (0
should be chosen, and the resultant solution is
(
x
1
,
x
2
)
1).
An example of such solution is indicated by the small, filled circle. Using the general
L
p
norm regularization, the solution is more likely to be sparse than the case of the
L
1
-normminimization. However, the computational burden for the general
L
p
norm
minimization is so high that it is seldom used in practical applications.
<
p
<
2.9.3 Problem with Source Orientation Estimation
When applying the
L
1
-norm regularization to the bioelectromagnetic source localiza-
tion, it has been known that the method fails in estimating correct source orientations.
The reason for this is described as follows: The components of the solution vector
x
is denoted explicitly as
s
1
,
s
z
N
T
s
y
s
y
s
y
s
1
,...,
s
j
,
s
j
,...,
s
N
,
x
=
1
,
j
,
N
,
,
s
y
s
j
are the
x
,
y
, and
z
components of the source at the
j
th voxel. When
the
j
th voxel has a source activity, it is generally true that
s
j
,
where
s
j
,
j
,
s
j
,
s
j
have nonzero
values. However, when using the
L
1
regularization, only one of
s
j
,
s
y
s
j
tends to
have nonzero value, and others tend to be close to zero because of the nature of a
sparse solution. As a result, the source orientation may be erroneously estimated.
To avoid this problem, the source orientation is estimated in advance using some
other method [
4
] such as the
L
2
-norm minimum-norm method. Then, the
L
1
-norm
method is formulated using the orientation-embedded data model in Eq. (
2.15
). That
is, we use
j
,
N
2
x
=
argmin
x
F
:
F
=
y
−
Hx
+
ʾ
1
|
x
j
|
.
(2.53)
j
=
In this case, the sparsity is imposed on the source vector magnitude,
s
1
,
s
2
,...,
s
N
,
and only a few of
s
1
,
s
N
have nonzero values, allowing for the reconstruction
of a sparse source distribution.
s
2
,...,
