Biomedical Engineering Reference
In-Depth Information
are shown. It can be seen in this figure that the
L
0
-norm constraint is approximated
by the
L
p
-norm constraint, and as
p
becomes closer to 0, the
L
p
-norm provides a
better approximation.
Let us see how the
L
p
-norm regularization causes sparse solutions when 0
≤
1. To do so, we consider a simplest estimation problem in which only two
voxels exist and the voxels have source intensity of
x
1
and
x
2
. We assume a noiseless
measurement using a single-sensor; the sensor data being represented by a scalar
y
.
The optimization for the
L
1
-norm solution is expressed in this case as
p
≤
x
=
argmin
x
(
|
x
1
|+|
x
2
|
)
subject to
y
=
h
1
x
1
+
h
2
x
2
,
(2.51)
T
, and
h
1
and
h
2
are the sensor lead field. For the sake of
comparison, we also argue the
L
2
-norm regularization whose optimization is given
as follows:
where
x
=
(
x
1
,
x
2
)
x
1
+
x
2
1
/
2
=
=
h
1
x
1
+
h
2
x
2
.
x
argmin
x
subject to
y
(2.52)
The optimization process is depicted in Fig.
2.3
.InFig.
2.3
a, the tetragon at the
center represents the
L
1
-norm objective function,
constant. The broken
line represents the
x
1
and
x
2
that satisfy the measurement equation
y
|
x
1
|+|
x
2
|=
h
2
x
2
.
Thus, as a result of the optimization in Eq. (
2.51
), the
x
1
and
x
2
on the broken line that
minimize
=
h
1
x
1
+
at which
the tetragon touches the broken line is chosen as the solution for the optimization.
Such solution is indicated by the small filled circle in Fig.
2.3
a. In this solution,
x
2
has a nonzero value but
x
1
is zero, i.e., a sparse solution is obtained. It can be seen
in this figure that in most cases, the point at which the tetragon touches the broken
|
x
1
|+|
x
2
|
should be chosen as the solution, i.e., the point
(
x
1
,
x
2
)
x
2
x
2
x
2
(b)
(c)
(a)
x
1
x
1
x
1
Fig. 2.3
The optimization process is depicted for the simple case in which a single sensor and
two voxels exist. Source magnitudes at the voxels are represented by
x
1
and
x
2
. The broken lines
represent the
x
1
and
x
2
that satisfy the measurement equation,
y
h
2
x
2
.The
filled black
circles
indicate an example of the solution for each case.
a
L
1
-norm regularization in Eq. (
2.51
).
The tetragon at the center represents the
L
1
-norm objective function
=
h
1
x
1
+
constant.
b
L
2
-norm regularization in Eq. (
2.52
). The circle at the center represents the
L
2
-norm objec-
tive function
x
1
+
|
x
1
|+|
x
2
|=
x
2
=
constant.
c
L
p
-norm regularization where 0
<
p
<
1
