Biomedical Engineering Reference
In-Depth Information
(a)
(b)
1000
800
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400
200
200
400
600
800
1000
200
400
600
800
1000
x
1
x
1
(c)
(d)
1000
800
600
400
200
200
400
600
800
1000
200
400
600
800
1000
x
1
x
1
Fig. 4.2
Plots of the constraint functions for the two-dimensional case.
a
The plot of
ˆ(
x
1
,
x
2
)
in
Eq. (
4.94
).
b
The plot of
ˆ(
x
1
,
x
2
)
in Eq. (
4.92
).
c
The plot of the constraint function for the
L
2
-norm
constraint:
ˆ(
x
1
,
x
2
)
=
x
1
+
x
2
.
d
The plot of the constraint function for the
L
1
-norm constraint:
ˆ(
x
1
,
x
2
)
=|
x
1
|+|
x
2
|
. The parameter
ʲ
was set to 1 when computing the plots in
a
and
b
4.10 Appendix to This Chapter
4.10.1 Derivation of Eq.
(
4.21
)
D
in Eq. (
4.20
) is rewritten as
1
x
k
H
T
y
k
2
x
k
H
T
H
x
k
−
ʲ
K
D
=
ʦ
+
ʲ
+
C.
(4.95)
k
=
1
Here, terms not containing
x
k
are expressed as
. We apply the completion of the
square with respect to
x
k
, i.e., we change Eq. (
4.95
)tohaveaform,
C
K
1
2
T
=
x
k
−
x
k
−
m
k
)
+
,
D
1
(
m
k
)
A
(
k
=
