Biomedical Engineering Reference
In-Depth Information
=
meaningless to impose the constraint
y
Fx
, so, instead of using the optimization
in Eq. (
2.22
), we should use
2
2
x
=
argmin
x
x
subject to
y
−
Fx
≤
d
,
(2.34)
2
where
d
is a positive constant. In Eq. (
2.34
), the condition
d
does not
require
Fx
to be exactly equal to
y
, but allow
Fx
to be different from
y
within a
certain range specified by
d
. Therefore, the solution
y
−
Fx
≤
x
is expected to be less affected
by the noise in the sensor data
y
.
Unfortunately, there is no closed-form solution for the optimization problem in
Eq. (
2.34
), because of the inequality constraint. Although we can solve Eq. (
2.34
)
numerically, we proceed in solving it by replacing the inequality constraint with
the equality constraint. This is possible because the solution of Eq. (
2.34
) generally
exists on the border of the constraint. Thus, we can change the optimization problem
in Eq. (
2.34
)to
2
2
x
=
argmin
x
x
subject to
y
−
Fx
=
d
.
(2.35)
Since this is an equality-constraint problem, we can use the method of Lagrange
multipliers. Using the Lagrange multiplier
ʻ
, the Lagrangian is defined as
d
2
2
L
(
x
,
c
)
=
x
+
ʻ
y
−
Fx
−
.
(2.36)
Thus, the solution
x
is given as
2
2
x
=
argmin
x
L
(
x
,
c
)
=
argmin
x
x
+
ʻ
y
−
Fx
.
(2.37)
In the above expression, we disregard the term
d
, which does not affect the results
of the minimization. Also, we can see that the multiplier
−
ʻ
ʻ
works as a balancer
between the
L
2
-norm
4
term
2
and the squared error term
2
.
x
y
−
Fx
To derive the solution of
x
that minimizes
L
(
x
,
c
)
, we compute the derivative of
L
(
x
,
c
)
with respect to
x
and set it to zero, i.e.,
y
T
y
x
T
x
L
(
x
,
c
)
1
∂
x
T
F
T
y
y
T
Fx
x
T
F
T
Fx
=
−
−
+
+
ʾ
∂
x
x
2
F
T
F
I
x
2
F
T
y
=−
+
+
ʾ
=
0
,
(2.38)
where we use 1
/ʻ
=
ʾ
. We can then derive
F
T
F
I
−
1
F
T
y
x
=
+
ʾ
.
(2.39)
4
A brief summary of the norm of vectors is presented in Sect. C.4 in the Appendix.
