Biomedical Engineering Reference
In-Depth Information
function equal to zero. Therefore, we cannot obtain an optimum solution of
x
based
only on the least-squares method.
A general strategy for overcoming this problem is to integrate a “desired property”
of the unknown parameter
x
into the estimation problem. That is, we choose
x
so as
to maximize this “desired property,” and also satisfy
y
Fx
. Quite often, a small
norm of the solution vector is used as this “desired property,” and in this case, the
optimum estimate
=
x
is obtained using
2
x
=
argmin
x
x
subject to
y
=
Fx
.
(2.22)
In the optimization above, the notation of “subject to” indicates a constraint,
(i.e., the above optimization requires that the estimate
x
be chosen such that
x
2
as well as satisfies
y
minimizes
Fx
.) To solve the constraint optimization
problem in Eq. (
2.22
), we use the method of Lagrange multipliers that can convert a
constrained optimization problem to an unconstrained optimization problem. In this
method, using an
M
x
=
1 column vector
c
as the Lagrange multipliers, we define a
function called the Lagrangian
×
L
(
x
,
c
)
such that
2
c
T
L
(
x
,
c
)
=
x
+
(
y
−
Fx
) .
(2.23)
The solution
x
is obtained by minimizing
L
(
x
,
c
)
above with respect to
x
and
c
—the solution
x
being equal to
x
obtained by solving the constrained optimization
in Eq. (
2.22
).
To derive an
x
that minimizes Eq. (
2.23
), we compute the derivatives of
L
(
,
)
x
c
with respect to
x
and
c
, and set them to be zero, giving
∂
L
(
x
,
c
)
F
T
c
=
2
x
−
=
0
,
(2.24)
∂
x
∂
L
(
,
)
x
c
=
−
=
.
y
Fx
0
(2.25)
∂
c
Using the equations above, we can derive
F
T
FF
T
−
1
y
x
=
.
(2.26)
The solution in Eq. (
2.26
) is called the minimum-norm solution, which is well known
as a solution for the ill-posed linear inverse problem.
2.7 Properties of the Minimum-Norm Solution
The minimum-norm solution is expressed as
F
T
FF
T
)
−
1
F
T
FF
T
)
−
1
Fx
F
T
FF
T
)
−
1
x
=
(
(
Fx
+
ʵ
)
=
(
+
(
ʵ
.
(2.27)
