Biomedical Engineering Reference
In-Depth Information
integrals of the form
E
=
F
(
x
,
y
,
Z
,
p
,
q
)
dx dy
.
E
has an extremum only if the Euler differential equation
F
z
−
∂
∂
x
F
p
−
∂
∂
y
F
q
=
0
is satisfied. If the solution is subject to the constraints
g
j
(
x
,
y
,
Z
)
=
0
,
j
=
1
,...,
k
,
then we have
j
=
1
λ
j
g
j
(
x
,
y
,
Z
)
.
k
G
=
F
+
Now the Euler equation is
k
F
z
−
∂
∂
x
F
p
−
∂
j
=
1
λ
j
∂
g
j
∂
y
F
q
+
∂
Z
=
0
.
(5.13)
The
λ
j
'
s
are called Lagrange multipliers. An example is provided in Section
5.3.2.1.
5.2.3.2 The Constraint Functions Used in SFS Models
When iterative algorithms are used for solving the SFS problem, constraints will
be proposed to secure a weak solution. The following constraints are examples:
(1) total squared brightness error [27]:
(
I
(
x
,
y
)
−
R
(
p
,
q
))
2
dxdy
.
G
0
=
(5.14)
(2) weak smoothness: After the tangent planes are available, the surface
Z
is reconstructed by minimizing the following functional:
(
p
x
+
p
y
+
q
x
+
q
y
)
dxdy
.
G
1
=
(5.15)
(3) integrability: Since
p
and
q
are considered independent variables, (
p
,
q
)
may not correspond to the orientation of the underlying surface
Z
, that
is, the surface
Z
cannot be derived by integrating
Z
x
=
p
,
Z
y
=
q
.
An
