Biomedical Engineering Reference
In-Depth Information
integrability constraint is then posed as
(
p
y
−
q
x
)
2
dxdy
,
G
2
=
(5.16)
or
(
Z
x
−
p
)
2
+
(
Z
y
−
q
)
2
dxdy
.
G
3
=
(5.17)
(4) depth [58]:
(
Z
(
x
,
y
)
−
d
(
x
,
y
))
2
dxdy
.
G
4
=
(5.18)
(5) minimal curvature:
(
Z
xx
+
2
Z
xy
+
Z
yy
)
dxdy
.
G
5
=
(5.19)
(6) strong smoothness [31]: Introduced in [31], this constraint is used to en-
force a stronger integrability and smoothness:
(
Z
xx
−
p
)
2
+
(
Z
yy
−
q
)
2
dxdy
.
G
6
=
(5.20)
A combination of the first three of the above constrains (5.14), (5.15), and
(5.16), that is,
k
=
1
λ
k
G
k
,
3
Eng
(
p
,
q
)
=
(5.21)
is commonly used to control the stability of iteration algorithms. Here
λ
k
,
k
=
1
,
2
,
3, are the Lagrange multipliers. The last three of the above constraints
are introduced to enforce the smoothness and convergence (of the depth con-
straint) of the approximation solution. We will demonstrate some examples in
Section 5.3.
An iterative scheme for solving the shape from shading problem has been
proposed by Horn
et al
. [27]. The method consists the following two steps.
Step 1
. A preliminary phase recovers information about orientation of the
planes tangent to the surface at each point by minimizing a functional
containing the image irradiance equation and an integrability constraint:
(
E
(
x
,
y
)
−
R
(
p
,
q
))
2
+
λ
(
p
y
−
q
x
)
2
dxdy
,
Eng
(
p
,
q
)
=
(5.22)
Step 2.
After the tangent planes are available, the surface
Z
is reconstructed
by minimizing the functional (5.17).
