Biomedical Engineering Reference
In-Depth Information
the energy function, which contains almost all the common constraints listed in
Section 5.2.3.2,
+
(
Z
xx
+
Z
xy
+
Z
yx
+
Z
yy
)
+
(
||
N
||
{
(
I
−
R
)
2
2
−
1)
+
((
Z
x
−
p
)
2
+
(
Z
y
−
q
)
2
)
+
((
R
x
−
I
x
)
2
+
(
R
y
−
I
y
)
2
)
}
dxdy
,
(5.48)
where
N
is defined as the surface normal,
I
is the input image,
R
is the re-
flectance map, (
x
,
y
) is an arbitrary pixel of the input image, and (
p
,
q
) is orien-
tation at pixel (
x
,
y
). The first term, (
I
−
R
)
2
, is called the brightness error term,
which is used to minimize the brightness error between the measured image
intensity and the reflectance function. The second tern, (
p
x
+
p
y
+
q
x
+
q
y
), is
called the regularization term which will always penalize large local changes in
the surface orientation and encourage the surface change gradually. The third
term, (
||
N
||
2
−
1), is called unit normal term and is used to normalize the con-
straints on the recovered normal by forcing the surface normal to be unit vectors.
The fourth term, ((
Z
x
−
p
)
2
+
(
Z
y
−
q
)
2
)
,
is called integrability term which is
used to ensure the valid surface. The last term, (
R
x
−
I
x
)
2
+
(
R
y
−
I
y
)
2
,isde-
fined as the intensity gradient term. It requires that the intensity gradient of the
reconstructed image be close to the intensity gradient of the input image in the
x
and
y
directions as much as possible. Sometimes, if an algorithm is designed
for a particular type of images, adequate constraints should be chosen to meet
some specific requirements.
In the following context we will introduce the most popular algorithm which
is based on the concept of optimization.
5.3.2.1 Zheng and Chellappa's minimization method
Zheng-Chellappa [70] chose the squared brightness error term (5.14), the inte-
grability term, and the intensity gradient term as their energy function, which is
defined to be
((
E
−
R
)
2
+
((
R
x
−
I
x
)
2
+
(
R
y
−
I
y
)
2
)
(5.49)
+
µ
((
Z
x
−
p
)
2
+
(
Z
y
−
q
)
2
))
dxdy
.
Recall that most of the traditional methods enforce the requirement that the
reconstructed (approximated) image should be close to the input (exact) image,
