Biomedical Engineering Reference
In-Depth Information
where
=
[0
,
1]
×
[0
,
1] with boundary conditions:
(
i
)
Z
(0
,
y
)
=
g
1
(
y
)
,
0
≤
y
≤
1
,
(
ii
)
Z
(1
,
y
)
=
g
2
(
y
)
,
0
≤
y
≤
1
,
(5.4)
(
iii
)
Z
(
x
,
0)
=
g
3
(
x
)
,
0
≤
x
≤
1
.
Here
g
i
,
i
=
1
,
2
,
3, are smooth functions.
An ideal Lambertian surface is one that appears equally bright from all view-
ing directions and reflects all incident light, absorbing none ( [29], p. 212). One
of the most interesting properties of a Lambertian surface is that the maximum
point of reflectance map is unique if it exists [51]. Assuming that the object has a
Lambertian surface and is illuminated by a planar wave of light, the Lambertain
reflectance map becomes
R
(
p
,
q
)
=
N
·
s
,
where
s
is the unit vector pointing to the light source, which is given.
A nonlinear shape from shading model is given by an ideal Lambertian sur-
face. In this case, the reflectance map has the well-known form:
1
+
p
0
p
+
q
0
q
1
+
p
0
+
q
0
1
+
p
2
R
(
p
,
q
)(
x
,
y
)
=
ρ
(5.5)
+
q
2
.
In a stereographic coordinate system, the stereographic coordinate (
f
,
g
)is
related to the Cartesian coordinate by
2
p
2
q
f
=
and
g
=
1
+
p
2
1
+
p
2
+
q
2
,
1
+
+
q
2
1
+
or conversely
4
f
4
−
f
2
4
g
4
−
f
2
p
=
and
q
=
−
g
2
.
−
g
2
In such a coordinate system, instead of using (
p
,
q
)
,
the reflectance map be-
comes
+
g
2
)
,
1
4
−
(
f
2
+
g
2
)
4
f
4
−
(
f
2
4
g
4
−
(
f
2
·
s
.
R
(
f
,
g
)
=
−
+
g
2
)
,
−
(5.6)
4
+
(
f
2
+
g
2
)
In summary, the shape from shading problems can be formulated by using
either
N
or (
p
,
q
)or(
f
,
g
). Together with adequate boundary conditions, the
shape from shading problem is to solve a linear or nonlinear partial differential
equation (PDE) of first order. In this chapter, we have limited our attention to
