Biomedical Engineering Reference
In-Depth Information
where
∂
Z
∂
x
,
∂
Z
(
p
,
q
)
=
(5.1)
∂
y
is the gradient field, the image irradiance (intensity function) of the surface
I
(
x
,
y
) and the reflectance map
R
(
p
,
q
) are related by the following image irra-
diance equation [29] (p. 218):
I
(
x
,
y
)
=
R
(
p
,
q
)
.
(5.2)
The reflectance map
R
(
p
,
q
) depends on the reflectance properties of the
surface and the distribution of the light sources. It could be linear or nonlin-
ear. An SFS problem is classified as a linear shape from shading problem if the
reflectance map is linear or otherwise it is a nonlinear shape from shading prob-
lem. For instance, the one commonly used to model the lunar surface—Maria
of the moon—is linear:
1
+
p
0
p
+
q
0
q
1
+
p
0
+
q
0
,
R
(
p
,
q
)
=
ρ
(5.3)
where
ρ,
the surface albedo, and
1
1
+
p
0
+
q
0
s
0
=
(
p
0
,
q
0
,
−
1)
T
,
the light source direction, are given. Solving the surface
Z
from (5.3) is a linear
shape from shading problem.
Equation (5.2) is sometimes called the Horn image irradiance equation since
it was first derived by Horn in 1970 in his thesis [26]. We would like to point out
that since Eq. (5.2) depends only on the partial derivatives (
p
,
q
) of the surface
Z
(
x
,
y
)
,
therefore without additional conditions, the uniqueness of the solution
is obviously not possible. These additional conditions are usually given by the
boundary conditions. Boundary conditions can be given in many different ways;
as an example, we consider the system
1
+
p
0
p
+
q
0
q
1
+
p
0
+
q
0
=
I
(
x
,
y
)
,
ρ
(
x
,
y
)
∈
,
