Biomedical Engineering Reference
In-Depth Information
5.2.2 The Characteristic Strip Method
Horn [29] established a method to find the solution of (5.2), the characteristic
strip method ( [29], p. 244). This method is to generate the characteristic strip
expansion for the nonlinear PDE (5.2) along a curve on the surface by solving a
group of five ordinary differential equations called characteristic equations:
Z
=
pR
p
+
qR
q
,
x
=
R
p
,
y
=
R
q
,
p
=
E
x
,
q
=
E
y
,
where the dot denotes differentiation along a solution curve. The characteristic
equation can be organized in a matrix format:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
x
y
Z
p
q
R
p
R
q
pR
p
+
qR
q
E
x
E
y
d
dt
=
.
(5.8)
The solution, (
x
,
y
,
Z
,
p
,
q
)
T
, to (5.8) forms a characteristic strip along the
curve. The curves traced out by the solutions of the five ordinary differential
equations are called characteristic curves, and their projections in the image
are called base characteristics. If an initial curve (with known derivative along
this curve) is given by a parametric equation:
r
(
η
)
={
x
(
η
)
,
y
(
η
)
,
Z
(
η
)
}
T
,
then we can derive the surface by integrating the equation
∂
Z
∂η
=
p
∂
x
∂η
+
q
∂
y
∂η
.
(5.9)
Example 2.
Consider an ideal Lambertian surface illuminated by a light
source close to the viewer at
(
p
0
,
q
0
,
1)
=
(0
,
0
,
1)
.
(
p
0
,
q
0
)
is the direction to-
ward the light source. In this case, the image irradiance equation is
1
1
+
p
2
I
(
x
,
y
)
=
+
q
2
,
where we have set
ρ
=
1
for simplicity.
