Biomedical Engineering Reference
In-Depth Information
where
P
(
x
,
y
)
=−
pI
(
x
,
y
)
+
ρ
p
0
,
Q
(
x
,
y
)
=−
qI
(
x
,
y
)
+
ρ
q
0
,
I
(
x
,
y
)
=
1
+
p
0
+
q
0
1
+
p
2
+
q
2
−
(
p
2
+
q
2
)
−
ρ.
Note that (5.32) is a first-order partial differential equation with nonconstant
coefficients
P
and
Q
and, therefore, the FDM can be used to solve it in the same
way as in Example 2.
5.2.4.2 Remarks
Remark 2.
About convergence of finite difference method.
Every numer-
ical method provides a formalism of generating discrete algorithms for ap-
proximating the solution of a PDE. Such a task could be done automatically
by a computer if there were no mathematical skills that require human in-
volvement. Consequently, it is necessary to understand the mathematics in
this black box which you put in your PDE for processing. This will involve
discussion on convergence, stability, and error analysis. However, these top-
ics are beyond the scope of this introductory chapter. We hope the loose ends
left here will stimulate your curiosity and further motivate your deep interest
in this subject. For the finite difference method used to solve the SFS model,
some results related to the issue of convergence can be found in Ulich [64].
For the linear problem, Ulich proved convergence for three methods: forward
difference method, backward difference method, and central and forward dif-
ference method for certain light directions. For the central and forward finite
difference method applied to the linear PDE derived from linearization of a
nonlinear shape from shading problem, she was able to prove convergence for
all light directions. For the case of application of FDM to the linear shape
from shading models, Wei et al. [65] discussed the convergence properties
for four explicit, two implicit, and four semi-implicit finite difference algo-
rithms. They also give comparisons of accuracy, solvability, stability, and
convergence of these schemes.
Remark 3.
About Multiscale methods.
Simple iterative methods (such as
the Jacobi method) tend to damp out high-frequency components of the er-
ror faster than any other method. Multiscale methods are introduced to
