Biomedical Engineering Reference
In-Depth Information
related algorithms and their implementations. The section ends with a discussion
about the advantages and disadvantages of the algorithms introduced in this
section. The last section attempts to introduce the recently developed wavelet-
based methods by using an example. A part of the section, however, is devoted
to a brief introduction of the basic facts of wavelet theory. In the hope that
readers will be able to extrapolate the elements presented here to initiate the
understanding of the subject on their own, the chapter concludes with some
remarks on other advanced methods. Finally, we include an intensive set of
references to make up whatever important spirits which the authors have indeed
hardly to touch in this short chapter.
5.2 Mathematical Background of SFS Models
Many problems of mathematical physics lead to PDEs. In general, PDEs are
classified in many different ways. However, in most mathematics literature,
PDEs are classified on the basis of their characteristics, or curves of information
propagation (see, for example, [60] and [19]). The irradiance equation (5.2) is
a first-order nonlinear equation. The general format of such an equation in the
two-dimensional space is given by
f
∂
Z
∂
y
,
z
,
x
,
y
∂
x
,
∂
Z
=
0
,
(
x
,
y
)
∈
.
(5.7)
Theoretically, a compactible boundary condition should be given as
Z
(
x
,
y
)
=
g
(
x
,
y
)
,
(
x
,
y
)
∈
,
where
is the boundary curve of the domain
.
In general, nonlinear PDEs are much more difficult than the linear equa-
tions, while the more the nonlinearity affects the higher derivatives, the more
difficult the PDE is. The irradiance equation (5.2) with a nonlinear reflectance
map (5.5) is a hyperbolic PDE of first order with severe nonlinearity. Although
the nonlinearity prevents the possibility of deriving any simple method to solve
the equation, there are still some techniques developed to obtain local informa-
tion of the solution to a certain extent. In this section, we briefly review some
basics about the irradiance equation, namely, the existence and uniqueness of
