Biomedical Engineering Reference
In-Depth Information
Remark 1.
The variational approach introduced in [27] does not necessarily
guarantee the existence of a solution of the problem. In fact, [10] has addressed
this crucial question and shown that the variational approach does not lead
to an exact solution of the SFS problem in general. For the discretization of
the Euler differential equation and some numerical methods used to solve it,
see Sections 5.2.5 and 5.3.
5.2.4 Numerical Methods for Linear
and Nonlinear SFS Models
Unfortunately, in practice, even with greatly simplified initial and boundary con-
ditions, the analytic solution for a nonlinear PDE is too difficult to obtain in a
closed form. A numerical technique is then employed to find a reasonable ap-
proximate solution. In this sense, it is more useful to know of such numerical
methods which provide us a technique to be actually used in everyday life.
When dealing with the shape from shading model, it becomes clear that the
analytic solutions to the irradiance equation (5.2) or the system of ordinary
equations (5.8) are practically impossible.
To obtain a numerical approximation for the solution, the first step is to
simplify the irradiance equation. The basic approaches for this purpose include:
direct method: discretizing the irradiance equation directly using Taylor
series or difference formula.
integral transform: using linear transforms, such as Fourier transform and
wavelet transform [13, 15].
projection method: approximating the solution by a finite set of basis func-
tions.
The second step is to choose a criterion to discretize the simplified irradiance
equation to get an algebraic equation. Then a numerical method is chosen to
solve the algebraic equation. Some of these steps can be done simultaneously.
5.2.4.1 Finite Difference Method
The FDM consists of two steps: (1) replacing the (partial) derivatives by
some numerical differentiation formulas to get a difference equation, that is,
