Shape From Shading Models - Biomedical Image Analysis - page 258

recovering a non-self-shadowing Lambertian surface with constant albedo. We
further assume that the object is illuminated by a single distant light source.
The earliest mathematical method to solve this problem, posed by Horn [28],
is based on the characteristic strip expansion (see next section). Like the idea
of dealing with any other nonlinear problems, linearization is the most common
and easiest approach to obtain an approximation to the exact solution. Taylor
expansion can be used to derive a linear equation associated with the original
equation. After the equation is linearized, some criteria are chosen to discretize
the linear PDE to get an algebraic equation. Such methods include, for example,
numerical differentiation and integral transform (see [13, 15]). Then a numerical
method is selected to find an approximation of the solution to the algebraic
problem numerically. Since there is no guarantee to the existence of the solution,
another approach is to search for optimization solution. This procedure includes
introduction of a satisfactory energy function and finding the solution of the
posed optimization problem numerically.
5.1.2 About this Chapter
This chapter is written for the purpose of introducing students and practitioners
to the necessary elements, including numerical methods and algorithms, in order
to understand the current methods and use them in dealing with some practical
problems. With a limited set of mathematical jargons and symbols, the emphasis
is given to kindle interest for the problem. This has been done by selecting those
methods which are easily understood and best demonstrate the idea of SFS
models. Of course, our selection of the techniques and numerical examples is
limited by the usual constraints: author prejudice and author limitation. Our
goal is to draw an outline or describe the framework for solving this problem
and leave the details to the readers for further study.
We conclude this section by giving an outline of the chapter. In this chapter,
we consider one of the reconstruction methods: shape from shading. The chapter
is organized as follows: the first section serves as a brief review of the SFS
models, their history, and recent developments. Section 5.2 provides certain
mathematical background related to SFS. It discusses some selected numerical
methods for solving discretized SFS problems. The emphasis is given to the well-
developed method—Finite difference method (FDM). Section 5.4 is devoted to
the illustration of numerical techniques for solving SFS problems. It concerns