Shape From Shading Models - Biomedical Image Analysis - page 260

the solution. We also describe a technique, characteristic strip method, which
leads to the solution of the equation.
5.2.1 The Uniqueness and Existence
It has been shown that surfaces with continuously varying surface orientation
give rise to shaded images. The problem of shape from shading is to recon-
struct the three-dimensional shape of a surface from the brightness or intensity
variation in a single black-and-white photographic image of the surface. For a
long time in history, the SFS model was believed ill-posed. However, it has been
shown that the problem in its idealized form is actually well posed or “partially”
well posed under a wide range of conditions ( [32, 42]).
The standard assumptions for the idealized surface are:
“Lambertian” reflectance—the surface is matte, rather than mirror-like and
reflects light evenly in all directions,
“Orthographic” projection—the illuminating light is from a single known
direction and that the surface is distant from the camera, and
“Nonocclusion”—all portions of the surface are visible.
If only one source of illumination is available, uniqueness can be proved. Fur-
ther Saxberg [51, 52] discussed conditions for existence of the solution. Olien-
sis [41, 42] has shown the following:
Proposition 1. For an image of a light region contained in a black back-
ground, if the reflectance map is known, as given in (5.2), then there is a
unique solution for a generic surface which is smooth and non-self-occluding.
Despite various existence and uniqueness theorems for smooth solutions
(see [14, 30, 34, 41, 42, 51, 52, 64]), in practice the problem is unstable, which
is catastrophic for general numerical algorithms [4, 18]. This is because the
reflectance map is, in general, given by its sampled data rather than an analytic
expression. This data may be sparse and contaminated by noise. We will not
go into the detailed discussion about the uniqueness and existence issue here;
the readers who are interested in this issue are referred to the excellent review
paper by Hurt [32] and references [14, 30, 34].