equation (5)), must remain constant. In order to partly
overcome this limitation, as well as to compensate for
surface reflectance properties different from the assu-
med model, a linear radiometric function accounting for
brightness and contrast differences of the various images
can be introduced. The resulting system of equations (7)
is then completed by adding equations for control infor-
mation with appropriate standard deviations and rewrit-
ten as a system of observation equations. In the most
simple case the weight matrix for the intensity value
observations is represented by the identity matrix. Since
the observation equations are nonlinear in the un-
knowns, the solution of the least squares adjustment is
found iteratively.
4. SHAPE FROM SHADING
Classical SFS refers to the problem of reconstructing the
surface of an object, given a single digital image by
relating the image intensity values directly to surface
inclinations relative to the direction of illumination. The-
se inclinations are then integrated to produce a geome-
tric model of the object surface. For the transformation
from object to image space the orthographic projection
is used. The parameters of exterior orientation are assu-
med to be known.
The basic equation of SFS is derived from equation (6)
by assigning a constant known object intensity value G
to the object surface. Looking at equation (5) this is
equivalent to assuming a constant known albedo p, and
the other parameters must have calibrated values.
Throughout the rest of this paper constant (changing)
object intensity values are assumed to result from con-
stant (changing) albedo only.
The surface normal vector 7 can be expressed in terms
of the object surface inclination:
n° = [-oZ/0X,—aZ/0Y,1] = [-Zx,-Zy,1]
(8)
Substituting the unit vector in the direction of illumina-
tion ass ” = [s1,s2, 53 ], equation (6) can be written as
sis yleG-zh-Zenis (9)
Zr Z7 x1
There are two unknowns, namely Zx and Zy, but only one
observation, namely g ( x , y ), for each point in object
space. Therefore, there exists an infinitive number of
solutions to equation (9). This is the fundamental inde-
terminability of SFS. It can be overcome by Working in
surface profiles /Horn 1970; Davis, Sonderblom 1984/ or
by introducing smoothness terms for the object surface
/Strat 1979; Ikeuchi, Horn 1981; Horn, Brooks 1986/.
Some of the rather strong assumptions of SFS can be
dropped, if more than one image is used simultaneously.
In binocular or in multi image SFS /Grimson 1984/ ima-
ges taken from different positions are analyzed. The
correspondence problem of image matching (there is in
general a need for conjugate points) must be overcome.
This is particularly complicated for constant albedo due
to the lack of intensity gradients. A solution for the
estimation of the parameters of exterior orientation is
given in de Graaf et al. /1990/.
In the method of photometric stereo /Woodham 1978;
Lee, Brady 1991/ images taken from the same position
under varying illumination directions are used. Thus, the
correspondence problem becomes trivial. Using two
images a unique determination of Zy and Zy is possible,
the use of three images allows in addition to solve for
variable and unknown albedo.
There are also ways to compute object surface heights
directly using SFS. Wrobel /1989/ suggests to introduce
a discrete geometric model in object space similar to the
one described in chapter 3 and to solve for the DTM
heights directly. Leclerc, Bobick /1991/ present a solu-
tion along the same lines. Horn /1990/ solves for surface
inclination and height simultaneously using coupled
partial differential equations. An implementation of this
approach is described in Szelinski /1991/. Thomas et al.
/1991/ investigate an approach combining SFS and ste-
reo radargrammetry to produce DTM from multiple
radar images. Kim, Burger /1991/ use a point light source
located near the object surface. The resulting variations
of the scene irradiance are used to compute surface
heights.
