. The light source vector is 
and the normal vector is 
as before.
However, we shall now consider unit vectors. (For two unit vectors, 
and 
, 
.) The albedo is 
as before. Then
If we know the direction of illumination and the reflectance function of a given surface, then this provides a constraint on the orientation of the surface normal (which gives us a possible route to shape from shading). To show how this works we shall assume a Lambertian surface illuminated by a distant point source of intensity . The light source vector is and the normal vector is as before.
However, we shall now consider unit vectors. (For two unit vectors, and , .) The albedo is as before. Then
where 
is a relative measure of surface brightness or image intensity.
Since 
then
This means that the vector defined by 
is perpendicular to the light source vector 
. the length of this vector can be calculated,
For convenience, we shall define this new vector as 
. Using
Equations 12 and 13 we have:
This situation can be illustrated in Figure 3, below. By Pythagoras's theorem, since 
,
the length of the vector along the light source direction is 
.
Figure 3: The vector triangle defining the surface normal.
and hence 
vector, which is the objective. Define the unknown vector 
and the known vector 
. Then from Equations 14 and 15, we have
Hence,
This has the form,
which is the equation of a conic section. This defines the locus of the tip of the surface normal. As an illustration, we could set the
light source directly behind the viewer, 
. Then we simplify Equation 18 to get the equation of a circle, as seen from the coincident light and viewing directions. This gives circular iso-brightness contours on a unit sphere, as shown in
Figure 9 on the top right hand side.
[ The Reflectance Map and the Image Irradiance Equation |
Photometric stereo ]
Comments to: Sarah Price at ICBL.