Given a reflectance map, then we can determine a unique value of radiance and hence image intensity from a known surface orientation . However, as shown in the previous section, we cannot determine a unique surface orientation from a single radiance value. In practice there are infinite surface orientations which give rise to the same brightness level (points on the iso-brightness contour), since brightness has only one degree of freedom and surface orientation has two. Surface orientation can only be determined uniquely in special cases, e.g. when for a Lambertian surface.

To determine local surface orientation, we need additional information; the simplest approach is to take two shaded images rather than one, but vary the position of the light source. This is photometric stereo. Then we obtain two values of image intensity, and at each point . In the previous section, we showed how each light source constrained the surface normal to a conic section, e.g. an ellipse. The intersection(s) of the conic sections provide the surface normal required. By building up the two dimensional pattern of surface normals we define surface shape.

In general, the image intensity values of each light source correspond to two points on the reflectance map, as follows

Thus we can determine the surface normal parameters from two images. Define the two light source vectors as and . Then, assume a particular reflectance function (See [References] - not Lambertian !) as given below.

Hence, re-arranging Equations 21, 22 and 23,

using the substitution . Provided , this gives a unique solution for surface orientation at all points in the image. Although we cannot determine absolute depth, from a known or assumed starting point we can build up incrementally a relative depth map from the orientation data. Figures 4 and 5 [Horn, 1986] illustrate the correspondence between a relief map and two shaded images, obtained from two diffferent positions of the source illumination.