Image Processing Reference
Figure 3.1: Correspondences between a reference image and an input image. The point-to-point cor-
respondences are shown as red lines going from the first image to the second one.
to explicitly exhibit its underlying ambiguities. More specifically, in the weak perspective case,
which will be defined below, a third of the degrees of freedom is unconstrained. By contrast, in
the full perspective case, there theoretically is only a scale ambiguity. However, for most realistic
scenarios, the same number of degrees of freedom as before are so poorly constrained as to be
unconstrained for all practical purposes.
Shape-from-shading techniques Horn and Brooks [ 1989 ] offer an alternative to shape-from-
correspondences for monocular shape recovery. However, despite many generalizations of
the original formulation to account for more realistic shading effects, such as interreflec-
tions Forsyth and Zisserman [ 1991 ], Nayar et al. [ 1991 ], specularities Oren and Nayar [ 1996 ], shad-
ows Kriegman and Belhumeur [ 1998 ], or non-lambertian materials Ahmed and Farag [ 2006 ], the
resulting solutions are only valid in specific environments. As a consequence, we will only discuss
techniques that rely on shading in conjunction with texture.
In the following analysis, we will assume the internal camera parameters to be known and, as indicated
above, we will distinguish between the behavior under the weak and full perspective camera models.
We therefore define them below.
Under the weak perspective model, the projection of a 3D point q i can be written as
d u i
A Rq i +
where A is the 2
2 matrix of camera internal parameters, R contains the first two rows of the full
camera rotation matrix, t is the 2
1 camera translation vector, and d is a scalar. In general, d is the
same for all the considered 3D points. In the case of the projection of a mesh, we can define a more