Fua - 8 
the network’s projections match image features in the view that was used but not necessarily in any 
other view because the fixed z values usually are erroneous. 
2. The height of the vertices is estimated by taking the network to be horizontal and searching through a 
range of z values, calculating the x and y values for each vertex so that the projection of the network 
remains the same in the view used in the previous step and retaining the z value that yields the optimal 
value of £/, the image energy of Equation 16. 
3. The 3-D positions of the network’s vertices are further refined by optimizing Ej with respect to all 
three degrees of freedom of the vertices simultaneously. 
As in the case of the 2-D networks, the optimization of Steps 1 and 3 can be performed using either 
steepest steepest gradient descent, conjugate gradient or constrained optimization. 
The number of degrees of freedom of generic 3-D networks can be reduced by forcing them to be planar. 
We do this either by defining a plane of equation 
z = ax + by + c (1~) 
and imposing that the vertices lie on such a plane or imposing planar constraints on sets of four vertices. 
In both cases, we replace the n degrees of freedom necessary to specify the elevation of each vertex by the 
three degrees of freedom required to define the plane. 
These 3-D networks can be further specialized to handle objects that are of particular interest in urban 
environments: trihedral corners found on building roofs and extruded objects that are used in R.CDE to 
model building outlines. 
Figure 5: Topology of a trihedral corner. It has four vertices and three edges that all share one vertex. 
The edges form 90 degrees angles, Two of them may be constrained to be horizontal. In 
this case the corner has only four degrees of freedom, three for the position of vertex 0 and 
1 for the orientation of the horizontal edges. 
Trihedral corners. They are modeled as networks with four vertices and three edges forming 90 degrees 
angle with each other, as shown by Figure 5. We typically impose the additional constraint that one edge 
be vertical while the two other are horizontal. Under such constraints, the trihedral corner has only four 
degrees of freedom: three for the position of the vertex that is shared by all three edges and one for rotation 
about the vertical axis. When optimizing using only one image, fixing the altitude removes one additional 
degree of freedom. In both cases, the optimization is much more constrained than for generic 3-D networks 
and, as a result, the convergence properties are substantially improved.