N of an initial 
1e terrain and 
ns of the two geo- 
em must solve the 
Equation 8, where 
| 2, y, or w vertex 
n exactly the same 
. The state vector 
d w and, at each 
erential equations, 
direct extension of 
orward. A network 
; 6 as before and a 
rand 1 < j <n}. 
(C) is computed as 
LU. (12) 
adients along the 
ngth. The snake is 
ascent or conjugate 
ount the fact that 
| views. As a result 
1 of the snake into 
996 
(f) 
Figure 2: Recovering the 3-D geometry of both terrain and ridges. (a) Refined ridgeline after 3-D optimization. (b) Shaded 
view of the terrain after refinement. (c) Side view of the ridgeline and terrain after independent optimization of 
each one. Note that the shape of the ridgeline does not exactly match that of the terrain. (d) Differences of 
elevation between the recovered ridgeline and the underlying terrain. The image is stretched so that black and 
white represent errors of minus and plus 80 feet, respectively. (e) Side view after optimization under consistency 
constraints. (f) Corresponding difference of elevation image stretched in the same fashion as (d). 
  
Figure 3: Snake topology. (a) A simple polygonal curve described by a sequential list of vertices v;, 1 € 
6 
  
IN 
5. (b) 
i A 
network described by a list of vertices v, 1 € i € 8, and a list of edges—((1 2) (2 3) (3 4) (45) (26) (3 7) (7 
8)). 
all the images, a list of visible edges. We compute this list 
by using the face-visibility methods embedded in RCDE as 
shown in Figure 4. 
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 (13) 
and imposing that the vertices lie on such a plane or im- 
posing planar constraints on sets of four vertices using the 
Inter 
constrained-optimization approach introduced in Section 3.1. 
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 ob- 
jects that are of particular interest in urban environments: 
trihedral corners found on building roofs and extruded ob- 
jects that are used in RCDE to model building outlines. In 
Figure 5, we show several buildings modeled by roughly en- 
tering their outlines within RCDE and optimizing the shapes 
national Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
PE NET ET EE TES 
  
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