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the extrinsic parameters of a camera and three for each part of 
a model, which in the following is refered to as model part or 
surface element. 
4 Constraint Description 
We propose a number of different constraints that are intended to 
judge properties of man-made regular objects such that subjec- 
tively good-looking models result. 
Every constraint is represented by a function c. The value of 
c corresponds to the deviation of the model from the respective 
restriction. An optimization module moves the parts to find the 
part positions which lead to the minimum sum of all constraint 
functions. 
Two different kinds of constraints are used: Internal and mea- 
surement constraints. Internal constraints describe properties 
within objects, i.e. they relate model parts to each other (e.g. 
perpendicularities or parallelisms). They are described in section 
4.1 through 4.4. 
Measurement constraints, on the other hand, integrate sensor data 
like depth maps and a scene segmentation into the model (section 
4.5 and 4.6). Hence internal properties and measured data are 
integrated in a uniform representation. 
  
Figure 5: Internal constraints used for the system 
4.1 Angle constraint 
Model parts can be arranged to form a certain angle between 
them. In figure 5 the angle «3 between the front and side wall 
of the model is intended to be 90 degrees. The use of the angle 
constraint is not restricted to 90 degrees, though. The constraint 
function is 
co = (ps — e(p)) :w,, (I) 
where q, denotes the angle as it should be. (p) describes 
the actual angle as function of the position vector p of the in- 
volved parts. w, is a weight that controls the relative influence 
of the constraints among each other. Every constraint function is 
595 
weighted with some weight w that empirically has to be deter- 
mined once. The function equals zero if the actual angle equals 
the intended angle. 
4.2 Parallel constraint for parts 
Two model parts can be constrained to be parallel. Extensions 
of buildings, for example, can be aligned parallel to larger walls, 
whose spatial orientation can be estimated more precisely from 
image data. In figure 5 the walls P, and P» are an example where 
an parallel constraint for parts could be used. This constraint is 
a special case of the angle constraint. For ps — 0 the constraint 
function is: 
Cpor — (p(p))” * Wpar. (2) 
4.3 Parallel constraint for edges 
In addition to model parts two model edges, which are the inter- 
sections of two parts, can be constrained to be parallel, e.g. two 
edges of a house should be parallel like edge Es to E4 or E» to 
Es in figure 5. The constraint function is the same as above ex- 
cept that the angle between the edges is used instead of the angle 
between the plane normals. 
4.4 Symmetry constraint 
Symmetries are important for subjectively good looking objects. 
Human observers are very sensitive to violations of expected geo- 
metric relations. For this purpose a symmetry constraint is intro- 
duced that judges the difference between two supposedly equal 
angles. In figure 5 this concerns the angles «1 and v», which 
means the slope of the roof should be identical for the front and 
back part. 
The cost function is: 
Csym = Wsa * (p1(P) — p2(p))”. (3) 
where p1 and p2 must equal each other to minimize the func- 
tion’s value to zero. 
We propose the following measurement constraints to incorpo- 
rate 2-D image features: 
45 Position constraint 
We introduce a position constraint to ensure that depth informa- 
tion (fig. 2) matches with the actual part position and orientation. 
À segmented image and a depth map are jointly used to estimate 
an initial orientation m;n;: and an initial center of gravity c;4;; for 
each part by plane regression. If the part is being moved by the 
optimization algorithm the actual position c(p) and n(p) may 
differ from their initial values (cf. figure 6). 
€ init 
N init 
  
Figure 6: Moving a surface element 
The cost function 
Cdir = We [Z* (init, (P)) + Wn - ((c(p) — Cini) - nau) 
(4) 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996