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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part B4. Beijing 2008 
x ; = < 
1 iff the i th straight line 
belongs to a roof contour; 
0 otherwise. 
contour. But, one can interpret that if both Fj and Fj are closed 
to the unit, then they are somehow near. It is also easy to note 
that the following properties hold: P(i, j) > 0 and P(i, j)= P(j, i). 
(5) In particular, P(i, j)= 0 if only if both straight lines (Fj and Fj) 
superpose the like parts of the projected LiDAR roof contour. 
The metric for the orientation between two straight lines Fj and 
Fj follows the same principles of the proximity metric and it is 
defined tanking into account the sigmoid function, i.e., 
The above rule gives rise to an n-dimensional discrete random 
vector, where n is the number of straight lines to be considered 
in the optimization process. This random vector is the unknown 
in the optimization process. Theoretically, the search space has 
2 n combinations to be considered in the global minimum 
computation of the energy function. The optimization algorithm 
used will be described later. 
Before proceeding with the development of the energy function, 
it is necessary to define two metrics, called the proximity and 
orientation metrics. Both metrics are the basis for defining the 
neighbour system for the problem under consideration. 
s*(ij) = 
2 
1 + exp[-/?.(# - # 0 )~ ] 
-1 
(7) 
where: 0 =0 y +0- 
0 x is the angle between the straight line Fj and the 
projected LiDAR straight line that is nearest to Fj 
0- is the angle between the straight line Fj and the 
projected LiDAR straight line that is nearest to Fj 
/? is a positive constant 
0 O is the optimal value (0° or 180°) of the parameter 
0 
The sigmoid function has some interesting properties: 1) it has 
only a minimum point at 0 = 0 Q ; 2) it takes value over [0; 1]; 3) 
it is symmetric around 0 = 6 Q ; and 4) the constant ft can be 
used to control the shape of the sigmoid function. The larger is 
the parameter /?, the harder is the penalization of deviations of 
0 from 0 O . 
Now, if Nj is the set of sites neighboring i, then any site j e N, if 
only if, 
Figure 1. Geometric elements for defining the proximity and 
orientation metrics 
The metric for the proximity between two straight lines Fj and 
Fj is defined as follows, 
i P<i " j) -‘P (8) 
where: t p and t s are the proximity and orientation thresholds, 
respectively 
P(i,j)=-(d‘+df+d'+dj) (6) 
2 
where: dj 1 andd^ are the distances between the endpoints of 
the straight line Fj and the projected LiDAR straight 
line that is nearest to Fj 
dj anddj are the distances between the endpoints of 
the straight line Fj and the projected LiDAR straight 
line that is nearest to Fj 
The energy function U(x) is elaborated based on three energy 
terms. The first term is an one-site click energy defined in such 
way to favor longer straight line, taking as reference the nearest 
projected LiDAR straight line. This energy term (U^x)) is 
expressed as follows, 
Uj(x) 
(9) 
Equation 6 is based on the principle that straight lines that are 
somehow interrelated and near to one another are perceived as 
belonging to a same unit. In this case, the unit is the reference 
roof contour, i.e., the projected LiDAR roof contour. The 
equation 6 is then an indirect proximity measurement between 
Fj and Fj, since it explicitly expresses the nearness between a 
pair of straight line (Fj and Fj) and the projected LiDAR roof 
where: n is the number of image straight lines 
L L is the length of the projected LiDAR straight 
F; 
line that is nearest to the i th image straight line (Fj) 
Lp is the length of the i ,h image straight line (Fj)