In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds). IAPRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
6 
4.3.2 Orthogonal connection We propose to promote like 
wise orthogonal frames. In fact, two rectangles are said orthogo 
nal. if one rectangle short side is opposite to the other rectangle 
long side (figure 11 (right)). That means, they satisfy the follow 
ing conditions: 
{ dc{i-j) < d Cm 
I d^(u,v) — I |< d UJm 
i + j = 0, 2, 4 or 6 
(21) 
By analogy with the expression (19) and (20). we define an en 
ergy component of a configuration x, related to orthogonal con 
nections ~orf.ii, as follows: 
Uart h ( X ) — 'forth ^ ^ Uorthi{?Cj'Xk) 
l<j<k<n(x) 0<?<3 
where 'forth is the weight of the described energy. Having spec 
ified the different types of connections that link buildings of an 
urban area, we can define an energy term that summarizes all 
the interactions between rectangles representing buildings as fol 
lows: 
U L,(*) = T.»e K,(x) + Ul, h (x)} (22) 
A new parameter 7related to rectangle interactions is intro 
duced in order to offset the value of the other parameters 7 ai and 
'forth which can be set to 1, since both alignment and orthogo 
nal connections are present in an equitable manner in the image 
lO(left). Hence, we are interested in estimating only one interac 
tion parameter which is the weight 7/ nt , by the SEM algorithm. 
Moreover, we initialize this interaction weight to 7°,, = 7^/2, 
since the empirical test showed that it must have a lower value 
than that of the data energy weight 7The SEM estimation al 
gorithm performed on the part of Amiens considered in the for 
mer test is very slow. It required 6 h and 21 min and contributed 
to the following estimates 7a = 30.0599 and 7= 16.8494. 
The result of the extraction phase depicted in figure 12 shows that 
the frames are closer and more aligned than those of the previous 
simulation. However, some pairs are still distant. In fact, the dis- 
Figure 12: Extraction results, favoring orthogonal and aligned 
rectangles: 83 rectangles (ß = 1000, 7d = 30.05, 7¡ nt = 
16.84, s = 0.3, do =4). 
tance between two remote objects cannot support a new object 
without overlapping with them. To avoid such a case, we can in 
crease the tolerated recovery rate s or we can employ object with 
variable size allowing the presence of small and large rectangles. 
In fact, in our tests we have limited the size of the occurred rect 
angles when we have set the object space parameters. 
5 CONCLUSION AND FUTURE WORK 
In this paper, we generalized the estimation method associated 
with a marked point process model to the case of multidimen 
sional shapes such as ellipses and rectangles. The proposed es 
timation method is based on the SEM algorithm. It showed its 
interest for estimating the parameters of an ellipse process within 
the framework of flamingo and tree crown extraction. Its appli 
cation to an image of boats in a seaport required the modification 
of both the data energy and the prior energy components of the 
proposed model. Moreover, in order to extract building footprint 
in a dense urban area, we introduced an energy term providing 
orthogonal and aligned frames. For that matter, we estimated by 
the SEM algorithm a new parameter related to this type of inter 
action. Nevertheless, incorporating ellipse or rectangle process in 
an iterative algorithm such as SEM algorithm is time consuming. 
Accelerating the proposed algorithm by optimization strategies 
could be a nice future work. The computation of the pseudo 
likelihood, representing more than 70% of the computation, could 
be parallelized using a multi-threaded program. Besides, the di 
rection of boats involved in the prior term was set manually. It 
could be interesting to estimate it automatically. 
ACKNOWLEDGEMENTS 
The authors would like to thank the French Space Agency (CNES), 
the French Forest Inventory (IFN) and Tour du Valat for provid 
ing the images presented in this paper. This research work has 
been partially supported by 1NR1A through a post-doctoral grant 
for the second author and partially by CNES through a contract. 
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