In: Paparoditis N.. Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds). IAPRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
A(x k ) =p(l ~p) (— + ) (4) 
\x k n-XkJ 
with p being the probability of regular edges occurrence. 
For practical purposes we fix the probability p to |: regular edges 
can appear everywhere in the region without any a priori. Fig 
ure 5 shows how this border compensation improves the process. 
Figure 3b shows a vertical intra-region energy profile: the first 
extreme is located between the ground-floor and the first one, sec 
ond extreme is located between the first floor and the second one. 
Split choice is then determined by Inter-Region Energy. 
3.3 Inter-Region Energy 
The inter-region energy evaluates the regular edges length along 
the split hypothesis under study. Equation 5 gives the inter-region 
of a vertical hypothesis Xk- Energy of horizontal hypotheses is 
computed in a similar way. 
Einter(Xk-Cv) = y^C v (i,j)Vi,j (5) 
j=1 
where ufj is the length of edge ihj. 
This split energy term let process be directed by main gradient lo 
cation. When some split hypotheses have analogous intra-region 
energy, inter-region energy is an accurate further selection crite 
rion. For instance on figure 3b, this information is essential. 
4 PERIODIC MODEL 
Dominant alignments is a geometric property in common with 
most of facade architecture style. (Burochin et al., 2009) has ver 
ified that these alignments pretty rightly directs first steps of a 
hierarchical segmentation. But facades contains an other very fre 
quent property that is actually missing in this previous approach: 
periodicity. We will see that accurate period hypotheses can be 
inferred from dominant alignments. 
We introduce in this section a periodic model that is to considered 
as a macro model. Indeed it lets the process operate a kind of 
other models factorization. It assumes sub-image Ik to be com 
posed of the periodic concatenation of a same sub-region that we 
name kernel of the model. A sub-figure composed of a window 
for instance could be the kernel of a model that represents a reg 
ular grid of similar windows (see figure 6). 
General rules of our repetitive pattern detection resemble (Wen 
zel et al., 2008) but we do not look for any hierarchy in the peri 
ods. We only select the most frequent in horizontal and vertical 
directions: this restriction is sufficient because the process is re 
cursive. We do not use the same interest points either: our points 
are intersections of dominant alignments. 
An important aspect of the periodic kernel concept is the fact that 
this kernel is not an irreducible region as described in (Muller 
et al., 2007). This kernel is possibly composed by sub-patterns. 
This case occurs by instance in figure 6: kernel is composed of 
three similar windows. We first generate period hypotheses. Then 
we select the best one to try to build a periodic kernel. 
Figure 6: Periodic model of a window grid pattern with perspec 
tive effects and local occlusions: Three horizontally aligned win 
dows framed in yellow constitute the kernel. 
4.1 Period hypothesis Generation 
Dominant alignments provide period hypotheses only with their 
coordinates w'hose distribution is partly regulated by main repet 
itive structures. We generate separately horizontal and vertical 
hypotheses. For vertical hypotheses, we accumulate all distances 
between horizontal dominant alignments to build a distance his 
togram. Figure 7 shows such a vertical distance histogram of 
region analyzed on figure 8. Horizontal distance histogram is 
computed in the same way. 
Figure 7: Distance histogram between horizontal dominant align 
ments of region displayed on figure 8. Each mode is related to the 
size of one repetitive pattern: matches proportions are mentioned. 
The best one is circled in red (floor size). 
Each mode of these histograms is related to the fixed size of one 
repetitive pattern. Small distances concern small patterns (such as 
balconies, windows or gabs between two windows if the kernel 
is a floor). We are looking for the size of macro patterns that is 
the sum of those small patterns sizes. These macro-patterns are 
supposed to exactly partition the considered region. 
4.2 Best Period selection 
We have now to select one best period hypothesis. Solution that 
we have found is to correlate a significant number of key point 
pairs that satisfy three conditions: 
1. homogeneous distribution in the region 
2. stable locations 
3. accurate matching measure