In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C., Tournaire O. (Eds). IAPRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
the membership of a wall to either of the two classes "wall 
exists" or "wall demolished". The fuzzy approach has been used 
in many works (Carlsson and Fullér, 1996; Sasikala and Petrou, 
2001; Kumar et al., 2009). We only describe how it is used for 
this work. 
Using training data, the fuzzy membership to the class “wall 
exists” at a given value of a measure is computed as the 
proportion of the number of existing walls to the total number 
of existing and demolished walls. The size of walls is also used 
as a weight. For example, if for all walls used in experiments, a 
total of 30 walls result to Lines Match Ratio of 0.9 with 25 of 
them from existing walls and 5 from demolished ones, then the 
fuzzy membership for “building exists” at Lines Match Ratio 
0.9 is 0.83 (25/30) and membership to “building demolished” 
is 0.17(5/30). 
Fuzzy membership curves for all measures have been derived 
from experiments as shown in Figure 9 and then curves have 
been generalized to fuzzy membership functions. 
4. COMBINING MEASURES FOR A WALL AND FOR 
A BUILDING 
4.1 Combining memberships for a wall 
The evidence theory was introduced by Dempster and Shafer 
(Dempster, 1967; Shafer, 1976) and has since then been studied 
and applied in many research works (Tahani and Keller, 1990; 
Rottensteiner et al., 2005; Kumar et al., 2009; Frigui et al.. 
2010). The general intention is to give a higher weight to more 
reliable measures and vice versa for the less reliable ones in 
order to reduce the conflict that arises between measures. The 
problem is usually on how to quantify the reliability when there 
is conflicting evidence from different sources. 
reliability of 0.4, which is normalized to 0.5 and then combined 
membership is 0.5. This is as expected because the 
memberships are very contradicting and we have no other 
reasons to select any one of them as correct or wrong. The 
result 0.5 implies complete ambiguity on whether the wall 
exists or is demolished. 
4.2 Combining wall memberships for a building 
The overall building verification is derived from memberships 
of different walls. For each wall a combined membership is 
obtained and then a summary measure is obtained as the mean 
of the wall memberships weighted by the wall sizes. 
5. RESULTS 
5.1 Fuzzy memberships for walls 
A fuzzy membership function for each of the measures has been 
derived from experiments with a number of walls (total 500 
walls) for existing and demolished (simulated) buildings (total 
17 buildings). Figure 9 show's different membership curves for 
different measures. For each measure the x-axis is the measure 
and y-axis is the membership to class “wall exists”. For 
example, the Lines Match Ratio curve show's for all values from 
0.2 to 1 the corresponding membership is 1. 
The measures have strengths and weaknesses that are used in 
obtaining the overall wall membership. Their characteristics can 
be generalized as follows. 
Lines Match Ratio is generally reliable because it clearly 
distinguishes existing and demolished walls. When walls exists 
the match ratios are high and when the w’alls are demolished the 
match is low. 
In our case we combine fuzzy memberships using weights. The 
weights, also here referred to as reliability is derived from 
memberships from different measures. A measure that returns a 
membership closer to 1 (exists) or closer to 0 (demolished) is 
assumed more reliable while close to 0.5 (equal membership to 
‘exists’ and ‘demolished’), assumed less reliable. To a decision 
maker, no conclusion can be made when the membership is 0.5 
for both classes “wall exists” and “wall demolished”. Reliability 
Wi is thus a deviation of a membership TTli from 0.5 and is 
computed with bias to memberships close to 0 or 1 by raising 
the deviation to a power /7 to obtainm; 
( 
I m-i — 0.5 
075 
For example, if for 4 walls we obtain membership from Lines 
Match Ratios as 1.0. 0.5, 0.0 and 0.1, using the power of 4, the 
reliability for each of these will be 1.0, 0.0, 1.0 and 0.4. The 
measures that classify more walls into one class (“wall exists” 
or “wall demolished”) are thus assigned more reliability. 
For each wall we obtain the reliability of each measure and then 
the combination is an aggregation of the memberships obtained 
from all measures weighted by the reliability (Yager, 1988; 
Sasikala and Petrou, 2001). 
For the number of measures (oi, a 2 ,a n ) with reliability 
(u>i, w-2, ...w n ) such that u),e[0,1] and normalized so that 
JZ" Wi — 1, the combined membership is 
F(ai,a 2 a») = a o w .v 
For example, if for a wall only two measures are available and 
result to memberships 0.9 and 0.1, then they are of equal 
With SIFT Match Ratio, most existing walls result to the ratio 
above 0. When it is around 0 the membership is close to 0.5, 
that is half of the walls exist and another half do not. It is 
reliable for match ratios above 0. 
Lines Direction Ratio has a big range of walls at around 0.5 
memberships, not clear whether the walls exist or not, meaning 
less reliability but when it is at 0 nearly all walls are 
demolished. This is useful for identifying demolished walls 
even when the Lines Match Ratio is not available. 
For Building Edges Ratio most existing and demolished walls 
result to 0.0 because many wall edges are not detected in the 
images. The Building Edges Ratio is then not reliable. This 
measure is therefore not useful for identifying demolished 
buildings. The measure is useful for identifying existing walls 
even when one image is available for an edge. The Correlation 
Coefficient Ratio is similar to the Lines Match Ratio while 
Correlation Coefficient -corners is similar to SIFT Match Ratio. 
The reliability of the different measures in different contexts 
has been used in arriving at combined memberships. Notice that 
when multiple images for one facade are not available the only 
measures available are Building Edges Ratio and Lines 
Direction Ratio. If a wall is demolished then the only useful 
measure is Lines Direction Ratio because Building Edges Ratio 
is reliable for existing walls. The Lines Direction Ratio is also 
not generally reliable as there may be lines in images that result 
to 3D vertical and horizontal lines when a wall does not exist. 
The Lines Direction Ratio is more useful when there are no