The International A) chives oj the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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In order to take advantages of the raster and vector 
representation that preexists and enhances the algorithm's 
performance and liability, first a geometric representation of the 
edge is extracted and then included in the LSM solution. 
2.3.1. Extraction of geometric representation of the edge 
We categorize edges as having two, three or four levels, 
depending on their type, based on radiometric representation 
(Phalke, 2005) (Fig. 2). 
Figure 2: two levels (left), three levels (middle), four levels 
(right) 
This corresponds to one, two or three edges on the image. We 
resample our window so that the edge is perpendicular to the x- 
axis of the window under examination. Then, we calculate the 
average of the first derivative of each column, at the X 
direction: 
g*(*O0 = 
dg(x,y) 
ÔX 
Y*gx( X >y) 
£*(*)=- 
У 
(8) 
(9) 
Clearly, for our set-up, the derivatives along the у-axis will be 
minimal. An analysis of the g x (x) graph in Fig. 3, by using 
the first and the second derivatives, reveals 3 (or less) maxima. 
Three criteria are introduced in the decision process to accept or 
reject a maximum that in essence corresponds to an edge. If we 
define шах 0 as the maximum of the three maxima found, then 
we compare this value with the existing maxima. 
. If max, > T x x max 0 then that edge is accepted because 
the high value of relative sharpness reveals a strong edge. This 
constant was set empirically to the value of 70%. 
.If max, <T 2 x max о then that edge is rejected because 
the low value of relative sharpness reveals a weak edge. This 
constant was set empirically to the value of 30%. 
. If T 2 x max 0 < max, < Г, x rnax 0 then we compare the 
second derivative, before and after the candidate maximum, by 
using the following criterion: 
Figure 3: averages of the first derivative in the x direction (top), 
edge template (bottom) 
If the above condition is satisfied, then the point is rejected, 
otherwise, it is included in the analysis. T 2 was set empirically 
to the value of 50%. 
The above mentioned percentage values may be viewed as 
general parameters of our approach. The values included here 
are results of empirical analysis. 
With the three above criteria, a "relative" check based on 
maximum value, compares the dominant edge with other 
gradient maxima to eliminate false responses. Before we 
finalize the levels, we perform a last check by introducing the 
first derivative in the Y direction. We claim that an edge should 
have high gray value variations perpendicular to itself, but at the 
same time low variations along itself. So in this step, we check 
for consistency along the Y-direction, but only where the edges 
(accepted maximums) that came from the previous step are not 
strong enough. The difference with the criteria used before is 
that this time absolute gray values are utilized, acting as 
thresholds, and not relative ones. During this process two 
thresholds are introduced: 
• The Agx, showing the difference in the gray values that 
might not imply an edge, 
• The Agy, expressing the difference in the gray values that 
might not imply homogeneity 
First we compare the accepted maximums of the previous step, 
with our Agx. If the condition 
max,. < Agx 
(П) 
(jc) + 1'V ^ (x + 1)|< Г 3 max 0 
is satisfied, there is the suspicion that local noise might exist or 
(IQ) the radiometric representation of the edge is not strong. To 
distinguish these two cases, we compute the first derivative at 
the Y direction, at the position where the candidate maximum is, 
\§X vv) and if it is smaller than the Agy, then that point is accepted. 
Trial and error Experiments showed that a value of 40 gray 
values should be assigned to Agx and a value of 7 gray values to 
Agy-