The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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2.3.2. Expressing Geometry Semantics through the 
Weight Matrix 
In the previous section, the analysis on prior raster information 
provided a description of the geometry of the edge that we are 
trying to match. This geometric information is inserted in our 
mathematical model by formulating accordingly the weight 
matrix. We claim that a combination of Gaussian distributions 
can be applied in order to assign higher weights at the accepted 
maxima. The formula of the Gaussian distribution is expressed 
as: 
cr l + <J 2 
d 
V— 2 In a 
(14) 
In order to compute the standard deviations of the Gaussians, 
another constraint should be introduced. Due to the fact that 
point B is the pre-extracted edge and point A is a computed 
peak, we do not want point A to influence the solution beyond 
point B. mathematically this is ensured if: 
G(x) = -== e ~ 2a2 (12) 
x/2 no 
In this formula p is the mean, s is the standard deviation and x is 
the coordinate of the pixel on the axis perpendicular to the edge. 
The main goal considered in designing the weight matrix is 
minimizing the effects of variations on our solution. In order to 
achieve this goal, we have to allow edge pixels to influence the 
solution of Equation 6 more than the rest of the template. This 
can be performed by manipulating the corresponding weight 
matrix P (of Equations 6 and 7). Indeed, by assigning high 
weight values to certain pixels, we allow them to influence the 
solution more than pixels with weight values approaching 0. 
Accordingly, we enhance the solution of the model presented 
earlier by incorporating local edge analysis in it. This 
transforms our template matching from a common area-based 
matching scheme to a content-based matching process, 
improving its performance potential. 
The formula used to formulate GD of the weights in the case of 
an edge represented by two levels corresponds to equation 12. 
In this equation, the mean expresses the position of the edge as 
defined by the older vector information. The standard deviation 
defines the uncertainty for existence of the edge on the position 
of the mean. The standard deviation depends on the resolution 
of an image, because in higher resolution the edge is expected to 
look sharper and in coarser resolution more blurry. In our case, 
for an average resolution of 1/10000, a threshold of 99% is 
considered. So the standard deviation should be one third of d/2 
in which d is the template size. Therefore, a standard deviation 
value of 4 pixels is assigned for template size of 21x21. 
When the edge is composed of three levels, we define d as the 
distance between two p. The GDs are used with their means 
JLl\ and // 2 > respectively. The mathematical representations for 
these distributions are: 
G{x) = e 
G-/;.) 2 
-2a, 2 
G(x) = e 
(13) 
In order to formulate the GDs and in essence of the weight 
matrix, we need to define the two values for the standard 
deviation 5, and S 2 -So, we need two equations. The first one 
is extracted from the fact that a point L should exist, where the 
two GDs intersected. Let d x and d 2 be the two distances from 
jU ] and jl 2 , respectively and a is the GD value of point L (Fig. 
4). The first equation is: 
S 2 >S j 
(15) 
On the other hand, that can statistically ensured by restricting 
the standard deviation. For a threshold of 99% for possibility of 
the observation to fall inside an interval, interval should be three 
times the standard deviation. So another equation is expressed 
as: 
3 s x =d 
(16) 
We substitute that in the first equation and we calculate S 2 . 
J 3-V-21n a 
cj 2 =d 
3V-21n a 
(17) 
Since (— 2 In a) should be positive, the value of a should 
always satisfy the equation: 
a< 1 (18) 
If the equations (15), (16), (17) and (18) are combined, we find: 
0.32 < ¿7 < 1 (19) 
Constant a expresses the weight of the intersecting point. If that 
weight is low, then the standard deviation of the second GD will 
be small, and weights will be denser near the point B. This will 
be applicable in the case of higher resolution, where the edges 
are distinct. On the other hand, if we choose a high intersecting 
value, then the GDs get wider and increase the contribution of 
surrounding pixel to point B. This is desirable in low resolution 
imagery, where edges are not so distinctive. Experiments 
showed that for an average resolution of 1/10000, a value close 
to 0.6 should be assigned to a. Finally, the weight distribution 
P(x) on the axes perpendicular to the vector edge is given by the 
formula: 
P{ x) = Max{G x (jc), G 2 (jc) } (20)