of 
-Compute all à such that f(x,a) = 0 and Ix xa) =0 
-Increment the corresponding accumulator cell: A(a) = A(a) + 1 
step3: After each edge pixel x has been considered, local 
maxima in the array A correspond to curves of f in the image. 
The advantage of this method is clear if we take into account that 
the gradient direction for each edge pixel is already known from 
the proceeding edge detection procedure. 
For example, the analytical form of the circle is given by: 
f(x,y) = (x-a)2+(y-b)2 = 12 (2) 
where (a,b) represents the center of the circle and r the radius. 
When we follow the edge pixels that lie on the circle the 
difference in gray level among the edge pixels is zero. That 
means that the magnitude of gradient is zero: 
ar 
ox 
en) eh at = 
x? (x-a) +2 (y-b) 5 =0 = (x-a) + (y-b} =0 
where 3 tan(®(x) RT 
and ®(x) is the gradient direction as indicated from the edge 
operator. 
Then: 
(x-a) + (y-b)* tan(D(x) - £ =0 => 
(x-a)*cos(P(x) - 2 + (y-b)* sin(P(x) - E) =0 (3) 
Using the equations (2) and (3), it is obvious that the parameter 
locus is reduced from a cone to a line. It can be concluded that 
when using the gradient direction information, the number of 
free parameters is reduced by one. 
The same algorithm can be used for all the analytical curves. For 
example an ellipse has 5 parameters. Using the gradient 
direction two of the parameters can be solved as a function of 
the other three. The computational effort in this case is O(e*d?) ; 
where e is the number of edge points and d the distinct values 
for each parameter. 
Concluding the reference to the Hough Transform it has to be 
highlighted that this powerful tool not only detects a predefined 
shape, but it also gives its size, orientation, and center. 
4. TARGET DETECTION USING HOUGH TRANSFORM 
An important task in Digital Photogrammetry is the performance 
of highly accurate measurements of target positions on the 
digital image in order to perform a transformation between 
image coordinate and object coordinate systems. These 
measurements can be done either by operator or automatically. 
In the latter case there is no need for an operator but the target 
must be first detected and its center must be precisely located 
using only the computer. Working in this direction [Trinder and 
Mikhail, 1982] introduced an algorithm for edge modeling based 
on least squares (edges,lines and cross targets), [Zhou, 1986] 
utilized an algorithm for locating ellipse centers based on the 
moment preserving method, [Trinder, 1988] proposed circles as 
targets utilizing again least squares, and [Mikhail, Akey and 
Mitchell, 1984] used Fourier descriptors and one dimensional 
moments for target location and recognition. 
In the present paper the Hough Transform is utilized to detect 
and accurately measure the following target: 
A white circle in a black background with four equally spaced 
diameters intersecting on its center (see figure 3). 
Fig. 3: Test target 
À circle is the most appropriate shape for a target because the 
image of a circle can only be a circle or an ellipse depending on 
the observation direction (see figures 4 and 5). 
  
Fig. 4: Digital image of the 
test target taken at an angle. 
Fig. 5: Gradient (Edge) digital 
image of the test target of Fig.4 
Hough Transform, as mentioned, cannot only detect circle and 
ellipse in an image but it can also directly give their centers 
which in the case of target detection is exactly what we are 
interested in. 
An alternative solution for the target center determination is first 
to detect the circle-ellipse image of the target using the Hough 
Transform and then to mask the rest of the image outside of the 
target, making in this way the detection of the four lines inside 
the circle easier using again the Hough Transform. The point of 
intersection of these lines is the center of the target. [Harjoko, 
1990] proposed the following algorithm for computing the 
intersection of lines that are detected using the Hough 
Transform: 
N lines in the image space form a cluster of lines with a common 
point of intersection ( xy). These lines correspond to N 
associated clusters in parameter space. It has already been 
mentioned that the measurements of the peakness of these 
clusters result in a set of N pairs of standard deviations and 
means. These quantities are further used to compute the center 
of the target ( x.y). The location of the target center is 
computed using least squares adjustment. For observations with 
different standard deviations each observation must be weighted 
  
  
  
  
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