Gamal Seedahmed 
  
The HT takes each of the points (xi, y;) to a sinusoidal curve in the (p,0) plane. The property that the 
Hough algorithm relies on is that curves that have common intersection points in the (p,0) plane belong 
to the same line in the (x,y) plane and they form a peak. This peak should be detected with its 
associated parameters (p,0). These parameters are used in a de-Houghing step in order to trace a line 
associated with these parameters. The polar representation is preferred over the slope-intercept to 
avoid the difficulties in vertical line detection. 
The HT can be implemented easily for any analytic curve by interchanging the role of the observations 
and parameters. For a circle with a known radius, the observations are the coordinates along the 
circumference and the parameters are the circle centroid. 
4 PRECISE LOCALIZATION 
The previous section addresses the problem of identification and approximate localization of analytical 
curves. With the HT technique we know where it is, but we have not made any effort yet to determine 
its position as accurately as possible. The success with identifying the major structures of the fiducial 
mark suggests taking a different approach to determine its center. That is, we compute the center from 
the structural elements rather than directly from the center pixels. In our implementation we used two 
concentric circles to indicate the center. The center will be determined by Least Squares adjustment 
(Schaffrin, 1997). The mathematical model for the two concentric circles is: 
(; 7x)! * (y - X)! - A se; 
(2) 
(x, x) +(y; y} -R =e, 
where (x;, ÿ; are the pixels on the outer circle with known radius Ry, (xj, yj) are the pixels on the inner 
circle whose radius is R,, and e;, e; are random errors. 
The proper handling of the stochastic properties of the model will be via condition equations with 
parameters. The model of condition equations with parameters states: 
2 p-1 
Brin) 20er) — €o(merya ) = 4 rene url er (0, Oo P ) (3) 
with rank (A) <min (m+r, m)=m. 
e  B: this matrix contains the partial derivatives with respect to the observations. 
e y:the observed pixel coordinates. 
e e: the true error vector. 
e A: this matrix contains the partial derivatives with respect to unknown parameters. 
e ¢&: the vector of the true unknown parameters (fiducial center). 
e P: weight matrix of the observations. 
For the above-mentioned stochastic model we need to introduce an estimation based on geometric 
or stochastic approach, such as least-squares adjustment to find an estimate for the unknown 
parameters. 
5 ALGORITHM 
Before the identification of the fiducial marks (FM), image patches of reasonable size that contains the 
FMs are extracted using a priori knowledge about their expected locations in the image, see Fig. (3). 
The process starts by running an edge operator over the image patches, e.g., Canny's edge algorithm 
with its associated parameters. For our experiments we use aerial photographs acquired by Wild RC10. 
Since the circular elements of the FMs are unique compared to their linear elements, the Hough space 
for the two circles using their known radii is generated first. After generating the Hough space for the 
two circles, a peak detection process is performed for the identification and an approximate 
localization, and then this step is followed by a de-Houghing to trace the pixels which belong to each 
circle in the edge image. Simply substituting the parameters of each circle in their corresponding 
  
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.