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.