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for the targets have been obtained. Results using both
techniques are shown in figure 4.
Figure 4. Target center determination using two
matching algorithms
2.5 Camera parameter estimation
First, a full math model was chosen to perform the geometric
calibration and estimate the sensor parameters. In this math
model, Ravi Samtaney (1999) tried to cover the significant
factors that might occur in geometric calibration. This model is
explored more below. The model relates the target coordinates
in object space and image space through the camera
parameters. This overdetermined and nonlinear system needs
an optimization criterion to be solved. Since we have some
initial values for a number of the camera parameters and their
uncertainty, we decided to use the unified least squares
algorithm to solve the system. Using the resulting distortion
parameters, plots were drawn to describe the radial and
decentering distortion behavior. Finally, the radial distortion
curve was equalized by small changes in corresponding
parameters. The results of the process for each sensor are
tabulated in the results section.
3. TARGET LOCATIONS IN IMAGE SPACE
In order to find the target locations in the image, two matching
approaches were used. First, the approximate locations are
obtained using the cross correlation matching. Second, we
refine the results of the first algorithm using LSQM.
3.1 Cross-correlation matching
Cross-correlation determines the similarity between two
corresponding patches. The conventional cross-correlation
approach cannot give the precise location of an object due to
many factors. Differences in attitude, distortion, and signal
noise are some examples that affect the correspondence in
geometry and radiometry (Mikhail, Bethel and McGlone, 2001;
Atkinson 1996). However, this algorithm usually gives an
approximate location of the correspondence within a few
pixels.
The ideal template will be passed through the image and the
matching function will be computed and recorded at the center
pixel of the patch. The match function, the normalized cross
correlation coefficient, ranges between +1 and -1. The
maximum value equals +1, which means they are identical.
Usually a threshold will be used to distinguish between
matches and non-matches. Some individual correlation results,
as shown in figure 3, are off from the center of the target only
by a pixel or less. These results enable us to use the least
squares refining technique directly.
3.2 Least squares matching (LSQM)
LSQM works as a powerful and an accurate technique to refine
an object’s coordinates in the image space based on the
correspondence between a image chip and a reference or
template chip (Mikhail, Moffitt and Francis, 1980; Atkinson,
1996). This technique utilizes the gradient in the x and y
directions in order to move the two patches with respect to
each other to get the best match. The match precision that we
are looking for with this technique is within a hundredth of a
pixel. The similarity between the two targets was only
geometrically modeled for this specific problem since the
radiometric differences were eliminated through some
preprocessing steps, as we will see below. The problem is to
match an ideal shape of the target with a small window from
the image containing the imaged target. The following steps
describe the automated procedure that was used for setting up
the two windows for matching:
1. Obtain the approximate location of the imaged target
using the first matching approach (cross correlation).
Those locations should be within a few pixels of the
exact location in order to make the geometric model
in LSQM converge and to produce accurate results.
2. Having the rough estimated location, a window
around that location from the image with adequate
size will be extracted for matching purposes. This is
all done systematically inside the code.
3. The ideal or template target is retrieved at this point.
Similarity in the intensity is enforced between the
two windows.
After specifying the two windows with the same size for
matching, the LSQM procedure takes place. Requiring
similarity in intensity between each of the two corresponding
pixels from the two windows is the basic condition for this
procedure. Since the two patches do not have the same
coordinate system, a 6-parameter geometric transformation is
used to relate them in the matching procedure (Atkinson,
1996). Those parameters will be corrected iteratively and will
be used to calculate the new coordinates x', y' in order to use
them in resampling the grid for the template window. We used
bilinear interpolation to resample the intensity values. The
whole procedure will be repeated as needed but using the new
template window with the new intensity values every time and
the parameters will be updated. The match will be achieved
when the system converges and those 6-parameters do not
change any further. The system might diverge if there is no
similarity between the two patches or the approximate location
of the match far off from the real one by more than several
pixels (Mikhail, Moffitt and Francis, 1980; Atkinson, 1996).
4. MATHEMATICAL MODEL FOR SELF-
CALIBRATION AND SOLUTION METHOD
4.1 Mathematical model
The mathematical model was chosen carefully in order to cover
all significant sources of geometric errors and estimate all
significant correction parameters for those errors. (Samtaney,
1999) explored this model in detail. It was derived from the
fundamental collinearity equations. This model relates two
