International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Imaging Inc.) the quality of the normalization process is
unknown.
It is, however, simple to perform a check of the epipolarity of
the stereo images. for the purposes of this study, groups of
conjugate points located at the extremities of the image were
matched using a geometrically unconstrained matching
algorithm. The matched points were then filtered to remove all
points with a normalized cross correlation coefficient of less
than 0.9. By comparing the y values (line numbers) of the
remaining well-matched points, it was possible to assess the
quality of the algorithm that has created the epipolar images.
The results of the matching of the four groups of points are
shown in table 1.
Group of Number of RMS y Mean y
points well-matched residual residual
points (pixels) (pixels)
Tartes 149 0.54 -0.40
image
Top right of 69 0.57 0.44
image : :
Bottom fof 109 0.65 -0.51
of image
Bottom right
E ? R En 4
ven 82 0.73 0.44
Table 1. Results of matching at extremities of the image pair.
The results presented in table 1 clearly show that the alignment
of the Melbourne stereopair to epipolar geometry is not exact:
there is apparently a systematic shift in the y direction of
approximately half a pixel between the two images. Thus,
throughout the rest of this study a more loosely defined epipolar
constraint has been used where the search space is limited not
to one single line, but to a two dimensional area extending to
one line on either side of the epipolar line. Since the matching
process is more loosely constrained, the processing time
necessarily increases.
4.3 Alignment of non-epipolar images to epipolar geometry
Since not all available Ikonos data is aligned to epipolar
coordinates (as is the case with the San Diego data used in this
study) the re-alignment of non-epipolar images to epipolar
geometry was investigated. There are various methods available
for achieving this, with some being more rigorous than others.
Three possible methods include using the satellite ephemeris
data (or strictly speaking, the image metadata), the fundamental
matrix, or a second order polynomial model.
Although the data supplier does not release detailed satellite
orientation data, a small amount of limited image metadata is
supplied with images. This data includes details such as the
time of acquisition, the sun azimuth and elevation, the
approximate scene location, but most importantly (to this study
at least) is the satellite azimuth and elevation. Using these
approximate values is it possible to estimate the angles of
rotation required to align the stereopair to epipolar coordinates.
However, the estimate is coarse and of unknown accuracy.
Additionally, it does not take into account that the images are
time-dependent linescanner images, and not frame photographs.
An alternative, and much more rigorous approach, is to
implicitly determine the relative geometry of the images and
hence derive the fundamental matrix, thus allowing a direct
mapping of points between the images (Luong and Faugeras,
1996). Although this technique of determination of epipolar
geometry is more commonly associated with computer vision
than photogrammetry, there is no reason why it cannot be
applied to Ikonos images. Unfortunately problems are not
uncommon during the process of deriving the fundamental
matrix, since instabilities in the solution can occur. Once again,
this method does not account for the linescanner nature of the
data.
A third method of approximating the epipolar geometry
between images, described by Zhang et al. (2002), is based ona
second order polynomial model proposed by Orun and
Natarajan (1994). The model assumes that during image
acquisition the pitch and roll angles remain constant, but the
yaw angle variation follows a second order polynomial. The
result is an epipolar curve, rather than an epipolar line, defined
by the quadratic polynomial:
y, md t0 t) tr (d. x T y, Td: s. @)
+X, tary, + IT
where (x, vj) and (x,, v,) are conjugate points in the left and
right images respectively, and ag to ag are parameters to be
determined. As can be seen, each unique point in the left image
can be associated with a unique epipolar curve in the right
image: knowledge of the coordinates of a point in the left image
allows an epipolar curve to be plotted, upon which the
conjugate point will lie. The parameters ao to ag have to be
determined in advance, which is done by substituting the
coordinates of known conjugate points into equation (2) and
solving as required.
In order to validate the quality of the second order polynomial
epipolar model, tests were carried out with the Melbourne
stereo data, which are known to be already aligned to epipolar
geometry. Initially 648 points were matched between the two
images using a non-constrained hierarchical matching strategy.
Of the 648 matched points, 108 were found to have cross
correlation coefficients of greater than 0.95, implying well
matched points. These 108 well matched points were then split
into two groups, one of which (54 points) was used to calculate
the parameters (ao to ag) of the second order polynomial model.
Although only nine points are required to obtain the solution
directly, 54 points provided a good deal of redundancy, and
allowed the parameters to be estimated by a least squares
method. The remaining 54 well matched points were used as
check points to verify the accuracy of the second order
polynomial model. By substituting the coordinates of the
matched point from the left image (x; v;) into the model, along
with the abscissa of the matched point from the right image (x,)
a new value of y, was calculated and compared to the expected
value (from the coordinates of the matched point in the right
image). This process was repeated for all 54 check points.
Consequently it was found that the root mean square
discrepancy was marginally less than one pixel, implying that
the second order polynomial epipolar model is sufficiently
accurate to be used as a geometric constraint for matching non
epipolar stereo images.
Confident that the Melbourne images could be successfully
aligned to epipolar coordinates, the San Diego images wert
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