The model used for LSQ, adopts the affine transformation for
geometric corrections with two additional parameters for
radiometric corrections and is identical to the model
described extensively in Baltsavias 1991 and Gruen 1996.
Supposing that the geometric transformation is:
X — 841 t à1»Xo + A21Y0
Y = bi +bi2Xo +b21Yo
(6)
where the unknowns are
qug
x = (da1,da;?,da;, ,db;;, dbi», db; ,r,,r]
The main equation for every observation (grey level
difference between right and left interpolated pixel) which
forms the A and | matrix:
0
f(x,y) —e(x,y) = a(x, y) + gyda; +gy X das» +g, Y , daz, ^
0
9yOb;; *gyXodbi5 *gyyodb», er, * g(x, y)r, (7)
where
Q(x, y) a(x, y)
6000 _og(x,y
gre Oy TE
are the partial derivatives along x and y axis respectively.
The matrix equation formed is Ax =1 and the solution being
x= (ATA) IAT (8).
If the corrections on dx and dy are still high, go to step 1
3. APPLICATION OF ELLIPSE MATCHING AND
COMPARISON WITH THE STANDARD SQUARE
TEMPLATE.
The main algorithm of the method is described in detail
above. At this time the matching software including the
algorithm is used as a learning tool for customisation and
optimisation. Hence there is a big number of parameters that
can be adapted or self-adapted. For simplicity and
comparison reasons it should be mentioned that both
algorithms are tested using
° Maximum template size 41x41 pixels. This means that
the automatic template size algorithm will check all
templates between 7x7 and 41x41 to find the best size for
a square template. If no template size is considered good
enough for matching, then the matching in this position is
being done using the maximum allowed (41x41 in this
case). Otherwise the matching will be done using the
template found. If matching with this template fails then
the matching will be attempted again with a bigger
template, actually the next template size will be
maxtempl — currenttempl (9)
2
This continues in case of failure until the maximum
template defined by the user is reached. A template of
41x41 is rather big, but even so in some cases of
nexttempl = currenttempl +
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
homogencous background it is useful. Of course both
methods start with the same template.
e The iterations stop if both dx and dy corrections are
lower than 0.2 pixels or if their number exceeds 12.
° Both methods use 8 unknowns, 6 geometric and 2
radiometric parameters.
e Correlation is being done prior to matching so that the
matching has initial approximations better than 1-2
pixels, which is the convergence radius for the LSQ
method. Since this technique is applied here, the starting
pixel (initial approximation) for both methods is the
same.
In order to test the initial motivation and the theoretical
background for the ellipse method four examples will be
presented, all for points along linear features.
3.1 Casel
This case is described to demonstrate that the square template
may return a “correct” match in an erroneous position, while
elliptical template returns the correct position.
In both cases the best template size was found to be 13x13
pixels, since the algorithm is invariant of the LSQM which
follows. Both methods start from the same initial
approximation (pixel in the right/search image), since this
point is provided using correlation.
Both methods return a “correctly” matched point. As shown
in figure 3, the square template method returned a wrongly
matched point, due to the aforementioned shift, which occurs,
in linear features. This phenomenon is particularly interesting
here, because the square template fails although it has rich
information (the dense shadow) just 3 pixels away. After
failing to match the square template of 13x13 pixels, the
algorithm used a 29x29 template, which returned after 3
iterations a matched point, which is obviously wrong (fig. 4).
The ellipse using 169 pixels (equivalent to 13x13) returned a
correct match after 3 iterations. The fact that the ellipse is
more accurate than the square is verified from the o, for the
gray level differences. In the square method c, is 25.14 while
the ellipse returned a much smaller c, of 12.73, indicating
that the match of the ellipse was much stronger.
It should be mentioned that due to the simplifications made
for the ellipse, in terms of shape and pixels used, the final
number of used pixels for matching was 169 and 167 in the
second and third iterations respectively. A deviation of 2
pixels in 169 pixels, or 1.2% is considered negligible and
certainly unable to affect the final result.
It should also be mentioned that in this case the ellipse
method was faster than the square one, not to mention that if
the algorithm was used with 13x13 template instead of the
self-adaptive, the square would have failed completely.
Figure 4. Comparison between the square and ellipse
template:Casel. From left to right: The
left/template image, the matched point in the right
(search) image using square, the matched point in
Ini
