mathematically-generated-triangle has three vertexes, gener-
ated-lengths, and also interior angles (Figure 1). Using
MGL
s
helps to solve the different end-points problem of line-seg-
ments mentioned earlier. In this case, it is possible to use the
geometrical-relations of
MGL
s (e.g., relative angles, the ratio of
generated-lengths) in matching. In addition, it is possible to
use the key-points of
MGL
s as the
GCP
s to solve the unknown
parameters of transformation between two spaces. These
points can be the vertexes or the mid-points of
MGL
s.
Using the intersections of each pair of
MGL
s, the coordinates
of the vertexes can be obtained. In this paper, they are called
mathematically generated points (
MGP
s) shown in Figure 1.
As the input information, there are two subsets of line-
segments,
P
= {
p
i
:
i
= 1,…,
n
} and
Q
= {
q
i
:
j
= 1,…,
n
} in
image and object spaces, respectively. The coordinates of the
end-points of line-segments
p
i
and
q
i
are {(
x
1
i
,
y
1
i
),(
x
2
i
,
y
2
i
)}, and
{(
X
1
j
,
Y
1
j
,
Z
1
j
),(
X
2
j
,
Y
2
j
,
Z
2
j
)}, respectively. The matching algorithm
is aimed to find corresponding couples of (
p
i
,
q
i
).
The last concept of this section is devoted to the geometric
relation between a triangle (formed by the extension of three
line-segments), and another extracted line-segments. These
extracted lines and triangle make three inner relative angles
(Figure 2). The geometric relation is only considered in cases
that all the relative angles are larger than a threshold (e.g., six
degrees). This threshold prevents the case where the line is
parallel with triangle sides. In this paper, these line-segments
are called as “crossing-lines” (Figure 2). The correspondence of
these crossing-lines are determined through the matching-phase.
Figure 2. The figure shows the selected conjugate patterns
(black dashed line) in image and object spaces. A pair of match
crossing-lines are shown as the continuous grey lines and their
conjugate crossing; MGLs as the grey dash lines. The angles of (
α,
β, γ
) and (
Α
, Β, Γ
) as well as the lengths of (
m
,
n
) and (
M
,
N
) are
the inner relative angles and generated-lengths of that crossing-
MGLs in both spaces. In addition, grey crosses are considered as
conjugate MGPs related to these crossing-MGLs.
Phase 1: High Quality Pattern Selection
In this phase, it is intended to select qualified triangles from
all the possible three-combinations of the line-segments in
the image space. Regarding to the large number of all pos-
sible triangles, three weighting criteria are designed to select
only the appropriate triangles and exclude the rest in favor of
computational time efficiency.
The triangles in image space are weighted based on the
probability to have correspondence in object space, their dis-
tributions, and the number of their crossing-lines.
Before weighting process, some initial elimination-proce-
dures are performed. As the first elimination-procedure, all
the small triangles are removed from the rest of the process.
To do so, only the line-segments longer than mean length (or
some more simplistic threshold) are used in the triangle pro-
duction. These line-segments are called “fine-quality-lines.”
In the second elimination-procedure, the triangles in
which the
MGP
s are outside the image space are also removed.
The third elimination-procedure focusses on the number
of crossing-lines where the triangles having a low number of
crossing-lines are also removed from the remainder to process.
In this research, the threshold of two crossing-lines is used.
After these optional eliminations, the remained triangles
are weighted based on the following three criteria:
•
First Criterion - the Distribution of the Patterns:
Distri-
bution of the control information (e.g., control-lines or
control-points) influences the geometrical strength of
the mathematical models. In this paper, two factors are
considered for the distribution weighting. The first factor
positively weights the equilateral triangles because these
triangles produce more well-distributed
MGP
s to solve the
coefficients of the models. This weighting is performed
based on the inner angles via the empirical Equation 1:
, ,
, ,
, ,
if
Quality Element
j
j
j
j
58
64
1
1
° ≤
(
)
≤ ° → −
=
ω ϕ κ
else if
Quality Element
else i
j
j
j
j
, ,
/
55
70
4 5
1
° ≤
(
)
≤ ° → −
=
ω ϕ κ
f
Quality Element
else if
j
j
j
j
, ,
/
50
80
3 5
40
1
° ≤
(
)
≤ ° → −
=
° ≤
ω ϕ
κ
ω ϕ
κ
ω ϕ κ
j
j
j
j
j
j
Quality Element
else if
/
(
)
≤ °→ −
=
° ≤
100
2 5
30
1
j
j
Quality Element
(
)
≤ °→ −
=
120
1 5
1
/
(1)
Here, (
ω
j
,
φ
j
,
κ
j
) are the three inner angles of the
j
th
triangle
(see Figure 1) and
Quality
–
Element
j
1
is its first weight. Based
on this experimental equation, well-distributed triangles gain
higher weights and more chance to be selected in
HQPS
-phase.
As the second factor, the above weighted patterns are
classified based on all their three generated-lengths. To do
so, the ranges between maximum and minimum generated-
lengths (sides) of all generated-triangles are divided into some
intervals (e.g., 20 intervals). Each
pattern is weighted based on the
position of its all three sides in
these intervals. Then a weight
(e.g.
Quality
–
Element
j
2
) is as-
signed to each triangle between 1
(for maximum generated-lengths)
to 1/20 (for minimum generated-
lengths) based on Figure 3.
Similar to the first factor,
Quality
–
Element
j
2
is also ap-
plied to peruse the distribution of
each pattern. On the other hand,
patterns with longer generated-
lengths are distributed better in
image space. So, higher weights
are dedicated to them.
Figure 3. The pseudo code of distribution checking based on generated-lengths.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
May 2016
367
