124
March 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Consider any three points that have been sampled for measur-
ing DQMs in Swath # 1 (
p
1
,
p
2
,
p
3
in Figure 8). Let these points
be selected from areas that have higher slopes. These three
points have corresponding neighboring points and three pla-
nar patches measured in Swath # 2. Let’s assume that these
three planar patches intersect at point P in Swath # 1(a virtual
point; Point P may not exist physically). Let’s assume that the
conjugate planar patches in Swath # 2 intersect at P´.
Let the equations (together called Equation 1) of the three
planes be:
a
1
X + b
1
Y + c
1
Z = d
1
a
2
X + b
2
Y + c
2
Z = d
2
(1)
a
3
X + b
3
Y + c
3
Z = d
3
Let N
p
=
a
1
b
1
c
1
a
2
a
1
a
1
a
3
b
3
c
3
If (
a
i
,
b
i
,
c
i
) are the direction cosines of the normal vectors of
the planar patches, then
d
i
is the signed perpendicular dis-
tance of the planes from the origin.
If the intersection point P is represented by (
x
,
y
,
z
)
p
=
X
p
then Equation 1 can be written as:
N
p
X
p
= D
1
p
(2)
Similarly, for the point P’ in Swath # 2, we have
N
p
´ X
P´
= D
1p´
If
Δ
X = X
p´
– X
p
,
Δ
N = N
P
´ – N
p
and
Δ
D = D
1
p´
– D
1
p
we get
(
N
p
+
Δ
N
)(
X
p
+
Δ
X
)
= D
1
p
+
Δ
D
(3)
Expanding Equation 3, we get:
N
p
X
P
+ N
p
Δ
X +
Δ
NX
P
+
Δ
N
Δ
X = D
1
p
+
Δ
D
However
NX
p
= D
1
p
(Equation 2), therefore we get
N
p
Δ
X +
Δ
NX
P
+
Δ
N
Δ
X =
Δ
D
(4)
Since we are interested only in Δ
X
, and there are no changes
to the normal vectors or displacement vectors when there is a
pure shift involved, the analysis can be further simplified by
shifting the origin to
X
p
(i.e.
X
p
= (0, 0, 0)).
Therefore Equation 4 becomes
N
p
Δ
X +
Δ
N
Δ
X =
Δ
D
(5)
Since we are measuring the discrepancy in calibrated point
clouds, the expectation is that Δ
N and
Δ
X
are small, and
hence the product can be neglected, Δ
N
Δ
X
, (Typically, Δ
N
≈
.01). Therefore Equation 5 becomes:
N
p
Δ
X =
Δ
D
(6)
Equation 6 is the equation of planes intersecting at Δ
X
, and at
a signed perpendicular distance Δ
D
from the origin. Since the
origin (the point
P
) lies on all three planar patches, and again
emphasizing the fact that we are testing calibrated data, Δ
D
values will be very close to the three measured point to plane
distance errors or DQMs. This assumption models the relative
errors as relative shift. In this scenario, the point to perpendic-
ular plane distance does not change if the point P and p1 lie on
the same plane, and the planes (PL1 in Swath #1 and Swath
#2) are near parallel. Therefore Equation 6 becomes
Np
Δ
X ≈
Δ
D
DQM
(7)
Considering the same analysis for all triplets of points of
patches with higher slopes, we replace Δ
X
with Δ
X
mean
,
N
p
with
N
(which is a
n
× 3 matrix containing normal vectors as-
sociated all
´
n
´
DQM measurements that have higher slopes),
and Δ
D
DQM
as the
n
× 1 vector of all DQM measurements.
This leads to the simple least squares equation for Δ
X
mean
N
Δ
X
mean
=
Δ
D
DQM
= DQM Measurements
(8)
Δ
X
mean
is easily calculated and the standard deviation for
Δ
X
mean
is also easily obtained by using standard error prop-
agation techniques. Δ
X
mean
provides a quantitative estimate
of the mean relative horizontal and vertical displacement of
features in the inter swath regions of the data. This process
can be viewed as a check of the quantification of vertical er-
rors in the data from the previous section
Geometric Quality Report
The Geometric Quality Report should provide a visual as well
as a tabular representation of the errors present in the data
set. The following are suggested to be used as criteria for
quantifying the quality of data:
•
The median of Discrepancy Angle values obtained at
sample points
•
Mean and RMSD of DQM measured in flat regions
•
Mean and RMSD of Horizontal Errors using DQM mea-
sured on sloping surfaces
The median angle of discrepancy is an indicator of residual
roll errors. The angle of discrepancy can be determined even
if there is only minimal overlap available, and hence can be
very useful for the data producers as well as data users as a
measure of quality of calibration or data acquisition. RMSD
measurements on flat regions provide an estimate of vertical
errors in the data the measurements made from sloping sur-
faces can be used to estimate horizontal errors.
PL
2
Point-to Plane DQM
distance
Figure8.pdf 1 2/16/2018 2:54:18PM
Figure 8 Three points the measured DQMs, the corresponding
planar patches (PL1, PL2, PL3) in Swath #1(green) and Swath # 2
(blue) and their virtual point of intersection (P’ and P) are shown.
