(
Δ
z
,
Δ
·
z
), and the third one having only
Δ
r
. These
correla-
tion groups
are determined based on grouping parameters
significantly correlated among each other and having similar
temporal decorrelation properties. Note that correlations only
exist within each correlation group, and that this defines
which of the associated covariance terms can be nonzero (e.g.
in
Σ
CU
1
,
σ
Δ
x
1
Δ
y
1
can be nonzero, but
σ
Δ
x
1
Δ
z
1
cannot). The adjust-
able parameters for the posts (
θ
1
,
θ
2
) are in their own correla-
tion group, and are only correlated within a
CU
.
The next step in calculating the output ground covariance
is to generate the partial derivatives of the model coordinates
(computed using the
groundToModel
() method, which is the
inverse of the
modelToGround
() method) with respect to the
adjustable parameters (component parameters in Equation
10). An example of the resulting Jacobian matrix,
A
, is shown
in Equation 15.
(14)
σ
σ
x
x
σ
σ
i
i
Σ
σ
σ
σ
σ
σ
σ
σ
CU
x
x y
x x
x y
y
y x
y
i
i
i
i
i
i
i
i
i
i
i
i
=
Δ
Δ Δ Δ Δ Δ Δ
Δ
Δ Δ Δ
2
2
0
0
0
Δ
Δ
Δ Δ
Δ
Δ
Δ Δ
y
y
y
z
z z
i
i
i
i
i
i
0
0
0
0
0
0
0
0
0
0
2
2
2
σ
sym
z
r
i
i
.
σ
σ
Δ
Δ
2
2
0
1 2
Δ
Δ
Σ
σ
σ
σ
θ
θ θ
P
ij
ij
ij
ij
Sym
=
Δ
Δ
θ
2
ij
1
2
2
.
( (
(
)))
A
n n
CU P
P CU P
P
n
3 14 2
1 2
1 11
1
2
21
1
× + +
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
X X
X X X
X
2
2
n
where
(
)
∂
∂
×
=
∂
∂
∂
∂
∂
∂
∂
∂
Δ Δ
∂
∂
∂
∂
∂
∂
Δ
X
CU
x
x
x
y
x
x
x
y
x
z
x
z
x
r
i
i
i
i
i
i
i
3 7
Δ Δ Δ
Δ
i
i
i
i
i
i
i
i
i
y
x
y
y
y
x
y
y
y
z
y
z
y
r
z
x
z
y
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
i
i
i
i
i
i
z
x
z
y
z
z
z
z
z
r
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
∂
∂
Δ
and
(
)
∂
∂
×
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
X
P
x x
y y
z
z
ij
ij
ij
ij
ij
ij
ij
3 2
1
2
1
2
1
2
θ
θ
θ
θ
θ
θ
(15)
The matrix of partial derivatives (
B
) of the model-space co-
ordinates (
x, y, z
) with respect to the ground-space (e.g.,
ECEF
)
coordinates is shown in Equation 16.
(
)
B
x
X
x
Y
x
Z
y
X
y
Y
y
Z
z
X
z
Y
z
Z
3 3
×
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
(16)
The ground-space covariance matrix (
Σ
G
) is then computed as
shown in Equation 17. In the equation,
Σ
mod
is the mensuration
error covariance matrix provided as input into modelToGround(),
and
Σ
ue
is the unmodeled error covariance matrix.
=
+
B A
(
)
Σ
σ σ σ
σ σ
σ
Σ
Σ Σ
G
a
T
mod ue
≡
+
−
X XY XZ
Y YZ
Z
sym
A
2
2
2
1
.
B
T
−
(
)
1
(17)
Ground-Space ULEM
Ground-Space
ULEM
can be used in situations when the
required metadata for Sensor-Space
ULEM
is not available,
or its management becomes impractical. For example, the
Ground-Space
ULEM
implementation can be applied in cases
where the complexity of the point cloud collection and
processing concept of operations precludes a straightforward
definition of a sensor-space projective model within the
final
point cloud (e.g., points from multiple overlapping passes are
voxelized, and the points within each voxel are replaced with
the mean of those points). In this specific case, Sensor-Space
ULEM
would often be implemented for the
initial
, contribut-
ing point clouds (e.g., per pass), with the final point cloud
product using pre-computed error covariance information as
defined by the Ground-Space
ULEM
implementation. Also, in
Ground-Space
ULEM
, there can be only one
CU
per dataset.
Parameterization
Ground-Space
ULEM
employs either one, or a combination of
two, adjustment models: the 3
DC
model and the Anchor Point
model. The 3
DC
model is similar to the sensor-space implemen-
tation, in that it can accommodate
CU
-wide transformation pa-
rameters. These adjustable parameters appear as a combination
of the seven parameters associated with a three-dimensional
conformal (3
DC
) coordinate transformation. They can include:
• ω
,
ϕ
, and
κ
(three sequential rotations about the
x, y,
and
z
axes, respectively), used to form rotation matrix,
M
• Δ
x,
Δ
y,
and
Δ
z
: three translations
• Δ
s
: a single scale factor correction.
These 3
DC
parameters are applied as shown in Equation
18 and illustrated in Figure 5. In the figure, arbitrary distance
d
is included to illustrate the effect of the scale factor correc-
tion (which is negative in the figure). Note that, as with the
coefficients in the Sensor-Space
ULEM
of
CU
-wide polynomi-
als, not all of the seven transformation parameters need to be
included as adjustable parameters. Those parameters that are
not included as adjustable parameters are considered zero
constants in the transformation model.
x
y
z
s M
x
y
z
x
y
z
= +
(
)
+
0
0
1
'
where
M
=
M
κ
M
ϕ
M
ω
(18)
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
July 2015
551
