In the local coordinate system,
Z
′
i
=
q
+ tan(
α
)
X
′
i
+ tan(
β
)
Y
′
i
(10)
= [1
X
′
i
Y
′
i
]
q
tan
tan
( )
( )
α
β
(11)
holds for the Z-coordinate of a point
′
i
, where
α
and
β
are the
angles between the plane’s normal and the Z-axis, and
q
is
the centroid’s position across the plane. With the Jacobian
J
i
=
[1,
X
′
i
,
Y
′
i
] for each observational equation the normal equation
matrix becomes
N w J J
i i
i
I
=
=
∑
1
(12)
or more explicit:
N w
X Y
X X X Y
Y X Y Y
i
i
i
i
i i
i
i i
i
i
I
=
=
∑
1
2
2
1
′
′
′ ′ ′ ′
′ ′ ′ ′
(13)
Please note that the matrix
N
is diagonal, i.e.,
N
= Diag([
N
11
,
N
22
,
N
33
])
(14)
=
′
′
(
)
=
∑
w
Diag
X Y
i
I
i
i
1
2 2
1,
,
(15)
implying that
∑
i
X
′
i
=
∑
i
Y
′
i
= 0 and
∑
i
X
′
i
Y
′
i
= 0 holds since
the origin of the coordinates is the centroid and the axes are
aligned with the principal components of the point cloud.
The theoretical covariance matrix is
Σ
=
N
–1
and therefore
diagonal, too. The variances of the estimated parameters
q
,
α
,
and
β
are:
σ
ˆ
q
N
wI
2
11
1
1
= =
−
(16)
w X
( )
σ σ
α
α
ˆ
ˆ
′
2 2
22
1
2
1
=
= =
−
tan
N
i
i
Σ
(17)
w Y
σ σ
β
β
ˆ
ˆ( )
′
2 2
33
1
2
1
=
= =
−
tan
N
i i
Σ
(18)
for small angles
α
and
β
.
For the reasoning, we need the uncertainty of the plane in
the global coordinate system as obtained by the transforma-
tion given in the next paragraph.
From Centroid to Homogeneous Representation
Given a plane
A
in the centroid representation (Equation 1),
the homogeneous representation reads:
A
=
=
−
A N
h
A D
0
(19)
with the normal
N
being the third column of the rotation ma-
trix
R
and the origin’s distance
D
=
N
X
0
to the plane
A
. We
refer to
A
h
and
A
0
as the homogeneous and the Euclidean part
of the homogeneous coordinates
A
of the plane. The covari-
ance matrix for the plane
A
′
= [0,0,1,0]
is then:
Σ
A
′
A
′
= Diag([
σ
2
α
,
σ
2
β
, 0,
σ
2
q
])
(20)
The point transformation
X
i
= H
X
′
i
with the motion matrix
H
=
R
o
X
0
1
(21)
leads to the plane transformation
A
= C
A
′
with the cofactor matrix
C
=
−
R
R
0
1
0
X
(22)
of H (see Förstner and Wrobel, 2016, p. 258). Thus, the covari-
ance matrix of
A
is
Σ
AA
=
C
Σ
A
′
A
′
C
(23)
In the following we interpret the entries in the normal
equation matrix (Equation 13) as moments. Assuming a con-
tinuous distribution function of the points defining a planar
surface patch, we consider the moments of polygons.
Moments of Polygons
Assuming that all points in a region have the same weight
and are uniformly distributed, the normalized moments of
order (
m,n
) of a region
R
are:
γ
m n
m n
R
A
x y x y
,
=
∫∫
1
d d
(24)
For
m = n =
0 we get
γ
0,0
= 1 since
A
is the area of the re-
gion
R
. The normalized centralized moments are:
µ
m n
m
n
R
A
x x y y x y
,
=
−
(
)
−
(
)
∫∫
1
0
0
d d
(25)
with the centroid coordinates
x
0
=
γ
1,0
and
y
0
=
γ
0,1
. The cen-
tralized second moments can readily be computed using:
μ
xx
=
μ
2,0
=
γ
2,0
–
γ
2
1,0
(26)
μ
yy
=
μ
0,2
=
γ
0,2
–
γ
2
0,1
(27)
μ
xy
=
μ
1,1
=
γ
1,1
–
γ
1,0
γ
0,1
(28)
Thus, it is not necessary to compute these quantities ex-
plicitly if the second moments are known (Steger, 1996b).
With the centralized second moments the normal Equation
Matrix 15 can be written as:
N
=
wI
Diag([1,
μ
xx
,
μ
yy
])
(29)
Using the equations for the moments of polygonal regions
(Steger, 1996b), we determine the two moments
μ
xx
and
μ
yy
.
This is done by applying Green’s theorem which allows re-
placing the area integrals by boundary integrals. By reducing
the surface integral to a curve integral along the borders of the
region
R
represented by a polygon
P
, we are able to compute
the integral as a function of the polygon’s vertices:
Let
P
(
x,y
) and
Q
(
x,y
) be two continuously differentiable
functions on the two-dimensional region
R
, and let
b
(
t
) be the
boundary of
R
. If
b
is piecewise differentiable and oriented
such that the interior is left of the boundary path, an integral
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
June 2018
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