1 0 0
0 1 0
0 0
1
1
1 2 3
4 5 6
7 8
c
−
= =
c c c
c c c
c c
A R
w
c
λ
γ
0
0 1
0 0
0
0
0
−
−
−
X
Y
Z
(8)
X
A A
A A
Y
A A
A A
Z
A A
A A
T
T
T
T
T
T
0
13
11
0
23
11
0
33
= −
(
)
(
)
= −
(
)
(
)
=
(
)
(
)
,
,
11
0
2
0
2
− −
X Y
(9)
In the next step, the inverse of the rotation matrix
R
c
w
(i.e.,
R
w
c
where
R
w
c
is the transpose of
R
c
w
) is directly estimated
through a quaternion-based approach. In this approach, a
geometric constraint is applied to align the image vector (
x
)
with the object vector (
X
), i.e., the vectors connecting the esti-
mated perspective center to the image point and the perspec-
tive center to the corresponding object point as illustrated in
Figure 1. This alignment can be mathematically described by
the constraint
X
i
=
R
w
c
x
i
+
e
i
, where
x
i
and
X
i
are the normal-
ized image and object vectors and
e
i
is the misalignment error
for the
i
th
normalized image/object vector. In order to estimate
the unknown rotation matrix
R
w
c
, least-squares adjustment is
employed to minimize the Sum of Squared Errors (
SSE
) for all
the involved
n
points as seen in Equation 10.
min
min
min
R i
n
i
T
i
R i
n
i
c
w
i
T
i
c
w
i
R
c
w
c
w
c
w
e e
R
R
=
=
∑ ∑
=
−
(
)
−
(
)
=
1
1
X x X x
i
n
i
T
i
i
T
i
i
T
c
w T
i
R
=
∑
+ −
1
2
X X x x x
X
( )
(10)
In Equation 10, the terms
X
i
T
X
i
and
x
i
T
x
i
are always positive
as they are the squared magnitudes of the
x
i
and
X
i
vectors.
Therefore, in order to minimize the
SSE
in Equation 10, the
rotation matrix has to be estimated in a way to maximize
the term
x
i
T
(
R
w
c
)
T
X
i
. This term can be formulated as the dot
product in Equation 11 and maximized using the quaternion
approach proposed by Horn (1987), where interested readers
can find more details. The pertinent quaternion basics to this
research are briefly explained here.
max
( )
.
R i
n
i
T
c
w T
i
R
i
n
c
w
i
i
c
w
R
R
max
=
=
∑
∑
=
1
1
x
X
x X
(11)
Quaternions have one real and three imaginary elements,
which are denoted in this paper by the symbol (
.
). A unit
quaternion
q
.
can represent any rotation in 3D space by a rota-
tion angle around an axis, which are defined by the real and
imaginary parts of the quaternion, respectively. According to
the quaternion properties, the rotation multiplication
R
w
c
x
i
is
equivalent to the quaternion multiplication
q
.
x
.
i
q
.
*
, where the
unit quaternion
q
.
corresponds to
R
w
c
and
q
.
*
is the conjugate
quaternion constructed by negating the imaginary part of
q
.
.
The term
x
.
i
is the quaternion form of
x
i
, which is nothing
but adding a zero as the real part and the three elements of
x
i
as the imaginary part, i.e.,
x
.
i
= (0,
x
i
). Using the quaternion
properties, Equation 12 can be derived, where
C
and
C
– are 4 ×
4 matrices that convert the quaternion-based multiplication to
a matrix-based multiplication, and
S
is a 4 × 4 matrix con-
structed using the components of
x
i
and
X
i
for all the available
image/object points. In order to maximize the term
q
.
T
S
q
.
while
maintaining the unity constraint of a quaternion rotation as
represented in Equation 13, one should use the Lagrange mul-
tiplier
λ
and maximize the target function
φ
in Equation 14. To
derive the desired quaternion, the target function
φ
should be
differentiated with respect to
q
.
as seen in Equation 15, which
yields the expression in Equation 16.
max
.
.(
)
q
i
n
i
i
q
i
n
i
i
q q
q
q
max
m
*
=
=
∑
∑
=
( )
=
1
1
x X
x X
ax
max
q
i
n
T
i
T
i
q
T
i
n
i
T
i
q C C q
q C C
=
=
∑
∑
( )
=
( )
1
1
x X
x X
( )
( )
q
(12)
max
q
T
q q q,
S
=
1
(13)
2
1
max
q
T
T
q q q q q (
)
φ
λ
( )
= −
−
S
(14)
q q
2 2
∂
∂
= − =
φ
λ
q
0
S
(15)
S
q q
=
λ
(16)
The expression in Equation 16 is satisfied if and only if
λ
and
q
.
are the corresponding eigenvalue and eigenvector of the
matrix
S
. In this case, the term
q
.
T
S
q
.
would reduce to
λ
since
the rotation defined by
q
.
is a unit quaternion (Equation 17).
Therefore, the term
q
.
T
S
q
.
is maximized when
λ
is the larg-
est eigenvalue of the matrix
S
and eventually the unknown
quaternion
q
.
is the eigenvector corresponding to the largest
eigenvalue. The rotation matrix
R
w
c
and corresponding rotation
angles (
ω
,
φ
,
κ
) can be derived from the quaternion
q
.
. This
approach can be used to derive the rotation matrix for other
SPR
methods that first estimate the position of the perspective
center, such as
DLT
or
P3P
.
q q q q q q
T
T
T
S
= = =
λ λ
λ
(17)
DLT-based Quaternion Approach
DLT
is a well-known approach to estimate the
EOP
s of the im-
age as well as the Internal Orientation Parameters (
IOP
s) for
an uncalibrated camera. The
DLT
relates a non-coplanar object
space to the image space using 11 coefficients
L
1
,
L
2
, …,
L
11
as represented in Equation 18, where
L
12
can be assumed to
be unity. The
DLT
method requires six or more non-coplanar
points to solve for these coefficients through which the
EOP
s/
IOP
s can be derived. To correlate the
DLT
coefficients with the
EOP
s/
IOP
s, the collinearity equations are expressed through
the form in Equation 19, where
x
p
and
y
p
are the principal
point coordinates,
c
x
and
c
y
represent the principal distances
in the X and Y directions, non-orthogonality of the image co-
ordinate system axes is denoted as
α
, and
K
is the calibration
matrix. Comparing Equations 18 and 19, one can derive Equa-
tion 20 for the position of the perspective center. In order to
derive the
IOP
s, the term
λ
KR
c
w
is substituted with
D
(Equation
21), and the required formulas to estimate the
IOP
s from the
DLT
coefficients can be estimated using the product
DD
T
. The
Figure 1. The required rotation
R
w
c
to co-align the vectors con-
necting the perspective center to the corresponding image and
object points.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
March 2015
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