are slower and require approximate values for the position
and orientation parameters, which are usually provided by a
closed-form approach. In this category, the Church method is
well-known within the photogrammetric society, which itera-
tively solves the
SPR
problem using three points (American
Society of Photogrammetry, 1980). In photogrammetric text
books such as Wolf
et al
. (2014), a non-linear solution for the
SPR
problem is explained. In this approach, the collinearity
equations are linearized and the Gauss-Newton optimization
is applied to recover the
EOP
s, by minimizing the back projec-
tion error in image space for all the available object points.
Yuan (1989) develops another iterative solution for three or
more points, using the homogenous form of the collinear-
ity equations. Lu
et al
. (2000) propose an approach for
n
≥
3
points in which the orientation and translation parameters are
successively updated. Zeng (2010) employs the Rodrigues ma-
trix instead of the Euler angles to improve the stability of the
non-linear
SPR
in case of poor initial values. Another alterna-
tive to represent a rotation in 3D space is quaternions. In Horn
(1987), quaternions are employed to solve the absolute ori-
entation problem. For the
SPR
problem, Jiang
et al
. (2007) use
quaternions to express the rotation matrix in the collinearity
equations in order to improve the stability of the non-linear
SPR
against poor initial values. Guan
et al
. (2008) recover the
position of the perspective center using a
P3P
approach, and
then estimate the orientation by adopting quaternions.
In this paper, we present three approaches that utilize qua-
ternions for solving the
SPR
problem. The first two approaches
are based on the projective transformation and
DLT
, which
transform the object space to the image using eight and eleven
coefficients when dealing with four or more coplanar points
and six or more non-coplanar points, respectively. The trans-
formation coefficients are used to solve for the position of
the perspective center first and the rotation matrix is directly
derived using quaternions afterwards. In order to have a more
general procedure for solving the
SPR
problem when dealing
with three or more points in any configuration, a quaternion-
based general approach is introduced, where the rotation
matrix is iteratively derived first. Then, the position of the
perspective center is computed by a direct solution.
The rest of this paper is organized as follows: In the next
section, the projective transformation is reviewed while
discussing the recovery of the rotation matrix using quater-
nions, followed by a brief description of the
DLT
method. The
quaternion-based general approach is presented in the next
section, followed by the experimental results from simulated
and real data. Finally, some conclusions and recommenda-
tions for future work are presented.
Projective-based Quaternion Approach
In this section, the proposed projective transformation ap-
proach to estimate the
EOP
s is explained. The projective trans-
formation relates a planar object space to image space using
eight coefficients (
c
1
,
c
2
, …,
c
8
) as expressed in Equation 1. To
estimate the coefficients, four or more planar object points
are required. To facilitate the derivation of the
EOP
s from the
projective transformation coefficients, the X and Y axes of the
object coordinate system can be chosen to be aligned along
the object space plane (i.e., Z = 0 for all the object points).
x
c X c Y c
c X c Y
y
c X c Y c
c X c Y
= + +
+ +
= + +
+ +
1
2
3
7
8
4
5
6
7
8
1
1
,
(1)
In this approach, the projective transformation coefficients
are estimated first, and the
EOP
s of the image are recovered
afterwards using the estimated coefficients. In this regard,
Kobayashi and Mori (1997) introduced a procedure to recover
the
EOP
s and the principal distance (
c
) from the projective
transformation coefficients. In their approach, the rotation
angles are recovered sequentially, which may cause ambigui-
ties in the recovery of the rotation angles. Using quaternions,
the rotation matrix is directly derived and thus avoiding any
possible ambiguities. The proposed procedure in this pa-
per begins by estimating the principal distance (
c
). For this
purpose, Equation 1 is reformulated to the form in Equation 2
where
γ
is a scale factor (Seedahmed and Habib, 2002).
x
y
c c c
c c c
c c
X
Y
1
1 1
1 2 3
4 5 6
7 8
=
γ
(2)
To establish the mathematical relationship between the
projective transformation coefficients and the unknown
EOP
s,
the collinearity equations are reformulated as in Equation 3,
where
R
c
w
represents the rotation matrix relating the object/
world coordinate system to the image coordinate system,
λ
is
a scale factor, (
X
O
,
Y
O
,
Z
O
) denotes the position of the perspec-
tive center, and
Z
= 0 for all the object points. The third row
of Equation 3 is multiplied by (-1/
c
) to reduce it to the form in
Equation 4.
x
y
c
R
X X
Y Y
Z Z
R
X X
Y Y
Z
w
c
o
o
o
w
c
o
o
o
−
=
−
−
−
=
−
−
−
λ
λ
=
−
−
−
λ
R
X
Y
Z
X
Y
w
c
o
o
o
1 0
0 1
0 0
1
(3)
x
y
R
X
Y
Z
w
c
o
o
o
1
1 0 0
0 1 0
0 0 1
1 0
0 1
0 0
=
−
−
−
−
λ
c
/
X
Y
1
(4)
Comparing Equations 2 and 4, one can derive Equation
5 that illustrates the parametric relationship between the
projective transformation coefficients and the
IOP
s/
EOP
s in
the collinearity equations. In order to estimate the principal
distance (
c
), the third row of Equation 5 is multiplied by –
c
and reduced to the form in Equation 6. Then, the orthogonal-
ity constraint between the first two columns of the rotation
matrix
R
c
w
is enforced. The orthogonality condition leads to
Equation 7 where
r
ij
is the element at the
i
th
row and
j
th
col-
umn of the rotation matrix .
γ
λ
c c c
c c c
c c
R
w
c
1 2 3
4 5 6
7 8
1
1 0 0
0 1 0
0 0 1
1 0
=
−
c
/
−
−
−
X
Y
Z
o
o
o
0 1
0 0
(5)
γ
λ
c c c
c c c
cc cc
r r
r X r Y r Z
o
o
1
2
3
4
5
6
7
8
11 12
11
12
13
− − −
=
− − −
c
o
o
o
o
o
o
o
r r
r X r Y r Z
r r
r X r Y r Z
21 22
21
22
23
31 32
31
32
33
− − −
− − −
(6)
r r r r r r c c c c c c c
c
c c c c
c c
11 12 21 22 31 32
1 2 4 5
2
7 8
1 2 4 5
7 8
0
+ + = + +
= ⇒ =
− −
(7)
Then, the perspective center coordinates (
X
O
,
Y
O
,
Z
O
) are
estimated using
c
1
,
c
2
, …,
c
8
by manipulating Equation 5 to
the form in Equation 8. Denoting the left hand side of that
equation as a matrix
A
, one can derive Equation 9 to produce
the perspective center coordinates, where (
A
T
A
)
ij
refers to the
element at the
i
th
row and
j
th
column of the product
A
T
A
.
210
March 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
