O I
O
i CP
Pj CP
r v
r
2
I I
Ii O
O M
1
i j
i i
Pj Pj
Pj Ii
p
r v
R r
r k
3 where,
i
= order of SPIs
j
= order of ground points
O CP
r
,
I
i P j
v
= observation and residual vectors of ground control point
I
i Pj
r
,
I
i P j
v
= observation and residual vectors of
th
j
SPI point of
th
i
SPI
Ii O
R
= rotation matrix from object frame to
th
i
SPI frame
2.4 Coplanar Condition and Essential Matrix
The relative orientation of two images I
k
and I
k-1
is obtained by the intersection of the projection rays of homologous image-
points. The base-vector and the two projection rays are coplanar. This relationship can be depicted in Fig. 4. In this situation, the
hexahedral volume made by the vectors, namely the light ray from I
1
camera, the light ray from I
2
camera and the base-vector is zero. Eq. 4 explains the concept of coplanar condition and it
can be rewritten as Eq. 5.
Figure 4. Concept of coplanar condition of SPIs
1 I1
1 2
2
R
O I
O O
P I
I I
r r
R
4
1 1
2 2
I I
O I
P O
I P
r R T R r
5
1 2
E=
I O
O I
R T R
6 where
1
I P
r
,
2
I P
r
= the light ray from I
1
and I
2
camera
1 2
O I
I
r
= the base line between I
1
and I
2
camera Essential Matrix describes the relationship between two images
I
k
and I
k-1
of a calibrated camera Scaramuzza, 2012. It contains the camera motion parameters up to an unknown scale
factor for the translation Sabzevari, 2014.To estimate Essential Matrix, there two common methods. One is using
normalized image coordinates, or else computing from Fundamental Matrix. For SPI case, the interior orientation is
nonlinear, so it is difficult to compute Essential Matrix from Fundamental Matrix.
The Essential Matrix can be computed from 2D to 2D feature correspondence using epipolar constraint. The minimal case
solution involves five 2D to 2D correspondences. Moreover, a simple and straightforward solution is the Longuet-
Higgins’ eight-point algorithm, which is summarized as each feature
match gives a constraint of the following form
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
ln ln
ln ln
ln ln
ln ln
ln
y y
y y
z ...
y y
y y
z
l r
l r
l r
l r
l r
l r
r l
r l
l r
rn rn
rn rn
rn rn
rn rn
rn
x x x y
x z x
y z
x z z
z E
x x x y
x z x
y z
x z z
z
7 where,
11 12
13 21
22 23
31 32
33 T
E e
e e
e e
e e
e e
Stacking the constraints from eight points gives the linear equation system
AE
, and by solving the system, the parameters of E can be computed. This homogeneous equations
can easily be solved using singular value decomposition SVD. Having more than eight points to an overdetermined system to
solve in the least-squares adjustment and provides a degree of robustness to noise. The SVD of A has the form
T
A UDV
, and the least-squares estimate of E with
1 E
can be found as the last column of V.
2.5 Camera Matrix and Ambiguity
In section 2.4, Essential Matrix can be derived with feature correspondence. Once Essential Matrix is known, the camera
matrices may be retrieved from E. It means that from the estimate of Essential Matrix, the rotation and translation parts
can be extracted up to scale. To extract the camera matrix, it needs to assume the first camera matrix is
[ | 0] P
I
8 where,
P
= camera matrix of first camera
I
= identity matrix Eq. 8 describes that the position of the first camera locates at
the origin and does not contain any rotation. With Eq. 6, where T is a skew-symmetric matrix that satisfies
1 2
O I
I
T v r
v
, for all vectors v, while R is an orthonormal matrix with
positive determinant, that is,
T
RR I
and
1 Det R
. The matrix S is singular, so Essential Matrix is also singular. Hence
one of the constraints on the elements of E is
Det E
. The problem addressed here is how to recover T, namely,
1 2
O I
I
r
and R given E Horn, 1990.
The ambiguity of the solution is illustrated in Fig. 1. Assume the first camera is at the origin of the coordinate system
[ | 0] P
I
, and the second camera
P
, is rotated and translated according to
i
P
. The correct solution for second camera is the transformation
matrix
i
P
which triangulates
a point
correspondence in front of both cameras. With the triangulation provided in the frame work, the correct solution can be
recognized. However, the situation become totally different in SPI case. The
approach for frame image is not suitable for the SPI since there is no definition of front side and back side with SPI. So there is
a novel method proposed to deal with SPI ambiguity problem. To select the correct solution in four possible solutions,
generate a virtual object point in the space first. Because the camera matrix of first camera and second camera have been
done already, the object coordinate in object frame can be solved.
Now we have the object coordinate in SPI frame
I P
r
, and object coordinate in object frame
O P
r
. The vector
I P
r
, can transform
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XLI-B6-251-2016
253
into object frame with rotation matrix
O I
R
, and the vector from SPI centre to object point
O I
P
r
, also can be computed. If both rotation and translation of second camera are correct, the
angle between
O I
I P
R r
, and
O I
P
r
will close to zero. This angle can be easily computed by inner product calculation. The main
concept of this algorithm is illustrated in Fig. 5.
Figure 5. Angle relationship of vectors Similar to the frame camera, there are four possible solutions
after extracting the rotation and translation form Essential Matrix. The rotation and translation ambiguity is illustrated as
Fig. 6. Fig. 6a is the correct solution since the angles of first camera and second camera are close to zero. Other cases shows
that at least
one camera contains the angle close to 180˚ which means either the rotation matrix of second camera is wrong or
the translation matrix of second matrix is wrong.
Figure 6. Rotation and translation ambiguity of SPI
2.6 Scale Factor and Ray Intersection
