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l (t)) , which needs to be minimised
with respect to the parameter t . Hartley and Zisserman ( 2003 ) state that this min-
imisation corresponds to the determination of the real-valued zero points of a sixth-
order polynomial function. As the estimated points
l (t))
can be formulated as d 2 ( S 1
x , S 1
d 2 ( S 2
x , S 2
˜
+
˜
x (e) exactly fulfil
the epipolar constraint, an exact, triangulation-based solution for the corresponding
projective scene point W
S 1
x (e)
S 2
˜
and
˜
x in the world coordinate system is obtained by inserting the
normalised coordinates (u (e)
1
˜
2 ,v (e 2 ) of S 1 x (e) and S 2 x (e) into ( 1.24 ).
The matrix G now has a zero singular value, to which belongs the singular vector
representing the solution for W
,v (e)
1
) and (u (e)
x .
Estimating the fundamental matrix F and, accordingly, the projective camera
matrices P 1 and P 2 and the projective scene points
˜
W
x i from a set of point cor-
respondences between the images can be regarded as the first (projective) stage of
camera calibration. Subsequent calibration stages consist of determining a metric
(Euclidean) scene reconstruction and camera calibration. These issues will be re-
garded further in Sect. 1.4.6 in the context of self-calibration of camera systems.
˜
1.3 The Bundle Adjustment Approach
In the following, the general configuration is assumed: K three-dimensional points
W x k in the world appear in L images acquired from different viewpoints, and the
corresponding measured image points are denoted by their sensor coordinates S i x k ,
where i =
1 ,...,K (Triggs et al., 2000 ; Hartley and Zisserman,
2003 ; Lourakis and Argyros, 2004 ).
A nonlinear function
1 ,...,L and k =
( C W T,
c j } i , W x ) is defined such that it yields the mod-
elled image coordinates by transforming the point W x in world coordinates into the
sensor coordinate system of camera i using ( 1.1 )-( 1.5 ) based on the camera pa-
rameters denoted by C W T and
Q
{
c j } i and the coordinates of the K three-dimensional
points W x k (Lourakis and Argyros, 2004 ; Kuhl et al., 2006 ) (cf. also Sect. 5.1 ). For
estimating all or some of these parameters, a framework termed 'bundle adjustment'
has been introduced, corresponding to a minimisation of the reprojection error
{
L
K
I i T 1 Q C W T,
c j } i , W x k
I i T 1 S i x k
S i
S i
2 ,
E BA =
{
(1.25)
i
=
1
k
=
1
which denotes the sum of squared Euclidean distances between the modelled and
the measured image point coordinates (Lourakis and Argyros, 2004 , cf. also Triggs
et al., 2000 ). The transformation by S i
I i T 1 in ( 1.25 ) ensures that the reprojection er-
ror is measured in Cartesian image coordinates. It can be omitted if a film is used for
image acquisition, on which Euclidean distances are measured in a Cartesian coor-
dinate system, or as long as the pixel raster of the digital camera sensor is orthogonal
( θ =
90 ) and the pixels are quadratic ( α u = α v ). This special case corresponds to
S i
I i T in ( 1.5 ) describing a similarity transform.
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