International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
or system equations (Mikhail et al., 2001, Equation B-27)
-NOOCTITA À
818-140
are solved and the unknown parameter vector w is updated
Uk] = Up + À. (3)
The process is repeated until the update A is small enough,
or the maximum number of iterations has been exceeded.
3 PROBLEM FORMULATIONS
3.1 Euler angles
The standard parameterisation of the rotation matrix M is
by the Euler angles for some specified sequence of rotation
axes. This results in a formulation with three unknowns
and no functional constraint. Thus, in each iteration, the
normal equations (1) would be solved.
3.2 Rotation matrix
An alternative formulation is to treat all rotation matrix el-
ements m411.m12...., mas as unknowns and use the six
functional constraints g;(w) — 0,2 — 1,....6, where
giu) —mim;-1349541,2,3. :ga(w) — mimo,
gs(w) = mi ms, go(w) = m4 ma, (4)
where M — [mi, m», m3]. This formulation has nine pa-
rameters and six functional constraints and replaces the
highly non-linear trigonometric functions with less non-
linear constraints. Furthermore, the rotation matrix is al-
ways well-defined for a given rotation, and is thus singular-
ity-free. However, the constraints are satisfied both for a
rotation matrix (| M |=+1) and a reflection matrix (|M |=-1).
For this parameterisation, the system equations (2) would
be solved in each iteration.
4 ALGORITHMS
Two versions of the bundle adjustment algorithm were im-
plemented in Matlab (The Mathworks, Inc.) with the pa-
rameterisations of sections 3.1 and 3.2.
4.1 Line search
Furthermore, the algorithms were implemented to work
with or without line search, a damping method common
within the optimisation community (Gill et al., 1981, Nash
and Sofer, 1996, Bjórck, 1996). With line search, the up-
date formula (3) is replaced by
Wk+1 = Wk d œ À, (5)
where the scalar œ is taken as the first value of the se-
quence 1, }, i, ... such that the updated estimate wy. of
the parameters are sufficiently better than the current es-
timate wy. (for details, see (Borlin et al., 2003)). E.g. for
$?
CM
g?!
cl
2. 12
I
15
LE
14
Figure 1: The Olympus subset of the Zürich data set with
camera stations indicated.
the unconstrained formulation, the line search method will
ensure that the next objective function value @(w;+1 ) is
smaller than the current $(w;.), i.e. that the residual sum
will decrease for each iteration. Since only objective func-
tion evaluations are used, the method is computationally
cheap.
Thus, the following bundle adjustment algorithms were
available:
Alg. 1U Euler angles, undamped (no line search). Corre-
sponds to classical bundle adjustment.
Alg. 1D Euler angles, damped (with line search). This al-
gorithm corresponds to the Gauss-Newton algorithm
in e.g. (Borlin et al., 2003).
Alg. 2U All nine rotation matrix elements and the six or-
thogonality constraints (4). Undamped. Corresponds
to bundle adjustment with functional constraints.
Alg. 2D All nine rotation matrix elements and the six or-
thogonality constraints (4). Damped. Corresponds
to the GNC (Gauss-Newton, Constrained) algorithm
in (Bórlin et al., 2003).
5 DATA SET
The Zürich City Hall data set (Streilein et al., 1999) was
selected to test the algorithms. The data set consists of 31
images taken with an Olympus C1400L and a Fuji DS300
camera. For this investigation, the 14 Olympus images
were selected. The difference in focal settings described
_ in (Rottensteiner et al., 2001) was ignored. From the Olym-
590
pus images, 33 camera pairs were identified with a suitable
number of homologous points, see Figure 1 and Table 1.
The smallest number of homologous points were 42.
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