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PROS AND CONS OF CONSTRAINED AND UNCONSTRAINED FORMULATION OF THE 
BUNDLE ADJUSTMENT PROBLEM 
Niclas BORLIN!, Pierre GRUSSENMEYER?, Jerry ERIKSSON!, Per LINDSTRÓM'! 
! Department of Computing Science, Umeá University, UME À, Sweden, Niclas. Borlin@cs.umu.se 
“National Institute of Applied Sciences of Strasbourg, MAP-PAGE UMR 694, STRASBOURG, France, 
pierre.grussenmeyer(@insa-strasbourg. fr 
KEY WORDS: Algorithms, Mathematics, Bundle, Generalisation, Modelling, Performance, Reliability. 
ABSTRACT 
Two implementations of the bundle adjustment problem were applied to a subset of the Zürich City Hall reference data 
set. One implementation used the standard Euler angle parameterisation of the rotation matrix. The second implemen- 
tation used all nine elements of the rotation matrix as unknowns and six functional constraints. The second formulation 
was constructed to reduce the non-linearity of the optimisation problem. The hypothesis was that a lower degree of 
non-linearity would lead to faster convergence. Furthermore, each implementation could optionally use the line search 
damping technique known from optimisation theory. 
The algorithms were used to solve the relative orientation problem for a varying number of homologous points from 33 
different camera pairs. 
The results show that the constrained formulation has marginally better convergence properties, with or without damping. 
However, damping alone halves the number of convergence failures at a minor computational cost. 
The conclusion is that except to avoid the singularities associated with the Euler angles, the preferred use of the constrained 
formulation remains an open question. However, the results strongly suggest that the line search damping technique should 
be included in standard implementations of the bundle adjustment algorithm. 
1 INTRODUCTION 
In photogrammetry, when maximum precision is required, 
the bundle adjustment problem is usually solved. The bun- 
dle adjustment problem is a non-linear least squares prob- 
lem, which must be solved iteratively. 
Within the optimisation community, it is well-known that 
the convergence properties of a non-linear problem depend 
on at least three factors; the quality of the starting approx- 
imation, the optimisation method used, and the degree of 
non-linearity of the problem, e.g. (Gill et al., 1981). 
Earlier papers have described modifications of the optimi- 
sation method in order to improve the convergence proper- 
ties, e.g. (Bórlin et al., 2003). In this paper, the discussion 
is extended to include different formulations of the bun- 
dle adjustment problem with and without functional con- 
straints. The re-formulations are done in order to reduce 
the degree of non-linearity of the problem. Earlier results 
on optical photogrammetry (Hágglund, 2004) indicate that 
the difference in formulations have a small impact on con- 
vergence properties if the problem has strong geometry 
and/or high redundancy. This investigation thus focuses 
on the relative orientation problem which may have weak 
geometry and/or low redundancy. 
2 BUNDLE ADJUSTMENT 
Bundle adjustment is based on the collinearity equations 
fima CX — Xo)--maa(Y -Yo)--maa(Z —Zo)) 
  
  
TI = may CX (— Xo)4-maa(Y — Yo )2-maa(Z — -Zu) 
U— 24 — f (mo1 (X — Xo)4-mo3(Y —Yo)4-m23(Z — Z0)) 
Y in = mai CX — Xo)4-maa2(Y —Yo)-maa(Z — Zo) 
589 
which describe the projection of a world point (X, Y, Z) 
onto an image point (x, y) of a camera with principal dis- 
tance f and principal point at (ro, yo). The camera is 
placed at (Xo, Yo, Zo) and its orientation is described by 
the rotation matrix M. 
The bundle adjustment method minimises the difference 
between projected object points and the corresponding mea- 
surements. For each world point 2 and camera j, define the 
residual v? to be 
  
iid T E fi my (Xs = Xp) +m, (Vi —Y, d) tmia(Zi- Zi) 
24 0 m: hm x; HY +m, (Y; —Y d)rmi, (Zi— — Zà) 
(X; XU (VS ian, UR —Z#) 
  
jm? 21 ( 
Yi - T f m3, (Xi—- Xl) m34(Yi - Yd) -mA4CZ;— Z2) 
where (.X;, Y;, Z;) are the object point coordinates, and 
(22,2) are the measured coordinates in camera j, cor- 
rected for lens distortion. 
Using the notation in (Mikhail et al., 2001, Appendix B), 
the value to be minimised is 
$ó — v Wo, 
where v is a vector containing all v7, and W is a weight: 
matrix. The minimisation is performed over a set of un- 
known parameters w. To emphasise that ¢ and v are func- 
tions of the unknown parameters, this will be written 
ó(w) — v(w)T Wow). 
In optimisation, the function ¢(w) is known as the objec- 
tive function. 
At each iteration k, the problem is linearised and the nor- 
mal equations (Mikhail et al., 2001, Equation B-19) 
NA=1 (1)