XV, Part B3. Istanbul 2004
ation
:
Bundle Adjustment and Incidence of Linear Features on the Accuracy of External Calibration Parameters
Franck JUNG, Didier BOLDO
MATIS Laboratory, IGN, 2-4 Av. Pasteur, 94165 Saint Mande Cedex, FRANCE
first. lastname@ ign.fr
KEY WORDS: Bundle Adjustment, Linear Features, Accuracy, Photogrammetry
ABSTRACT:
In this paper we investigate the influence of linear features (segments) in a bundle adjustment. Bundle adjustment is the problem of
refining a set of parameters (internal calibration, external calibration, 3D model). The refinement is performed by the minimization of
a cost function. In usual photogrammetric applications, this cost function is based on image tie points and 3D control points. The cost
function measures the distance between the observed data and the model. Linear features are especially important when dealing with
architectural or terrestrial images, since they do not require segment extremity to match. So, they are easier to detect automatically
within particular scenes (buildings, landscapes...). First, we present the basic concepts of bundle adjustment and the integration of
linear features. Using the same concept, the linear feature model is based on the distance between the 3D line re projection in the image
and the detected image segments. We describe an algorithm for the resolution of this non-linear least-square problem under constraints.
Second, we study the influence of these features on two cases. The first case is a calibration polygon, with a large overlap between
images. The second one is a facade of a building. We compare a statistical evaluation of the reliability of the estimated parameters to
the theoretical bounds calculated with the eigenvalues and eigenvectors of the Hessian matrix associated to our problem. We stress that
segments can be relevant features and can highly increase precision.
1. INTRODUCTION
Accurate parameter estimation is paramount for many image
based applications. Feature matching and 3D feature reconstruc-
tion are intimately linked to camera parameter estimation. Hence,
the knowledge of the viewing parameters permits to face the 3D
structure reconstruction problem. Conversely, accurate feature
matching can provide a refinement of the viewing parameter esti-
mation. For instance, viewing parameter estimation is a problem
that both the computer vision community and the photogramme-
try community have tackled for the last decades. Moreover, both
communities are currently able to provide a solution for a rough
estimation of numerous parameters using different techniques.
On the one hand, linearity of geometric relations using projec-
tive geometry provides simple expressions for many parameters
(Heuel, 2001). Hence, robust algorithms providing viewing pa-
rameters (Faugeras et al., 2001),(Xu and Zhang, 1996), (Torr and
Davidson, 2003) are widespread in the computer vision commu-
nity. These robust estimations often lack accuracy or metric inter-
pretation. On the other hand, many problems in the photogram-
metric community can be faced thanks to a rough estimation of
the viewing parameters (using GPS, INS). Thus, given a rough
estimation of the parameters, bundle adjustment provides jointly
a refinement of the 3D structure and of the viewing parameters
((Triggs et al., 2000), (Hartley and Zisserman, 2002), (Kraus,
1993)). Usually, only points are used for the process. But, in
man-made environments, linear features are often easier to detect.
So, it is of importance to include them in the bundle adjustment
process, as well as to quantify their contribution. The quantifi-
cation can be done by estimating the accuracy improvement of
each parameter. If the measurements used for bundle adjustment
are Gaussian, we can use error propagation results in order to as-
sociate an uncertainty to each estimated parameter. Correlation
between different parameters can also be estimated (Triggs et al.,
2000), (Hartley and Zisserman, 2002), (Fórstner, 2004).
2. BUNDLE ADJUSTMENT
2.1 Introduction
Bundle adjustment is the well known process used to refine a
set of parameters such as camera positions and orientations, 3.D
point and line positions. It relies on the minimization of a cost
function based on all available observations such as Ground Con-
trol Points (GCP), tie points and tie lines. In this section, we suc-
cessively describe the parameters, the cost function and, finally,
the observations.
2.2 Parameters
Parameters are representative of the problem modeling. In this
problem, three kinds of parameters are to be retrieved : camera's,
points' and linear features'.
Camera Camera position is obviously represented by three co-
ordinates (a, y, z) in a true 3D system. Multiple representations
exist for the camera orientation. We have chosen to use a unitary
quaternion to represent it. This representation is non-minimal but
simple and never degenerate. Unitary quaternions are represented
as a quadruplet of value (go, q1, q2, q3), with constraint :
dodi di-4i—1-0 (1)
and there is a simple mapping taking unitary quaternions to ro-
tation matrices (Chou and Kamel, 1991). The system has 7 un-
knowns and 1 constraint for each camera.
Point feature A point is represented by three coordinates
(2,y,z) in a true 3D system. So the system has 3 unknowns
for each point.
Linear feature A linear feature can be rather tricky to repre-
sent. Both computer vision and photogrammetry communities
have tried to introduce these features for camera parameter esti-
mations (Habib, 1999), (Xu and Zhang, 1996). We have chosen a
simple model: linear features are considered infinite in 3D, and
their image measurement is simply the position of their extremi-
ties (Kumar and Hanson, 1994).
3D linear features have only 4 degrees of freedom, but represen-
tations using a minimal set of parameters can be rather complex.
So we have chosen to use a simple 6 parameter representation :
a line is represented by a start point P and a direction &. Any
point M on the line can be expressed as M = P + kd, with
& € IR. To get a unique representation, P and 4 are subject to
two constraints :
Iz" — 1-0 (2)
and
GPs 3)
where O has coordinates (0,0, 0). The system has 6 unknowns
and 2 constraints for each line.
