EXPERIMENTAL RESULTS ON THE DETERMINATION OF THE TRIFOCAL
TENSOR USING NEARLY COPLANAR POINT CORRESPONDENCES
Camillo RESSL
Institute of Photogrammetry and Remote Sensing
University of Technology, Vienna, Austria
car@Qipf.tuwien.ac.at
Working Group 11/1
KEY WORDS: Orientation, Algorithms, Performance Test, Orientation, Calibration, Bundle block adjustment .
ABSTRACT
In this article we examine the computation of the trifocal tensor from different view points: the minimization of algebraic
or reprojection error, the consideration of the internal constraints, and the effect of nearly coplanar object points. It is
shown using synthetic data, that a correct solution for the trifocal tensor can be obtained as long as the object points
deviate from a common plane by at least 1 % of the viewing distance. Using real data it is shown that the orientation
parameters derived from the tensor can be successfully used to initialize a subsequent bundle block adjustment.
1 INTRODUCTION
Projective geometry is widely used in Computer Vision
because it enables linear and simple representations for
several orientation methods; e.g. the trifocal tensor for the
relative orientation of three uncalibrated images [Hartley
1997].
These alternative representations based on projective ge-
ometry, however, are still not very popular in Photogram-
metry, because there are several drawbacks associated with
them:
e The linearity of these representations is achieved by
over-parameterization, i.c. more parameters than
the actual degrees of freedom (DOF) are used. Conse-
quently certain non-linear constraints among the pa-
rameters must be satisfied; and the direct linear solu-
tion does not satisfy them in general.
e The linearity is further achieved by considering the
images to be uncalibrated, i.e. the information of
an a-priori given interior orientation can not be used
directly (or the linearity gets lost).
e The linearity of these representations can be used ef-
fectively only if the so-called algebraic error is min-
imized instead of the so-called reprojection error (i.e.
the error in the original image measurements).
e The solution of the respective linear system of equa-
tions for determining the alternative parameters fails
if the image correspondences originate from exactly
planar object points or lines.
e [n projective geometry only linear mappings are dealt
with. Consequently, unknown non-linear image dis-
tortion can not be handled directly in this frame-
work; known non-linear image distortion could be re-
moved from the images in a preprocessing step.
In contrast to all these drawbacks, however, we have the
linearity of these alternative representations, which is a
huge benefit, as it liberates us from the requirement to pro-
vide approximate values for the exterior (and sometimes
also interior) orientation parameters, which are inevitable
for a conventional bundle block adjustment.
With the benefit and drawbacks mentioned above, a rea-
sonable strategy would consist of two steps: a) perform
the orientation using a convenient alternative representa-
tion and b) to use the orientation parameters obtained
thereby as approximate values for a subsequent bundle
block adjustment to further refine the result by consider-
ing a-priori known interior orientation and modelling the
non-linear image distortion.
To successfully apply this strategy it is essential, that the
linear orientation in step a) does not fail. The main reason
for the linear orientation to fail are coplanar object points.
Due to the noise in the images this will not only happen for
mathematically exact coplanar object points, but already
for nearly coplanar points. The required deviation from
a common plane is further increased by the other draw-
backs; i.e. the negligence of the internal constraints, the
unknown interior orientation and the unknown non-linear
image distortion.
Therefore, it is interesting to investigate the effects of the
drawbacks mentioned above on the linearly obtained ori-
entation parameters in the case of nearly coplanar object
points; e.g. what minimum deviation from a common plane
is necessary for a successful solution if the internal con-
straints are considered or neglected and if algebraic error
or reprojection error is minimized? For this investigation
we will consider the trifocal tensor, which is made up of 27
elements (with 18 DOF) and linearly describes the relative
orientation of three uncalibrated images. Compared with
the other linear representations of two (the fundamental
matrix with 9 elements and 7 DOF, [Luong and Faugeras
1996]) and four images (the quadfocal tensor with 81 el-
ements and 29 DOF, [Hartley 1998]), the trifocal tensor
is more robust than the fundamental matrix, due to the
third image, and not as complex as the quadfocal tensor
(9 vs. 52 internal constraints).
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