**L. Baumela, L. de Agapito, P. Bustos and I. Reid Email **

In the Proceedings of the International Conference on Pattern Recognition, 3rd - 8th September 2000

* We consider the motion estimation problem in the case of very closely spaced views. We revisit the differential epipolar equation providing an interpretation of it.
On the basis of this interpretation we introduce a cost function to estimate the parameters of the differential epipolar equation, which enables us to compute the
camera extrinsics and some intrinsics. In the synthetic tests performed we compare this continuous method with traditional discrete motion estimation and contrary
to previous findings cannot perceive any computational advantage for the continuous method.
*

**L. de Agapito, E. Hayman and I. Reid Email **

Technical Report OUEL 0225/00, Department of Engineering Science, University of Oxford

* In this paper we describe the theory and practice of
self-calibration of cameras which are fixed in location and may freely
rotate while changing their internal parameters by zooming. The basis
of our approach is to make use of the so-called {\it infinite
homography constraint} which relates the unknown calibration matrices
to the computed inter-image homographies. In order for the calibration
to be possible some constraints must be placed on the internal
parameters of the camera.
We present various self-calibration methods. First an iterative
non-linear method is described which is very versatile in terms of the
constraints that may be imposed on the camera calibration: each of the
camera parameters may be assumed to be known, constant throughout the
sequence but unknown, or free to vary. Secondly, we describe a fast
linear method which works under the minimal assumption of zero camera
skew or the more restrictive conditions of square pixels (zero skew
and known aspect ratio) or known principal point. We show
experimental results on both synthetic and real image sequences (where
ground truth data was available) to assess the accuracy and the
stability of the algorithms and to compare the result of applying
different constraints on the camera parameters. We explore the
degenerate configurations and identify that pure pan-tilt motion
exhibits degeneracies when minimal constraints are imposed on the
intrinsic parameters. We also identify some near-ambiguities that
arise under rotational motions showing that coupled changes of certain
parameters are barely observable making them indistinguishable.
We then derive an optimal Maximum Likelihood estimator for the
calibration and the motion parameters. Prior knowledge about the
distribution of the estimated parameters (such as the location of the
principal point) may also be incorporated via Maximum a Posteriori
estimation. Finally we study the negative effect of radial distortion
in the self-calibration process and point out some possible solutions
to it.
*

**R.
Hartley, E. Hayman, L. de Agapito and I. Reid
Email
**

In the Proceedings of the International Conference on Computer Vision, Sept 1999.

* This paper considers the problem of self-calibration of a camera from
an image sequence in the case where the camera's
internal parameters (most notably focal length) may change.
The problem of camera self-calibration from a sequence of images
has proven to be a difficult one in practice, due to
the need ultimately to resort to non-linear methods, which have
often proven to be unreliable. In a stratified approach
to self-calibration, a projective reconstruction is obtained first
and this is successively refined first to an affine and then to
a Euclidean (or metric) reconstruction. It has been observed
that the difficult step is to obtain the affine reconstruction,
or equivalently to locate the plane at infinity in the projective
coordinate frame. The problem is inherently non-linear and requires
iterative methods that risk not finding the optimal solution.
The present paper overcomes this difficulty by
limiting the search for the plane at infinity to a
3-dimensional cubic region of parameter space.
It is then possible to carry out a dense search over this cube
in reasonable time. For each hypothesised placement of the
plane at infinity, the calibration problem is reduced to one
of calibration of a non-translating camera, for which
fast non-iterative algorithms exist.
A cost function based on the result of the trial calibration is used to
determine the best placement of the plane at infinity.
Because of the simplicity of each trial, speeds of over 10,000
trials per second are achieved on a 256Mhz processor.
It is shown that this dense search allows one to avoid areas of local
minima effectively and find global minima of the cost function.
*

**L. de Agapito, R.
Hartley and E. Hayman Email **

In the Proceedings of the Computer Vision and Pattern Recognition Conference, June 1999.

* A linear self-calibration method is given for computing the
calibration of a stationary but rotating camera. The internal
parameters of the camera are allowed to vary from image to image,
allowing for zooming (change of focal length) and possible variation
of the principal point of the camera. In order for calibration to be
possible some constraints must be placed on the calibration of each
image. The method works under the minimal assumption of zero-skew
(rectangular pixels), or the more restrictive but reasonable
conditions of square pixels, known pixel aspect ratio, and known
principal point. Being linear, the algorithm is extremely rapid, and
avoids the convergence problems characteristic of iterative
algorithms. *

**L. de Agapito, E. Hayman and I. Reid Email **

In Proceedings of the ninth British Machine Vision Conference, Sept 1998.

* We present a method for self-calibration of a camera which is free to
rotate and change its intrinsic parameters, but which cannot
translate. The method is based on the so-called infinite homography
constraint which leads to a non-linear minimisation routine to find
the unknown camera intrinsics over an extended sequence of images. We
give experimental results using real image sequences for which ground
truth data was available.
*

**L. de Agapito, D.Q. Huynh and M.J. BrooksEmail **

in Proceedings of the sixth International Conference on Computer Vision, 1998. Bombay (India).

* We show that the self-calibration of a stereo head from
corresponding points in an image pair is in certain circumstances
prone to considerable error. A novel error analysis reveals that the
automated determination of relative orientation and focal length is
adversely affected when the cameras verge inwards a similar amount,
and when the principal point locations have a horizontal error. This
analysis is facilitated by the adoption of closed-form solutions for
self-calibration from previous work of Brooks et al. It is also shown
that estimation of the fundamental matrix associated with a stereo
head image pair is improved when a domain-specific parameterization
and associated computational techniques are adopted. Experiments
conducted with such image pairs suggest that, given cognisance of
sensitive configurations and adoption of the revised method of
fundamental matrix estimation, robust reconstructions are attainable.
This is demonstrated on the problem of metrically reconstructing a
scene from two pairs of images obtained by an uncalibrated stereo head
undergoing unknown ground-plane motion. *

**M.J. Brooks, L. de Agapito, D.Q. Huynh and L. Baumela Email **

submitted.

* We consider the problem of metrically reconstructing a scene
viewed by a moving stereo head. The head features two cameras with
coplanar optical axes arranged on a lateral rig, each camera being
free to vary its angle of vergence. Under various constraints, we
derive novel explicit forms for the epipolar equation, and show that a
static stereo head constitutes a degenerate camera configuration for
carrying out self-calibration in the sense of Hartley. The situation
is retrieved by consideration of a stereo head undergoing ground plane
motion. New closed-form solutions for self-calibration are thereby
obtained. It is also shown that the quality of scene reconstruction is
much enhanced by the adoption of domain-specific techniques for
computation of the fundamental matrix. An error analysis reveals that
reconstruction is adversely affected by inward-facing camera vergence
angles that are similar in value, and by a principal point location
whose horizontal component is in error. Experiments conducted with
dynamic stereo head images suggest that, given cognisance of these
sensitivities, robust reconstructions are attainable. *

**M.J. Brooks, L. de Agapito, D.Q. Huynh and L. Baumela Email **

* in Proc. Fourth European Conference on Computer Vision, Cambridge,
UK. April 1996.Lecture Notes in Computer Science, 1065, Vol. II, pages 415-426.
Springer-Verlag, B. Buxton and R. Cipolla (eds.)*.

* In this paper we consider the self-calibration problem in the
special context of a stereo head, perhaps the most commonly adopted
binocular camera configuration in robotics. Here two cameras are
arranged on a lateral rig with coplanar optical axes, each being free
to vary its angle of vergence. Under various constraints, we derive
explicit forms for the epipolar equation, and show that a static
stereo head constitutes a degenerate camera configuration for carrying
out self-calibration in the sense of Hartley. The situation is
retrieved by consideration of a special kind of motion of the stereo
head in which the baseline remains confined to a plane. A new direct
method for self-calibration is thereby obtained, inspired by an
earlier discrete analysis approach by Zhang et al. Key factors in
this approach are the development of explicit, analytical forms of the
fundamental matrix, and the use of the vergence angles in the
parameterisation of the problem. *

Lourdes de Agapito Vicente Last modified: Thu Mar 21 13:06:18 GMT 2002