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240 J. Gaspar et al.
simultaneous observation of bearings and distances to floor landmarks. Given
distances and bearings, the pose-computation is simplified to the calculation
of a 2D rigid transformation.
The fact that the pose-computation is based on feature locations, implies
that they contain errors, propagated from the feature tracking process. To
overcome this, we propose a complimentary pose-computation optimisation
step, based on a photometric criterium. We term this optimisation fine pose
adjustment, as opposed to the pose-computation based on the features which
is termed coarse pose computation. It is important to note that the pose-
estimation based on features is important for providing an initial guess for
the fine pose adjustment step.

Image Dewarpings for Scene Modelling
Images acquired with an omni-directional camera, e.g. based on a spherical
or hyperbolic mirror, are naturally distorted. Knowing the image formation
model, we can correct some distortions to obtain Panoramic or Bird’s Eye
Views.
The panoramic view groups together, in each scan line, the projections of
all visible points, at a constant angle of elevation. The bird’s eye view is a
scaled orthographic projection of the ground plane. These views are advanta-
geous e.g. for extracting and tracking vertical and ground plane lines.
Panoramic and Bird’s Eye Views are directly obtained by designing cus-
tom shaped mirrors. An alternative approach, as described next, is to simply
dewarp the omnidirectional images to the new views.
Panoramic View: 3D points at the same elevation angle from the axis of
the catadioptric omnidirectional vision sensor, project to a 2D circle in the
image. Therefore, the image dewarping is defined simply as a cartesian to
polar coordinates change:

I(α, R)= I 0 (R cos(α)+ u 0 ,R sin(α)+ v 0 )
where (u 0 ,v 0 ) is the image centre, α and R are the angle and radial coordi-
nates. The steps and range of α and R are chosen according to the resolution,
and covering all the effective area, of the omnidirectional image. One rule for
selecting the step of α is to make the number of columns of the panoramic
image equal to the perimeter of the middle circle of the omnidirectional image.
Hence inner circles are over-sampled and outer circles are sub-sampled. This
rule gives a good tradeoff between data loss due to sub-sampling and memory
consumption for storing the panoramic view.
Bird’s Eye View: In general, 3D straight lines are imaged as curves in the
omnidirectional image. For instance, the horizon line is imaged as a circle.
Only 3D lines that belong to vertical planes containing camera and mirror
axis project as straight (radial) lines.