Page 265 - Innovations in Intelligent Machines

258 J. Gaspar et al.
The remainder of this section shows how to obtain a 3D reconstruction from
this information.

Using Back-projection to form Perspective Images

In this section, we derive a transformation, applicable to single projection
centre omnidirectional cameras that obtain images as if acquired by perspec-
tive projection cameras. This is interesting as it provides a way to utilize
methodologies for perspective cameras directly with omnidirectional cameras.
In particular, the interactive scene reconstruction method (described in the
following sections) follows this approach of using omnidirectional cameras
transformed to perspective cameras.
The acquisition of correct perspective images, independent of the scenario,
requires that the vision sensor be characterised by a single projection centre
[2]. The unified projection model has, by definition, this property but, due to
the intermediate mapping over the sphere, the obtained images are in general
not perspective.
In order to obtain correct perspective images, the spherical projection
must be first reversed from the image plane to the sphere surface and then,
re-projected to the desired plane from the sphere centre. We term this reverse
projection back-projection.
The back-projection of an image pixel (u, v), obtained through spherical
projection, yields a 3D direction k · (x, y, z) given by the following equations
derived from Eq. (1):
2
2
a =(l + m),b =(u + v )

2 2
x = la − sign(a) a +(1 − l )b u (25)
y a + b v
2

2
z = ± 1 − x − y 2
√
where z is negative if |a| /l > b, and positive otherwise. It is assumed,
without loss of generality, that (x, y, z) is lying on the surface of the unit
sphere. Figure 17 illustrates the back-projection. Given an omnidirectional
image we use back-projection to map image points to the surface of a sphere
10
centred at the camera viewpoint .
At this point, it is worth noting that the set M = {P : P =(x, y, z)} inter-
preted as points of the projective plane, already define a perspective image.
By rotating and scaling the set M one obtains specific viewing directions and

10
The omnidirectional camera utilized here is based on a spherical mirror and there-
fore does not have a single projection centre. However, as the scene depth is large
as compared to the sensor size, the sensor approximates a single projection cen-
tre system (details in [33]). Hence it is possible to find the parameters of the
corresponding unified projection model system and use Eq. (25).