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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
or error can be calculated in dependency on their image co-
ordinates (image space based). No information about the
scene structure are needed. An example for such an pro-
jection is the general used model for the optical distortion.
Strictly, it has not a single viewpoint, but the accuracy of
this approximation is well enough.
3. Class 3 is formed by projections with non single viewpoints
which do not preserve straight lines. The projection rays are
no straight lines and they do not intersect in one point. But
their envelope forms a locus of viewpoint in three dimen-
sions which is called caustic surface or just caustic (Swami-
nathan et al., 2001). The resulting image distortions are
called caustic distortions. Their exact determination bases
on the position of the observed feature in object space (ob-
ject space based). Therefor information about the scene
structure are necessary to determine the influence of the dis-
tortions. Imagesystems like wide-angle, fish- eye and cata-
dioptric cameras with a spherical and conical reflector based
design (Nayar et al., 2000), camera clusters, strict model of
objective distortion and multi media geometry (e.g. air and
water) belong to this class. In section 2 we will present the
caustic of a multi-media system.
4. The combination of a non single viewpoint with an invari-
ance of straight lines is under the valid physical laws not
possible.
The influence of image distortion using imaging systems with
non single viewpoints is object space based, that means it cannot
be determined or corrected without any information of the scene
structure. If no information about the scene structure are given,
it is necessary to make some assumption about the scene struc-
ture (e.g. (Swaminathan et al. 2003)). For the mapping process
between the object and the image space, special algorithms are
needed. For example the iterative algorithms for the multi media
geometry in (Maas, 1995), which could be very complex.
Another method is to replace the non single viewpoint by a sin-
gle viewpoint, so that the mapping process can be modeled with-
out any information about the object space. Swaminathan et al.
presented in (Swaminathan, 2001) a method to determine a sin-
gle viewpoint by estimation the best location to approximate the
caustic by a point for catadioptric cameras. This methods based -
on the determination of singularities of the caustic.
A method which is used here to define a single viewpoint is first
mentioned in (Wolff and Fórstner 20004 and was published in
more detail in (Wolff and Fórstner 2001): the explicit strict phys-
ical model with non single viewpoints is replaced and approxi-
mated by a less complex projective mapping with a single view-
point. Therefor no pre-informations about the scene structure are
needed. The estimation of the approximation is posed as the
minimization of the back projection error in image space. The
introduced approximation is applicable for all kinds of optical,
non projective mappings. The degree of approximation can be
augmented by partitioning the object space into small segments
and calculating a local approximation for every part of the object
space separately. For this partitioning we need the extension of
the observing area approximately. The method was presented in
(Wolff and Fórstner 2001) used for a matching process based on
the trifocal tensor.
1.2 Goal of this paper
In the context of non projective projections, the paper makes the
following key contributions:
607
e Under the background of the taxonomy of imaging systems
we survey the non projective multi media geometry (project-
ing rays passes different media e.g. air, perspex and water).
It belongs to class 3 with a caustic as a non single viewpoint.
e We presend a new image point matching algorithm for a 3D
reconstruction using multiple views, based on geometrically
constraints alone. The method uses all images simultane-
e. Sida bbss EEE a vir-
tual, projective camera is used for the image point matching
process for multiple views with multi media geometry. As
we will see , this is implemented without loosing the quality
of the strict model significantly.
e Different quality tests for the approximation and the point
matching algorithm are realized.
1.3 Projective Geometry
We use multiple-view geometry as it has been developed in recent
years and is documented in (Hartley and Zisserman 2003).
Assuming straight lines preserving mappings, the projection of
object points X to image points x' can be modeled with the direct
linear transformation (DLT):
X 2 KR(J| - Z)X
for object points X: represented in Plücker coordinates. P is the
projection matrix, K the calibration matrix, R the rotation matrix
and Z the projection center of the camera.
2 GEOMETRY OF IMAGING SYSTEMS WITH NON
SINGLE VIEWPOINTS
2.1 Caustics as Loci of Viewpoints
For the modeling of point projection we need two relations:
I. A projection relation predicting the image point x’ of a given
object point X.
2. An inverse projection relation, giving the mapping ray L in
the object space. In case of projective mapping a light ray is
build by the projection center and the image point. In case
of non projective mappings only that part of the broken ray
is important, which intersects the object point.
For Class 1 and 2 of our classification the realization of these two
relations is geometrically trivial. The mapping ray is built by the
object point or rather the image point and the projection center.
In the case of image distortion a correction of the image points
can be calculated image space based.
For class 3 relation 2 is also trivial. The projecting light rays
change their direction because of refraction and reflection (see
Fig. 1). These changes can be directly determined using the
Snell’s refraction law and reflection law. Relation 1 is not as
trivial like the others, because the direction of the ray coming
from the object point is not directly determinable if the object
point and the physical pupil of the lense is given alone. But, as
seen in Fig. 1, the envelope of the rays, which do not intersect in
one point, forms a locus of viewpoints in three dimensions, the
so called caustic. The light rays in object space are the tangent
on this surface. Each point on the caustic surface represents the
three-dimensional position of a viewpoint and its viewing direc-
tion. Thus, the caustics completely describes the geometry of the
catadioptric camera (Swaminathan et al., 2001).
