From a classification point of view the extracted
regions are the candidates for the identification
step. The task is to extract proper features from
the candidates and to compare them with the fea-
tures of the known objects. A result of the motion
based segmentation of the image is the border
line. This closed contour can be considered to be
the observed geometric quantity of the unknown
object. The problem is now that the appearance
of the object in the image, and this applies to
the border line as well, is distorted by the per-
spective projection of the object. If it is possible
to find descriptive quantities which are not affec-
ted by the projection then object recognition is
possible without knowing the rotation and trans-
lation between a camera centered and an object
centered coordinate system. These quantities are
called invariants (Forsyth et al., 1990).
Depending on the complexity of the object and
on the formulation for deriving the invariants the
theory of invariants defines exactly the number of
invariant features with respect to a specific geo-
metric transformation. For solving the recogni-
tion task by automatic pattern recognition often a
few invariant features are sufficient. For the sim-
ple and planar geometric figures of the objects of
interest suitable invariant features have to be fo-
und. In our application the object models for the
traffic signs are restricted to triangles, rectangles
and circles.
Usually, in photogrammetry the mapping from
object space to image space is postulated to be
a general perspective projection. The perspec-
tive projection can be approximated by an af-
fine transformation for planar objects and if the
distance between the object and the camera is
large and the field of view is narrow (Costa et al.,
1989). For an affine mapping it is easier to de-
termine invariant features than for a perspective
transformation. In the next section first some dif-
ferent procedures for the determination of affine-
invariant quantities from points, lines and areas
are discussed. For the simple objects only a few
invariant features can be determined. Further
the figures possess specific symmetries. Thus for
these objects it is very interesting to find out
which of the procedures are suitable for the deter-
mination of affine-invariant features at all. The
result is a pre-selection of the possible procedures
for the determination of affine-invariant features
from a closed border line. Because a procedure
should be as reliable as possible the sensitivity of
the affine-invariant features with respect to noise
in the border line is evaluated. A quality measure
for the sensitivity is the probability for a wrong
classification. This probability can be used for
the assessment of the separability of parametric
models based on the observations.
3.1 Determination of affine-invariant
features
Various procedures are proposed for the deter-
mination of affine-invariant features using points,
lines or areas. These features are affine-invariant
coordinates (Costa et al., 1989) for the points,
affine-invariant curvatures (Cyganski et al., 1987)
or affine-invariant Fourier descriptors (Arbter,
1989) for the lines and affine-invariant moments
(Hu, 1962) for the areas. The procedures will be
shortly discussed in the following. The detailed
mathematical formulations can be found in Gei-
selmann (1992) or in the references quoted above.
3.1.1 Affine-invariant coordinates
At least 4 points (a,b,c,d) are necessary for
the determination of affine-invariant coordinates.
With 3 points, for example a,b,c, a basic tri-
angle is defined. Then the affine-invariant coor-
dinates of point d are determined by the quoti-
ent of the area F of the triangles b,c,d or a,c,d
and a, b,c, respectively. Affine mapping using a
transformation matrix T' changes the area F of
the triangles according to F' — det(T)F. Ob-
viously the quotient of two triangular areas is
affine-invariant. Unfortunately this simple pro-
cedure has a considerable drawback. For the de-
termination of affine-invariant coordinates of the
object and of its image the identical basic triangle
has to be used. This equals the need of establis-
hing the correspondence of 3 points of the object
and the image. In consequence, the procedure
does not fit the aim of real-time identification.
3.1.2 Affine-invariant curvatures
Similar to the standard formulation of the curva-
ture of a planar curve the affine-invariant curva-
ture x(7) is defined by
K(T) = ü(r)ö(r) — ü(r)ö(r) .
With (u(r),v(r)) a parametric formulation of
the curve is described as a function of the af-
fine length 7 (Naas and Schmid, 1972). As in
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