refer to triangles, rectangles and circles with an
edge size of 20 pixels.
Concerning the separability of rectangle and
circle some problems may occur because the only
non-zero invariant of both figures is I; (cf. table
1). The advantage of this procedure is that the
calculation of the moments and invariants is very
simple and that no threshold or other parameter
has to be defined a priori.
3.2 Empirical investigations for choosing a
suitable procedure
The practical suitability of the different procedu-
res described in section 3.1 for classification will
now be investigated with the help of simulations.
With respect to real-time mapping the procedure
should be as simple as possible. On the other
hand, the reliability of the procedure must be
.satisfactory. In this case a high reliability mea-
sure is identified with a small probability for a
misclassification. With the formulas for the de-
termination of affine-invariant features the proba-
bility for a misclassification could be theoretically
determined by using the law of error propagation.
Because the formulas are very extensive the pro-
bability is determined by using simulations. For
this purpose the border lines of the figures are dif-
ferently distorted and noise is added. Then the
affine-invariant features are calculated and used
to assess the probability for misclassification by
the help of the Bhattacharyya distance (Funku-
naga, 1972).
For this experiments we restrict on the form fac-
tor k which is derived from the affine-invariant
Fourier descriptors and on the 4 affine-invariant
features 11,12, I3, I3 which are derived from the
moments up to order 3. Different probabilities
for misclassification are received depending on the
edge length of the figure and on the amount of
noise in the border line. Such a number gives,
for example, the probability that a triangle was
classified as a circle.
The results show that the largest probability of
a misclassification is received for short edges and
between the figures rectangle and circle. The dia-
grams (figure 5) show the probabilities for these
both figures. The results of a noise free border
line and a noisy border line (Gaussian noise with
a standard deviation of 2) are plotted for compa-
rison. Both procedures are equally suitable for a
Propability for misclassification
De noise g - 0 (rectangle, circle)
25 —A— Fourier descriptor
-@- moments
6 8 10 12 14 16 18 20 22 24 28 28 30 32 34 36 38 40
edge length
Propability for misclassification
D6] noise o 22
50
(rectangle, circle)
40% ? -A— Fourier descriptor
304
A
200 SAAL
A
® A
10- es A AA
e
-69- moments
AAA
eee À
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
edge length
0
Figure 5: Probability for misclassification
border line without noise. The differences which
are obtained for the noisy border line are bigger.
The moment based invariants show a better sepa-
rability than the Fourier descriptor based shape
factor. The reason is that for the affine-invariant
quantities found by the moments the area is used
and not the line information.
As a result one can say that the calculation of the
affine-invariant features determined by the mo-
ments supplies the best results concerning the
separability. Therefore the recognition process
should rely on these features. The measured com-
puting time (VAX 3500) for the feature extraction
of a 80 x 80 region of interest amounts to 0.2 sec
(Fourier descriptor approach) and 0.4 sec (mo-
ment approach), i.e. in practice, it is fast enough
for real-time mapping applications.
468
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