anbul 2004
tures The
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yying a dis-
xij leading
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reach Zi.
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were gen-
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rners in the
'oordinates
(7)
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Analysis of points under image noise. To generate noisy im-
age data, the reference images 7; were contaminated with zero
mean Gaussian white noise
n ^ N(0,02),
the noise variance being varied from on = 0.1 grey values ([gr])
to on, = 12 [gr] in steps of V2 [gr].
N= 100 test images I vss. duet were generated for each
image /; and each noise level on. The point extraction was ap-
plied to each test image, for each point zm leading to a set
po — ten | id Xm - x ec e}
of observations (7)
Xm.
Using eq. 4, the bias b and the covariance matrix 3,4 of the
observations were estimated point-wise for each pc ^) over all
noise levels Gn.
4.3 Provisional results
First results of our experiments are plausible, indicating that the
proposed methods for reference data definition may be success-
fully used in characterizing image processing algorithms.
4.3.1 Noise characteristics of corner extraction. Results con-
cerning the noise sensitivity of the corner extraction are illus-
trated in fig. 5 and fig. 6.
As to be expected, the empirical standard deviations 64 and 6,
of extracted corners in x- and y-direction and the resulting mean
error Gp = 4/02 + 62 increase with increasing image noise (cf.
fig. 5). For most corners, the increase of 65, 6, and 6 is stronger
than linear and thus stronger than to be expected. This may be
caused by the fact that to detect all desired corners, the smoothing
parameter c of the corner extraction was adapted linearly to the
standard deviation o;, of the image noise, reaching from e, —
D 7{pell for on .i[gr|to.04 - 0.9 [pell for o4. — 10[gr].
As smoothing deteriorates the quality of the point localization
(cf. (Canny, J.F., 1983)), the loss of precision may thus be partly
caused by enlarging the smoothing filter for images with a larger
amount of noise.
o, [pel] 9, [pel]
1.4 V4 rrr reer 1.4
RootOf(a +6?) [pel]
Figure 5: Empirical precision of extracted corners on noisy im-
ages. Each curve represents the uncertainty of a single point.
Left: Empirical standard deviation o, in x-direction. Center:
Empirical standard deviation o, in y-direction. Right: Mean lo-
calization error op = 4/02 + 03.
Also the estimated bias | b | of extracted points increases with
increasing image noise (cf. fig. 6), which will be mainly due to
1065
b xU b, [pel Ibi pel
Figure 6: Estimated bias of extracted corners on noisy images,
each curve representing the bias of a single corner. Left: Bias b,
in z-direction. Center: Bias b, in y-direction. Right: Norm | b |
of the bias.
enlarging the smoothing filter dependent on the image noise. As
to be expected, the different behavior of the bias components 5,
and b, of different points indicates that the bias depends on the
perspective under which a corner is observed.
Shortening over different resolutions
8 T —_—
[— Point No35 ||
ze +
o
I Perm
w 4} ALL frere tee a a 4
i
2 À À 1
1 2 3
8 T T
— Point No.36 ||
E'| ; I |
& E. un
ear | Lex ll 1
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2 1 i L j
1 2 3
8 T T en)
{-— Point No.37 | |
m 6 3 4
o i
4b l Treen ly
I Fat J
2 i 1
1 3
Pyramid level
Figure 7: Shortening of straight lines and edges on different res-
olutions.
4.3.2 Shortening of edges at junctions The results concern-
ing the shortening of extracted lines and edges at junctions are
depicted in the figs. 7, 8 and 9.
For three junctions in a single image, fig. 7 shows the one-sided
shortening of the junction branches dependent on the image reso-
lution. The results were drawn from the three lowest levels of an
image pyramid. The mean shortening of extracted lines reaches
from 3 to 5 pixels. It decreases with decreasing image resolution.
For a single junction, in fig. 8 the mean and the variance of the
shortening of edges is depicted over all images. The shortening
varies over different images, depending on the perspective under
which the junction is observed and on its illumination. The short-
ening is large especially in situations with low contrast at edges.
Fig. 9 shows for each junction the mean and the standard devi-
ation of the shortening of adjacent edges, with the mean and the
standard deviation being taken over all images. The one-sided
shortening reaches up to 10 pixels. Again the worst results are
obtained for junctions with low contrast at edges.
5 CONCLUSIONS AND OUTLOOK
This paper proposes two methods for generating reference data in
the context of characterizing image processing algorithms. The
