ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
The Lambert W-function is defined implicitly by (c.f. (Cor-
less et al., 1996))
LambertW(x) - exp(LambertW(x)) = x.
0 20 40 60 80 100 0 ‘001 0015 002 0025 0.03 0.035 0.04 0.045
SNR- s
Figure 4: Resolving power in lines/mm for aerial images
With a pixel size of 15 jm as a function of SNR (left, o =
1) and of the width c of the PSF (right, SNR=10)
Figure 4 shows the resolving power of our ideal edge de-
tector in lines/mm for aerial images as a function of the
signal to noise ratio and of the width c of the point spread
function. The resolving power increases with increasing
SNR and reaching 25-30 lines/mm for good SNRs. It de-
creases with increasing blur, falling below 10 lines/mm for
9 2 45ym. These results are reasonable, as they are con-
firmed by practical experiences with digital aerial images
(c.f. (Albertz, 1991)).
2.5 Contrast, Gradient and Local Scale
We now derive a simple relation between the contrast, the
gradient and the local scale, which we will use to determine
the local scale at an edge. We assume an edge in an image
to be a blurred version of an ideal edge. In case the PSF is
a Gaussian G, (x) the edge follows
s(z)- erfo(r) = kZerf (£) +m
where m is the mean intensity and k is the contrast. Fol-
lowing (Fuchs, 1998) the contrast can be determined from
the standard deviation c, of the signal around the edge,
k = 20,. The gradient magnitude of the edge is given by
the first derivative of the edge function, which in our case
is kG,(0) 2 k/(V/2z0). Thus we have the relation
k
IVg| — osa
From this and k — 2c, we can easily derive
2 0;
Oo =
7 |Vg|
The practical procedure determines the variance of the sig-
nal from
o; 7 E(g?) - (E(g) = 9? x Ga — (9 + Ga)
A - 208
where the kernel width / is chosen to be large enough to
grasp the neighbouring regions. We use a kernel size of | =
20. The gradient magnitude should be estimated robustly
from a small neighborhood. We use a Gaussian kernel with
o = 1 for estimating the gradient magnitude.
2.6 Blind estimating the PSF from a single image
We are now prepared to develop a procedure for blindly
estimating the PSF from a single image. Blind estimation
means, we do not assume any test pattern to be available.
As the PSF is derived via the sharpness of the edges, and
the PSF is the image of an ideal point, a ó-function, we
need to assume that the image contains edges which in the
original are very sharp, thus close to ideal step-edges. This
can e. g. be assumed for images of buildings or other man-
made objects, as the sharpness of the edges in object space
is much higher than the resolution of the imaging system
can handle. Formally, if the image scale is 1 : S, the width
o; of the image of the sharp edge would be o; = o, / S and
we assume that this value is far beyond what the optics or
the sensor can handle.
Now, for each edge we obtain a single value o,. In case
it would be the image of an ideal edge in object space it
can be interpreted as an edge with the expected mean fre-
quency 1/0. in the MTF in that direction. Thus we obtain
a histogram from all edges with
1 COS 1 COS
Ue = — d and Ue = —— oso
G. sin $ d. sin ¢
where the direction vector points across the edge. We use
two values, as we do not want to distinguish between edges
having different sign.
In case the edge is already fuzzy in object space, the es-
timated value c, of the edge will be larger, thus the 1 /0e
will be smaller. Therefore we search for the ellipse which
contains all points we and has smallest area. This ellipse is
an estimate for the shape of the ellipse uw Xu = 1, thus
for 3 of the PSF.
3 EXPERIMENTAL RESULTS
The following examples want to show the usefulness of the
approach. In detail we do the following:
1. Using an ideal test image (Siemens star) with known
sharpness we compare our estimation with given gro-
und truth (cf. fig. 5).
2. Using the same test image but with noise we check
the sensitivity of the method is with respect to noise
(cf. fig. 6).
3. Using real images with known artificial blur we check
whether the method works in case the edge distribu-
tion is arbitrary (cf. fig 7).