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).