Table 1 
COMPARISON OF PRECISIONS OF MATCHING OBTAINED BY LEAST SQUARES 
AND CROSS-CORRELATION METHODS 
Feature 
Spread function 25.0 um 
Pixel size 12.5 um 
Quantisation level 
Cross 100 x 10 Um 
8 bits/pixel. 
Precisions in terms of pixel size x 1000 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Method of Scale Variations Rotation (degrees) 
matching 
1.0 0.9 0.8 0 10 20 30 
LSQR 38 47 42 38 31 47 180 
Cross- 52 75 120 52 83 170 * 
Correlation 
* - no results 
Table 2 
COMPARISON OF MATCHING PRECISIONS OBTAINED FOR DIFFERENT SHAPED FEATURES 
FOR THE LEAST SQUARES METHOD 
Pixel size 
12.5 um 
Precisions in terms of pixel size x 1000 
TYPE 
OF 
FEATURE 
SCALE 
CIRCLE 
CROSS 
ELLIPSE 
* - no results 
Table 1 reveals that the cross-correlation method, for images 
in which there are no scale distortions or rotations, will result 
in marginally worse results than the least squares method. 
However, as the scale distortions and rotations increase, the 
precisions obtained by cross-correlation rapidly deteriorate and 
are a factor 3 to 4 larger than those obtained by the least 
squares method, while matching fails for a rotation between 
the two images of 309. This result confirms the widely held 
views amongst photogrammetrists that the cross-correlation 
method will be significantly affected by distortions between 
the two images, and indeed may result in completely erroneous 
matches. 
In Table 2 are shown comparisons between 3 different 
features subject to scale variations and rotations, and different 
levels of blur. As previously demonstrated, this table 
confirms that better results are generally obtained for a cross 
than for a circle or ellipse, and also the small influence that 
image quality has on the precision of least squares matching. 
Foerstner (1982) has given a formula for the theoretical 
precision of matching 2 digital images by the least squares 
method as follows: 
c? 
aad rr 
X © N.SNRZ c. 
8 
where o2 is the variance of estimating the shift 
  
822 
ROTATION 
(degrees) 
10 
SPREAD FUNCTION 
(um) 
30 10 25 
* * 
44 
36 
50 
38 
32 
48 
31 
68 
parameter in the matching, 
N is the number of pixels containing the 
information about the feature, 
SNR is the signal to noise ratio in the image, 
o; is the variance of the signal, 
o?, is the variance of the gradient of the image. 
8 
Adopting N=50 for a circle of 100um with pixel size of 
12.5um, SNR=10, o2 -3600, and o? 
g g' =900 for an image 
quantized to 8 bits, Ox is calculated to be 0.04 pixel, as 
indeed has been obtained for the feature in Figure 1. 
Similarly, for a larger circular feature of 200um, the precision 
will be a factor 2 less or 0.02 pixel which also agrees with that 
shown in Figure 2. As stated above, a blurred image will be 
larger than a sharp image of the same nominal size because of 
the effects of blur, and therefore, the factor N of a blurred 
image in equation 1 will be larger than for a sharp image, 
while G,' will be slightly smaller. This appears to ex, 
the better results obtained for blurred images than for shi. p 
images. These results demonstrate an agreement between the 
theoretical precisions and those obtained by simulation.