0 
720.1 
430.2 
10.3 
   
  
  
0. 0.20.4 0.89/0.8 1 1.21.4 0 0.20.40.60.9 1. 1.21.4 
  
Figure 7: Systematic error in the velocity estimate of a 
moving random pattern using the least squares differential 
method (left) of (Lucas and Kanade, 1981) and the total 
least squares (tensor method) of (Haussecker and Jähne, 
1997). Signal to noise ratio is one. 
CCD cameras show significant large-scale and small- 
scale spatial variations in the order of about 196, which 
cannot be neglected and may show quite different spa- 
tial variations for different cameras. Since these pat- 
terns are static, they are superimposed to the real mo- 
tion in the sequence. In parts of the sequence where 
the local contrast is low, the static patterns dominate 
the structure and thus a lower or even zero velocity is 
measured. 
The influence of such static patterns can nicely be 
demonstrated for moving objects with low contrast 
such as the slightly textured elephant in 6. Dirt on 
the glass window of the CCD sensor causes spatial 
variations in the responsivity of the sensor. At the 
edges of the speckles, the smallest eigenvalue of the 
structure tensor shows high values indicating motion 
discontinuities. The motion field indeed shows drops 
at the positions of the speckles. If a simple two-point 
calibration is performed using the measured respon- 
sivity and an image with a dark pattern, the influence 
of the speckles is no longer visible both in the smallest 
eigenvalues and the motion field. 
7. RESULTS 
Figure 7 shows a comparison of the least squares ap- 
proach of Lucas and Kanade (1981) and the total least 
squares (structure tensor) technique. While the least 
squares technique shows a bias for increasing veloci- 
ties, the total least squares technique does not show a 
bias and performs better for all velocities within the 
range of the temporal sampling theorem. 
In order to prove the theoretical error limits of the 
technique in real sequences, accuracy tests have been 
performed. Defining the local noise portion (LNP) 
as the ratio between the noise variance 9? and the 
local variance varg(g,) = B(gn)? —(Bgn)? of the noisy 
image gj 
  
2 2 
ape Io Bow OLIM 11 
varg(gn) varg(g)+ 0? 
we get a normalized measure quantifying the presence 
of noise within a local neighborhood. A value of LNP 
— 0.5 means that the noise variance lies in the same 
710 
  
displacement [Mframe] 
5E-4 0,005 0,05 0,5 
10 FF TA AT WATT EEE ES 
  
     
  
  
  
noise level (LNP): 
e 0.00 
eo 0.37 
^ 0.70 
v 0.90 
   
  
  
- 
"T 
  
  
  
o 
= 
computed displacement [pixels/frame] 
  
  
  
- e] A dedii à. A Add 
0,01 0,1 1 10 
displacement [pixels/frame] 
  
  
  
Figure 8: Subpixel accuracy of the tensor method. A si- 
nusoidal pattern with a wavelength of 20 pixels has been 
shifted from 0.01 pixels/frame up to the theoretical limit 
of 10pixels/frame given by the temporal sampling theo- 
rem. The contribution of noise to the signal within the 
spatiotemporal region of support of the filters is quantified 
by the local noise portion, LNP. The dashed lines indicate 
relative errors of +5 %. 
order as the local variance varg(g) of the noise-free 
pattern g. 
Figure 8 shows the results of the averaged optical flow 
for a sinusoidal test pattern with additive noise of dif- 
ferent LNPs. The result exhibits the high subpixel 
accuracy of the structure tensor technique. The struc- 
ture tensor technique was applied to a variety of appli- 
cation examples from oceanography (IR ocean surface 
images), botany (growth processes) and traffic scenes. 
It proved to work well without any adaption to the 
image content. Figure 9 shows examples of such se- 
quences. As a nice detail the pedestrian in figure 9 is 
about to lift his right leg which is clearly visible in the 
optical flow field. 
8. CONCLUSIONS 
The structure tensor technique allows to compute 
dense displacement vector fields from extended im- 
age sequences with high subpixel accuracy. Addition- 
ally, it yields a measure of certainty as well as a mea- 
sure for the type of motion quantifying the presence of 
an aperture problem. The structure tensor technique 
proved to work well for a variety of applications. The 
measures of coherency and aperture problem allow 
the technique to automatically adapt to local spatio- 
temporal image structures without additional param- 
eters. 
ACKNOWLEDGEMENTS: We gratefully ac- 
knowledge financial support by the ‘Deutsche 
Forschungsgemeinschaft’, DFG, within the frame of 
the research unit ‘Image Sequence Analysis to Inves- 
tigate Dynamic Processes’. 
  
  
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