original coherency 
     
corner 
  
  
  
Figure 3: Illustration of the coherency and type measures 
for a moving grid on a linear positioning table. 
4.4 Coherency and type measures 
Although the rank of the structure tensor proves to 
contain all information necessary to distinguish differ- 
ent types of motion it can not be used for practical 
implementations because it does not constitute a nor- 
malized measure of certainty. Additionally it is only 
defined for integer values 0,...,3. In real sequences 
usually mixtures between the types of motion occur. 
In this section we will introduce two normalized mea- 
sures to quantify the confidence and type of motion 
which are suited for practical implementations. In 
contrast to the rank they yield multivalued numbers 
between 0.0 and 1.0. 
Coherency (coh): In order to quantify the overall 
certainty of displacement estimation we define the nor- 
malized coherency measure 
AS Aso" 
coh = mm 5 (13) 
with A; and As denoting the largest and smallest eigen- 
value of the structure tensor, respectively. Table 1 
shows the values of coh for different cases of motion. 
For both types of apparent motion, i. e. aperture prob- 
lem and no aperture problem, coh is equal one. If 
no displacement can be computed it is identical zero. 
With increasing noise level coh approaches zero for all 
different types of motion. 
Edge measure (edge): While the coherency coh 
gives a normalized estimate for the certainty of the 
computation it does not allow to identify areas of ap- 
parent aperture problem. In order to quantify the 
presence of an aperture problem we define the edge 
measure ; 
Al = Am 
edge = (zem) (14) 
708 
where À,, denotes the medium eigenvalue of the struc- 
ture tensor. Table 1 shows the values of edge for dif- 
ferent cases of motion. Only if an aperture problem is 
present edge reaches its maximum value of coh. For all 
other types of motion it is equal zero. Note that the 
edge measure is normalized between 0.0 and coh since 
it is not possible to detect the presence of an aperture 
problem more reliably than the overall certainty. 
  
  
motion type coh | edge | rank 
homogeneous brightness | 0 0 0 
aperture problem 1 1 1 
no aperture problem 1 0 2 
no coherent motion 0 0 3 
  
  
  
  
  
  
Table 1: Rank, coherency and edge measure for dif- 
ferent types of motion. 
Corner measure (corner): From the two indepen- 
dent measures coh, and edge, a corner measure can 
be computed by corner = coh - edge. It constitutes 
the counterpart of edge, quantifying the absence of 
an aperture problem, i. e. selecting points where both 
components f; and fa of the optical flow f can be 
reliably computed. 
Figure 3 illustrates the coherency and type measures 
for a moving calibration target on a linear positioning 
table. The coherency measure shows the entire grid 
without regions of homogeneous brightness. These ar- 
eas split up into the edges and crossing points of the 
grid for the edge and corner measure, respectively. 
The importance of coherency and type measures for 
quantitative analysis are illustrated in Figure 4. The 
optical flow field of a moving ring pattern shows ran- 
dom flow vectors in homogeneous regions with addi- 
tive noise. With the coherency measure these regions 
can be identified. Further knowledge about the pres- 
ence of an aperture problem allows to reconstruct the 
real flow field using the local regularization technique 
proposed in (Haussecker and Jähne, 1997). 
Confidence measures for the local least squares ap- 
proach have also been defined by Barron et al. (1994) 
and Simoncelli (1993). Both try to quantify the singu- 
larity of the matrix A in (3) by analyzing the eigenval- 
ues of A. While Simoncelli (1993) proposes to thresh- 
old the sum of eigenvalues, Barron et al. (1994) ar- 
gue that the smallest eigenvalue proved to be more 
reliable in practical implementations. The matrix À 
constitutes the structure tensor of a two-dimensional 
subspace of the spatiotemporal neighborhood and rep- 
resents local orientation in two-dimensional images 
(Bigün and Granlund, 1987, Knutsson, 1998). The 
two-dimensional equivalent to the three-dimensional 
coherency measure is the difference of the two eigen- 
values, normalized to the sum of the eigenvalues 
(Jàhne, 1993, 1997). Rather than thresholding parts 
of the information, this measure quantifies the con- 
  
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