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Optical flow is calculated by the generalized gradient method based on spatio-temporal filtering.
Let 
denotes brightness at a point 
in an image at
time t. If this point moves to a point 
at time t+dt,
the following equation holds:
Taylor expansion of the right side of the equation (1) is
where 
,
,
กค
and e is the high order terms of dx,dy,dt.
Assuming that e is negligible, we obtain the next equation:
where 
, 
is flow vector.
This is the constraint equation of the gradient method.
By applying two different spatial filters g,h to the input image
,
the following two constraint equations are derived.
where 
represents convolution.
The flow vector 
is obtained by solving these simultaneous
equations.
The filters used in the equaiton 4 shold have following features.
A smoothing filter is applied to garantee the capability of differentiation in temporal direction.
We define the reliability of the flow by the angle between two
lines corresponding to equations
(4) as shown in figure 1. The
reliability 
is given as:
where 
and 
.
Note that the right side of this equation is the absolute value of the
determinant of the coefficient matrix of the flow vector normalized by
the spatial gradient of brightness.
If the reliability 
is small, the flow vector is not calculated.
Figure 1: reliability of optical flow
If the contrast is small, the flow vector is not reliable, either.
The reliability 
of the contrast is given as:
If the reliability 
is small, the flow vector is not calculated at
the point.
A Result of extracting optical flow is shown in Figure 2. Lines stands for flow vectors, and colors stands for its direction. In the area which doesn't have gray points, flow vectors aren't calculated bacause of its unreliability.
