A ®n*al,1587 ) 
Stlmate s optical 
uatl °ns obtained 
la § e as a system 
^nations are 
r M- More 
^ s ystem to 
ation equations. 
)e equal to nxn 
' P ixe l in small 
We can get „» 
'ermineti linear 
small temporal 
ve can obtain n 
straints metbob 
overdetermined 
ares solution in 
nethod. 
2.3 Imposition of the Smoothness Condition 
Another approach of solving the gradient constraint 
equation is imposition ofcondition, that is; 
(a) spatial smoothness of optical flow (spatial global 
optimization method) (Barron, Fleet and Beauchemin, 
1994, Beauchemin and Barron, 1997, Horn and 
Schunck, 1981, Schunck, 1984); 
(b) temporal smoothness (temporal global optimization 
method); 
(c) their combination. 
One way to express the additional condition is to minimize 
the square of the magnitude of the gradient of the optical 
flow velocity: 
du 
dx 
du 
dy 
— and 1 — 1 + 
dv 
dx 
' dv 
The total error, E, to be minimized as 
X y 
/ 
(9) 
+ a 
du 
dx 
2 ( du ' 2 
dy 
dv N 
2 \ 
(10) 
The minimization is to be accomplished by finding suitable 
values for the optical flow velocity (u, v). Using the 
calculus of variation, following equations are obtained. 
I x 2 u + I x I y v = a 2 W 2 u - I x I t 
I x I y u + I x 2 v - a 2 V 2 v - I y I t 
(11) 
However, it would be very costly to solve these equations 
simultaneously by one of the methods, such as Gauss- 
Jordan elimination. So, these equations should be solved 
by iterative method that is Gauss-Seidel method. 
In the temporal optimization method, smoothness 
condition is expressed as 
(12) 
The total error is minimized in the same way as spatial 
global optimization method. 
3. EXPERIMENTS 
In this chapter, various gradient-based approaches 
described in Chapter 2 are applied to real sequential 
images of traffic scene. The size of the frame in the 
sequential images is 720 by 480 pixels. And time interval 
is 1/30 second. Figure 1 shows the frame of sequential 
images used in the optical flow estimation. The vehicles 
in Figure 1 move from upper right to bottom left. 
Velocities of the vehicles in this image were measured, and 
the results were about (1) 20 pixels/frame, (2) 3 
pixels/frame and (3) 2 pixels/frame, respectively. These 
values were used as measurements for comparison among 
the several gradient-based approaches. 
Figure 1: Image of Traffic Scene. 
Spatial neighborhood was defined as 5 by 5 pixels, and 
temporal neighborhood was defined as 3 frames. A 
constant optical flow over these neighborhoods was 
assumed. In global optimization method, iterative 
number was 100, and coefficient a was defined as 100. 
Figure 2 through Figure 9 show optical flow estimated by 
each method at a frame. Estimated optical flow is 
depicted as segment at an interval of 20 pixels, and the 
length of segments is ten times as long as estimated value. 
Figure 2 shows the result which was solved by spatial local 
optimization method. The magnitude of one of the 
estimated flow vectors in each vehicle was (1) 6.4 
pixels/frame, (2) 2.0 pixels/frame and (3) 1.5 pixels/frame, 
respectvely. While there were precise flow vectors in the 
vehicles (2), (3), even in the same vehicles flow vectors 
could not be obtained at many pixels. This problem will 
be described later. In the vehicle (1), there were many 
flow vectors, however inaccurate flow vectors were 
included. 
Figure 2: Spatial Local Optimization Method.