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Proceedings International Workshop on Mobile Mapping Technology
Li, Rongxing

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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
(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:
— and 1 — 1 +
' dv
The total error, E, to be minimized as
X y
+ a
2 ( du ' 2
dv N
2 \
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
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
The total error is minimized in the same way as spatial
global optimization method.
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
Figure 2: Spatial Local Optimization Method.