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.