I(x, y, t). Now consider what happens when the pattern 
moves. The brightness of a particular point in the pattern 
is constant, so that 
"-0. 
dt 
(1) 
The differentiation (1) can be expanded in a Taylor series 
If we let 
and then, 
dl dx dl dy dl 
+ — +— = 0. 
dx dt dy dt dt 
(2) 
dx , dy 
u = — and V = — , 
dt dt 
(3) 
I x u +1 V + I t = 0 
(4) 
Flinchbaugh, 1990, Mitiche, Wang and Aggarwal, 1987, 
Woodham, 1990); 
(d) by their combination. 
The spatial local optimization method estimates optical 
flow by solving a group of observation equations obtained 
from a small spatial neighborhood of the image as a system 
of linear equations. Two observation equations are 
sufficient to arrive at unique solution for (u, v). More 
than two equations may be included in the system to 
reduce the effects of errors in the observation equations. 
Let the small spatial neighborhood, Q ? be equal to nxn 
pixels. The observation equation at each pixel in small 
spatial neighborhood can be obtained. We can get n 2 
observation equations: 
hx U + 
I 2x U + 
I ly v + I it = 0 
12y V + ^21 = 0 
lixU+iiyV + iit =0 
(5) 
2J impositi 0 " 
Another apP rc 
equation is 1,n P 
Schunck,15 
(b) temporal i 
method); 
(c) their combii 
One way to ex[ 
the square of t 
flow velocity: 
/fl« 
(obi 
The total error, 
where we have also introduced the additional abbreviations 
I x , I y and 7, for the partial derivatives of image brightness 
with respect to x, y and t, respectively, that is 
a/ 
dt' 
(5) 
This equation expresses a plane, which have normal vector 
(w,v, 1), and a measured point (7 X , 7 y , 7 t ) is on the plane. 
Due to single linear equation in the two unknowns u and v, 
the parameter u and v, that is x-component and y- 
component of optical flow respectively, cannot be 
determined. As a consequence, the optical flow (u, v) 
cannot be computed locally without introducing additional 
constraints. 
In order to solve this problem, various methods have been 
proposed. The basic methods of gradient-based approach 
is compiled in following section. 
HI 
Converting to vector notation, overdetermined linear 
equation system (5), is given by 
Gf=-b (6) 
where 
Iu 
V 
'V 
G = 
lix 
h 
II 
u 
, b = 
i it 
1 2 
I 2 
V 
I 2 
n X 
n‘y 
n t 
has the least squares solution 
+a 2 
i 
The minimizati 
values for the 
calculus of van 
However, it wi 
simultaneously 
Jordan elimina 
by iterative me 
2.2 Increase in the Number of Observation Equations 
One of the approaches of solving the gradient constraint 
equation is increase in the number of observation 
equations: 
(a) by the assumption that a constant velocity over each 
spatial neighborhood (spatial local optimization 
method) (Barron, Fleet and Beauchemin, 1994, 
Kearney, Tompson, and Boley, 1987, Lucas and Kanade, 
1981); 
(b) by the constant velocity over temporal neighborhood 
(temporal local optimization method) (Kearney, 
Tompson and Boley, 1987, Nomura, Miike and Koga, 
1991); 
(c) by use of three channels (RGB, HSI) of each pixel 
(multispectral constraints method) (Markandey and 
/ =[G T G)~ l G T b 
provided that the inverse of G T G exists. 
(8) 
In the temporal local optimization method, small temporal 
local neighborhood is n frames, and then we can obtain n 
observation equations. Multispectral constraints method 
has three observation equations. These overdetermined 
linear equation systems have the least squares solution in 
the same way as spatial local optimization method. 
Ì № cha, 
Ascribed in 
"Hages of tra 
P2-7-2